POTENTIAL THEORY ON LIPSCHITZ DOMAINS IN ...faculty.missouri.edu › ~mitream › poiss18.pdfPOTENTIAL THEORY ON LIPSCHITZ DOMAINS IN RIEMANNIAN MANIFOLDS: SOBOLEV-BESOV SPACE RESULTS

POTENTIAL THEORY ON LIPSCHITZ DOMAINS

IN RIEMANNIAN MANIFOLDS:

SOBOLEV-BESOV SPACE RESULTS AND THE POISSON PROBLEM

Marius Mitrea1

Department of Mathematics, University of MissouriColumbia, Missouri 65211

and

Michael Taylor2

Department of Mathematics, University of North CarolinaChapel Hill, North Carolina 27599

1Partially supported by NSF grant DMS-98700182Partially supported by NSF grant DMS-9600065

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1. Introduction

There has been a substantial amount of work in the area of elliptic, constantcoefficient PDE’s in Lipschitz domains via layer potential methods. In particular,the classical Dirichlet and Neumann boundary value problems for the flat-spaceLaplacian ∂2

1 + · · · + ∂2n with boundary data in Lp spaces for optimal ranges of p

have been solved. There are also sharp results for the inhomogeneous Dirichlet andNeumann problems. See [Ve], [DK], [JK2], [FMM], [Za] and the references thereinfor more on this subject. Similar issues are studied in [AP] for the flat-space bi-Laplacian in Euclidean Lipschitz domains.

In [MT] and [MMT] the authors initiated a program to extend the applicabilityof the layer potential theory from the setting of the flat-space Laplacian on Lipschitzdomains in Euclidean space to the setting of variable coefficients, and more generallyto the context of Lipschitz domains in Riemannian manifolds. The paper [MT] dealtwith the scalar Laplace-Beltrami operator, and [MMT] treated natural boundaryproblems for the Hodge Laplacian acting on differential forms.

Quite recently, more progress in this direction has been made in [MT2], wherea sharp Lp theory for scalar layer potentials associated with the Laplace-Beltramioperator in Lipschitz domains has been developed. Let us now recall the generalsetting and describe some of the main results from [MT] and [MT2].

Consider a smooth, connected, compact Riemannian manifold M , of real dimen-sion dim M = n ≥ 3, equipped with a Riemannian metric tensor g =

∑

j,k gjk dxj⊗dxk whose coefficients are Lipschitz continuous. The Laplace-Beltrami operator onM is then given in local coordinates by

(1.1) ∆u := div (grad u) = g−1/2 ∂j(gjkg1/2 ∂ku),

where we use the summation convention, take (gjk) to be the matrix inverse to(gjk), and set g := det (gjk). For V ∈ L∞(M) we introduce the second order,elliptic differential operator

(1.2) L := ∆− V.

Assume V ≥ 0 and, also, V > 0 on a set of positive measure in M . This impliesthat

(1.3) L : Lp1(M) −→ Lp

−1(M)

is an isomorphism for each p ∈ (1,∞), where Lps(M) denotes the class of Lp-Sobolev

spaces on M . See [MT].Throughout this paper, we let Ω ⊂ M be a connected open set that is a Lipschitz

domain, i.e., ∂Ω is locally representable as the graph of a Lipschitz function. We

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will always assume that V > 0 on a set of positive measure in each connectedcomponent of M \ Ω. Consider the Dirichlet boundary problem

(1.4) Lu = 0 in Ω, u∣

∣

∂Ω = f,

and the Neumann boundary problem

(1.5) Lu = 0 in Ω, ∂νu∣

∣

∂Ω = g,

where ν denotes the outward unit normal to ∂Ω, and ∂ν = ∂/∂ν is the normalderivative on ∂Ω. When the boundary data are from Lp(∂Ω) (for appropriate p)then, in (1.4)–(1.5), the boundary traces are taken in the nontangential limit senseand natural estimates (involving the nontangential maximal function) are sought.More specifically, if γ(x)x∈∂Ω is a family of nontangential approach regions (cf.[MT] for more details) and u is defined in Ω then u∗, the nontangential maximalfunction of u, if defined at boundary points by

(1.6) u∗(x) := sup |u(y)| : y ∈ γ(x), x ∈ ∂Ω.

Then the natural accompanying estiamtes for (1.4) and (1.5) are, respectively,

(1.7) ‖u∗‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω)

and

(1.8) ‖(∇u)∗‖Lp(∂Ω) ≤ C‖g‖Lp(∂Ω).

In the case when Ω is a Lipschitz domain in the Euclidean space and L = ∂21+· · ·+∂2

nis the flat-space Laplacian, these problems were first treated in [Da], [JK] by meansof harmonic measure estimates. Shortly thereafter, building on [FJR] where thecase of C1 domains was treated, a new approach using layer potential techniqueswas developed in [Ve], [DK]. In these latter papers, a key ingredient was the L2

boundedness of Cauchy type integrals on Lipschitz surfaces due to [CMM] (followingthe pioneering work of [C]).

In [MT] we have extended such operator norm estimates on Cauchy integrals toa variable coefficient setting, allowing for an analysis of potential type operators inthe manifold setting described above. To be more specific, denote by E(x, y) theintegral kernel of L−1, so

(1.9) L−1u(x) =∫

M

E(x, y)u(y) dVol(y), x ∈ M,

where dVol is the volume element on M determined by its Riemannian metric.Then, for functions f : ∂Ω → R, define the single layer potential

(1.10) Sf(x) :=∫

∂Ω

E(x, y)f(y) dσ(y), x /∈ ∂Ω,

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and define the double layer potential by

(1.11) Df(x) :=∫

∂Ω

∂E∂νy

(x, y) f(y) dσ(y), x /∈ ∂Ω.

Here dσ denotes the canonical surface measure on ∂Ω induced by the Riemannianmetric on M . The following results on the behavior of these potentials, whichextend previously known results for the flat Euclidean case, were demonstrated in[MT].

Define Ω+ := Ω and Ω− := M \ Ω; note that Ω± are Lipschitz domains. Givenf ∈ Lp(∂Ω), 1 < p < ∞, we have, for a.e. x ∈ ∂Ω,

(1.12) Sf∣

∣

∂Ω±(x) = Sf(x) :=

∫

∂Ω

E(x, y)f(y) dσ(y),

(1.13) Df∣

∣

∂Ω±(x) = (± 1

2I + K)f(x),

and

(1.14)∂Sf∂ν

∣

∣

∣

∂Ω±(x) = (∓ 1

2I + K∗)f(x),

where, for a.e. x ∈ ∂Ω,

(1.15) Kf(x) := P.V.∫

∂Ω

∂E∂νy

(x, y) f(y) dσ(y).

Here P.V.∫

∂Ω indicates that the integral is taken in the principal value sense. Specif-ically, we fix a smooth background metric which, in turn, induces a distance functionon M . In particular, we can talk about balls and P.V.

∫

∂Ω is defined in the usualsense, of removing such small geodesic balls and passing to the limit. Moreover, fora.e. x ∈ ∂Ω,

(1.16) ∂νSf∣

∣

∂Ω±(x) = (∓ 1

2I + K∗)f(x),

where K∗ is the formal transpose of K.The operators

(1.17) K, K∗ : Lp(∂Ω) −→ Lp(∂Ω), 1 < p < ∞,

and

(1.18) S : Lp(∂Ω) −→ Lp1(∂Ω), 1 0 so that the operators

(1.19)± 1

2I + K : Lp(∂Ω) −→ Lp(∂Ω), 2− ε 0 ona set of positive measure in each connected component of M \ Ω) the operators

(1.20)

12I + K : Lp(∂Ω) −→ Lp(∂Ω), 2− ε 0 on a set of positive measure in Ω, then

(1.21)

− 12I + K : Lp(∂Ω) −→ Lp(∂Ω), 2− ε < p < ∞,

− 12I + K∗ : Lp(∂Ω) −→ Lp(∂Ω), 1 < p < 2 + ε,

− 12I + K : Lp

1(∂Ω) −→ Lp1(∂Ω), 1 < p < 2 + ε,

are isomorphisms, while, if V = 0 on Ω, then

(1.22)

− 12I + K : Lp(∂Ω)/C −→ Lp(∂Ω)/C, 2− ε < p < ∞,

− 12I + K∗ : Lp

0(∂Ω) −→ Lp0(∂Ω), 1 < p < 2 + ε,

− 12I + K : Lp

1(∂Ω)/C −→ Lp1(∂Ω)/C, 1 < p < 2 + ε,

are isomorphisms, where C is the set of constant functions on ∂Ω and Lp0(∂Ω)

consists of elements of Lp(∂Ω) integrating to zero.Given these results, solutions to the Dirichlet, Regularity and Neumann problems

for the operator L in Lipschitz subdomains of M can be produced for Lp boundarydata, for appropriate (sharp ranges of) p and optimal nontangential function esti-mates; cf. [MT2]. The paper [MT2] also contains invertibility results in the localHardy space h1(∂Ω), its dual bmo (∂Ω), and spaces of Holder continuous functionsCα(∂Ω), for small α > 0. Parenthetically, let us point out that real interpolationand (1.21)–(1.22) also allow one to solve the Dirichlet and the Neumann problemswith boundary data in Lorentz spaces Lp,q(∂Ω) for appropriate ranges of indices.

One of our primary goals in this paper is to continue this line of work and toprove invertibility results for the boundary layer potentials in the right sides of(1.12)–(1.14) in the class of Besov spaces Bp

s (∂Ω) for the optimal range of the

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parameters p and s. We mention a sample result. Let Ω ⊂ M be an arbitraryLipschitz domain. There exists ε ∈ (0, 1] so that, if p ∈ [1,∞] and s ∈ (0, 1) satisfyone of the following three conditions

(1.23)

21 + ε

< p <2

1− εand 0 < s < 1;

1 ≤ p <2

1 + εand

2p− 1− ε < s < 1;

21− ε

< p ≤ ∞ and 0 < s <2p

+ ε,

then, with q ∈ [1,∞] denoting the conjugate exponent of p, the operators

(1.24) 12I + K : Bp

s (∂Ω) −→ Bps (∂Ω), S : Bq

−s(∂Ω) −→ Bq1−s(∂Ω),

are invertible. For a complete statement see §11. This extends work in [FMM]where the same issue has been studied in the flat-space setting. Our approachparallels that in [FMM] but the setting of Lipschitz domains in Riemannian mani-folds introduces significant additional difficulties, which require new techniques andideas to overcome. On the analytical side, one major concern is understanding thebehavior of the fundamental solution E(x, y) near the diagonal in M ×M in thecontext of a metric tensor whose coefficients have a rather limited amounted ofsmoothness; here we build on results in [MT], [MMT] and [MT2].

Even though our main results are for scalar functions, we are occasionally leadto consider differential forms of higher degrees. Given this, we develop mappingproperties of layer potentials in this context. See §§6–8. To obtain such invertibilityresults, in the present paper we shall assume more regularity on the metric tensorg than the Lipschitz condition mentioned above. We say more about this below.

Having these invertibility results, we then proceed to solve the Poisson problemfor the Laplace-Beltrami operator with Dirichlet and Neumann boundary conditionsfor data in Sobolev-Besov spaces. Concretely, consider

(1.25)

Lu = f ∈ Lps+ 1

p−2(Ω),

Tr u = g ∈ Bps (∂Ω),

u ∈ Lps+ 1

p(Ω),

and

(1.26)

Lu = f ∈ Lq1q−s−1,0(Ω),

∂fν u = g ∈ Bq

−s(∂Ω),

u ∈ Lq1−s+ 1

q(Ω).

The technical definition of ∂fν u is given in §12. Here we would like to point out that

∂fν u can be thought of as a “renormalization” of the normal gradient of u on ∂Ω, in a

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fashion that depends decisively on f . Assume that p ∈ (1,∞) and s ∈ (0, 1) satisfyone of the three conditions listed in (1.23) and that q is the conjugate exponentof p. Then (1.25) and (1.26) are solvable with naturally accompanying estimates.Certain limiting cases of (1.25)–(1.26) are also considered; see §§9–10.

Our main results regarding the solvability of (1.25)–(1.26) extend previous workin [JK2], [FMM], [Za] where similar problems for the flat-space Laplacian have beenconsidered. As in [FMM] we employ the method of layer potentials. In additionto establishing invertibility results, like (1.24), we also need to understand themapping properties of the operators (1.10)–(1.11), sending functions from Sobolev-Besov spaces on ∂Ω into functions in appropriate Sobolev-Besov spaces in Ω. Wedo this in §§6–8.

We outline the structure of the rest of this paper. In §2, following [MT] and[MMT], we include a discussion of the nature of the main singularity of the fun-damental solution E(x, y). In §3 we establish some useful interior estimates fornull-solutions of the operator L. In §§4–5 we collect basic definitions and severalimportant properties of Sobolev and Besov spaces, as well as atomic Hardy spaces.Mapping properties for Newtonian potentials are presented in §6. In §§7–8 we thenestablish a number of useful results for single and double layer potential operatorson Sobolev-Besov spaces.

Two endpoint results for the Poisson problem for L (with Neumann and Dirichletboundary conditions, respectively) are studied in §§9–10. Then, in §11, invertibilityresults for boundary layer potentials, including (1.24), are presented. Subsequently,these are used in §§12–13 to solve the general Poisson problem for L, respectivelywith Neumann and Dirichlet boundary conditions, for data in Sobolev-Besov spaces.Applications to complex powers of the Laplace-Beltrami operator (with either ho-mogeneous Dirichlet or homogeneous Neumann boundary conditions) in an arbi-trary Lipschitz domain are presented in §14. These extend results on the flat,Euclidean case given in [JK2], [JK3] and [MM].

As mentioned above, some of our arguments require more regularity of the metrictensor than the Lipschitz condition. For example, most of our scalar results areobtained under the assumption that

(1.27) gjk ∈ C1+γ , some γ > 0.

Some of our arguments, as in [MMT], need

(1.28) gjk ∈ Lr2, some r > n,

and we shall occasionally require that

(1.29) gjk ∈ Lr2, some r ≥ 2n.

We would like to stress that, for all our main results, the assumption (1.27) suffices.Nonetheless, we will explicitly state which of these hypotheses we need in eachsegment of the analysis.

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In closing, let us point out that, for the results that we have in mind, such as thewell posedness of (1.22)–(1.23), some smoothness of the metric tensor is required.More specifically, the condition gjk ∈ L∞ (∀ j, k) is, generally speaking, not enoughin order to guarantee, e.g., the well-posedness of the Lp-Poisson problem (1.22)(with s := 1 − 1/p) for p in some a priori given open interval containing 2 (evenwhen the underlying domain is smooth). This can be seen from an adaptation ofan example given in §5 of [Me]; we omit the details here. The interested reader isalso referred to Propositions 1.7–1.10 in Chapter 3 of [Ta2] for some related results.

2. Estimates on the fundamental solution

Retain the hypotheses made in §1 on M . As for the metric tensor

(2.1) g =∑

j,k

gjk dxj ⊗ dxk,

unless otherwise stated, in this section we will assume that the coefficients (gjk)satisfy (1.28), i.e.,

(2.2) g ∈ Lr2(M, Hom(TM ⊗ TM,R)), for some r > n = dim M,

where TM is the tangent bundle of M .As mentioned in the introduction, we will primarily study the Laplace opera-

tor on scalar functions, but some constructions involving the Hodge Laplacian ondifferential forms will arise, and in this section we work in the context of forms.

Let Λ`TM , ` ∈ 0, 1, ..., n, stand for the `-th exterior power of the cotangentbundle T ∗M . Its sections consist of `-differential forms. If (x1, ..., xn) are localcoordinates in an arbitrary coordinate patch U on M and u ∈ Λ`TM then u|U =∑

|I|=` uI dxI , where the sum is performed over ordered `-tuples I = (i1, ..., i`),1 ≤ i1 < i2 < · · · < i` ≤ n and, for each such I, dxI := dxi1 ∧ · · · ∧ dxi` . Here, ofcourse, the wedge stands for the usual exterior product of forms, while |I| denotesthe cardinality of I.

The Hermitian structure in the fibers on TM extends naturally to T ∗M bysetting 〈dxj , dxk〉x := gjk(x). The latter further induces a Hermitian structureon Λ`TM by selecting ωI|I|=` to be an orthonormal frame in Λ`TM providedωj1≤j≤n is an orthonormal frame in T ∗M (locally). Note that

(2.3) 〈dxI , dxJ 〉x = det(

(gij(x))i∈I,j∈J)

.

Recall that the exterior derivative operator d given locally by

(2.4) d =n

∑

j=1

∂∂xj

dxj ∧ ·

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and denote by δ its formal adjoint with respect to the metric induced by (2.3). Thelatter acts on a differential form u ∈ Λ`TM , locally written as u|U =

∑

|I|=` uI dxI ,by δu =

∑

|I|=`−1(δu)I dxI where, for each I,

(2.5) (δu)I := −∑

|J|=`

∑

j,k

gjkεJjI

∂uJ

∂xk+

∑

|K|=`−1

∑

j,k,r,s

εrKjI gjkΓs

rkusK in U.

Here, for 1 ≤ j, k, s ≤ n,

(2.6) Γsjk := 1

2

∑

t

gst(

∂gtj

∂xk+

∂gtk

∂xj− ∂gjk

∂xt

)

are the Christoffel symbols. Also, for any two ordered arrays J , K, the generalizedKronecker symbol εJ

K is given by

(2.7) εJK :=

det ((δj,k)j∈J,k∈K) , if |J | = |K|,0, otherwise,

where, as usual, δj,k := 1 if j = k, and zero if j 6= k.Denote by ∆` := −dδ − δd the Hodge-Laplacian on `-forms, ` ∈ 0, 1, ..., n and

fix some positive, not identically zero function V ∈ C∞(M). Recall from [MMT]that under these conditions, the operator L := ∆`−V , acting on suitable spaces of`-forms has an inverse, (∆`−V )−1, whose Schwartz kernel, E`(x, y), is a symmetricdouble form of bidegree (`, `).

In local coordinates, in which the metric tensor is given by (2.1), we can set

(2.8) e`0(x− y, y) := Cn

(∑

j,k

gjk(y)(xj − yj)(xk − yk))−(n−2)/2

Γ`(x, y)

for appropriate C = Cn, where Γ` is the double form of bidegree (`, `) given by

(2.9) Γ`(x, y) :=

∑

|I|=`

∑

|J|=` det ((gij(y))i∈I,j∈J) dxI ⊗ dyJ , if ` ≥ 1,

1, if ` = 0.

Note that e`0(z, y) is smooth and homogeneous of degree −(n− 2) in z ∈ Rn \ 0 and

C2+γ in y, for some γ > 0. Then define the remainder e`1(x, y) so that

(2.10) E`(x, y)√

g(y) = e`0(x− y, y) + e`

1(x, y).

In the theorem below we collect useful estimates for e`1(x, y) and its derivatives.

These estimates follow from Propositions 2.5 and 2.8 of [MMT].

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Theorem 2.1. For each ` ∈ 0, 1, ..., n and each ε ∈ (0, 1), the remainder e`1(x, y)

satisfies

(2.11) |∇jx∇k

ye`1(x, y)| ≤ Cε|x− y|−(n−3+j+k+ε),

for each j, k ∈ 0, 1.Furthermore, for ` = 0 the same conclusion is valid under the (weaker) hypothesis

(1.27) on the metric tensor.

In the sequel, we shall also need information about the “commutators” betweend, δ on the one hand and the forms E`(x, y) on the other hand. This is made precisein the proposition below, which is taken from §6 of [MMT].

Proposition 2.2. There exists a double form R`(x, y) of bidegree (`, ` +1) so that

(2.12)R` ∈ C1+γ

loc

(

(M ×M \ diag) ∪ (x, y) : x /∈ supp dV )

, some γ > 0,

|∇jx∇k

yR`(x, y)| ≤ C|x− y|−(n−4+j+k), 0 ≤ j, k ≤ 1,

and so that

(2.13) δx(E`+1(x, y)) = dy(E`(x, y)) + R`(x, y).

Furthermore,

(2.14) dx(E`(x, y)) = δy(E`+1(x, y))−R`(y, x).

We shall occasionally also use R` to denote the boundary integral operator withkernel R`(x, y), that is

(2.15) R`f(x) :=∫

∂Ω

〈R`(x, y), f(y)〉 dσ(y), f ∈ Lp(∂Ω,Λ`+1TM).

Also, set Rt` for the analogous operator with kernel R`(y, x).

Next, let Ω be a Lipschitz domain in M and denote by S` the single layer potentialoperator on ∂Ω with kernel E`(x, y), i.e.,

(2.16) S`f(x) :=∫

∂Ω

〈E`(x, y), f(y)〉 dσ(y), x ∈ M \ ∂Ω,

where f ∈ Lp(∂Ω,Λ`TM). Note that at the level of scalar functions, i.e. when` = 0, this agrees with (1.12). Also, (∆` − V )S`f = 0 in M \ ∂Ω. Finally, setS`f := S`f |∂Ω for ` = 0, 1, ..., n.

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3. Interior estimates

In this section we derive interior estimates (Lp-style) for elliptic systems ofPDE’s. General standard references are [ADN], [DN]. Here we obtain quite sharpestimates for operators whose coefficients have the limited regularity described by(3.2) below.

To proceed, let L be a second order, elliptic differential operator in a compactRiemannian manifold M , locally given by

(3.1) Lu =∑

j,k

∂jAjk(x)∂ku +∑

j

Bj(x)∂ju− V (x)u.

Above, Ajk :=(

aαβjk

)

α,β, Bj :=

(

bαβj

)

α,βand V :=

(

vαβ)

α,β are matrix-valued

functions with entries satisfying

(3.2) aαβjk ∈ Lip, bαβ

j ∈ Lr1, vαβ ∈ Lr, for some r > n.

We note that the Laplace-Beltrami operator on scalar functions satisfies (3.2) whenthe metric tensor is Lipschitz. Also, the Hodge Laplacian ∆` on `-forms, for 1 ≤` ≤ n− 1, satisfies (3.2) if the metric tensor satisfies (1.28).

Our first proposition is a local regularity result that sharpens part of Proposition3.3 in [MT].

Proposition 3.1. Let O be an open subset of M , and assume r/(r−1) 0, vαβ ∈ L∞,

then (3.3) holds for any q ∈ (1,∞) with the hypothesis u ∈ Lploc(O) weakened to

u ∈ L1loc(O).

Proof. We first treat (3.3) when the hypothesis on u is strengthened to u ∈ Lpτ,loc(O)

for all τ < 1. For notational simplicity, we use Lps in place of Lp

s,loc(O), etc. Recallthat Sm

ρ,δ stands for Hormander’s classes of symbols. Since we find it necessary towork with symbols which only exhibit a limited amount of regularity in the spatialvariable (while still C∞ in the Fourier variable) we define

(3.5)p(x, ξ) ∈ LipSm

ρ,δ ⇐⇒ |Dβξ p(x, ξ)| ≤ Cβ〈ξ〉m−ρ|β|, and

‖Dβξ p(·, ξ)‖Lip(Rn) ≤ Cβ〈ξ〉m−ρ|β|+δ.

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We also denote by OPLipSmρ,δ the corresponding class of pseudodifferential opera-

tors p(x,D) with symbols p(x, ξ) ∈ LipSmρ,δ, and by ØP LipSm

ρ,δ the class of operatorsq(D,x) with symbols q(ξ, y) = p(y, ξ) when p(x, ξ) ∈ LipSm

ρ,δ.Turning to the problem at hand, we use a symbol decomposition on the first-

order differential operator Aj :=∑

k Ajk(x)∂k, of a sort introduced by [KN] and[Bo] (cf. also the exposition in [Ta]). Picking δ ∈ (0, 1), write

(3.6) Aj = A#j + Ab

j , where A#j ∈ OPS1

1,δ and Abj ∈ OPLipS1−δ

1,δ .

Then set L# :=∑

∂jA#j ∈ OPS2

1,δ, elliptic, and denote by E# ∈ OPS−21,δ a

parametrix of L#. Make all pseudodifferential operators properly supported. Wehave, on O,

(3.7) u = E#f −∑

j

E#∂jAbju−

∑

j

E#Bj∂ju + E#V u, mod C∞,

where f := Lu. Note that the hypotheses imply E#f ∈ Lq2.

Next we shall need mapping properties for pseudodifferential operators whosesymbols have a limited amount of smoothness in the spatial variables. Concretely,the result that is relevant for us here reads as follows. If 1 < p < ∞ and δ ∈ (0, 1)then

(3.8) p(x, ξ) ∈ LipSm1,δ =⇒ p(x,D) : Lp

s+m → Lps , −(1− δ) < s ≤ 1.

When s < 1, this is a special case of results in [Bou] (cf. also Proposition 2.1.E in[Ta]). The case when s = 1 requires a separate analysis, which we now give.

To prove (3.8) with s = 1 it suffices to take m = −1. We need to show that

(3.9) p(x, ξ) ∈ LipS−11,δ , u ∈ Lp =⇒ Dj(p(x,D)u) ∈ Lp.

If Dj falls on u, the estimate is clear from standard results. If Dj falls on thecoefficients of p(x,D), we need only note that

(3.10) p(x, ξ) ∈ L∞S−a1,δ , a > 0 =⇒ p(x,D) : Lp → Lp, 1 ≤ p ≤ ∞,

which follows from elementary integral kernel estimates. This finishes the justifica-tion of (3.8).

Going further, based on the discussion above we have

(3.11)∀ v ∈ Lp

τ , 0 < τ ≤ 2− δ =⇒ Abjv ∈ Lp

τ−1+δ

=⇒ E#∂jAbjv ∈ Lp

τ+δ.

In order to continue, we need a multiplication result to the effect that

(3.12) Lr1 · L

pτ−1 → Lp∗

τ−1

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whenever

(3.13) max 0, n(1/p + 1/r − 1) < τ ≤ 1, n < r < ∞, and 1 < p∗ < p < ∞.

This is a special case of Theorem 1 on p. 171 in [RS]. Hence, from (3.2) and (3.12),it follows that

(3.14) u ∈ Lpτ , ∀ τ < 1 =⇒ E#Bj∂ju ∈ Lp

τ+1, ∀ τ < 1.

Indeed, we only need to check that τ ∈ (0, 1) can be chosen so that τ > n(1/p +1/r − 1), i.e., that 1/n > 1/p + 1/r − 1. Nonetheless, the largest right side in thislast inequality occurs when p 1 and r n, and that is 1/n. A direct inspectionshows that the limiting case of (3.14) corresponding to τ = 1 is also true.

Going further, u ∈ Lpτ and V ∈ Lr imply that V u ∈ Lq with 1/q := 1/p + 1/r−

τ/n and, hence, E#V u ∈ Lq2 → Lp

τ+δ, as long as δ ∈ (0, 1) and r > max p, n.Thus, at this stage we see that the right side of (3.7) belongs to Lp

τ+δ and so does u.This is an improvement of regularity for u and the argument can be iterated a finitenumber of times, to produce the conclusion in (3.3) in the case under discussion.

To treat the case τ = 0, i.e., when u ∈ Lp, we apply symbol smoothing to the firstorder differential operator

∑

j ∂jAjk, which this time we denote Ak = Ak(D,x) ∈ØPLipS1

cl. Symbol smoothing produces

(3.15) Ak = A#k + Ab

k, A# ∈ ØPS11,δ = OPS1

1,δ, Abk ∈ ØPLipS1−δ

1,δ .

Note that ØPLipSm1,δ consists of adjoints of elements of OP LipSm

1,δ, so instead of(3.8) we have

(3.16) p(ξ, y) ∈ LipSm1,δ =⇒ p(D,x) : Lp

s → Lps−m, −1 ≤ s < 1− δ.

This time, let E# ∈ OPS−21,δ denote a parametrix for

∑

A#k ∂k. Then, if u ∈ Lp and

Lu = f ∈ Lq we have

(3.17) u = E#f −∑

k

E#Abk∂ku−

∑

j

E#Bj∂ju + E#V u, mod C∞.

Now

(3.18) u ∈ Lp =⇒ Abk∂ku ∈ Lp

−2+δ =⇒ E#Abk∂ku ∈ Lp

δ .

To treat terms like E#Bj∂ju, note that ∂ju ∈ Lp−1 and Bj ∈ Lr

1 for each j. Now,the inclusion

(3.19) Lp−1 · Lr

1 → Lp−1

14

is true for r/(r− 1) n. Thus, E#Bj∂ju ∈ Lp1. Also, V u ∈

Lp∗ where 1/p∗ := 1/r + 1/p, by Holder’s inequality so that E#V u ∈ Lp∗

2 → Lp1,

by Sobolev’s embedding theorem.We see that the right side of (3.17) then belongs to Lp

δ , if δ ∈ (0, 1), so u ∈ Lpδ for

δ < 1. This reduces our problem to the case already treated, so (3.3) is established.The rest of the proposition, with stronger hypotheses on the coefficients of L

and weaker hypotheses on u, has a similar proof. In fact, to get started, it sufficesto assume u ∈ L1+ε

−δ,loc, for some (small) ε, δ > 0.

We now produce some weighted estimates, which will be very useful for subse-quent analysis.

Proposition 3.2. Let Ω be a Lipschitz domain in M and take s ≤ 1, r/(r − 1) <p < r. For u ∈ Lp

2,loc(Ω) satisfying Lu = 0 in Ω we have, under the hypothesis(3.2),

(3.20)

∫

Ω

dist(x, ∂Ω)(1−s)p |∇2u(x)|p dVol(x)

≤ C∫

Ω

dist(x, ∂Ω)−sp |∇u(x)|p dVol(x) + C∫

Ω

|u|p dVol.

Proof. Denote by Qr(x) the cube in Rn centered at x and whose side length is r.We claim that for each arbitrary, fixed x0 ∈ Ω and 0 < ρ < 16

9 n−1/2 dist (x0, ∂Ω)there holds

(3.21)∫

Qρ(x0)

|∇2u|p dVol ≤ Cρ−p∫

Q9ρ/8(x0)

|∇u|p dVol + C∫

Q9ρ/8(x0)

|V u|p dVol.

Passing from (3.21) to (3.20) is done by multiplying both sides with an appropriatepower of ρ and summing over cubes in a Whitney decomposition of Ω. Specifi-cally, let Qii∈I be an open covering of Ω with cubes Qi ⊆ Ω so that 9

8Qi ⊆ Ω,dist (Qi, ∂Ω) ≈ ρi := diam Qi and the collection 9

8Qii∈I has the finite intersec-tion property; cf. [St]. Then, applying (3.21) to each Qi, multiplying both sidesby ρ(1−s)p

i and summing up the resulting inequalities yields a version of (3.20) withthe only difference that, in the current scenario, the last integral is ‖V u‖p

Lp(Ω).To further transform this we use Holder’s inequality and a Sobolev-type estimate

to write(

∫

Ω

|V u|p dVol)1/p

≤ C(

∫

Ω

|u|p∗dVol

)1/p∗

≤ C(

∫

Ω

|∇u|p dVol)1/p

+(

∫

Ω

|u|p dVol)1/p

(3.22)

15

where 1/p∗ := 1/p − 1/r. Note that for the second estimate to hold we need1 ≤ p ≤ p∗ ≤ ∞ and 1/p − 1/p∗ < 1/n (cf., e.g., Lemmas 7.12 and 7.16 in [GT]).These are true since we are assuming p ∈ (1, r) and r > n. Now, based on theabove discussion, (3.22) can be used to conclude (3.20).

To prove (3.21), use x0 as the center of a coordinate system and, as in [MT],[MMT], introduce the dilation operators vρ(x) := v(ρx), x ∈ Q9/8 := Q9/8(x0).Thus, if Lu = 0, then uρ satisfies the equation

(3.23) ∂jAjkρ ∂kuρ + ρBj

ρ∂juρ = fρ,

where

(3.24) Ajkρ (x) :=

(

aαβjk (ρx)

)

α,β , Bjρ(x) :=

(

bαβj (ρx)

)

α,β , Vρ(x) :=(

vαβ(ρx))

α,β ,

and

(3.25) fρ := ρ2Vρuρ.

Furthermore, the equation (3.23) holds on Q9/8, for ρ in some interval (0, ρ0], onwhich the collection Ajk

ρ is uniformly Lipschitz and ρBjρ is bounded in Lr

1, sincer > n.

The regularity result in Proposition 3.1 then gives, as long as p ∈ (r/(r − 1), r),

(3.26)∫

Q1

|∇2uρ|p dVol ≤ Cp

∫

Q9/8

[

|fρ|p + |uρ|p]

dVol,

for any solution to (3.23) on Q9/8. Now if uρ solves (3.23), so does uρ − a for anyconstant (vector) a, so (3.26) holds with uρ replaced by uρ − a, for any a. Take ato be the average of uρ on Q9/8 and use Poincare’s inequality

(3.27)∫

Q9/8

|uρ − a|p dVol ≤ Cp

∫

Q9/8

|∇uρ|p dVol,

to get

(3.28)∫

Q1


∫

Q9/8

|fρ|p dVol + Cp

∫

Q9/8

|∇uρ|p dVol,

for any solution uρ to (3.23) on Q9/8. Now, if Lu = 0, then uρ satisfies (3.23) withfρ given by (3.25), so we have

(3.29)∫

Q1


∫

Q9/8

|∇uρ|p dVol + Cpρ2p∫

Q9/8

|Vρuρ|p dVol.

In turn, this estimate is equivalent to (3.21) and the proof of (3.20) is finished.

16

Proposition 3.3. If we strengthen (3.2) to

(3.30) aαβjk ∈ Lip, bαβ

j ∈ Lr1, vαβ ∈ Lr, for some r ≥ 2n,

then the conclusion in Proposition 3.2 holds for 1 ≤ p < r.

In passing, let us point out that (3.30) is automatically fulfilled by the Hodge-Laplacian ∆`, 1 ≤ ` ≤ n, if the metric tensor satisfies (1.29), and by ∆ − V onscalars given (1.27).

Proof. Let 1 ≤ p ≤ r/(r−1) and let q ∈ (r/(r−1), r) be arbitrary. In this case, weknow by Proposition 3.1 that u ∈ Lq

loc(Ω), Lu ∈ Lqloc(Ω) ⇒ u ∈ Lq

2,loc(Ω). Holder’sinequality plus a quantitative form of this implication give

(3.31)(∫

Q1

|∇2uρ|p dVol)1/p

≤ C(

∫

Q9/8

[

|fρ|q + |uρ|q]

dVol)1/q

,

for any solution to (3.23). Again we can replace uρ by uρ − a. This time use

(3.32)(

∫

Q9/8

|uρ − a|q dVol)1/q

≤ C(

∫

Q9/8

|∇uρ|q∗dVol

)1/q∗

,

provided

(3.33) 1 ≤ q∗ ≤ q ≤ ∞ and1q∗− 1

q<

1n

(see, e.g., Lemmas 7.12 and 7.16 in [GT]), to get that, for any solution to (3.23),

(3.34)(

∫

Q1

|∇2uρ|p dVol)1/p

≤ C(

∫

Q9/8

|∇uρ|q∗dVol

)1/q∗

+ C(

∫

Q9/8

|fρ|q dVol)1/q

.

Thus, if (3.25) holds, we have(

∫

Q1

|∇2uρ|p dVol)1/p

≤C(

∫

Q9/8

|∇uρ|q∗dVol

)1/q∗

+ Cρ2(

∫

Q9/8

|Vρuρ|q dVol)1/q

.(3.35)

Now use(

∫

Q9/8

|Vρuρ|q dVol)1/q

≤ Cρ−n/r(

∫

Q9/8

|uρ|q dVol)1/q

≤ Cρ−n/r(

∫

Q9/8

|∇uρ|p + |uρ|p dVol)1/p

,(3.36)

17

where 1/q := 1/q − 1/r and the last inequality is Sobolev’s. For this to hold, weneed

(3.37) 1 ≤ p ≤ q ≤ ∞ and1p− 1

q<

1n

.

Granted (3.37) and the possibility of choosing q so that

(3.38) p = q∗

we get

(3.39)∫

Q1

|∇2uρ|p dVol ≤ C∫

Q9/8

|∇uρ|p dVol + Cρ(2−n/r)p∫

Q9/8

|uρ|p dVol,

and, further,

(3.40)∫

Qρ

|∇2u|p dVol ≤ Cρ−p∫

Q9ρ/8

|∇u|p dVol + Cρ−np/r∫

Q9ρ/8

|u|p dVol.

As before, this latter inequality leads to

(3.41)

∫

Ω

dist(·, ∂Ω)(1−s)p |∇2u|p dVol

≤ C∫

Ω

dist(·, ∂Ω)−sp |∇u|p dVol + C∫

Ω

dist(·, ∂Ω)(1−s−n/r)p |u|p dVol

and, further, to (3.20), via the elementary estimate

(3.42)∫

Ω

dist (·, ∂Ω)−αp|F |p dVol ≤ C∫

Ω

|F |p dVol + C∫

Ω

|∇F |p dVol,

which is valid for 1 ≤ p < ∞ and α < 1/p. For this program to work it suffices tohave

(3.43)nr

<1p.

The conclusion is that (3.20) holds for 1 ≤ p ≤ r/(r− 1) if, given such a p, one canchoose q ∈ (r/(r−1), r) so that (3.33), (3.37), (3.38) and (3.43) are satisfied. Someelementary algebra shows that this is indeed the case provided r ≥ 2n.

For later reference we record below two more related estimates.

18

Proposition 3.4. Assume p ∈ (r/(r − 1), r) and let u ∈ Lploc(Ω) satisfy Lu = 0

in Ω. Here L is as in (3.1) and its coefficients are as in (3.2). Then, for any ballBρ(x) with x ∈ Ω, 0 < ρ < 1

2 dist (x, ∂Ω), there holds

ρ‖∇u‖L∞(Bρ(x)) ≤ C( 1

ρn

∫

B2ρ(x)|u(z)− u(x)|p dVol(z)

)1/p

+ Cρ2−n/r|u(x)|.(3.44)

In particular, for p as above, for any α ∈ R we have

(3.45)∫

Ω

dist(x, ∂Ω)α+p |∇u(x)|p dVol(x) ≤ C∫

Ω

dist(x, ∂Ω)α |u(x)|p dVol(x).

These results also hold for 1 ≤ p < ∞ if the coefficients of L satisfy (3.4).

Proof. Note that under the current smoothness assumptions for the coefficients ofL, in each coordinate patch O, Proposition 3.1 gives

(3.46) u ∈ Lploc(O), Lu ∈ Lq

loc(O) ⇒ u ∈ C1+εloc (O) some ε > 0, if q >

n2

.

A natural estimate accompanies (3.46) also. As before, we apply (3.46) to theequation

(3.47) ∂jAjkρ ∂k(uρ − a) + ρBj

ρ∂j(uρ − a) + ρ2Vρ(uρ − a) = fρ,

where, this time, fρ := aρ2Vρ and a ∈ R. Note that ρ2Vρ; ρ ∈ (0, ρ0] is boundedin Lr and that ‖fρ‖Lq(O) ≤ C a ρ2−n/r. Choosing a := u(x) yields (3.44) after arescaling.

Finally, (3.45) is a simple consequence of (3.44), and the last part of the statementfollows similarly with the help of (the last part in) Proposition 3.1.

4. Sobolev and Besov spaces on Lipschitz domains

The theory of Besov spaces and Sobolev spaces on Lipschitz domains in Euclideanspace has been thoroughly treated in [JK2]. Other references for various aspects ofthe theory include [BL], [Pe], [BS], [Tr], and [Gr]. We recall some of these resultshere, to fix notation, and we indicate extensions to the manifold setting.

We recall that, if Ω is a Lipschitz domain in Rn, then, for 1 ≤ p, q < ∞ and0 < s < 1, one definition of the Besov space Bp,q

s (∂Ω) is the collection of allmeasurable functions f on ∂Ω such that

(4.1)

‖f‖Bp,qs (∂Ω)

:= ‖f‖Lp(∂Ω) +

∫

∂Ω

(

∫

∂Ω

|f(x)− f(y)|p

|x− y|(n−1+sq)p/q dσ(y))q/p

dσ(x)

1/q

< ∞.

19

In the special situation when p = q, it is customary to simplify the notation a bitand write Bp

s (∂Ω) in place of Bp,ps (∂Ω).

The case when p = ∞ for the “diagonal” scale corresponds to the non-homogeneousversion of the space of Holder continuous functions on ∂Ω. More precisely, B∞

s (∂Ω),0 < s < 1, is defined as the Banach space of measurable functions on f so that

(4.2) ‖f‖B∞s (∂Ω) := ‖f‖L∞(∂Ω) + supx,y∈∂Ω

x 6=y

|f(x)− f(y)||x− y|s

< ∞.

Also, recall that Bp−s(∂Ω) := (Bq

s(∂Ω))∗ for each 0 < s < 1, 1 < p ≤ ∞ andq = (1− 1

p )−1. Let Lip (∂Ω) denote the collection of all Lipschitz functions on ∂Ω.Next, we consider the case when Ω is a Lipschitz domain in the compact Rie-

mannian manifold M . It is then natural to say that f belongs to Bp,qs (∂Ω) for

some 1 ≤ p, q ≤ ∞ and 0 < s < 1 if (and only if) for any smooth chart (O, Φ)and any smooth cut-off function θ ∈ C∞comp(O), (fθ) Φ−1 ∈ Bp,q

s (Φ(O ∩ ∂Ω)).The following elementary lemma allows for transferring many results about Besovspaces originally proved in the classical Euclidean setting to the case of boundariesof Lipschitz domains in Riemannian manifolds.

Lemma 4.1. Let (Oi, Φi)i∈I be a finite set of local bi-Lipschitz charts so that∂Ω ⊆ ∪i∈IOi and Φi(∂Ω ∩Oi) = Rn−1. Also, consider (χi)i∈I a partition of unitysubordinate to (Oi)i∈I . Finally, for a measurable function f on ∂Ω denote by Fi

the extension by zero outside its support of (fχi) Φ−1i , for each i ∈ I.

Then, for 1 ≤ p, q ≤ ∞ and 0 < s < 1, the following two statements areequivalent:

(1) f ∈ Bp,qs (∂Ω);

(2) Fi ∈ Bp,qs (Rn−1) for each i ∈ I.

Moreover, if these conditions hold, then

(4.3) ‖f‖Bp,qs (∂Ω) ≈

∑

i∈I

‖Fi‖Bp,qs (Rn−1).

To illustrate the point made before the statement of this result we note thatfrom well known interpolation results in Rn the following can be proved. First,recall that [·, ·]θ and (·, ·)θ,p denote, respectively, the brackets for the complex andreal interpolation method (cf., e.g., [Ca], [BL], [BS]).

Proposition 4.2. For 0 < θ < 1, 1 ≤ p0, p1, q0, q1 ≤ ∞ and 0 < s0, s1 < 1, thereholds

(4.4) [Bp0,q0s0

(∂Ω), Bp1,q1s1

(∂Ω)]θ = Bp,qs (∂Ω)

where 1p := 1−θ

p0+ θ

p1, 1

q := 1−θq0

+ θq1

and s := (1− θ)s0 + θs1.

20

A similar result is valid for −1 < s0, s1 < 0, and for the real method of interpo-lation.

Another result of interest for us is an atomic characterization of the Besov spaceB1

s (∂Ω). First, we shall need a definition. For 0 < s < 1, a B1s (∂Ω)-atom is a

function a ∈ Lip(∂Ω) with support contained in Br(x0) ∩ ∂Ω for some x0 ∈ ∂Ω,r ∈ (0, diamΩ], and satisfying the normalization conditions

(4.5) ‖a‖L∞(∂Ω) ≤ rs−n+1, ‖∇tana‖L∞(∂Ω) ≤ rs−n.

Here ∇tan stands for the tangential gradient operator on ∂Ω and, throughout thepaper, distances on M are considered with respect to some fixed, smooth back-ground Riemannian metric g0. The atomic theory of [FJ] lifted to ∂Ω gives thefollowing.

Proposition 4.3. Let 0 < s < 1 and f ∈ B1s (∂Ω). Then there exist a sequence of

B1s (∂Ω)-atoms ajj and a sequence of scalars λjj ∈ `1 such that

(4.6) f =∑

j≥0

λjaj ,

with convergence in B1s (∂Ω), and

(4.7) ‖f‖B1s(∂Ω) ≈ inf

∑

j≥0

|λj | : f =∑

j≥0

λjaj , aj B1s (∂Ω)-atom, (λj)j ∈ `1

.

In the sequel, we shall also need to work with the Besov spaces B1−s(∂Ω), s ∈

(0, 1). Inspired by the corresponding atomic characterization from [FJ], set

(4.8) B1−s(∂Ω) := C +

f =∑

j≥0

λjaj : aj B1−s(∂Ω)-atom, (λj)j ∈ `1

,

where the series converges in the sense of distributions, and C is the space of constantfunctions on ∂Ω. In this context, a B1

−s(∂Ω)-atom, 0 < s < 1, is a function a ∈L∞(∂Ω) with support contained in Br(x0)∩∂Ω for some x0 ∈ ∂Ω, 0 < r < diam Ω,and satisfying

(4.9)∫

∂Ω

a dσ = 0, ‖a‖L∞(∂Ω) ≤ r−s−n+1.

Furthermore, for f ∈ B1−s(∂Ω), 0 < s < 1,

(4.10) ‖f‖B1−s(∂Ω) := inf

‖g‖L∞ +∑

j≥0

|λj | : f = g +∑

j≥0

λjaj

21

where g ∈ C, aj ’s and (λj)j are as in (4.8). Parenthetically, observe that ∇tan :B1

s (∂Ω) → B1s−1(∂Ω) is a bounded operator; this is trivially checked using the

above atomic decompositions.Next, we include a brief discussion of the Besov and Sobolev classes in the interior

of a Lipschitz domain Ω ⊂ M . First, for 1 ≤ p, q ≤ ∞, s > 0, the Besov spaceBp,q

s (M) is defined by localizing and transporting via local charts its Euclideancounterpart, i.e., Bp,q

s (Rn) (for the latter see, e.g., [Pe], [BL], [BS], [Tr], [JW]).Going further, Bp,q

s (Ω) consists of restrictions to Ω of functions from Bp,qs (M).

This is equipped with the natural norm, i.e., defined by taking the infimum of the‖ · ‖Bp,q

s (M)-norms of all possible extensions to M . The spaces Bp,ps (Ω) will be

abbreviated as Bps (Ω).

Using Stein’s extension operator and then invoking well known real interpolationresults (cf., e.g., [BL]), it follows that for any Lipschitz domain Ω ⊂ M ,

(4.11) (Bp0,q0s0

(Ω), Bp1,q1s1

(Ω))θ,p = Bp,qs (Ω)

if 1p = 1−θ

p0+ θ

p1, 1

q = 1−θq0

+ θq1

, s = (1−θ)s0 +θs1, 0 < θ < 1, 1 ≤ p0, p1, q0, q1 ≤ ∞,s0 6= s1, s0, s1 > 0.

A similar discussion applies to the Sobolev (or potential) spaces Lps(Ω), this time

starting with the potential spaces Lps(M) (lifted to M from Rn via an analogue of

Lemma 4.1; for the latter context see, e.g., [St], [BL]). For s ∈ R we define thespace Lp

s,0(Ω) to consist of distributions in Lps(M) supported in Ω (with the norm

inherited from Lps(M)). It is known that C∞comp(Ω) is dense in Lp

s,0(Ω) for all valuesof s and p.

Recall (cf. [JW]) that the trace operator

(4.12) Tr : Lps(Ω) −→ Bp

s− 1p(∂Ω)

is well defined, bounded and onto if 1 < p < ∞ and 1p < s < 1 + 1

p . Thisalso has a bounded right inverse whose operator norm is controlled exclusivelyin terms of p, s and the Lipschitz character of Ω. Similar results are valid forTr : Bp,q

s (Ω) → Bps− 1

p(∂Ω). In this latter case we may allow 1 ≤ p ≤ ∞; cf. [BL].

Next, if 1 < p < ∞ and 1p < s < 1 + 1

p , the space Lps,0(Ω) is the kernel of the

trace operator Tr acting on Lps(Ω). This follows from the Euclidean result [JK2;

Proposition 3.3]. In fact, for the same range of indices, Lps,0(Ω) is the closure of

C∞comp(Ω) in the Lps(Ω) norm.

For positive s, Lp−s(Ω) is defined as the space of linear functionals on test func-

tions in Ω equipped with the norm

(4.13) ‖f‖Lp−s(Ω) := sup

|〈f, g〉| : g ∈ C∞comp(Ω), ‖g‖Lqs(M) ≤ 1

where tilde denotes the extension by zero outside Ω and 1p + 1

q = 1. For all valuesof p and s, C∞(Ω) is dense in Lp

s(Ω). Also, for any s ∈ R,

(4.14) Lq−s,0(Ω) = (Lp

s(Ω))∗ and Lp−s(Ω) =

(

Lqs,0(Ω)

)∗.

22

For later reference, let us point out that

(4.15) Lps−1+ 1

p(Ω) =

(

Lq−s+ 1

q(Ω)

)∗, Lq

−s+ 1q(Ω) =

(

Lps−1+ 1

p(Ω)

)∗,

for 0 < s < 1 and 1 < p, q < ∞ with 1p + 1

q = 1. These follow from (4.14) andthe fact that Lp

α,0(Ω) = Lpα(Ω) for 0 ≤ α < 1

p , 1 < p < ∞ (the latter can be easilydeduced from Proposition 3.5 in [JK2]).

We shall also need the fact that the exterior derivative operator

(4.16) d : Lps(Ω,Λ`TM) −→ Lp

s−1(Ω, Λ`+1TM)

is well defined and bounded for 0 ≤ ` ≤ n, 1 < p < ∞ and s ≥ 0. See, e.g., [FMM]for a discussion. A similar result holds for δ, the formal adjoint of d, provided themetric tensor on M is sufficiently smooth. Here, Lp

s(Ω, Λ`TM) stands for the spaceof `-forms on Ω with coefficients in Lp

s(Ω).As for Lp

s(∂Ω), 1 < p < ∞, −1 ≤ s ≤ 1, define this to be Lp(∂Ω) when s = 0and f ∈ Lp(∂Ω); ∇tanf ∈ Lp(∂Ω) when s = 1. The case when 0 < s < 1 can behandled by defining Lp

s(∂Ω) by complex interpolation:

(4.17) Lps(∂Ω) := [Lp(∂Ω), Lp

1(∂Ω)]s.

Finally, when −1 ≤ s < 0, set

Lps(∂Ω) :=

(

Lq−s(∂Ω)

)∗,

1q

+1p

= 1.

Going further, it is well known that Lps(Ω), Lp

s,0(Ω), Lp−s(Ω), Lp

−s,0(Ω), Lps(∂Ω)

and Lp−s(∂Ω) are complex interpolation scales for 1 < p < ∞ and nonnegative s

(in the case of the last two scales we also require that s ≤ 1). Also, the Besov andSobolev spaces on the domain are related via real interpolation. For instance, wehave the formula

(4.18) (Lp(Ω), Lpk(Ω))s,q = Bp,q

sk (Ω)

when 0 < s < 1, 1 0,

(4.19)1 < p ≤ 2 =⇒ Bp,p

s (Ω) → Lps(Ω) → Bp,2

s (Ω),

2 ≤ p < ∞ =⇒ Bp,2s (Ω) → Lp

s(Ω) → Bp,ps (Ω).

Also, the same inclusions hold with Ω replaced by ∂Ω.A more detailed discussion and further properties of these spaces, as well as

proofs for some of the statements in this paragraph for Euclidean domains canbe found in [BL], [BS], [JK2]. We consider next yet other characterizations ofmembership in the classes of Sobolev and Besov spaces, together with some of theirconsequences.

23

Proposition 4.4. Let Ω be a Lipschitz domain in M and fix 1 ≤ p ≤ ∞, k ∈ 0, 1and a function u in Ω so that ∇ju ∈ Lp

loc(Ω) for 0 ≤ j ≤ k+1. Then, for 0 < s < 1,

(4.20) dist (·, ∂Ω)1−s|∇k+1u|+ |∇ku|+ |u| ∈ Lp(Ω) =⇒ u ∈ Bpk+s(Ω).

Suppose next that 1 2 assume further thats 6= 1/p. Then

(4.21) dist (·, ∂Ω)1−s|∇k+1u|+ |∇ku|+ |u| ∈ Lp(Ω) =⇒ u ∈ Lpk+s(Ω).

Finally, if 0 < s < 1p < 1, then

(4.22) u ∈ Bpk+s(Ω) =⇒ dist (·, ∂Ω)−s|∇ku|+ |u| ∈ Lp(Ω).

Moreover, naturally accompanying estimates are valid in each case.

Proof. The implications (4.20)–(4.21) are proved in [JK2]. This was done in theEuclidean setting but it can easily be adapted to manifolds. Also, the implica-tion (4.22) follows by observing that u ∈ Bp

k+s(Ω) entails ∇ku ∈ Bps (Ω) and then

invoking [Gr], Theorems 1.4.2.4 and 1.4.4.4.

We conclude this section with a discussion of membership in the classes ofSobolev and Besov spaces for null-solutions of elliptic PDE’s. First, we need afew lemmas.

Lemma 4.5. Let L be as in (3.1) and assume that its coefficients satisfy (3.2).Assume u ∈ Bp

s (Ω), for some p ∈ (r/(r − 1), r), s ∈ (0, 1), and Lu = 0 in Ω. Then

(4.23) dist (·, ∂Ω)1−s|∇u| ∈ Lp(Ω).

The same conclusion holds if p = ∞. Also, if the coefficients of L satisfy (3.4) then(4.23) is actually true for 1 ≤ p ≤ ∞.

Proof. Note that it suffices to estimate dist (x, ∂Ω)1−s|∇u(x)| for x near ∂Ω. If0 < ρ < dist (x, ∂Ω) then, working in local coordinates, Proposition 3.4 gives

dist (x, ∂Ω) |∇u(x)| ≤C(

ρsp∫

Bρ(x)

|u(z)− u(x)|p

|z − x|n+sp dz)1/p

+ C dist (x, ∂Ω)2−n/r|u(x)|(4.24)

so that

(4.25)(

dist (x, ∂Ω)1−s|∇u(x)|)p ≤ C

∫

Bρ(x)

|u(z)− u(x)|p

|z − x|n+sp dz + C|u(x)|p.

Since u ∈ Bps (Ω), this readily yields a local version of (4.23), near ∂Ω. Away from

∂Ω, the desired conclusion follows from (3.45).Next, if p = ∞, then (4.23) is a simple consequence of (3.44). The last part of

the statement of the lemma can be proved in a similar fashion.

24

Lemma 4.6. Let L be as in (1.2), with Lipschitz metric tensor, and assume thatu ∈ Lp

s(Ω), with 1 < p < ∞ and 0 ≤ s ≤ 1, satisfies Lu = 0 in Ω. Then

(4.26) dist (·, ∂Ω)1−s|∇u| ∈ Lp(Ω).

Proof. First note that (4.26) is obvious when s = 1 and that (the last part in)Proposition 3.4 furnishes (4.26) when s = 0. In particular, the assignment u 7→ ∇uis continuous from H ∩ Lp(Ω) and H ∩ Lp

1(Ω) into the weighted Lebesgue spaceLp(Ω, dist (·, ∂Ω)p dVol) and Lp(Ω), respectively, where we set

(4.27) H = u ∈ Lploc(Ω) : Lu = 0,

a space that is independent of p ∈ (1,∞), by (the last part in) Proposition 3.1.Interpolating with change of measure (cf. [SW]) gives that the map

(4.28) ∇ : [H ∩ Lp(Ω),H ∩ Lp1(Ω)]s −→ Lp(Ω,dist (·, ∂Ω)(1−s)p dVol)

is continuous for 0 ≤ s ≤ 1. At this stage, the desired result is a consequence ofthe fact that

(4.29) [H ∩ Lp(Ω),H ∩ Lp1(Ω)]s = H ∩ Lp

s(Ω), 0 ≤ s ≤ 1.

This concludes the proof of the lemma, modulo that of (4.29) which is treatedseparately below.

The result (4.29) is proven by an argument similar to one used in the proofof Theorem 4.2 in [JK2], for L = ∆0, the flat-space Laplacian. We present theargument here, in order to make clear what properties we need on the metrictensor.

Lemma 4.7. Whenever the metric tensor is Lipschitz, the result (4.29) is true forall p ∈ (1,∞).

Proof. To begin with, note that the claim about (1.3) extends, via (the last partin) Proposition 3.1, to include the fact that

(4.30) L : Lp2(M) −→ Lp(M) is an isomorphism ∀ p ∈ (1,∞).

Next, for j = 0, 1, let E : Lpj (Ω) → Lp

j (M) denote Stein’s extension operator, andlet R : Lp

j (M) → Lpj (Ω) denote restriction. Then set

(4.31) D = LE : Lpj (Ω) → Lp

j−2(M), G = RL−1 : Lpj−2(M) → Lp

j (Ω).

To justify the mapping property stated for G, note that (4.30) implies L−1 :Lp(M) → Lp

2(M), so by duality L−1 : Lp−2(M) → Lp(M), 1 < p < ∞.

One sees that GD = I on Lpj (Ω), while Q := DG satisfies Q2 = Q and f ∈

Lpj−2(M) ⇒ Qf = f on Ω, i.e.,

(4.32) I −Q : Lpj−2(M) −→ Lp

j−2,0(M \ Ω), j = 1, 2.

Also, given u ∈ Lpj (Ω) we have u ∈ H if and only if Du ∈ Lp

j−2,0(M \Ω) for j = 1, 2.With this set-up, Theorem 14.3 of [LM] applies to give (4.29).

Finally, we present the following result, which for the flat-space Laplacian wasestablished in §4 of [JK2]:

25

Proposition 4.8. Let L be as in (1.2), with Lipschitz metric tensor, and considera function u so that Lu = 0 in Ω. Then, if 1 2, assume further thats 6= 1/p. Then

(4.34) u ∈ Bps (Ω) =⇒ u ∈ Lp

s(Ω).

Proof. This is a direct consequence of Lemmas 4.5–4.6 and Proposition 4.4.

In order to state our last result in this section, recall that the nontangentialmaximal operator acts on a section u defined in the Lipschitz domain Ω, by

(4.35) u∗(x) := sup |u(y)| : y ∈ γ(x), x ∈ ∂Ω.

Here, γ(x) ⊂ Ω is a suitable nontangential approach region with “vertex” at x ∈ ∂Ω.Our aim is to relate L2-estimates on the nontangential maximal function to

membership in the Sobolev space L21/2 for null-solutions of L. We have

Proposition 4.9. Consider L as in (3.1), a second order strongly elliptic, formallyself-adjoint operator so that

(4.36) aαβjk ∈ C1+γ , γ > 0, bαβ

j ∈ Lr1, vαβ ∈ Lr, r > n.

Then, for each u so that Lu = 0 in Ω, there holds

(4.37) u ∈ L21/2(Ω) ⇐⇒ u∗ ∈ L2(∂Ω).

Also, naturally accompanying estimates are valid.

Proof. We start with the right-to-left implication in (4.37). As a consequence ofthe well-posedness of the Dirichlet problem for L− λ (where the real parameter λis assumed to be large) and the mapping properties of the layer potential operatorsinvolved in the integral representation of the solution, the estimate

(4.38)∫

Ω

dist(x, ∂Ω) |∇u(x)|2 dVol(x) ≤ C∫

∂Ω

|u∗|2 dσ,

uniformly in u satisfying Lu = 0 in Ω, has been established in [MMT]. See, e.g.,(3.55) and Corollary 3.2 in that paper. In concert with (4.19), this readily yieldsthe right-to-left implication in (4.37).

As for the opposite implication, since the problem has local character, there isno loss of generality in assuming that Ω is small, e.g., contained in some coordinate

26

patch. Moreover, so we claim, it suffices to consider the case when u, satisfyingLu = 0 in Ω, also belongs to C0(Ω)∩L2

1(Ω) and, for each ε > 0, prove the estimate

(4.39)∫

∂Ω

|u|2 dσ ≤ Cε

∫

Ω

(dist(x, ∂Ω) |∇u(x)|2 + |u(x)|2) dVol(x) + ε∫

∂Ω

|u∗|2 dσ.

Hereafter we shall work in local coordinates and the constant Cε is supposed todepend exclusively on the Lipschitz nature of Ω, ε and L.

To justify the claim, let us show how (4.39) can be used to finish the proof ofthe left-to-right implication in (4.37). Indeed, granted (6), Lemma 4.5 and a simplelimiting argument which utilizes an approximating sequence Ωj Ω with boundedLipschitz character lead to

(4.40) supj‖u‖L2(∂Ωj) ≤ Cε‖u‖L2

1/2(Ω) + ε‖u∗‖L2(∂Ω).

In order to continue, we invoke an estimate from [MMT], to the effect that

(4.41) ‖u‖L2(∂Ω) ≈ ‖u∗‖L2(∂Ω),

uniformly for null-solutions u ∈ C0(Ω) of L, with constants depending only on theLipschitz nature of Ω and L. This is a consequence of Theorem 3.1 in [MMT] (cf.especially the estimate (3.5) there) and has been established under the requirementthat the operator L satisfies the non-singularity hypothesis:

(4.42) ∀D(⊆ M) Lipschitz domain, u ∈ L21,0(D), Lu = 0 =⇒ u = 0 in D.

Let us assume for now that this is the case and explain how the proof of theproposition can be finished.

With Ωj in place of Ω, (4.41) readily gives that, in our current setting, ‖u∗‖L2(∂Ω) ≈supj ‖u‖L2(∂Ωj). Utilizing this back in (4.40) and choosing ε small enough, estab-lishes the estimate ‖u∗‖L2(∂Ω) ≤ C(∂Ω)‖u‖L2

1/2(Ω), as desired.Finally, (4.39) can be established as in §2 of [DKPV]. The approach devised in the

aforementioned paper is sufficiently flexible, as it rests on the following ingredients:(i) the assumptions of symmetry and strong ellipticity for L, (ii) interior estimatesfor L, (iii) the existence of an adapted distance function (see p. 1432-1433 in [DKPV]for details), and (iv) integrations by parts, Schwarz’s inequality and the standardCarleson estimate. All these are available in our case (cf. §3 for (ii)) and theargument in [DKPV] can be adapted to our situation to yield (4.39). We leave thedetails to the interested reader.

At this point we return to (4.42) and indicate how one can dispose off thisextra assumption. The first step is to observe that if L is negative-definite on Mthen (4.42) is automatically verified. Indeed, if D ⊂ M is a Lipschitz domainand u ∈ L2

1,0(D) is such that Lu = 0 in D then, with tilde denoting extension

27

by zero outside D, we see that u ∈ L21(M) satisfies supp Lu ⊆ ∂D. In particular,

〈Lu, u〉 = 0 due to support considerations. Since we are assuming L < 0 on M ,this forces u = 0, as desired.

In the second step we shall show how L can be altered off Ω so that it becomesnegative-definite on M . In the process, we can (and will) assume that Ω is verysmall (relative to L, in a sense made more precise below). To this end, fix someconstant A ∈ (0,∞) so that, in the current setting, L − A is negative-definite onL2(M). Also, pick B > A and set Bj = Bj(x) = B(1− χΩj ) where, for some fixedp ∈ Ω, Ωj = x ∈ M : dist (x, p) < 1/j.

Here is the claim which finishes the proof of the proposition.

(4.43) L−Bj is negative-definite on M for large j.

Since this is a consequence of 0 < (−L+Bj +A)−1 < A−1 (all operators consideredon L2(M)) and standard functional analysis, it suffices to prove that

(4.44)(L−Bj −A)−1 → (L−B −A)−1,

in L2-operator norm, as j →∞.

To see this, given f ∈ L2(M) set uj = (L−Bj −A)−1f and u = (L−B −A)−1f .We have ‖uj‖L2 ≤ ‖f‖L2 . Also, the fact that

(4.45) (L−A)uj = Bjuj + f

is bounded in L2(M) entails ‖uj‖L21≤ C0‖f‖L2 . Now (L−A)(u−uj) = Bu−Bjuj =

B(u − uj) + (B − Bj)uj which implies (L − A − B)(u − uj) = (B − Bj)uj and,further,

(4.46) ‖u− uj‖L2 ≤ B−1‖(B −Bj)uj‖L2 .

Note that, generally speaking,

(4.47) ‖(B −Bj)v‖L2 ≤ δjB‖v‖L21,

with δj → 0 as j →∞. Hence

(4.48) ‖u− uj‖L2 ≤ B−1‖(B −Bj)uj‖L2 ≤ C0δj‖f‖L2 ,

and (4.44) is proven.

5. The Banach envelope of atomic Hardy spaces

28

Fix an arbitrary Lipschitz domain Ω in M and for each p ∈ ((n − 1)/n, 1) sets := (n − 1)( 1

p − 1). Note that 0 ≤ s < 1. A function a ∈ L∞(∂Ω) is called anHp(∂Ω)-atom if

∫

∂Ω a dσ = 0 and, for some boundary point x0 ∈ ∂Ω and some0 < r < diamΩ,

(5.1) supp a ⊆ ∂Ω ∩Br(x0), ‖a‖L∞(∂Ω) ≤ r−(n−1)/p.

Recall that C is the space of constant functions on ∂Ω. For (n− 1)/n < p < 1 (andp = 1, respectively) we recall that the atomic Hardy space Hp(∂Ω) is defined as thevector subspace of

(

B∞s (∂Ω)/C

)∗(and L1(∂Ω), respectively) consisting of all linear

functionals f that can be represented as

(5.2) f =∑

j≥0

λjaj , (λj)j ∈ `p, aj ’s are Hp(∂Ω)-atoms,

in the sense of convergence in(

B∞s (∂Ω)

)∗(and L1(∂Ω), respectively). We equip

Hp(∂Ω) with the (quasi-)norm

(5.3) ‖f‖Hp(∂Ω) := inf

(∑

j≥0

|λj |p)1/p

: f =∑

j≥0

λjaj as in (5.2)

.

We shall also need the localized version of Hp(∂Ω), i.e., hp(∂Ω) ⊆ (B∞s (∂Ω))∗

defined by

(5.4) hp(∂Ω) := Hp(∂Ω) + C = Hp(∂Ω) + Lq(∂Ω), ∀ q > 1.

As is well known,

(5.5) (hp(∂Ω))∗ =

B∞s (∂Ω), if p < 1,

bmo (∂Ω), if p = 1.See, e.g., [CW] for a more detailed account. It is also well-known that

(5.6)hp(∂Ω)(n−1)/n<p≤1 ∪ L

p(∂Ω)1<p<∞

is an interpolation scale for the complex method.

See the discussion in [KM].Next we discuss the “minimal enlargement” of hp(∂Ω) to a Banach space, its so

called Banach envelope. To define this properly, we digress momentarily for thepurpose of explaining a somewhat more general functional analytic setting. A goodreference is [KPR].

Let V be a locally bounded topological vector space, whose dual separates pointsand fix U a bounded neighborhood of the origin. Then, with co A standing for theconvex hull of a set A ⊆ V , the functional given by

(5.7) ‖x‖V := inf γ > 0; γ−1x ∈ co(U)defines a (continuous) norm on V (any other such norm corresponding to a differentchoice of U is in fact equivalent to (5.7)). Then, V, the Banach envelope of V, is thecompletion of V in the norm (5.7). Thus, V is a well defined Banach space, uniquelydefined up to an isomorphism. Also, the inclusion V → V is continuous and has adense image. Furthermore, V and V have the same dual; see [KPR]. Another usefulobservation is contained in the next lemma (whose proof is an exercise).

29

Lemma 5.1. Let V1, V2 be two topological vector spaces as above and consider abounded linear operator T : V1 −→ V2. Then T extends to a bounded linear operatorT : V1 −→ V2 with

(5.8) supx‖T (x)‖V2

≤ ‖x‖V1inf ε > 0 : T (U1) ⊂ εU2 ,

where Uj ⊂ Vj is the bounded neighborhood with respect to which the norm ‖ · ‖Vj

has been defined, j = 1, 2.If, moreover, T is an isomorphism then so is T .

Our main result in this section relates hp(∂Ω) to the atomic Besov spaces B1−s(∂Ω),

defined by (4.8)–(4.10). It is an improvement of Theorem 5.4 in [FMM].

Proposition 5.2. Let Ω be a bounded Lipschitz domain in M and fix some in-dex n−1

n < p < 1. Then hp(∂Ω), the Banach envelope of hp(∂Ω), coincides withB1−s(∂Ω), where s = (n− 1)( 1

p − 1).

Proof. The crucial observation is that hp(∂Ω), as a subspace of (B∞s (∂Ω))∗, is given

by

(5.9) C +

f =∑

j≥1

λjaj in (B∞s (∂Ω))∗ : (λj)j ∈ `1, aj ’s are Hp(∂Ω)-atoms

endowed with the norm(5.10)

f 7→ inf

‖g‖L∞(∂Ω) +∑

|λj | : f = g +∑

λjaj , g ∈ C, λj ’s, aj ’s as in (5.9)

.

For Euclidean domains this has been proved in [FMM] and the proof given therereadily adapts to our current setting. Now the desired conclusion follows by notingthat each Hp(∂Ω)-atom is in fact a B1

−s(∂Ω)-atom, granted that s = (n−1)( 1p −1).

We mention that Proposition 5.2 plus previously noted duality results yield(B1

−s(∂Ω))∗ = B∞s (∂Ω), a special case of duality results that are contained in

Theorem 2.11.2 of [Tr].

6. Newtonian potentials on Sobolev and Besov spaces

Let the manifold M , the Lipschitz domain Ω and the potential V be as in §1.The goal is to study the mapping properties on scales of Sobolev-Besov spaces forthe associated Newtonian potential operator

(6.1) Π`f(x) :=∫

Ω〈E`(x, y), f(y)〉 dVol(y), x ∈ Ω.

30

In (6.1), f is a differential form of degree ` ∈ 0, 1, ..., n in Ω, and E`(x, y) is theSchwartz kernel of

(6.2) (∆` − V )−1 : L2−1(M, Λ`TM) −→ L2

1(M, Λ`TM),

as in §2. In other words, Π` = R(∆` − V )−1E, where E denotes extension by 0 offΩ and R denotes restriction to Ω.

Proposition 6.1. Assume the metric tensor satisfies (1.28). For each ` ∈ 1, ..., n,

(6.3)Π` : (Lq

s+1(Ω, Λ`TM))∗ −→ Lp1−s(Ω,Λ`TM),

∀ s ∈ [−1, 1], r/(r − 1) < p, q < r, 1/p + 1/q = 1

and

(6.4)Π` : (Bq

s+1(Ω, Λ`TM))∗ −→ Bp1−s(Ω, Λ`TM),

∀ s ∈ (−1, 1), r/(r − 1) < p, q < r, 1/p + 1/q = 1,

are well defined, bounded operators.For ` = 0 and under the (weaker) assumption that the metric tensor is Lipschitz,

(6.3)–(6.4) are valid for the full range 1 < p, q < ∞.

Proof. Given our assumption on the metric, if s = −1 then (6.3) holds for 1 < p < r;see Theorem 2.9 in [MMT]. The full range −1 ≤ s ≤ 1 is then easily seen from this,duality and interpolation (note that Π` is formally self-adjoint).

The assertion on Besov spaces is a corollary of the preceding result and repeatedapplications of the method of real interpolation (together with the correspondingduality and reiteration theorems).

Finally, the last part in the proposition follows much as before; this time, how-ever, we invoke (4.30).

7. Single layer potentials on Sobolev and Besov spaces

In this section we continue to retain the same hypotheses made in §1 for M , Ωand V .

Recall the fundamental solution E`(x, y) for the operator ∆` − V and the singlelayer potential operator

(7.1) S` : (Lip(∂Ω,Λ`TM))∗ −→ C1+γloc (Ω,Λ`TM), some γ > 0,

defined by

(7.2) S`f(x) := 〈E`(x, ·)∣

∣

∂Ω, f〉, x ∈ Ω,

for each f ∈ (Lip(∂Ω, Λ`TM))∗. Hereafter, 〈·, ·〉 stands for the natural dualitypairing between a topological vector space and its dual (in the case at hand, betweenLip(∂Ω, Λ`TM) and (Lip(∂Ω,Λ`TM))∗).

Our main result in this section summarizes the mapping properties of the oper-ator (7.1) on scales of Sobolev-Besov spaces.

31

Theorem 7.1. Assume the metric tensor satisfies (1.29) Then, for 1 ≤ p ≤ ∞,0 < s < 1 and 1 ≤ ` ≤ n, the operator

(7.3) S` : Bp−s(∂Ω, Λ`TM) −→ Bp

1+ 1p−s(Ω, Λ`TM)

is well-defined and bounded. In fact, if 1 < p < ∞, then

(7.4) S` : Bp−s(∂Ω, Λ`TM) −→ Lp

1+ 1p−s(Ω, Λ`TM)

is also a bounded linear mapping, so that ∀ s ∈ (0, 1), ∀ p ∈ (1,∞),

(7.5)‖S`f‖Bp

1+ 1p−s

(Ω,Λ`TM) + ‖S`f‖Lp

1+ 1p−s

(Ω,Λ`TM)

≤ C(Ω, s, p)‖f‖Bp−s(∂Ω,Λ`TM),

uniformly for f ∈ Bp−s(∂Ω,Λ`TM).

In particular, if Tr stands for the trace map on ∂Ω, the operator S` := TrS` hasthe property

S` : Bp−s(∂Ω,Λ`TM) −→ Bp

1−s(∂Ω,Λ`TM),

for 1 ≤ p ≤ ∞, 0 < s < 1 and 1 ≤ ` ≤ n.Finally, if ` = 0, the same results are valid for S` under the assumption that the

metric tensor satisfies (1.27).

Handling S` on Besov spaces utilizes size estimates for the integral kernel (andits derivatives). A general statement to this effect is formalized and proved in theLemmas 7.2–7.3 below. Since these lemmas have some interest of their own, westate (and prove) them in a slightly more general version than what is needed here.

Lemma 7.2. Let Ω be a Lipschitz domain in M and consider a positive integerN . For k(x, y) defined on M ×M \ diagonal and satisfying

(7.6) |∇ix∇j

yk(x, y)| ≤ κ dist (x, y)−(n−2+i+j), ∀ i = 0, 1, ..., N, ∀ j = 0, 1,

for some κ > 0 independent of x, y, introduce

(7.7) Kf(x) := 〈k(x, ·)∣

∣

∂Ω, f〉, x ∈ Ω.

Then, for 0 < s < 1, this operator satisfies the estimates

‖dist (·, ∂Ω)s−1+i|∇1+iKf | ‖L1(Ω) + ‖∇iKf‖L1(Ω)

+ ‖Kf‖L1(Ω) ≤ C(Ω, κ, s)‖f‖(B∞s (∂Ω))∗ , ∀ i = 0, 1, ..., N − 1,(7.8)

32

uniformly in f ∈ (B∞s (∂Ω))∗.

Proof. Let us estimate the leading term in the left side of (7.8); the argument forthe remaining terms is simpler. The point is to establish the estimate

(7.9)∥

∥

∥

∫

Ω

dist (x, ∂Ω)s−1+i∇1+ix k(x, ·)g(x) dVol(x)

∥

∥

∥

B∞s (∂Ω)≤ C(Ω, κ, s)‖g‖L∞(Ω),

uniformly for g ∈ L∞comp(Ω). To this end, for a fixed, arbitrary g ∈ L∞comp(Ω), weshall focus on establishing(7.10)

∣

∣

∣

∫

Ω

g(x) dist (x, ∂Ω)s−1+i(∇1+ix k(x, p)−∇1+i

x k(x, q)) dVol(x)∣

∣

∣ ≤ C dist (p, q)s,

uniformly for p, q ∈ ∂Ω. Now, fix two arbitrary boundary points p, q ∈ ∂Ω and, fora large constant C, bound the integral above by(7.11)

∫

dist (x,p)<Cdist (p,q)|g(x)| dist (x, ∂Ω)s−1+i|∇1+i

x k(x, p)| dVol(x)

+∫

dist (x,p)<Cdist (p,q)|g(x)|dist (x, ∂Ω)s−1+i|∇1+i

x k(x, q)| dVol(x)

+∫

dist (x,p)>Cdist (p,q)|g(x)|dist (x, ∂Ω)s−1+i|∇1+i

x k(x, p)−∇1+ix k(x, q)| dVol(x)

=: I + II + III.

To deal with I, in the light of (7.6), by localizing and pulling back to Rn+, it suffices

to consider

(7.12)∫

|x′−p′|+|t+ϕ(x′)−ϕ(p′)|<C|p′−q′|

ts−1+i

(|x′ − p′|+ |t + ϕ(x′)− ϕ(p′)|)n−1+i dt dx′,

where ϕ : Rn−1 → R is a Lipschitz function and x = (x′, t + ϕ(x′)), p = (p′, ϕ(p′)),q = (q′, ϕ(q′)). Accordingly, we seek a bound of order |p′ − q′|s. To this effect, wenote that the integral (7.12) is majorized by

(7.13)

C∫

|x′−p′|+t<C|p′−q′|

ts−1+i

(|x′ − p′|+ t)n−1+i dt dx′

≤ C(

∫ ∞

0

ts−1+i

(1 + t)n−1+i dt)(

∫

|x′|<C|p′−q′|

1|x′|n−1−s dx′

)

≤ Cn,s|p′ − q′|s,

and the last bound has the right order. Similar arguments also apply to II in(7.11) since dist (x, q) < dist (x, p) + dist (p, q) < C dist (p, q). Thus, we are left

33

with estimating III. For this, an application of the mean-value theorem togetherwith (7.6) and a change of variables allow us to write

III ≤ C∫

|x′−p′|+|t+ϕ(x′)−ϕ(p′)|>C|p′−q′|

ts−1+i|p′ − q′|(|x′ − p′|+ |t + ϕ(x′)− ϕ(p′)|)n+i dx′dt

≤ C∫

|x′−p′|+t>C|p′−q′|

ts−1+i|p′ − q′|(|x′ − p′|+ t)n+i dx′dt.

(7.14)

Making x′ 7→ x′′ := (x′ − p′)/|p′ − q′| and t 7→ t′ := t/|p′ − q′| in the last integralabove readily leads to a bound of order |p′− q′|s, i.e., the proper size. This finishesthe proof of (7.8).

Lemma 7.3. Retain the same hypotheses as in Lemma 7.2 and recall the operatorK introduced in (7.7). Then, for 0 < s < 1, this operator satisfies the estimates

‖dist (·, ∂Ω)s+i|∇1+iKf | ‖L∞(Ω) + ‖∇iKf‖L∞(Ω)

+ ‖Kf‖L∞(Ω) ≤ C(Ω, κ, s)‖f‖B∞−s(∂Ω), ∀ i = 0, 1, ..., N − 1,(7.15)

uniformly in f ∈ B∞−s(∂Ω) := (B1

s (∂Ω))∗.

Proof. Consider the leading term in the left side of (7.15); all the others can behandled similarly. Here the idea is prove that

(7.16) ‖∇1+ix k(x, ·)‖B1

s(∂Ω) ≤ C(Ω, s) dist (x, ∂Ω)−s−i, ∀ i = 0, 1, ..., N − 1,

uniformly in x ∈ Ω. Clearly, this suffices in order to conclude (7.15). The remainderof the proof, modeled upon [FMM], consists of a verification of (7.16).

The problem localizes and, hence, it suffices to prove the estimate

(7.17)∫

∂Ω

∫

∂Ω

|∇1+ix k(x, p)−∇1+i

x k(x, q)||p− q|n−1+s dσ(p) dσ(q) ≤ C dist (x, ∂Ω)−s−i,

uniformly for x ∈ Ω, in the case when Ω is the domain above the graph of a Lipschitzfunction ϕ : Rn−1 → R.

Now, for a fix, sufficiently large C > 0, split the inner integral according towhether |x− p| < C|p− q| or |x− p| > C|p− q|. Thus, it suffices to treat

∫

∂Ω |I| dσ,∫

∂Ω |II| dσ and∫

∂Ω |III| dσ, where

(7.18)

I :=∫

|x−p|<C|p−q|

|∇1+ix k(x, p)|

|p− q|n−1+s dσ(p),

II :=∫

|x−p|<C|p−q|

|∇1+ix k(x, q)|

|p− q|n−1+s dσ(p),

III :=∫

|x−p|>C|p−q|

|∇1+ix k(x, p)−∇1+i

x k(x, q)||p− q|n−1+s dσ(p).

34

To this end, note first that a change of variables based on the representationsx = (x′, ϕ(x′) + t), p = (p′, ϕ(p′)) and q = (q′, ϕ(q′)) gives

(7.19)

|I|

≤ C∫

|x′−p′|+|t+ϕ(x′)−ϕ(p′)|<C|p′−q′|

|p′ − q′|−n+1−s

(|t + ϕ(x′)− ϕ(p′)|+ |x′ − p′|)n−1+i dp′

≤ C∫

|x′−p′|+t<C|p′−q′|

dp′

|p′ − q′|n−1+s(t + |x′ − p′|)n−1+i .

Substituting x′ − p′ = th in the last integral above and then integrating against∫

Rn−1 dq′ yields

(7.20)

∫

∂Ω

|I| dσ

≤ C1ti

∫

Rn−1

(

∫

|h|+1≤C|x′−th−q′|/t

dh(|h|+ 1)n−1+i|x′ − th− q′|n−1+s

)

dq′.

Substituting again, this time first x′ − q′ = tw and then w − h = rω, r > 0,ω ∈ Sn−2, we may further bound the last integral in (7.20) by

(7.21)

C1

ts+i

∫

Rn−1

∫

|h|+1≤C|w−h|

dh dw(|h|+ 1)n−1+i|w − h|n−1+s

=C

ts+i

∫

1(|h|+ 1)n−1

(

∫ ∞

|h|+1

drrs+1

)

dh = Cn,s,i t−s−i,

which is a bound of the right order for∫

∂Ω |I| dσ. The same arguments work tobound

∫

∂Ω |II| dσ, by observing that |x − p| < C|p − q| ⇒ |x − q| < C ′|p − q| andusing Fubini’s theorem.

As for∫

∂Ω |III| dσ, we note first that since |x − z| ≥ C|x − p| uniformly forz ∈ [p, q],

(7.22)∫

∂Ω

|III| dσ ≤ C∫

∂Ω

∫

|x−p|>C|p−q|

1|p− q|n−2+s|x− p|n+i dσ(p) dσ(q).

As before, pulling back everything to Rn−1, it is enough to bound

(7.23)∫

Rn−1

∫

|x′−p′|+t>C|p′−q′|

1|p′ − q′|n−2+s(|x′ − p′|+ t)n+i dp′ dq′.

35

Substituting x′ − p′ = th and then x′ − q′ = tw gives

(7.24)

1ti+1

∫

Rn−1

(∫

|h|+1>C|p′−q′|/t

1(|h|+ 1)n+i|q′ − x′ + ht|n−2+s dh

)

dq′

=1

ts+i

∫

Rn−1

1(|h|+ 1)n+i

(

∫

|h|+1>C|w−h|

dw|w − h|n−2+s

)

dh

= Cn,s,it−s−i,

as desired. This finishes the proof of (7.16) and, with it, the proof of the lemma.

We are now ready to present the

Proof of Theorem 7.1. Consider first the single layer potential (7.1) on the scale ofBesov spaces and recall the decomposition (2.10). This and (2.11) show that thekernel of S` satisfies the estimates (7.6) and, hence, Lemmas 7.2–7.3 apply. FromLemma 7.2 (with N = 1) we get, dropping the dependence of the various norms onthe exterior power bundle,

(7.25) ‖dist (·, ∂Ω)s−1|∇S`f | ‖L1(Ω) + ‖S`f‖L1(Ω) ≤ C(Ω, s)‖f‖(B∞s (∂Ω))∗ ,

uniformly in f ∈ (B∞s (∂Ω))∗. Since (∆` − V )S`f = 0 in Ω, it follows from (7.25)

and Proposition 3.3 that

‖dist (·, ∂Ω)s|∇2S`f | ‖L1(Ω) + ‖∇S`f‖L1(Ω)

+ ‖S`f‖L1(Ω) ≤ C(Ω, s)‖f‖(B∞s (∂Ω))∗ ,(7.26)

uniformly in f ∈ (B∞s (∂Ω))∗. Up to this point, the hypothesis (1.28) on the metric

sufficed. However, it is here that the hypothesis (1.29) on the metric tensor isneeded for the first time. To be more precise, (1.29) is needed when ` > 0; for thecase ` = 0, (1.27) suffices.

Let us digress momentarily and point out that we could have arrived at (7.26)solely based on Lemma 7.2 in which we take N = 2. However, this approachrequires the pointwise control of (mixed) derivatives of order three for the kernelE`(x, y); cf. (7.6). Under the present approach, this would require a version of(2.11) with (mixed) derivatives of order three placed on the residual part e`

1(x, y)which, in turn, would require a metric tensor of class Lr

3, r > n, (for the techniquesof [MT], [MMT] to apply unaltered). This is why we choose to establish (7.25) byusing Lemma 7.2 with N = 1 and then, further, invoke the interior estimates of §3in order to obtain (7.26). In this latter scenario, the hypothesis (1.29) suffices.

Returning to the mainstream discussion, note that (7.26) in concert with Propo-sition 4.4 give that the operator

(7.27) S` : (B∞s (∂Ω))∗ −→ B1

2−s(Ω), s ∈ (0, 1),

36

is well-defined and bounded. Observe that, by Proposition 5.2, this is a strongerresult than the one corresponding to p = 1, 0 < s < 1 in Theorem 7.1.

Going further, Lemma 7.3 (with N = 1) applied to the single layer operatorgives

(7.28) ‖dist (·, ∂Ω)s|∇S`f | ‖L∞(Ω) + ‖S`f‖L∞(Ω) ≤ C(Ω, s)‖f‖B∞−s(∂Ω),

uniformly in f ∈ B∞−s(∂Ω). Now the conclusion in the first part of Theorem 7.1

follows from (7.26) and (7.28) by virtue of Proposition 4.4 and interpolation.Turning our attention to the second part, i.e., when the range of S` is taken

on the scale of Sobolev spaces, note that (7.25), (7.28) and Stein’s interpolationtheorem for analytic families of operators give that for 1 ≤ p ≤ ∞, 0 < s < 1,

(7.29) ‖dist (·, ∂Ω)s−1/p|∇S`f | ‖Lp(Ω) + ‖S`f‖Lp(Ω) ≤ C(Ω, p, s)‖f‖Bp−s(∂Ω),

uniformly in f ∈ Bp−s(∂Ω). Now, this already leads to the desired conclusion when

s−1/p ≥ 0, thanks to (4.21) in Proposition 4.4. In the case when s−1/p < 0, we firstinvoke Proposition 3.3 (here the hypothesis (1.29) is used again when ` > 0; when` = 0, (1.27) suffices) and, proceeding as before, we arrive at the same conclusionif 1 < p < r, s ∈ (0, 1/p).

Finally, interpolation between this range and the one treated earlier, covers thefull unit square, i.e. s ∈ (0, 1), 1/p ∈ (0, 1). This finishes the proof of the Theorem7.1.

Parenthetically, let us point out that (7.3)–(7.4) and real interpolation also givethat for 1 < p, q < ∞,

(7.30)‖S`f‖Bp,q

1+ 1p−s

(Ω,Λ`TM) ≤ C(Ω, s, p, q)‖f‖Bp,q−s (∂Ω,Λ`TM),

‖S`f‖Bp,q1−s(∂Ω,Λ`TM) ≤ C(Ω, s, p, q)‖f‖Bp,q

−s (∂Ω,Λ`TM),

uniformly for f ∈ Bp,q−s (∂Ω, Λ`TM), ∀ s ∈ (0, 1).

Our next result deals with the case when the domain of S` is in the scale ofSobolev spaces. To state it, recall that for any two real numbers a, b we set a∨ b :=max a, b.

Theorem 7.4. Assume that the metric tensor satisfies (1.28). Then for 1 < p <∞, 0 ≤ s ≤ 1 and 1 ≤ ` ≤ n, the operator

(7.31) S` : Lp−s(∂Ω, Λ`TM) −→ Bp,p∨2

1−s+1/p(Ω,Λ`TM)

is well defined and bounded. When ` = 0, this holds under hypothesis (1.27).

Proof. Note that it suffices to treat only the cases when s = 0 and s = 1 since therest follows by complex interpolation (cf. [BL]). In fact, we shall only consider thesituation when s = 0, the rest being similar.

37

Recall the residual kernel e`1(x, y) from §2 and denote by B, B the integral oper-

ators

(7.32)

Bf(x) :=∫

∂Ω

〈∇xe`1(x, y), f(y)〉 dσ(y), x ∈ Ω,

Bf(x) :=∫

∂Ω

〈∇ye`1(x, y), f(y)〉 dσ(y), x ∈ Ω.

Then, if the metric tensor satisfies (1.28), or (1.27) in case ` = 0,

(7.33) B : Lp(∂Ω) −→ Bp,ps (Ω), 0 < s < 1, 1 < p < r,

and

(7.34) B : Lp(∂Ω) −→ Lp1(Ω), 1 < p < ∞,

are bounded operators. This follows from Lemmas 2.11–2.12 in [MMT]. In partic-ular,

(7.35) B, B : Lp(∂Ω) −→ Bp,p1/p(Ω), ∀ p ∈ (1,∞),

are also bounded. For B, this is contained in (7.33) if 1 0 small.

The corresponding statement for B (in (7.35)) is clear from (7.34).To continue, denote by CµSm

cl the class of classical symbols q(ξ, x) of order mwhich are Cµ in x, for some µ ∈ [0,∞], while still smooth in ξ ∈ Rn\0. The problemat hand localizes and, granted the fact that the operators (7.35) are bounded,when Ω is an Euclidean Lipschitz domain it suffices to show the following. Ifq(ξ, x) ∈ CµS−2

cl , µ ≥ 1, has a principal symbol that is even in ξ, then the Schwartzkernels of ∂jq(D, x), q(D, x)∂j ∈ ØPC0S−1

cl are all kernels of operators mappingLp(∂Ω) boundedly into Bp,p∨2

1/p (Ω) for each 1 < p < ∞.Take for instance the case of the Schwartz kernel k(x − y, y) of ∇xq(D,x) ∈

ØPC0S−1cl , for some q(ξ, x) ∈ CµS−2

cl , µ ≥ 1, whose principal symbol is even in ξand denote by K the corresponding integral operator, i.e.,

(7.37) Kf(x) :=∫

∂Ω

k(x− y, y)f(y) dσ(y), x ∈ Ω.

By performing a decomposition in spherical harmonics (cf. [MT] for details insimilar circumstances), there is no loss of generality in assuming that µ = ∞.

38

Next, fix f ∈ Lp(∂Ω) and set u := Kf in Ω. Analogously to [JK2], [Ve2], we usethe fact that ‖u‖Bp,q

1/p(Ω) is controlled by a finite sum of expressions of the type

(∫ r

0tq−q/p

(

∫

Sr

∫ r

t|∇u(x′, ϕ(x′) + s)|p dx′ds

)q/p dtt

)1/q

+(

∫ r

0tq−q/p

(

∫

Sr

∫ r

t|u(x′, ϕ(x′) + s)|p dx′ds

)q/p dtt

)1/q=: I + II.

(7.38)

Here ϕ : Rn−1 → R is a Lipschitz function used to describe ∂Ω locally and Sr ⊆ ∂Ωis a surface ball of fixed radius r > 0.

Our aim is to bound I and II by ‖f‖Lp(∂Ω) in the case when 1 < p < ∞ andq := p ∨ 2. The first observation is that II in (7.38) can easily be controlled using

(7.39) ‖u∗‖Lp(∂Ω) = ‖(Kf)∗‖Lp(∂Ω) ≤ C‖f‖Lp(∂Ω),

where the last estimate is proved in [MT] (recall that (·)∗ has been introduced in§1). As for I, following [JK2] we invoke Hardy’s inequality (cf., e.g., Appendix Ain [St]) in the case 1 < p ≤ 2 plus Minkowski’s inequality in order to write

(7.40)

|I| ≤ C(

∫ r

0

(

∫

Sr

|s∇u(x′, ϕ(x′) + s)|p dx′)q/p ds

s

)1/q

≤ C(∫

Sr

(

∫ r

0|s∇u(x′, ϕ(x′) + s)|q ds

s

)p/qdx′

)1/p.

Note that we can arrive at same majorand for I as above in the case 2 ≤ p < ∞simply by using Fubini’s theorem since, in this case, p = q.

At this point observe that matters are reduced to proving the Lp-boundedness,1 < p < ∞, of the Lq((0, r), ds/s)-valued operator T given by the assignment:

(7.41) Lp(Sr) 3 f 7→ s∇Kf(x′, ϕ(x′) + s) ∈ Lp(Sr, Lq((0, r), ds/s)).

It is preferable to deal first with the Hilbert space setting, i.e., when q = 2, since inthis case the vector-valued Calderon-Zygmund theory works. Concretely, setting

(7.42) ks(x′, y′) := s∇1k((x′ − y′, ϕ(x′)− ϕ(y′) + s), (y′, ϕ(y′)), x′, y′ ∈ Rn−1,

the estimates

(7.43) |∇i+11 ∇j

2k(x, y)| ≤ C|x− y|−(n−i−j), i, j ≥ 0,

39

readily imply that

(7.44)(

∫ r

0|∇i

x′∇jy′ ks(x′, y′)|2

dss

)1/2≤ C|x′ − y′|−(n−1+i+j), 0 ≤ i + j ≤ 1.

In turn, these express the fact that the kernel of T in (7.41) is standard. Theboundedness of the operator T when p = 2 follows from

(7.45)

‖Tf‖L2(Sr,L2((0,r),ds/s)) ≤ C(

∫

Sr

∫ r

0s|∇u(x′, ϕ(x′) + s)|2 ds dx′

)1/2

≤ C(

∫

Ω

dist (x, ∂Ω)|∇u(x)|2 dx)1/2

≤ C‖f‖L2(∂Ω).

The crucial step in (7.45) is the last inequality and this has been proved in Theorem1.1 of [MMT]. This finishes the proof of the Lp-boundedness of T when q = 2 andtakes care of the 1 < p ≤ 2 part in the statement of the theorem.

Next, consider the case when p ≥ 2. When f has small support, contained in anopen subset of ∂Ω where ∂Ω is given as the graph of ϕ : x′ ∈ Rn−1 : |x′| < r → Rand x0 = (x′0, ϕ(x′0)) ∈ ∂Ω is an arbitrary boundary point, then

(7.46)

s|∇u(x′0, ϕ(x′0) + s)|

≤∫

|y′|<r

s|∇k((x′0, ϕ(x′0) + s), (y′, ϕ(y′))| · |f(y′, ϕ(y′))| dy′

≤ CMf(x0),

uniformly in s, where M denotes the Hardy-Littlewood maximal operator on ∂Ω.The last inequality in (7.46) is a consequence of fact that the expression

(7.47) s|∇k((x′, ϕ(x′) + s), (y′, ϕ(y′))|

behaves like the Poisson kernel for the upper-half space; see, e.g., Theorem 2,pp. 62–63 in [St]. Thus,

(7.48) sups∈(0,r)

s|∇u(x′0, ϕ(x′0) + s)| ≤ CMf(x0).

Using this and the fact that M is bounded on Lp(∂Ω), 1 2 now follows by interpolating the end-point results

corresponding to q = 2 and q = ∞. (Note that here we use the fact that Bp,q1s →

Bp,q2s for 1 ≤ q1 < q2 ≤ ∞.) This completes the proof of Theorem 7.4.

The last result of this section deals with the mapping properties of layer potentialoperators associated with general elliptic, second order differential operators onLipschitz domains. As such, it further auguments the results in §2 of [MMT] wheresome partial results in this direction where first proved.

Theorem 7.5. Let E ,F → M be two (smooth) vector bundles over the (smooth)compact, boundaryless manifold M , of real dimension n. It is assumed that themetric structures on E, F and M have coefficients in L2

r for some r > n.Let

(7.50) Lu =∑

j,k

∂jAjk(x)∂ku +∑

j

Bj(x)∂ju− V (x)u,

where Ajk = (aαβjk ), Bj = (bαβ

j ) and V = (dαβ) are matrix-valued functions, be anelliptic, second-order differential operator mapping C2 sections of E into measurablesections of F . It is assumed that, when written in local coordinates, the coefficientsof L and L∗ satisfy

(7.51) aαβjk ∈ L2

r, bαβj ∈ L1

r, dαβ ∈ Lr, for some r > n.

Moreover, suppose that L is invertible as a map from H1,2(M, E) onto H−1,2(M,F)and denote by E the Schwartz kernel of L−1.

Let Ω be an arbitrary Lipschitz domain in M . For a first order differentialoperator P ∈ Diff1(F ,F) with continuous coefficients, consider the integral operatorD with kernel (Idx ⊗ Py)E(x, y), i.e.,

(7.52) Df(x) :=∫

∂Ω〈(Idx ⊗ Py)E(x, y), f(y)〉 dσ(y), x ∈ Ω.

Then, for each 1 < p < ∞,

(7.53) D : Lp(∂Ω,F) −→ Bp,p∗

1/p (Ω, E)

is a bounded operator.Consider next the integral operator S which is constructed as before but, this time,

in connection with the kernel E(x, y). Then, for each 1 < p < ∞ and 0 ≤ s ≤ 1,

(7.54) S : Lp−s(∂Ω,F) −→ Bp,p∗

−s+1+1/p(Ω, E)

is a bounded operator. Moreover, if 1 ≤ p ≤ ∞ and 0 < s < 1, then

(7.55) S : Bp−s(∂Ω,F) −→ Bp

−s+1+1/p(Ω, E)

41

is a bounded operator also. Finally, the same conclusion applies to

(7.56) S : Bp−s(∂Ω,F) −→ Lp

−s+1+1/p(Ω, E)

provided 1 < p < ∞ and 0 < s < 1.

Proof. This follows much as Theorem 7.4 and Theorem 7.1, via a decomposition inspherical harmonics; cf. [MT] for details in similar circumstances.

8. Double layer potentials on Sobolev and Besov spaces

We shall work with our usual set of hypotheses on M , Ω and V made in §1.The aim is to describes the action of the operators (1.11) and (1.15) on scales ofSobolev-Besov spaces. We begin with the case when the domain of the operatorsis the scale of Besov spaces.

Theorem 8.1. Assume that the metric tensor satisfies (1.27). Then, for 1 ≤ p ≤∞ and 0 < s < 1, the operator

(8.1) D : Bps (∂Ω) −→ Bp

s+ 1p(Ω)

is well defined and bounded. In fact, if 1 < p < ∞ and 0 < s < 1, then

(8.2) D : Bps (∂Ω) −→ Lp

s+ 1p(Ω)

is also bounded. Furthermore,

(8.3) K : Bps (∂Ω) −→ Bp

s (∂Ω)

is well defined and bounded for 1 ≤ p ≤ ∞ and 0 < s < 1.

This is based on a series of lemmas, which we now formulate and prove. Wedebut with the following Holder result which, on the Besov scale, corresponds top = ∞.

Lemma 8.2. Assume that the metric tensor on M satisfies (1.27). Then, for0 < s < 1, D is a bounded linear map from B∞

s (∂Ω) into B∞s (Ω).

Proof. This has been proved in §7 of [MT2] as a consequence of the estimate

(8.4) supx∈Ω

(

dist (x, ∂Ω)1−s|∇Df(x)|)

≤ C(Ω, s)‖f‖B∞s (∂Ω),

uniformly for f ∈ B∞s (∂Ω), if 0 < s < 1. Cf. (7.24) of [MT2].

We next turn attention to the case p = 1 on the Besov scale. The main result inthis regard is the lemma below.

42

Lemma 8.3. Assume that the metric tensor satisfies (1.27). Then the operator Dmaps B1

s (∂Ω) linearly and boundedly into B1s+1(Ω) for each 0 < s < 1.

Proof. Since Df is a null-solution for L in Ω, by Proposition 3.3 and Proposition4.4, it suffices to show that

(8.5)∫

Ω

dist (x, ∂Ω)−s|∇Df(x)| dVol(x) ≤ C‖f‖B1s(∂Ω),

uniformly for f ∈ B1s (∂Ω). Fix an arbitrary f ∈ B1

s (∂Ω), 0 < s < 1. Note that, inthe left side of (8.5), the contribution away from the boundary is easily estimated,e.g., by the interior estimates in §3. Thus, we may replace Ω by C ∩ Ω, where Cis a small collar neighborhood of ∂Ω in M . Furthermore, via a partition of unity,there is no loss of generality in assuming that f has small support, contained in acoordinate patch.

To continue, let us assume for a moment that V = 0 on Ω. In this case, since ∇Dannihilates constants, we can replace f in (8.5) by f−f(π(x)) where π : C∩Ω → ∂Ωis some Lipschitz continuous map so that dist (x, ∂Ω) ≈ dist (x, π(x)), uniformly forx ∈ C∩Ω. Next, since the metric satisfies (1.27), the decomposition (2.10) in concertwith Theorem 2.1 give that the kernel k(x, y) of ∇D satisfies

(8.6) |k(x, y)| ≤ C dist (x, y)−n, ∀x ∈ Ω, ∀ y ∈ ∂Ω.

Thus, we need to estimate

(8.7)∫

C∩Ω

∫

∂Ω

dist (x, ∂Ω)−sdist (x, y)−n|f(y)− f(π(x))| dσ(y)dVol(x).

Note that max

dist (x, ∂Ω),dist (y, π(x))

≤ C dist (x, y). Also, set t := dist (x, ∂Ω)and x := π(x). Then, the integral in (8.7) is bounded by I + II where

(8.8) I :=∫ ∞

0

∫

∂Ω

∫

dist (y,x)≥Cty∈∂Ω

t−sdist (y, x)−n|f(y)− f(x)| dσ(y)dσ(x)dt

and

(8.9) II :=∫ ∞

0

∫

∂Ω

∫

dist (y,x)≤Cty∈∂Ω

t−s−1dist (y, x)−n+1|f(y)− f(x)| dσ(y)dσ(x)dt,

where C > 0 is a fixed, sufficiently large constant. To continue, pull-back everythingto Rn−1 and denote by (ωg)(x) := ‖g(x + ·) − g(·)‖L1(Rn−1) the L1-modulus of

43

continuity of an arbitrary function g : Rn−1 → R. Then, changing the order ofintegration and integrating first with respect to t, we obtain

|I| ≤ C∫ ∞

0

1ts

∫

|z|>Ct

(ωf)(z)|z|n

dzdt

≤ C∫

Rn−1

(ωf)(z)|z|n−1+s dz ≤ C‖f‖B1

s(Rn−1),(8.10)

which has the right order. In a similar manner,

|II| ≤ C∫ ∞

0

1t1+s

∫

|z|<Ct

(ωf)(z)|z|n−1 dzdt

≤ C∫

Rn−1

(ωf)(z)|z|n−1+s dz ≤ C‖f‖B1

s(Rn−1).(8.11)

This concludes the proof of the lemma in the case when V = 0 on Ω.To treat the general case, let D0 be the double layer potential corresponding to

the choice of a potential V0 which vanishes in Ω. By the previous discussion, itsuffices to prove that

(8.12) ‖dist (·, ∂Ω)−s|∇(D −D0)f | ‖L1(Ω) ≤ Cs‖f‖L1(∂Ω),

for each s ∈ (0, 1), uniformly for f ∈ L1(∂Ω). Now, by (2.10) and Theorem 2.1, theintegral operator in the left side of (8.12) has a kernel k(x, y) satisfying, for eachε > 0,

(8.13) |k(x, y)| ≤ Cε dist (x, ∂Ω)−s dist (x, y)−(n−1+ε), ∀x ∈ Ω, ∀ y ∈ ∂Ω.

This and elementary estimated then readily yield (8.12).

Let us pause for a moment and discuss an alternative approach to the Lemma8.3. While this works for differential forms of higher degree, the metric tensorwould have to satisfy (1.29) in place of (1.27). Nonetheless, this result (or ratherits proof) will be useful for us in the sequel.

To get started, let us consider a generalization of (1.11) at the level of differentialforms of arbitrary degrees. Specifically, for each ` ∈ 0, 1, ..., n, we introduce thedouble layer potential of a `-form f in, say, L2(∂Ω, Λ`TM) by setting

(8.14) D`f(x) :=∫

∂Ω

〈(ν(y) ∨ dy − ν(y) ∧ δy)E`(x, y), f(y)〉 dσ(y), x ∈ Ω,

where ∧, ∨ are the usual exterior and interior, respectively, products of forms. Its(nontangential) boundary trace is

(8.15) D`f |∂Ω = ( 12I + K`)f, a.e. on ∂Ω,

where

(8.16) K`f(x) := P.V.∫

∂Ω

〈(ν(y) ∨ dy − ν(y) ∧ δy)E`(x, y), f(y)〉 dσ(y), x ∈ ∂Ω.

See §6 of [MMT] for more on these. Note that when ` = 0, the operators (8.14),(8.16) reduce precisely to (1.11) and (1.15), respectively.

44

Lemma 8.4. Assume that the metric tensor satisfies (1.29). Then, for 0 < s < 1and ` ∈ 0, 1, ..., n, the operators

(8.17) dD` : B1s (∂Ω, Λ`TM) −→ B1

s (Ω, Λ`+1TM)

and

(8.18) δD` : B1s (∂Ω, Λ`TM) −→ B1

s (Ω, Λ`−1TM)

are well defined and bounded.

Proof. We deal only with (8.17) since the case of (8.18) follows from this via anapplication of the Hodge star isomorphism.

A basic ingredient in the proof is an identity to the effect that

(8.19)

dD`f(x) =∫

∂Ω

〈dxdyE`(x, y), ν(y) ∧ f(y)〉 dσ(y)

+∫

∂Ω

〈dxR`−1(y, x), ν(y) ∨ f(y)〉 dσ(y)

= dδS`+1(ν ∧ f)(x)−R`(ν ∧ f)(x) + dRt`−1(ν ∨ f)(x)

= − δdS`+1(ν ∧ f)(x)−R`(ν ∧ f)(x)− V S`+1(ν ∧ f)(x)

+ dRt`−1(ν ∨ f)(x)

= δS`+2(ν ∧ df)(x)− δRt`+1(ν ∧ f)(x)−R`(ν ∧ f)(x)

− V S`+1(ν ∧ f)(x) + dRt`−1(ν ∨ f)(x).

Recall that S` is the single layer potential on `-forms, i.e., the boundary integraloperator with kernel E`(x, y); cf. §2. Also, for the last equality see §6 in [MMT].

Now, if f is a `-form with components B1s (∂Ω)-atoms, then (componentwise)

ν ∧ df ∈ hp(∂Ω) with p ∈ ((n− 1)/n, 1), s ∈ (0, 1) related by 1− s = (n− 1)( 1p − 1),

and ‖ν ∧ df‖hp(∂Ω) ≤ C with C independent of f . This follows from (4.5). Inparticular,

(8.20) ‖ν ∧ df‖(B∞1−s(∂Ω))∗ ≤ C.

Now the desired conclusion about δS`+2(ν∧df) follows from Proposition 4.3, (8.20)and Theorem 7.1 (it is at this point that the hypothesis (1.29) is used).

The remaining terms after the last equality sign in (8.19) are residual and canbe handled more directly. In fact, it suffices to assume (1.28), which we shall do.To this end, it helps to note that B1

s (∂Ω) → Lq(s)(∂Ω) for 1/q(s) := 1− s/(n− 1).Take, for instance, the case of δRt

`+1(ν ∧ f); it suffices to show that

(8.21) Rt`+1 : Lq(s)(∂Ω, Λ`+1TM) −→ B1

1+s(Ω, Λ`TM), 0 < s < 1,

45

is bounded. Indeed, as we shall see momentarily, we even have

(8.22) Rt`+1 : Lp(∂Ω, Λ`TM) −→ Lp

2(Ω, Λ`+1TM), ∀ p ∈ (1, r).

To justify (8.22), let us introduce

(8.23) R`+1g(x) :=∫

Ω

〈R`+1(x, y), g(y)〉 dVol(y), x ∈ Ω,

and observe that when the metric tensor satisfies (1.28),

(8.24)R`+1 : Lq′

−2,0(Ω) = (Lq2(Ω))∗ −→ Lq′

2 (Ω),

r/(r − 1) < q, q′ < r, 1/q + 1/q′ = 1,

is bounded. This follows by invoking an observation made in §6 of [MMT] to theeffect that R`+1 = (∆`−V )−1(dV ∨)(∆`+1−V )−1 and by appealing to the mappingproperties of the Newtonian potentials established in §6 of the present paper. Thepoint is that (8.22) is a consequence of (8.24) and duality, upon noticing that

(8.25)∫

Ω

〈Rt`+1f, g〉 dVol =

∫

∂Ω

〈Tr (R`+1g), f〉 dσ

for any reasonable forms f , on ∂Ω, and g, on Ω. Specifically, since

(8.26) Tr : Lq′

2 (Ω) −→ Lp(∂Ω), ∀ p ∈ (1,∞), ∀ q′ >n2

,

is bounded, (8.25) entails

(8.27)∣

∣

∣

∫

Ω

〈Rt`+1f, g〉 dVol

∣

∣

∣ ≤ C‖g‖(Lq2(Ω,Λ`+1TM))∗‖f‖Lp(∂Ω,Λ`TM),

for 1 n/2. This last condition can be arranged giventhe validity range for (8.24) and the fact that we are assuming r > n. For sucha q′, (8.27) proves the membership of Rt

`+1f to Lq2(Ω, Λ`+1TM) whenever f ∈

Lp(∂Ω,Λ`TM) with 1 < p < ∞, plus natural estimates. Thus, (8.22) holds for pclose to 1.

In fact, a similar reasoning as above but with (8.26) replaced by

(8.28) Tr : Lq′2 (Ω) −→ Lq′(∂Ω), ∀ q′,

shows that (8.22) holds for any p ∈ (r/(r− 1), r). Then the full range 1 < p < r in(8.22) follows by interpolation.

46

Hence, the proof of (8.22) is finished and this, in turn, completes the proof of(8.21). As the remaining terms in the right side of the last equality in (8.19) aretreated similarly, the proof of the lemma is finished.

We are now ready for the

Proof of Theorem 8.1. Let us first deal with the operator (8.1). The case p = ∞,0 < s < 1 is contained in Lemma 8.2 whereas Lemma 8.3 covers the case p = 1,0 < s < 1. The full range then follows by interpolation.

Turning now attention to the operator (8.2), recall from the proof of Lemma 8.3that

(8.29) ‖dist (·, ∂Ω)−s|∇Df | ‖L1(Ω) + ‖Df‖L1(Ω) ≤ C(Ω, s)‖f‖B1s(∂Ω)

holds uniformly for f ∈ B1s (∂Ω). Then, (8.29) in concert with the Holder estimate

(8.4) and Stein’s interpolation theorem for analytic families of operators yield theestimate

(8.30) ‖dist (·, ∂Ω)1−s−1/p|∇Df | ‖Lp(Ω) + ‖Df‖Lp(Ω) ≤ C(Ω, p, s)‖f‖Bps (∂Ω),

uniformly for f ∈ Bps (∂Ω) for each 1 ≤ p ≤ ∞, 0 < s < 1.

To continue, we distinguish two cases. First, if 1 − s − 1/p ≥ 0 then (8.30)and Proposition 4.4 yield that Df ∈ Lp

s+1/p(Ω) plus natural estimates. Second, if1 − s − 1/p < 0 we proceed similarly and arrive at the same conclusion as before.The only difference is that we involve Proposition 3.3 (cf. the comment following itsstatement) before invoking Proposition 4.4. This completes the proof of the claimregarding the operator (8.2).

Finally, the last point in the statement of the theorem is a consequence of (8.1)and the trace formula (1.13).

In the remainder of this section we analyze the action of the double layer potentialon the scale of Sobolev spaces.

Theorem 8.5. Assume that the metric tensor satisfies (1.28). Then, for 1 < p <∞ and 0 ≤ s ≤ 1, the operator

(8.31) D : Lps(∂Ω) −→ Bp,p∨2

s+1/p(Ω)

is bounded.

Proof. For (8.31) with s = 0, arguments similar to the ones used in the proof ofTheorem 7.4 apply. Matters can again be reduced to the same pattern in the cases = 1, thanks to the identity (8.19). Note that, in this later situation, we needestimates like

(8.32) Rt`+1 : Lp(∂Ω,Λ`TM) −→ Bp,p

1+1/p(Ω, Λ`+1TM), ∀ p ∈ (1,∞).

In turn, the estimate (8.32) follows easily from (8.22) and embedding results (sincewe are assuming r > n). Now the claim about (8.31) is obtained by interpolation.

For our next theorem, the following observation is useful. To state it, recall thenontangential maximal operator (·)∗ from §1.

47

Proposition 8.6. If the metric tensor satisfies (1.28) and 1 < p < ∞, then

(8.33) f ∈ Lp1(∂Ω) =⇒ (∇Df)∗ ∈ Lp(∂Ω),

plus a natural estimate.

Proof. This follows from (8.19), with ` = 0, plus estimates on the single layerpotential S` established in §2 of [MMT].

Our next result contains an improvement of (8.31) and (7.31) with ` = 0 in therange 2 ≤ p < ∞.

Theorem 8.7. Assume that the metric tensor satisfies (1.28). Then, for 2 ≤ p <∞ and 0 ≤ s ≤ 1, the operators

(8.34) D : Lps(∂Ω) −→ Lp

s+1/p(Ω)

and

(8.35) S : Lp−s(∂Ω) −→ Lp

1−s+1/p(Ω)

are well defined and bounded.

Remark. When 0 < s < 1 and 2 ≤ p < ∞, (8.34) also follows from (4.19) and(8.2). Similarly, for the same ranges of indices, (8.35) is a consequence of (4.19)and (7.4). Note that, in this scenario, the hypothesis (1.27) on the metric tensorsuffices. Thus, the main novel points addressed by Theorem 8.7 are s = 0 ands = 1.

Proof of Theorem 8.7. Our proof builds on an idea from [JK2]. Assume first thats = 0. In this case, granted (1.20), proving (8.34) is the same as proving such aresult with D replaced by T := D ( 1

2I + K)−1, in the range 2 ≤ p < ∞. Based on[MMT], we have that

(8.36) T : L2(∂Ω) −→ L21/2(Ω)

is bounded. Also, since T is the solution operator for the Dirichlet problem (1.4),the maximum principle gives that

(8.37) T : L∞(∂Ω) −→ L∞(Ω)

is bounded. Now, the inclusion

(8.38) [L∞(Ω), L21/2(Ω)]θ → Lp

1/p(Ω), 0 < θ < 1, p := 2/θ,

together with (8.36), (8.37) and interpolation allow one to conclude that

(8.39) D : Lp(∂Ω) −→ Lp1/p(Ω), 2 ≤ p < ∞,

48

is bounded. Observe that by interpolating (8.39) with D : L21(∂Ω) → L2

3/2(Ω)which, in turn, follows from (8.31), we arrive at the conclusion that

(8.40) D : Lps(∂Ω) −→ Lp

s+1/p(Ω), 2 ≤ p ≤ 2/s, 0 < s ≤ 1,

is bounded.Consider next f ∈ Lp

1(∂Ω), 2 ≤ p < ∞, and set u := Df . Our aim is to provethat u ∈ Lp

1+1/p(Ω) plus estimates. Introducing v := Xu, where X ∈ T ∗M is anarbitrary vector field with smooth coefficients, it suffices to show that

(8.41) v ∈ Lp1/p(Ω) plus a natural estimate.

By Proposition 8.6 we have v∗ ∈ Lp(∂Ω). If we next set w := v − Π0(v) in Ω.it follows that Lw = 0 in Ω and w∗ ∈ Lp(∂Ω), at least if p is large. Indeed, by§6, Π0(v) ∈ Lp

2(Ω) → C0(Ω) if p is large. Thus, from the integral representationfor the solution of the Lp-Dirichlet problem with p ≥ 2 (established in [MT2]) and(8.39) we may conclude that w ∈ Lp

1/p(Ω) if p is large. Since Π0(v) ∈ Lp2(Ω), (8.41)

follows. This proves (8.34) for s = 1 and p large. Now, interpolating what we havejust proved with (8.40) gives (8.34) for the full range of indices specified in thestatement of the theorem.

The argument for (8.35) is similar. The starting point is to consider the solutionoperator T := S S−1 plus the fact that S−1 : Lp

1(∂Ω) −→ Lp(∂Ω) is an isomor-phism for 2 ≤ p < ∞ ([MT2]). Proceeding as before yields (8.35) for s = 1. Finally,(8.35) with s = 0 is handled as before by taking this time u := Sf , f ∈ Lp(∂Ω),2 ≤ p < ∞. This finishes the proof of the theorem.

In closing, let us point out that (1.19) and Theorem 8.7 (or rather its proof) givethat the solution u of the boundary problem (1.4) & (1.7) satisfies

(8.42) ‖u‖Lp1/p(Ω) ≤ C(Ω, p)‖f‖Lp(∂Ω) if 2 ≤ p < ∞.

On the other hand, by Theorem 8.5 and (1.20),

(8.43) ‖u‖Bp,p∨21+1/p(Ω) ≤ C(Ω, p)‖f‖Lp

1(∂Ω) if 1 < p ≤ 2.

As for the solution u of the boundary problem (1.5)& (1.8), Theorem 7.4, in concertwith (1.21)–(1.22), yields that

(8.44) ‖∇u‖Lp1/p(Ω) ≤ C(Ω, p)‖g‖Lp(∂Ω) if 1 < p ≤ 2.

9. An endpoint Neumann problem

Retaining the hypotheses of §1 on M , V and the metric, including (1.27), here westudy a limiting case of the Neumann problem. Specifically, for a Lipschitz domainΩ ⊂ M , we shall be concerned with the case when the boundary data belong toB1−s(∂Ω) for s ∈ (0, s0), where s0 ∈ (0, 1) depends on the domain. First we need a

couple of technical results.

49

Lemma 9.1. Let ψ ∈ B∞s (∂Ω) for some s ∈ (0, 1). Then there exists ψ ∈ C0(Ω),

locally Lipschitz in Ω, so that ψ∣

∣

∂Ω = ψ and dist (·, ∂Ω)1−s∇ψ ∈ L∞(Ω).

Proof. The problem localizes and, via a bi-Lipschitz change of variables can betransported to the Euclidean upper-half space. There, the Poisson extension doesthe job; see, e.g., [St].

Lemma 9.2. There exists s0 = s0(Ω) > 0 so that whenever 0 < s < s0,

(9.1) ‖dist (x, ∂Ω)s−1|∇u| ‖L1(Ω) ≤ C(Ω, s)‖u‖B12−s(Ω)

uniformly for u ∈ B12−s(Ω) with Lu = 0 in Ω.

The proof of this lemma is postponed until the next section; cf. (10.7).Assume that 0 < s < s0, where s0 is as in Lemma 9.2. We will be concerned

with the Neumann boundary problem with boundary data in B1−s(∂Ω), i.e.

(9.2)

Lu = 0 in Ω,∂u∂ν

= f ∈ B1−s(∂Ω),

u ∈ B12−s(Ω).

The boundary condition in (9.2) is interpreted as the equivalent of

(9.3)∫

Ω

〈∇u,∇ψ〉 dVol = 〈f, ψ〉, ∀ψ ∈ B∞s (∂Ω),

where tilde is the extension operator introduced in Lemma 9.1, and 〈·, ·〉 in the rightside stands for the natural pairing between B1

−s(∂Ω) ⊆ (B∞s (∂Ω))∗ and B∞

s (∂Ω).It is to be noted that, by Lemmas 9.1–9.2, the integral in the left side of (9.3) isabsolutely convergent.

An observation made first (in the flat Euclidean setting) in [FMM] and whichalso applies to the present context is that the space of natural boundary data in(9.2) is indeed B1

−s(∂Ω) and not the larger space (B∞s (∂Ω))∗. This is supported

by the observation that even though − 12I + K∗ is, as we shall see momentarily, an

isomorphism of the larger space (B∞s (∂Ω))∗ for small s, and even though S maps

the latter space boundedly into B∞2−s(Ω), the natural jump formula

(9.4) ∂νSf = (− 12I + K∗)f

necessarily fails for general f ∈ (B∞s (∂Ω))∗. This is because, as will be shown in

Theorem 10.1, the normal derivative of Sf ∈ B12−s(Ω) always belongs to a smaller

subspace of (B∞s (∂Ω))∗, namely B1

−s(∂Ω).Our main result in this section is the theorem below, which extends Theorem

7.2 of [FMM]. Recall that C stands for the collection of all constant functions on∂Ω.

50

Theorem 9.3. There exists s0 ∈ (0, 1) depending only on Ω so that, if 0 < s < s0

and f ∈ B1−s(∂Ω) (with the extra condition 〈f, C〉 = 0 imposed if V = 0 in Ω) then

there exists a unique (modulo constants, if V = 0 in Ω) solution u to the Neumannproblem (9.2). Moreover, u satisfies the estimate

(9.5) ‖u‖B12−s(Ω) + ‖dist (·, ∂Ω)s−1|∇u| ‖L1(Ω) + ‖u‖L1(Ω) ≤ C(Ω, s) ‖f‖B1

−s(∂Ω) .

In particular, when V > 0 on a set of positive measure in Ω,

(9.6) ‖Tru‖B11−s(∂Ω) ≤ C(Ω, s) ‖f‖B1

−s(∂Ω) .

When V = 0 in Ω, then Tru in (9.6) should be considered modulo constants.

Note that this result implies that, under the same hypotheses, the Neumann-to-Dirichlet operator for L is bounded from B1

−s(∂Ω) into B11−s(∂Ω).

In proving Theorem 9.3, the following result is very useful.

Lemma 9.4. There exists s0 ∈ (0, 1) such that, when V > 0 on a set of positivemeasure in Ω,

(9.7) − 12I + K∗ : B1

−s(∂Ω) −→ B1−s(∂Ω)

is an isomorphism for 0 < s < s0.When V = 0 in Ω, then the same conclusion holds if we restrict to the subspace

of B1−s(∂Ω) consisting of functionals that annihilate C.

Proof. Granted that V > 0 on a set of positive measure in Ω, it has been shownin Theorems 7.6–7.7 of [MT2] that there exists ε > 0 so that − 1

2I + K∗ is anisomorphism of hp(∂Ω) for 1−ε < p ≤ 1. Our result then follows from this, Lemma5.1 and Proposition 5.2. The case when V = 0 in Ω follows by a minor variation ofthis argument.

We are now ready to present the

Proof of Theorem 9.3. We give the proof when V > 0 on a set of positive measurein Ω; the easy modifications needed when V = 0 in Ω are left to the reader.

Let s0 be as in Lemma 9.4, and s ∈ (0, s0). Then Lemma 9.4 gives that g :=(− 1

2I + K∗)−1f exists in B1−s(∂Ω) and ‖g‖B1

−s(∂Ω) ≤ C‖f‖B1−s(∂Ω). If we now set

u := Sg, it follows that Lu = 0 in Ω and, by invoking Theorem 7.1 and Proposition4.4, u ∈ B1

2−s(Ω) and (9.5) holds. To see that u actually solves (9.2) we shall provethat u satisfies (9.3). Indeed, this is a consequence of the general identity

(9.8)∫

Ω

〈∇Sh,∇ψ〉 dVol = 〈(− 12I + K∗)h, ψ〉, ∀ψ ∈ B∞

s (∂Ω),

which we claim is valid for arbitrary h ∈ B1−s(∂Ω) (recall that the tilde operator

is as in Lemma 9.1). In turn, this is easily checked on B1−s(∂Ω)-atoms and, hence,

51

extends by density to the whole B1−s(∂Ω) thanks to (7.25). This completes the

proof of the existence part.Turning to uniqueness, assume that u ∈ C1

loc(Ω) solves the homogeneous versionof (9.2) so that, in particular,

(9.9)∫

Ω

〈∇u,∇ψ〉 dVol = 0, ∀ψ ∈ B∞s (∂Ω).

Let Ωj Ω be a sequence of C∞ subdomains approximating Ω and, for some fixedpoint x0 ∈ Ω, consider the Neumann function Nj for L in Ωj with pole at x0, i.e.,

(9.10) Nj(x) := E(x0, x)− Sj(

(− 12I + K∗

j )−1(∂νj E(x0, ·)∣

∣

∂Ωj

)

for x ∈ Ωj .

Also, hereafter, Sj , K∗j , etc., will denote operators similar to S, K∗, etc., but

constructed in connection with ∂Ωj rather than ∂Ω. Take a smooth function 0 ≤φ ≤ 1 which vanishes identically near x0 and is identically 1 near ∂Ω. Green’sformula and an integration by parts then give

(9.11)

u(x0) =∫

∂Ωj

Nj∂u∂νj

dσj =∫

∂Ωj

φNj∂u∂νj

dσj

=∫

Ωj

〈∇(φNj),∇u〉 dVol.

The key step is to prove that, as j →∞,

(9.12)∫

Ωj

〈∇(φNj),∇u〉 dVol −→∫

Ω

〈∇(φN),∇u〉 dVol = 0,

where N is the Neumann function for L in Ω with pole at x0. Then, passing to thelimit in (9.11) will give u ≡ 0 in Ω as desired.

Turning our attention to (9.12), first we shall prove that the second integralvanishes. Indeed, it has been proved in [MT2] that Nj ∈ Cα(Ωj \ x0) andN ∈ Cα(Ω \ x0) for some α = α(Ω) > 0 independent of j. Consequently,

(9.13)dist (·, ∂Ωj)1−α|∇Nj |, dist (·, ∂Ω)1−α|∇N | ≤ C

away from x0, uniformly in j,

by the Holder theory in [MT2]. In particular,

(9.14)dist (·, ∂Ωj)1−α|∇(φNj)|, dist (·, ∂Ω)1−α|∇(φN)| ≤ C

in Ωj , uniformly in j.

52

With this in hand, the second integral in (9.12) vanishes on account of (9.9) if0 < s < α which we will assume for the remaining part of the proof.

At this point we are left with proving the convergence in (9.12) which we tacklenext. For j ≥ k we write

(9.15)

∫

Ωj

〈∇(φNj),∇u〉 dVol =∫

Ωj\Ωk

〈∇(φNj),∇u〉 dVol

+∫

Ωk

〈∇(φNj)−∇(φN),∇u〉 dVol

+∫

Ωk

〈∇(φN),∇u〉 dVol

=: Ij,k + IIj,k + IIIk.

Now, by (9.9), (9.14) and Lebesgue’s dominated convergence theorem, limk→∞ IIIk =0. Further,

(9.16)

|Ij,k| ≤(

supx∈Ωj\Ωk

dist (x, ∂Ωj)1−s|∇(φNj)(x)|)

×∫

Ωj\Ωk

dist (·, ∂Ωj ∪ ∂Ωk)s−1|∇u| dVol.

The first factor in the right side of (9.16) is bounded uniformly in j, k, by (9.14)and our assumption on s. Also, by Lemma 9.2, the second factor in (9.16) is≤ C‖u‖B1

2−s(Ωj\Ωk), i.e., small if j, k are large enough.To conclude the proof, we only need to show that, for a fixed k, |IIj,k| is small

if j is large enough. Thus, if we set fj := (−12I + K∗

j )−1(∂νj E(x0, ·)∣

∣

∂Ωj) and

f := (− 12I + K∗)−1(∂νE(x0, ·)

∣

∣

∂Ωj), it suffices to prove that

(9.17) ∇(φSjfj)|Ωk → ∇(φSf)|Ωk in L2(Ωk) as j →∞.

This, in turn, follows from Lebesgue’s dominated convergence theorem. Somewhatmore specifically, if Λj : ∂Ω → ∂Ωj is a natural bi-Lipschitz homeomorphism, itcan be proved that fj Λj → f in L2(∂Ω). This gives pointwise convergence. Thedomination is trivially given by |∇(φSjfj)| ≤ C on Ωk uniformly in j. The proofof uniqueness is therefore complete.

Note that (9.5) and (9.6) follow from Lemma 7.2, Lemma 9.4 and the integralrepresentation of the solution.

53

10. An endpoint Dirichlet problem

We continue to assume our standard hypotheses on M , Ω, V and the metrictensor, including (1.27). The aim of this section is to study the Dirichlet problemwith boundary data in the Besov space B1

1−s(∂Ω). Besides establishing the wellposedness of this problem for small s, our approach also shows that the solutionhas a normal derivative in B1

−s(∂Ω). Moreover, this is accompanied by a naturalestimate. Specifically, we have the following extension of Theorem 5.8 of [JK2] andTheorem 7.1 of [FMM].

Theorem 10.1. There exists s0 = s0(Ω) > 0 so that for 0 < s < s0 and f ∈B1

1−s(∂Ω), the Dirichlet problem

(10.1)

Lu = 0 in Ω,

Tru = f on ∂Ω,

u ∈ B12−s(Ω),

has a unique solution. The solution satisfies

(10.2) ‖dist (·, ∂Ω)s−1|∇u| ‖L1(Ω) + ‖u‖L1(Ω) ≈ ‖u‖B12−s(Ω) ≈ ‖f‖B1

1−s(∂Ω)

for constants that depend only on Ω and s. Furthermore, ∂νu ∈ B1−s(∂Ω).

More specifically, there exists g ∈ B1−s(∂Ω) such that u is also a solution of the

Neumann problem (9.2) with boundary datum g. In addition, there holds

(10.3)∥

∥

∥

∂u∂ν

∥

∥

∥

B1−s(∂Ω)

≤ C(Ω, s)‖f‖B11−s(∂Ω).

As a corollary of this and Theorem 9.3 (cf. also the remark following its state-ment), we see that under the hypotheses of the above theorem the Dirichlet-to-Neumann operator for L is actually an isomorphism of B1

1−s(∂Ω) onto B11−s(∂Ω) if

V > 0 on a set of positive measure in Ω (in fact a similar results also holds whenV ≡ 0 in Ω).

Proof of Theorem 10.1. We start with the existence part. As in [FMM], we developan approach which will eventually give us information about the normal derivativeof the solution.

Choose s0 as in Theorem 9.3 and fix an arbitrary s ∈ (0, s0). First, we shallprove a technical result to the effect that there exists C = C(Ω) > 0 such that, if

p ∈ ((n− 1)/n, 1) is so that s = (n− 1)(

1p − 1

)

, then

(10.4) ‖S−1a‖hp(∂Ω) ≤ C, ∀ a B11−s(∂Ω)-atom.

54

In the above, B11−s(∂Ω)-atoms are regarded as elements in L2

1(∂Ω) so that S−1abelongs to L2(∂Ω) ⊂ hp(∂Ω). In order to prove (10.4), the key step is to establishthe estimate

(10.5) ‖f‖hp(∂Ω) ≤ C‖∇tanSf‖hp(∂Ω) + C‖Sf‖L1(∂Ω),

for 1 − ε 0 is independent of f ∈ hp(∂Ω). Indeed,choosing f := S−1a in (10.5) yields (10.4) at once in view of (4.5). Note that ssmall guarantees that |p− 1| < ε.

As for (10.5), we note that, for p = 1, this follows from Theorem 6.3 of [MT2]together with Proposition 3.2 of [MT2] via the usual jump-relations. Also, ∇tanSis a bounded mapping of the complex interpolation scale consisting of hp(∂Ω) for(n − 1)/n < p ≤ 1 and Lp(∂Ω) for 1 < p < ∞ into itself (cf. Proposition B.6in [MT2]). Consequently, by [KM], an estimate like (10.5) is stable under smallperturbations of the parameter p near 1. This proves (10.5) and, hence, concludesthe proof of (10.4).

Returning to the main line of reasoning, fix an arbitrary f ∈ B11−s(∂Ω). By

Proposition 4.3, there exists a sequence of scalars (λj)j ∈ `1 and a sequence (aj)jof B1

1−s(∂Ω)-atoms such that

(10.6) f =∑

j≥0

λjaj and∑

j≥0

|λj | ≤ 2‖f‖B11−s(∂Ω).

If we now set uj := S(S−1aj) in Ω then, so we claim, u :=∑

j≥0 λjuj in Ω solves(10.1). To justify this observe that Lu = 0 and

(10.7)

‖u‖B12−s(Ω) + ‖dist (·, ∂Ω)s−1|∇u| ‖L1(Ω) + ‖u‖L1(Ω)

≤ C∑

j≥0

|λj | ≤ C‖f‖B11−s(∂Ω) = C‖Tru‖B1

1−s(∂Ω)

≤ C‖u‖B12−s(Ω),

by Theorem 7.1, (7.25) and (10.4). Let us point out that the estimate (10.7) takescare of Lemma 9.2, stated without proof in the previous section.

It is also implicit in the calculation above that∑m

j=0 λjuj → u in B12−s(Ω) as

m →∞ so that, by the continuity of the trace operator,

(10.8) Tr u =∑

j≥0

λjTr uj =∑

j≥0

λjaj = f,

in B11−s(∂Ω), which concludes the proof of the existence part.

Turning to uniqueness, let us assume that u solves the homogeneous version of(10.1). Adapting an idea from [JK2] consider Ωj Ω an approximating sequence

55

of smooth subdomains of Ω and, since u ∈ B12−s,0(Ω), take uj ∈ C∞comp(Ωj) approx-

imating u in the norm of B12−s(Ω). Thus, with Trj standing for the trace operator

on ∂Ωj ,

(10.9) ‖Trj u‖B11−s(∂Ωj) = ‖Trj (u− uj)‖B1

1−s(∂Ωj) ≤ C‖u− uj‖B12−s(Ω) −→ 0,

as j → ∞. Let us next assume for a moment that Ω is a smooth domain anddenote by Gj(x, y) is the Green function for L in Ωj . Then ‖∇yGj(x, ·)‖Lp(∂Ωj) ≤C(x, p) < +∞ uniformly in j, for each p > 1. This, (10.9) and the integral repre-sentation formula

(10.10) u(x) =∫

∂Ωj

∂νj,yGj(x, y)(Trju)(y) dσj(y), x ∈ Ωj ,

allow us to conclude, upon letting j → ∞, that u(x) = 0. Thus, since x wasarbitrary, u vanishes in Ω and this concludes the proof of the uniqueness part whenΩ is smooth.

Returning now to the general case of an arbitrary Lipschitz domain, by what wehave proved so far (i.e., existence and estimates in Lipschitz domains plus unique-ness in smooth domains) we deduce that, in each Ωj , the estimate

(10.11) ‖u‖B12−s(Ωj) ≤ C‖Trj u‖B1

1−s(∂Ωj)

holds with a constant independent of j. Now the desired conclusion follows from(10.11) by passing to the limit in j and invoking (10.9).

In order to show that u constructed above solves a Neumann problem (in thesense of §9) with an appropriate boundary datum in B1

−s(∂Ω) = hp(∂Ω) it sufficesto check that

(10.12)∂uj

∂ν∈ hp(∂Ω) and

∥

∥

∥

∂uj

∂ν

∥

∥

∥

hp(∂Ω)≤ C, uniformly in j.

However, this easily follows from the integral representation of each uj , the resultsin §6 and the boundedness of K∗ on hp(∂Ω) (cf. [MT2], Proposition B.6). Finally,(10.3) also follows in light of (10.12) and (10.6).

11. Invertibility of boundary integral operators on Besov spaces

Again, we assume the standard hypotheses on M , Ω, V and the metric tensor,including (1.27). To state the main result of this section, recall that C stands forthe collection of all constant functions on ∂Ω and set

(11.1) Bp−s(∂Ω) := f ∈ Bp

−s(∂Ω) : 〈f, χ〉 = 0, ∀χ ∈ C.

for 1 ≤ p ≤ ∞, 0 < s < 1.

56

Theorem 11.1. There exists ε ∈ (0, 1] with the following significance. Let 1 ≤p ≤ ∞ and 0 < s < 1 be so that one of the conditions (I)–(III) below are satisfied:

(11.2)

(I) :2

1 + ε< p <

21− ε

and 0 < s < 1;

(II) : 1 ≤ p <2

1 + εand

2p− 1− ε < s < 1;

(III) :2

1− ε< p ≤ ∞ and 0 < s <

2p

+ ε.

Also, let q ∈ [1,∞] denote the conjugate exponent of p. Then the operators listedbelow are invertible:

[1] 12I + K : Bp

s (∂Ω) −→ Bps (∂Ω);

[2] 12I + K∗ : Bq

−s(∂Ω) −→ Bq−s(∂Ω);

[3] S : Bq−s(∂Ω) −→ Bq

1−s(∂Ω).If V = 0 in Ω, then

[4] ± 12I + K : Bp

s (∂Ω)/C −→ Bps (∂Ω)/C;

[5] ± 12I + K∗ : Bq

−s(∂Ω) −→ Bq−s(∂Ω);

[6] S : Bq−s(∂Ω) −→ Bq

1−s(∂Ω)/C,are also invertible. Finally, when V > 0 on a set of positive measure in Ω, then

[7] − 12I + K : Bp

s (∂Ω) −→ Bps (∂Ω);

[8] − 12I + K∗ : Bq

−s(∂Ω) −→ Bq−s(∂Ω),

are invertible.These results are sharp in the class of Lipschitz domains. However, if ∂Ω ∈ C1

then we may take 1 ≤ p ≤ ∞ and 0 < s < 1.

For each 0 ≤ ε ≤ 1 consider the region Rε ⊂ R2 which is the interior of thehexagon OABCDE, where O = (0, 0), A = (ε, 0), B = (1, 1−ε

2 ), C = (1, 1),D = (1 − ε, 1), E = (0, 1+ε

2 ). Then the “invertibility” region described in (11.2)simply says that (s, 1/p) belongs to Rε or, possibly, to the (open) segments OA,CD. Note that the region encompassed by the parallelogram with vertices at (0, 0),(1, 1

2 ), (1, 1) and (0, 12 ) is common for all Lipschitz domains, and that Rε can be

thought as an enhancement of it. Also, for ε = 1, Rε simply becomes the standard(open) unit square in the plane. The sense in which this result is optimal is thatfor each ε > 0 and for each point (s, 1

p ) ∈ (0, 1)× (0, 1)\Rε, there exists a Lipschitzdomain Ω such that (1)–(8) in Theorem 11.1 fail.

The proof of Theorem 11.1 uses interpolation and several special cases of interestare singled out below.

Proposition 11.2. There exists s0 = s0(Ω) > 0 such that for 0 < s < s0 theoperators

(11.3) S : B1−s(∂Ω) −→ B1

1−s(∂Ω),

57

(11.4) S : B∞s−1(∂Ω) −→ B∞

s (∂Ω),

(11.5) 12I + K : B1

1−s(∂Ω) −→ B11−s(∂Ω),

are isomorphisms.

Proof. Note that, by the results in §7 and §8, they are well-defined and bounded.Consider some f ∈ B1

1−s(∂Ω) which has an atomic decomposition of the formf =

∑

λiai, where (λi)i ∈ `1 and the ai’s are B11−s(∂Ω)-atoms. By (10.4), we can

find hi ∈ hp(∂Ω) so that ‖hi‖hp(∂Ω) ≤ C and Shi = ai. Then∑

λihi convergesin hp(∂Ω) to an element g which S should send into f . By Proposition 5.2, f ∈B1−s(∂Ω). Thus, S in (11.3) is onto.To see that S is also one-to-one, take some f ∈ B1

−s(∂Ω) = hp(∂Ω) so thatSf = 0. It follows from the uniqueness for the Dirichlet problem in Ω± with datain B1

1−s(∂Ω) that Sf must vanish identically both in Ω+ := Ω and in Ω− := Rn \ Ω.See §10. Now, since f is the jump of ∂νSf across ∂Ω, we infer that f = 0. Thisproves that the operator in (11.3) is invertible.

The invertibility of the operator (11.4) follows by duality from what we havejust proved. Finally, Lemma 7.2, (11.3) and the fact that S intertwines K∗ and Kimply that the operator (11.5) is also an isomorphism, given that

(11.6) 12I + K∗ : B1

−s(∂Ω) −→ B1−s(∂Ω)

is an isomorphism. The proof of this is similar to, but simpler than, the proof of(9.7).

Let us digress for a moment and point out that a proof of Proposition 11.2can also be given based on the atomic theory from [MT2] plus a recent functionalanalytic result from [MM2]. In order to be more specific, we need some notation.

Call f an atom for Hp1(∂Ω), for (n− 1)/n < p ≤ 1, if it is supported in a surface

ball of radius r ∈ (0, diam Ω] and ‖∇tanf‖L2(∂Ω) ≤ r(n−1)(1/2−1/p). Then Hp1(∂Ω) is

defined as the `p-span of such atoms (and is equipped with the natural quasi-norm).Also, for (n−1)/n 1. Then there exists ε = ε(Ω) > 0 so that

(11.7) S : hp(∂Ω) −→ hp1(∂Ω), 1

2I + K : hp1(∂Ω) −→ hp

1(∂Ω)

are isomorphisms for 1 − ε < p ≤ 1. Indeed, when p = 1 this follows as in [DK],granted the results of [MT2]. It then further extends to a small interval about p = 1by general stability results for complex interpolation scales of quasi-Banach spacesfrom [KM].

The second ingredient we need is a result from [MM2] to the effect that

(11.8) F p,qα (Rn−1) = B1,1

α+(n−1)(1−1/p)(Rn−1), ∀ p ∈ (0, 1), q ∈ (0,∞), α ∈ R.

58

Here F p,qα (Rn) is the class of Triebel-Lizorkin spaces in Rn (cf., e.g., [Tr]), and hat

denotes the Banach envelope (cf. §5). Our interest in the Triebel-Lizorkin scaleF p,q

α stems from the identifications Hp = F p,20 and Hp

1 = F p,21 valid for 0 < p ≤ 1

(cf. the discussion in [MM2]). When (n − 1)/n < p < 1 and 0 ≤ α ≤ 1, thesame results remain true with Rn−1 replaced by the boundary of a (n-dimensional)Lipschitz domain Ω. Thus, applying the “hat” to (11.7) (in effect, invoking Lemma5.1) yields (11.3) and (11.5) at once.

We now return to the task of presenting the

Proof of Theorem 11.1. For simplicity, assume that V > 0 on a set of positivemeasure in Ω; the remaining cases require only minor alterations and are left to thereader.

We deal first with the operator 12I + K. The segments (E, O) and (B,C) cor-

responding to invertibility on Lp(∂Ω) for 0 < 1/p < (1 + ε)/2, and on Lp1(∂Ω) for

(1 − ε)/2 < 1/p < 1, respectively, have been treated in [MT2]. Also, the segment(O, A), corresponding to invertibility on B∞

s (∂Ω) for s small, has been taken careof in [MT2], while invertibility on the segment (C,D) is covered by Proposition11.2. Notice that the interior of the convex hull of these segments is precisely theregion Rε. Now, the desired result follows by repeated applications of the real andcomplex methods of interpolation together with a routine check that the inversesTs,p of 1

2I + K coincide on the intersections of the various spaces just considered,so that both 1

2I + K and Ts,p can simultaneously be interpolated.By duality, we obtain results for 1

2I + K∗ on the corresponding dual scales.Similar interpolation arguments apply to yield the statements made about − 1

2I+Kand − 1

2I + K∗ from what has been established before. Furthermore, a similarreasoning applies to the operator S, given Proposition 11.2 and the results in §7 of[MT2] on S, namely

(11.9) S : Lp(∂Ω) ∼−→ Lp1(∂Ω), 1 < p < 2 + ε,

plus, by duality,

(11.10) S : Lq−1(∂Ω) ∼−→ Lq(∂Ω), 2− ε < q < ∞.

The only novel point here is to check that B1−s(∂Ω) interpolates “well” with the

scale Bp−s(∂Ω), p ∈ (1,∞), s ∈ (0, 1). That is, we need

(11.11)[

B1−s0

(∂Ω), Bp−s1

(∂Ω)]

θ = Bpθ−sθ

(∂Ω), θ ∈ (0, 1),

for sθ := (1 − θ)s0 + θs1 and 1/pθ := (1 − θ) − θ/p, whenever 0 < s0, s1 < 1,1 < p < ∞. Note that the space in the left side is reflexive, as the intermediatespace between two Banach spaces one of which is reflexive. Thus, by the dualitytheorem for the complex interpolation method ([BL]),

(11.12)[

B1−s0

(∂Ω), Bp−s1

(∂Ω)]

θ =([

B∞s0

(∂Ω), Bqs1

(∂Ω)]

θ

)∗=

(

Bq/θsθ

(∂Ω))∗

,

59

where 1/p + 1/q = 1. From this, the desired conclusion follows.The fact that these results are sharp in the class of Lipschitz domains is discussed

in [FMM]. Finally, that ∂Ω ∈ C1 allows us to take 1 ≤ p ≤ ∞ and 0 < s < 1 followsfrom the results in [MT], [MMT], [MT2] via the same interpolation patterns.

Remark. We mention that the result [1] in Theorem 11.1, on the invertibility of12I + K, was established in (7.16) of [MT2] for (s, 1/p) in the smaller region Pε,the interior of the parallelogram with vertices OBCE. There, the metric tensorwas assumed to be Lipschitz. In view of Theorem 7.1 of [MT3] this result holdseven when the metric tensor is Holder continuous. Similarly the other resullts inTheorem 11.1 hold for (s, 1/p) ∈ Pε, when the metric tensor is Holder continuous.

12. The general Poisson problem with Neumann boundary conditions

Again, retain the standard hypotheses on M , Ω, V and the metric tensor, in-cluding (1.27). The first order of business is to properly formulate the Poissonproblem for the operator L with Neumann boundary conditions in the Lipschitzdomain Ω ⊂ M . We commence by defining the normal component of any 1-form Fwith components in Lq

−s+ 1q(Ω) for 0 < s < 1 and 1 < p, q < ∞, 1

p + 1q = 1.

Concretely, for an (arbitrary) extension f ∈(

Lps+ 1

p(Ω)

)∗= Lq

−s− 1p ,0(Ω) of

the distribution δ F ∈(

C∞comp(Ω))′

(as usual, 〈δ F , φ〉 = 〈F , dφ〉, for each φ ∈C∞comp(Ω)), we denote by νf · F the (scalar) normal component of F , with respectto the extension f . This is defined as the linear functional in Bq

−s(∂Ω) =(

Bps (∂Ω)

)∗

given by

(12.1) 〈νf · F, φ〉 := 〈f , φ〉+ 〈F , dφ〉, ∀φ ∈ Bps (∂Ω),

where φ ∈ Lps+ 1

p(Ω) is an extension (in the trace sense) of φ. The second pairing in

the right side of (12.1) is understood in the sense of (4.15) and is well defined sincedφ ∈ Lp

s+ 1p−1(Ω). In turn, this membership is a consequence of our assumptions

and (4.16).It is not difficult to check that the definition is correct and that

(12.2) ‖νf · F‖Bq−s(∂Ω) ≤ C‖F‖Lq

−s+ 1q(Ω) + C‖f‖(

Lp

s+ 1p

(Ω))∗ .

Note that for any function u ∈ Lq1−s+ 1

q(Ω) and any extension f of Lu = δ du−V u,

considered first as a distribution in Ω to an element in Lq−1−s+ 1

q ,0(Ω), the “normal

derivative” ∂fν u can be defined (with respect to the extension f), in the sense of

(12.1), as νf+V u · du.

60

For further reference we also note the integral formulas

(12.3) u = Π0f +D(Tr u)− S(∂fν u),

valid for arbitrary u ∈ Lq1−s+ 1

q(Ω) with Lu extendible to f ∈

(

Lps+ 1

p(Ω)

)∗. Here,

1 < p < ∞, 0 < s < 1 and Π0 is just the Newtonian potential on scalar-valuedfunctions. Also,

(12.4) 〈∂fν Π0(f), φ〉 = 〈f,Dφ〉, ∀φ ∈ Bp

s (∂Ω),

is valid for any f ∈(

Lps+ 1

p(Ω)

)∗for 1 < p < ∞, 0 < s < 1. They can be easily

justified starting from (12.1), using a limiting argument and invoking the mappingproperties of Π0, S, D established in §§6–8.

The main focus of this section is the boundary problem

(12.5)

Lu = f ∈ Lq1q−s−1,0(Ω),

∂fν u = g ∈ Bq

−s(∂Ω),

u ∈ Lq1−s+ 1

q(Ω).

When V = 0 in Ω, this is subject to the (necessary) compatibility condition

(12.6) 〈f, 1〉 = 〈g, 1〉.

In this regard, our main result is the following.

Theorem 12.1. Assume that the metric tensor satisfies (1.27). There exists ε =ε(Ω) > 0 having the following property.

Suppose that p ∈ (1,∞) and s ∈ (0, 1) are such that one of the conditions (I)–(III) in (11.2) is satisfied. Also, let q be the conjugate exponent of p. Then, ifV > 0 on a subset of positive measure in Ω, the Poisson problem with Neumannboundary condition (12.5) has a unique solution. In fact,

(12.7) u = Π0(f) + S(

− 12I + K∗)−1(

g − ∂fν Π0(f)

)

and there exists a positive constant C which depends only on Ω, p, s, such that

(12.8) ‖u‖Lq

1−s+ 1q(Ω) ≤ C‖f‖Lq

1q−s−1,0

(Ω) + C‖g‖Bq−s(∂Ω).

A similar set of results is valid when V = 0 in Ω. In this case, the compatibilitycondition (12.6) is assumed and the solution is unique modulo additive constants(also, (12.8) must be modified accordingly).

61

Finally, if ∂Ω ∈ C1 then we may take ε = 1. That is to say, the conclusionshold for all p ∈ (1,∞), s ∈ (0, 1).

Remark. It should be noted that similar results are valid for the scales of Besovspaces, i.e., when f ∈

(

Bps+ 1

p(Ω)

)∗. In this case the solution u belongs to Bq

1−s+ 1q(Ω)

and the second pairing in the right side of (12.1) remains meaningful because ofTheorem 1.4.4.6 and Corollary 1.4.4.5 in [Gr].

Proof of Theorem 12.1. Assume first that V does not vanish in Ω; the other case ishandled similarly. In view of Proposition 6.1 and (12.4), subtracting Π0(f) reducesthe problem to solving (12.5) with f = 0 and g := g − ∂f

ν Π0(f) ∈ Bq−s(∂Ω). For

this latter problem, a solution is given by S[

(− 12I + K∗)−1g

]

; cf. Theorem 7.1and Theorem 11.1. This finishes the proof of the existence part. Note that (12.8)follows from the integral representation formula (12.7) of the solution and mappingproperties of layer potentials.

It remains to establish uniqueness. To this end, if u ∈ Lq1−s+ 1

q(Ω) solves the

homogeneous version of (12.5), then taking the boundary trace in (12.3) readilygives that (− 1

2I + K)(Tru) = 0. The important thing is that the region Rε isinvariant to the transformation (s, 1

p ) 7→ (1 − s, 1 − 1p ) and that Tr u ∈ Bq

1−s(∂Ω).Thus, on account of Theorem 11.1, Tr u ≡ 0. Utilizing this back in (12.6) yieldsu ≡ 0 in Ω, as desired.

The argument for C1 domains is similar and, hence, omitted.

An important particular case, corresponding to s = − 1p , is singled out below.

Corollary 12.2. There exists ε = ε(Ω) > 0 so that, if 32 − ε < p < 3 + ε, then

for any f ∈ Lp−1,0(Ω) and any g ∈ Bp

− 1p(∂Ω), satisfying the compatibility condition

(12.6) if V = 0 in Ω, the Neumann problem

(12.9)

Lu = f in Ω,

∂fν u = g on ∂Ω,

u ∈ Lp1(Ω),

has a unique (modulo additive constants, if V = 0 in Ω) solution u. Moreover, ∇usatisfies the estimate

(12.10) ‖∇u‖Lp(Ω) ≤ C(Ω, p)(

‖f‖Lp−1,0(Ω) + ‖g‖Bp

− 1p

(∂Ω)

)

.

If ∂Ω ∈ C1, this holds for all p ∈ (1,∞).

Proof. One only needs to observe that ( 1p , 1 − 1

p ) ∈ Rε for p in a neighborhood ofthe interval [ 32 , 3].

In view of the counterexamples in [FMM], the results in this section are sharpin the class of Lipschitz domains.

62

Remark. Since ∂fν u is defined for a class of functions u for which the notion of the

trace of ∂νu is utterly ill defined, it is appropriate to explain that ∂fν u is not an

extension of the operation of taking the trace of ∂νu; perhaps it is useful to regardit as a “renormalization” of this trace, in a fashion that depends strongly on thechoice of f . Recall that, for u ∈ Lq

1−s+ 1q(Ω), Lu is naturally defined as a linear

functional on the space Lps+ 1

p ,b(Ω), taken by definition to be the closure of C∞0 (Ω)

in Lps+ 1

p(Ω). The choice of f is the choice of an extension of this linear functional

to en element of (Lps+ 1

q(Ω))∗ = Lq

1q−s−1,0(Ω).

As an example, consider u ∈ L21(Ω) and suppose that actually u ∈ L2

2(Ω), so∂νu ∈ L2

12(Ω) is well defined. In this case, Lu ∈ L2(Ω) has a “natural” extension

f0 ∈ L2−1,0(Ω). Any other extension f1 ∈ L2

−1,0(Ω) differs from f0 by an element ofL2−1(M) supported on ∂Ω. We have

(12.11) ∂f0ν u = ∂νu,

but if f1 = f0 + ω and ω 6= 0 is supported on ∂Ω, then ∂f1ν u is not equal to ∂νu;

we leave its computation as an exercise.

13. The general Poisson problem with Dirichlet boundary conditions

We once again retain the usual set of hypotheses on M , Ω, V and the metrictensor, including (1.27). In this section we shall deal with the Poisson problemfor L with Dirichlet boundary conditions and data in Sobolev-Besov spaces. Assuch, this extends previous work for constant coefficient operators in [JK2] and[FMM]. Sharpness of the range of (s, p) for which the following theorem holds wasestablished, for the class of Lipschitz domains in Euclidean space, in [JK2].

Theorem 13.1. Assume that the metric tensor satisfies (1.27). Then there existsε = ε(Ω) > 0 with the following property. If p ∈ (1,∞) and s ∈ (0, 1) are such thatone of the conditions (I)–(III) in (11.2) are satisfied, then for any f ∈ Lp

s+ 1p−2(Ω)

and any g ∈ Bps (∂Ω) the Dirichlet problem

(13.1)

Lu = f in Ω,

Tru = g on ∂Ω,u ∈ Lp

s+ 1p(Ω),

has a unique solution. Also, there exists C > 0 depending only on Ω, p, s, suchthat the solution satisfies the estimate

(13.2) ‖u‖Lp

s+ 1p

(Ω) ≤ C‖f‖Lp1p +s−2

(Ω) + C‖g‖Bps (∂Ω).

63

Moreover, if Π denotes the Newtonian potential for L on M , we can write

(13.3)u = Π(f)|Ω +D

(

( 12I + K)−1(g − TrΠ(f))

)

= Π(f)|Ω + S(

S−1(g − TrΠ(f)))

in Ω,

where f is an extension of f to an element in Lps+ 1

p−2(M).

In particular, if f ∈ Lp1p +s−2,0(Ω), then the solution has a normal derivative in

Bps−1(∂Ω) (in the sense discussed in §12) and

(13.4) ‖∂fν u‖Bp

s−1(∂Ω) ≤ C(

‖f‖Lp1p +s−2,0

(Ω) + ‖g‖Bps (∂Ω)

)

.

Similar results are valid on the scale of Besov spaces, i.e., when f ∈ Bp1p +s−2(Ω).

In this case, the solution u belongs to Bps+ 1

p(Ω).

Finally, if ∂Ω ∈ C1 then we can actually take ε = 1, i.e., the conclusions holdfor all p ∈ (1,∞), s ∈ (0, 1).

Proof. Fix an arbitrary f ∈ Lp1p +s−2(Ω) =

(

Lq1+ 1

q−s,0(Ω))∗

. Since Lq1+ 1

q−s,0(Ω) can

be identified (via extension by zero outside the support and restriction to Ω) withψ ∈ Lq

1+ 1q−s(M) : supp ψ ⊆ Ω, we may invoke Hahn-Banach’s extension theorem

to produce f ∈ Lp1p +s−2(M) so that f |Ω = f and the norm of f is controlled by

that of f . Now, clearly, (13.3) solves (13.1). Note that (13.2) and (13.4) also followfrom (13.3) and the mapping properties of the operators involved. Uniqueness canbe established by mimicking the argument already utilized in the proof of Theorem12.1.

Finally, invoking [MT], [MMT] and proceeding as before, it is clear that we maytake ε = 1 if ∂Ω ∈ C1.

Corollary 13.2. There exists ε = ε(Ω) > 0 so that if 32 − ε < p < 3 + ε then for

any f ∈ Lp−1(Ω) and any g ∈ Bp

1− 1p(∂Ω) the Dirichlet problem

(13.5)

Lu = f in Ω,Tru = g on ∂Ω,

u ∈ Lp1(Ω),

has a unique solution. This satisfies

(13.6) ‖u‖Lp1(Ω) ≤ C‖f‖Lp

−1(Ω) + C‖g‖Bp

1− 1p

(∂Ω)

and, if f ∈ Lp−1,0(Ω),

(13.7) ‖∂fν u‖Bp

− 1p

(∂Ω) ≤ C(

‖f‖Lp−1,0(Ω) + ‖g‖Bp

1− 1p

(∂Ω)

)

.

64

Similar results are valid on the scale of Besov spaces. Finally, if ∂Ω ∈ C1 then wecan actually take ε = 1.

In the last part of this section, we elaborate on the connection between Pois-son problems and Helmholtz decompositions (for the latter topic see also [FMM],[MT2]). The observation we wish to make is contained in the proposition below.

Proposition 13.3. Let Ω be an arbitrary (connected) Lipschitz subdomain of Mand fix 1 < p, q < ∞ with 1/p + 1/q = 1. Then the following are equivalent:

[i] The Lp-Helmholtz decomposition

(13.8) Lp(Ω, Λ1TM) = dLp1(Ω) ⊕ ω ∈ Lp(Ω, Λ1TM); δω = 0, ν ∨ ω = 0

holds (where the direct sum is topological).[ii] The Lp

1-Poisson problem for the Laplace-Beltrami operator with homoge-neous Neumann boundary conditions

(13.9)

∆u = f ∈ Lp−1,0(Ω), 〈f, 1〉 = 0,

∂fν u = 0 on ∂Ω,

u ∈ Lp1(Ω)/R,

is well posed.[iii] The Lq-Helmholtz decomposition (analogous to (13.8)) holds.[iv] The Lq

1-Poisson problem for the Laplace-Beltrami operator with homoge-neous Neumann boundary conditions (analogous to (13.9)) holds.

In particular, the range of p’s for which the Lp-Helmholtz decomposition (13.8)as well as the Lp

1-Poisson problem (13.9) are valid is always an interval which isinvariant under taking conjugate exponents.

Proof. We proceed to show [ii] assuming that [i] is valid. To this end, let q denote theconjugate exponent of p and let f ∈ Lp

−1,0(Ω) be such that 〈f, 1〉 = 0, otherwise arbi-trary; thus, f ∈ (Lq

1(Ω)/R)∗. Note that, by Poincare’s inequality, d maps Lq1(Ω)/R

isomorphically onto a closed subspace of Lq(Ω, Λ1TM). From the Riesz and Hahn-Banach theorems we may then conclude that there exists w ∈ Lp(Ω,Λ1TM) sothat

(13.10) 〈f, v〉 =∫

Ω〈w, dv〉 dVol, ∀ v ∈ Lq

1(Ω).

Next, decompose w = du+ω, according to (1). Then, based on (13.10), it is easy tocheck that the just constructed function u solves (13.9). Uniqueness and estimatesfor (13.9) follow from the corresponding uniqueness and estimates for (13.10).

Conversely, assume now that [ii] is well posed; our goal is to show that [i] holds.Indeed, if w ∈ Lp(Ω, Λ1TM) is arbitrary and fixed then v 7→

∫

Ω〈w, dv〉 dVol definesa linear functional on Lq

1(Ω)/R. Denoting by f ∈ (Lq1(Ω)/R)∗ this functional it

65

follows that (13.10) holds. Let now u ∈ Lp1(Ω) solve the Poisson problem (13.9)

for this datum f and set ω := w − du ∈ Lp(Ω, Λ1TM). Then it follows that∫

Ω〈ω, dv〉 dVol = 0 for each v ∈ Lq1(Ω). In turn, this readily implies that, first,

δω = 0 in Ω and, second, that ν ∨ ω = 0 on ∂Ω. Thus, w = du + ω is the desireddecomposition. Everything else is as before.

Going further, let

(13.11) T :(

L21(Ω)

)∗= L2

−1,0(Ω) −→ L21(Ω)

be the (well defined, linear and bounded) solution operator for the problem (13.10)with p = 2, i.e. T (f) = u. Since, by Green’s formula, T is self-adjoint it follows thatT extends as a bounded mapping of Lp

−1,0(Ω) into Lp1(Ω) if and only if T extends

to a bounded mapping of Lq−1,0(Ω) into Lq

1(Ω), for 1/p + 1/q = 1. This proves that[ii] and [iv] are equivalent.

The fact that [iv] is, in turn, equivalent to [iii] is already contained in [i]⇔[ii].Finally, the last part in the statement of the proposition follows from what we haveproved so far and interpolation.

In closing, let us point out that a similar result is valid for the Helmholtz de-composition

(13.12) Lp(Ω, Λ1TM) = dLp1,0(Ω) ⊕ ω ∈ Lp(Ω,Λ1TM); δω = 0

and the Poisson problem for the Laplace-Beltrami operator with homogeneousDirichlet boundary conditions

(13.13)

∆u = f ∈ Lp−1(Ω),

u ∈ Lp1,0(Ω).

We omit the details.Of course, counterexamples to the well posedness of the Poisson problems (13.9)

and (13.13) translate in the failure of the Helmholtz decompositions (13.8) and(13.12), respectively.

14. Complex powers of the Laplace-Beltrami operator

If ∆0 =∑n

j=1 ∂2/∂x2j stands for the usual space-flat Laplacian in Rn then

∇(−∆0)−1/2 corresponds, modulo a normalization constant, precisely to (Rj)nj=1,

the system of Riesz transforms in Rn. In particular, from the classical Calderon-Zygmund theory, the (vector Riesz transform) operator

(14.1) ∇(−∆0)−1/2 : Lp(Rn) → Lp(Rn)

66

is bounded for any 1 < p < ∞. See [St1].In this section, our aim is to study the analogous problem when Rn is replaced

by Ω, a connected Lipschitz domain in a Riemannian manifold M . In this context,we shall work with the associated Laplace-Beltrami operator ∆, although similarresults hold for the Schrodinger operator L := ∆ − V . The hypotheses that wemake on M and Ω are those of §1.

It is natural to impose boundary conditions and we shall consider ∆D and ∆N ,the Laplace-Beltrami operators equipped, respectively, with homogeneous Dirichletand Neumann conditions in Ω. Thus, the natural question is whether

(14.2) (−∆D)−1/2 : Lp(Ω) → Lp1,0(Ω)

and

(14.3) (−∆N )−1/2 : f ∈ Lp(Ω) :∫

Ωf = 0 → Lp1(Ω)

are bounded operators. For arbitrary domains, it has been recently shown in [CD]and [DMc] that this is indeed the case for any 1 < p ≤ 2. At the other extreme, ifΩ has a smooth boundary, then well known techniques based on pseudo-differentialoperators and Calderon-Zygmund theory allow one to take 1 < p < ∞. For Lip-schitz domains in the flat Euclidean space, the optimal range of p’s turns out tobe (1, 3 + ε); see the discussion in [JK2], [JK3], [MM]. Here we present a variablecoefficient extension of such results.

In fact, we shall deal with more general complex powers of

(14.4) A := (−∆D)1/2 and B := (−∆N )1/2.

In order to state our main results, let us introduce the region R ⊂ R2 given by

(14.5) R :=

(r, t) : max r/3, r − 1 ≤ t < 1, 0 ≤ r < 2

.

Theorem 14.1. Assume that the metric tensor satisfies (1.27). There exists aneighborhood ˜R of R in (r, t) : 0 ≤ r < 2, 0 < t < 1, depending on Ω, such thatfor any (r, 1/q) ∈ ˜R and δ ∈ R, the operators

(14.6) A−r+iδ : Lq(Ω) ∼−→ Lqr,0(Ω),

(14.7) B−r+iδ : Lq(Ω) ∼−→ Lqr(Ω)/R

are isomorphisms on the indicated spaces.

Here, for a space X of functions in Ω, X := f ∈ X :∫

Ω f = 0 and X/R is thespace X modulo constants.

67

Proof. The proof rests on three basic ingredients: the estimates for (−∆D)−1,(−∆N )−1 from §§12–13, Stein’s complex interpolation theorem (cf. [SW]) and Lp-bounds for purely imaginary powers of the operators (−∆D)1/2, (−∆N )1/2 (cf.[St2]). It parallels arguments in §7 of [JK2], supplemented by arguments in [MM],where this program is carried out in detail in the flat Euclidean setting. To givesome of the flavor, we sketch the opening arguments for (14.6). Basic Hilbert spacetheory gives

(14.8) A : L21,0(Ω) ∼−→ L2(Ω), A : L2(Ω) ∼−→ L2

−1(Ω).

We then have

(14.9)As+iγ : L2(Ω) ∼−→ L2

−s(Ω), 0 ≤ s ≤ 1,

A−32+δ+iγ : L2(Ω) ∼−→ L2

32−δ,0(Ω), 0 < δ ≤ 1

2 ,

the first by Stein interpolation, the next by applying A−2 and using the p = 2 caseof Theorem 13.1. A very general result of Stein gives

(14.10) Aiγ : Lp(Ω) ∼−→ Lp(Ω), 1 < p < ∞,

with an exponential bound. Then another application of Stein interpolation gives

(14.11) A−1+iγ : Lp(Ω) ∼−→ Lp1,0(Ω), 3

2 < p < 3.

The argument proceeds with further interpolation arguments and applications ofTheorem 13.1. In particular, (14.11) is extended to

(14.12) A−1+iγ : Lp(Ω) ∼−→ Lp1,0(Ω), 1 < p < 3 + ε.

We refer to [JK2] and [MM] for further details.

The next corollary follows from Theorem 14.1 much as in [JK2], [JK3], [MM], sowe shall state it here without further proof. For the flat-space Euclidean Laplacian,similar results have also been obtained (via a different approach which in the caseof (14.15)–(14.16) yields smaller intervals of p’s) in [AT].

Corollary 14.2. Assume that the metric tensor satisfies (1.27). There exists ε =ε(Ω) > 0 so that

(14.13) A−1 : Lq(Ω) ∼−→ Lq1,0(Ω)

and

(14.14) B−1 : Lq(Ω) ∼−→ Lq1(Ω)/R

68

are isomorphisms for 1 < q < 3 + ε. In particular,

(14.15)∫

Ω

|∇f |q dVol ≤ Cq

∫

Ω

|√

−∆D f |q dVol, ∀ f ∈ Lq1,0(Ω), q ∈ (1, 3 + ε),

and

(14.16)∫

Ω

|∇g|q dVol ≤ Cq

∫

Ω

|√

−∆N g|q dVol, ∀ g ∈ Lq1(Ω), q ∈ (1, 3 + ε).

Furthermore, the operators

(14.17) A : Lq1,0(Ω) −→ Lq(Ω)

and

(14.18) B : Lq1(Ω) −→ Lq(Ω)

are bounded for 1 < q < ∞. That is, there exists C = C(Ω, q) > 0 so that

(14.19)∫

Ω

|√

−∆D f |q dVol ≤ Cq

∫

Ω

|∇f |q dVol, ∀ f ∈ Lq1,0(Ω), q ∈ (1,∞),

and

(14.20)∫

Ω

|√

−∆N g|q dVol ≤ Cq

∫

Ω

|∇g|q dVol, ∀ g ∈ Lq1(Ω), q ∈ (1,∞).

Finally, this result is sharp in the class of Lipschitz domains. If, however, ∂Ω ∈ C1,then (14.15)–(14.16) are also valid for 1 < q < ∞.

The last corollary contains a variable coefficient extension of a result of B. Dahlberg([Da2]) concerning estimates for Green potentials in Lipschitz domains. In addition,we also treat the case of the Neumann potential.

Corollary 14.3. Assume that the metric tensor satisfies (1.27), and considerG(x, y) and N(x, y), the Green and Neumann functions of the Laplace-Beltramioperator in an arbitrary Lipschitz domain Ω ⊂ M . Also, denote by G and N the op-erators sending f , respectively, to

∫

Ω G(·, y)f(y)dVol(y) and∫

Ω N(·, y)f(y)dVol(y).Then, for some ε = ε(Ω) > 0,

(14.21)(

∫

Ω

|∇Gf(x)|q dVol(x))1/q

≤ C(

∫

Ω

|f(x)|p dVol(x))1/p

, f ∈ Lp(Ω),

69

and

(14.22)(

∫

Ω

|∇Ng(x)|q dVol(x))1/q

≤ C(

∫

Ω

|g(x)|p dVol(x))1/p

, g ∈ Lp(Ω),

provided 1 < p < q < 3 + ε and 1q = 1

p −1n . This result is sharp in the class of

Lipschitz domains.

Proof. To prove (14.21), we note that G = (−∆D)−1 = A−2, and

(14.23) A−1 : Lp(Ω) −→ Lp1,0(Ω) → Lq(Ω), A−1 : Lq(Ω) −→ Lq

1(Ω),

under the given hypotheses on p and q, by Theorem 14.1, indeed by (14.12). Theproof of (14.22) is similar.

Remark. It is perhaps worth explaining why (14.21) is not an immediate corollaryof Theorem 13.1. That is, given f ∈ Lp(Ω), write u = ∆−1

D f as u = v + w, wherev ∈ Lp

2(M) ⊂ Lq1(M) solves ∆v = f on a neighborhood of Ω (with f extended by

0 off Ω) and w solves

∆w = 0 on Ω, w∣

∣

∂Ω = h = −v∣

∣

∂Ω.

We have h ∈ Bq1− 1

q(∂Ω), and hence Theorem 13.1 applies, but only for 3

2 − ε < q <

3 + ε. This argument fails to treat the case 1 < q ≤ 32 − ε.

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