(page 75)

Atomic Decomposition of Hp SpacesAssociated with Some Schrodinger Operators

Jacek Dziubanski

Abstract. Let {Tt}t>0 be the semigroup generated by aSchrodinger operator −A = ∆−V , where V is a nonnegativepolynomial on Rd. We say that f is in Hp

A associated with theoperator A if the maximal function Mf(x) = supt>0 |Ttf(x)|belongs to Lp(Rd). We characterize elements of the space Hp

A

for 0 < p ≤ 1 by a special atomic decomposition.

1. Introduction. Let A be a Schrodinger operator on Rd which has theform

(1.1) A = −∆ +V,

where V (x) =∑β≤αaβx

β is a nonnegative nonzero polynomial on Rd, α =

(α1,α2, . . . ,αd), ∈ Nd, N = {0,1,2, . . .}. These operators were studied by anumber of authors, cf. [Fe], [HN], [Z].

We say that f is in the space HpA if the maximal function

(1.2) Mf(x) = supt>0|Ttf(x)|

is in Lp(Rd), where {Tt}t>0 is the semigroup generated by −A. The quasi-normin Hp

A is defined by

(1.3) ‖f‖pHpA

= ‖Mf‖pLp(Rd).

Let∫∞

0 λdEA(λ) be the spectral resolution of A. For a function ϕ from the

75

Indiana University Mathematics Journal c©, Vol. 47, No. 1 (1998)

76 J. Dziubanski

Schwartz class S([0,∞)), ϕ(0) 6= 0, we define the maximal operator Mϕ setting

(1.4) Mϕf(x) = supt>0|ϕ(tA)f(x)|,

where ϕ(tA)f =∫∞

0 ϕ(tλ)dEA(λ)f.We shall show that the quasi-norms ‖Mf‖pLp and ‖Mϕf‖pLp are equivalent.Our aim is to present an atomic characterization of the spaces Hp

A forp ∈ (0,1].

Operators of the form (1.1) appear as images of certain homogeneousoperators on homogeneous nilpotent Lie groups which, in fact, are differentialoperators if V is a sum of squares of polynomials, cf. e.g. [DHJ], [HJ], andreferences there. These methods are also used here. For p = 1 similar resultshave been obtained in [DZ1].

We define an auxiliary function m(x,V ), see [Z], by

(1.5) m(x,V ) =∑β≤α|DβV (x)|1/(|β|+2),

where |β| = |(β1,β2, . . . ,βd)| = β1 +β2 + · · ·+βd.Since V is a nonzero polynomial, there is a constant c > 0 such that

c ≤ m(x,V ) for every x ∈ Rd.We set

B0 = {x ∈ Rd : c ≤ m(x,V ) < 1},(1.6)

Bn = {x ∈ Rd : 2(n−1)/2 ≤ m(x,V ) < 2n/2} for n = 1,2,3 . . . .

We have Rd =⋃∞n=0Bn. We will denote by B(x,r) the ball in Rd with the

center at x and radius r.We say that a function a is an atom for the space Hp

A associated to a ballB(x0,r) if

(1.7) supp a ⊂ B(x0,r),

(1.8) ‖a‖L∞ ≤ (volB(x0,r))−1/p,

(1.9) if x0 ∈ Bn, then r ≤ 21−n/2,

(1.10) if x0 ∈ Bn and r ≤ 2−1−n/2, then∫xβa(x)dx = 0

for all |β| ≤ d(

1p− 1).

Hp Spaces Associated with Some Schrodinger Operators 77

The atomic quasi-norm in the space HpA is defined by

(1.11) ‖f‖pHpA atom = inf

{∑j

|cj |p},

where the infimum is taken over all decompositions f =∑j cjaj , with aj being

HpA atoms and cj being scalars.

Our main result in this paper is as follows:

Theorem 1.12. Let ϕ ∈ S([0,∞)), ϕ(0) 6= 0. Then for every p ∈ (0,1],there is a constant C > 0 such that

(1.13) ‖Mϕf‖Lp ≤ C‖Mf‖Lp ,

(1.14) ‖Mf‖Lp ≤ C‖f‖HpA atom,

(1.15) ‖f‖HpA atom ≤ C‖Mϕf‖Lp .

Our spaces HpA are of a different nature than the classical Hardy spaces

Hp(Rd), which may be thought of as the spaces Hp−∆(Rd), ∆ being the ordinary

Laplacian on Rd. It follows from Theorem 1.12 and properties of the sets Bn,see Section 2, that Hp(Rd) is a proper subspace of the space Hp

A. Every elementof Hp

A can be decomposed into atoms that are supported on small balls, butfor certain atoms no moment condition is required. Actually our Hp

A atoms arescaled local atoms in the sense of Goldberg, cf. Section 8. The idea of the proofof Theorem 1.12 is based on the fact that the kernels of the operators Tt looklike the classical heat kernels multiplied by (1 + t1/2m(x,V ))−b, which have fastenough decay when t is large (cf. (3.18′)). The kernels of the operators ϕ(tA) havea similar feature. If A is the Hermite operator, this type of decay of the kernelsTt can be read from Mehler’s formula. In order to get appropriate estimates forthe kernels ϕ(tA), where A is a Schrodinger operator with a positive polynomialpotential, we use the fact that A can be obtained as ΠP , where P is a regularkernel on a special nilpotent Lie group G and Π is a unitary representation ofG, cf. Section 3. The theory of regular kernels on nilpotent Lie groups, whichwas developed by P. G lowacki, turns out to be crucial here, cf. Section 3.

Acknowledgments. The author is greatly indebted to Andrzej Hulanickifor his remarks. He also wishes to express his gratitude to the referee for hisseveral helpful comments.

2. Resolution of identity associated with Bn. The following twolemmas below present important properties of the sets Bn.

78 J. Dziubanski

Lemma 2.1. There is a constant C such that for every R > 2 and everyn, if x ∈ Bn, then

{n′ : B(x,2−n/2R)∩Bn′ 6= Ø} ⊂ [n−C log2R, n+C log2R].

Proof. Assume that y ∈ B(x,2−n/2R)∩Bn′ , where x ∈ Bn. By thedefinition of Bn′ there is β ≤ α such that (1/C)2(n′−1)(|β|+2)/2 ≤ |DβV (y)|.Applying the Taylor formula and the fact that x ∈ Bn, we obtain

1C

2(n′−1)(|β|+2)/2 ≤∣∣∣∑γ≤α

1γ!Dγ+βV (x)(y−x)γ

∣∣∣ ≤ C2(|β|+2)n/2R|α|,

which implies n′ ≤ n+C log2R. In the same manner we can see that n ≤n′+C log2R.

Lemma 2.2. There is a constant C and a collection of balls B(n,k) =

B(x(n,k),21−n/2), n = 0,1,2, . . ., k = 1,2, . . ., such that x(n,k) ∈ Bn, Bn ⊂⋃kB(x(n,k),2−n/2), and #{(n′,k′) : B(x(n,k), R2−n/2)∩B(x(n′,k′), R2−n

′/2) 6=Ø} ≤ RC for every (n,k) and R ≥ 2.

Proof. For fixed n let B(x(n,k),2−2−n/2) be a countable sequence of balls

such that x(n,k) ∈ Bn, B(x(n,k),2−2−n/2)∩B(x(n,k′),2−2−n/2) = Ø for k 6= k′,

and Bn ⊂⋃kB(x(n,k),2−n/2). It is now easy to check, using Lemma 2.1, that the

family of balls B(x(n,k),21−n/2), n = 0,1, . . ., k = 1,2, . . . satisfies the conclusionof Lemma 2.2.

As a consequence of Lemma 2.2, we obtain the following result:

Lemma 2.3. There are nonnegative functions ψ(n,k) such that

(2.4) ψ(n,k) ∈ C∞c (B(x(n,k),21−n/2)),

(2.5)∑(n,k)

ψ(n,k)(x) = 1,

(2.6)∥∥∥∥ ∂|β|∂xβ

ψ(n,k)

∥∥∥∥L∞≤ Cβ2|β|n/2.

3. Functional calculus of operators on homogeneous groups andSchrodinger operators. Let G be a homogeneous nilpotent group equippedwith a family of dilations δt (cf. [FS]). A distribution P on G is called a regular

Hp Spaces Associated with Some Schrodinger Operators 79

kernel of order r > 0, if P coincides with a smooth function away from the originand

(3.1) 〈P,f ◦ δt〉 = tr〈P,f〉.

It was proved in [D2] that for every Schrodinger operator of the form (1.1)there exist a homogeneous nilpotent Lie group G, a unitary representation Π ofG, and a regular symmetric kernel P of order 2 such that

(3.2) ΠP = A.

We shall denote by the same letter P the convolution operator f 7→ f ∗P .The kernel P satisfies the following maximal subelliptic estimates proved by P.G lowacki (cf. [G], [D2]); for every left-invariant homogeneous differential operator∂ on G and every N ≥ |∂|/2 there is a constant C > 0 such that

(3.3) ‖∂f‖L2(G) ≤ C‖(1 +P )Nf‖L2(G), for f ∈ C∞c (G),

where |∂| is the degree of homogeneity of ∂.Topologically G can be written as G = X ⊕Y = Rd⊕RD, where

D = (α1 + 1)(α2 + 1) . . . (αd + 1). The construction of the group G and therepresentation Π is such that for every multi-index β ∈ Nd there is a left-invariantdifferential operator ∂ on G such that

(3.4)∂|β|

∂xβ= Π∂ .

Let {St}t>0 be the semigroup of linear operators on L2(G) generated by theessentially self-adjoint operator −P . The estimates (3.3) and the homogeneityof P imply that the semigroup {St}t>0 is of the form

Stf = f ∗ qt, qt(g) = t−Q/2q1(δt−1/2g), where qt ∈ C∞(G)∩L2(G),

Q denotes the homogeneous dimension of G. Moreover, the following estimatesfor qt hold (cf. [G], [D2]): for every left-invariant homogeneous differentialoperator ∂ on G there is a constant C > 0 such that

(3.5) |∂qt(g)| ≤ Ct(t1/2 + |g|)−Q−|∂|−2,

where |g| is a fixed homogeneous norm on G and |∂| is the degree of homogeneityof ∂.

80 J. Dziubanski

For a positive integer k let W k(G) denote the set of all functions F ∈ Ck(G)such that

supg∈G|∂F(g)(1 + |g|)k| <∞

for every left-invariant differential operator ∂ on G of degree ≤ k.Similarly, we define W k(Rd×RD) to be the set of all functions F =

F (x,ξ) ∈ Ck(Rd×RD) such that

sup(x,ξ)∈Rd×RD

∣∣∣∣∣ ∂|β|∂xβ∂|β′|

∂ξβ′F (x,ξ)(1 + |x|+ |ξ|)k

∣∣∣∣∣ <∞for all β ∈Nd, β′ ∈ ND such that |β|+ |β′| ≤ k.

Obviously, cf. [FS], for every k > 0 there exists k′ such that W k′(G) ⊂W k(Rd×RD) and W k′(Rd×RD) ⊂W k(G).

For an integer m > 0 let us denote by Sm0 ([0,∞)) the subspace of allfunctions ϕ from S([0,∞)) such that

dj

dλjϕ(0+) = 0 for j = 1,2,3, . . . m.

We set S∞0 ([0,∞)) =⋂∞m=1Sm0 ([0,∞)).

Let∫∞

0 λdEP (λ) be the spectral resolution of the positive definite operatorP . The theorem below was actually proved in [D1].

Theorem 3.6 (cf. [D1]). For every k > 0 there is an integer m > 0 suchthat for ϕ ∈ Sm0 ([0,∞)) there is a function F[ϕ] ∈W k(G) such that

ϕ(P )f = f ∗F[ϕ],

where ϕ(P )f =∫∞

0 ϕ(λ)dEP (λ)f .Moreover, for every left-invariant differential operator ∂ on G of degree ≤ k

there is a constant C such that for ϕ ∈ Sm0 ([0,∞)) we have

|∂F[ϕ](g)|(1 + |g|)k ≤ Cm∑j=0

supλ≥0

{(1 +λ)m

∣∣∣∣ djdλj ϕ(λ)∣∣∣∣} .

Corollary 3.7. If ϕ ∈ S∞0 ([0,∞)), then there is a Schwartz class functionF[ϕ] on G such that

ϕ(P )f = f ∗F[ϕ].

Moreover, ϕ(tP )f = f ∗F[ϕ]t , where F[ϕ]

t (g) = t−Q/2F[ϕ](δt−1/2g).

Hp Spaces Associated with Some Schrodinger Operators 81

We shall need the following result:

Proposition 3.8. Assume that ψ ∈ S([0,∞)), 0 /∈ suppψ. Let F[ψ] be theconvolution kernel of ψ(P ). Then

(3.9)∫G

xβF[ψ](x)dx = 0

for every multi-index β ∈ Nd+D.

Proof. For a multi-index β let N be a positive integer such that N >

‖β‖+Q, where ‖β‖ is the homogeneous degree of the monomial xβ on G. Thenby the spectral theorem

(3.10) F[ψ](x) = PNF[ω](x),

where ω(λ) = λ−Nψ(λ). Observe that the operator PN is a convolution with aregular kernel of order 2N . Therefore, xβ ∗PN is well defined, and∫

G

F[ψ](x)xβ dx =∫G

(F[ω] ∗PN )(x)xβ dx =∫G

F[ω](x)(xβ ∗PN )dx.

Note that xβ ∗PN is a C∞ function on G homogeneous of degree ‖β‖− 2N < 0.Thus xβ ∗PN ≡ 0, which implies (3.9).

The properties of the kernels F[ϕ]t of the operators ϕ(tP ) on the group G

allow us to obtain suitable estimates for the integral kernels Φt(x,w) of theoperators ϕ(tA) on Rd. One can deduce from the equality ΠP = A that

(3.11) Πϕ(tP ) = ϕ(tA) for ϕ ∈ S([0,∞)).

The construction of G, Π, and P is such that if F ∈ L1(G), then the integralkernel F (x,w) of the of the operator ΠF on Rd is given by

F (x,w) = F (w−x,V (x), . . . ,DβV (x), . . .),(3.12)

F (x,ξ) = (FYF)(x,ξ),

where FYF is the partial Fourier transform of F with respect to Y variables (cf.[D2], [DHJ]). Consequently, if Ft(g) = t−Q/2F(δt−1/2g), then the integral kernelFt(x,w) of the operator ΠFt is of the form

(3.13) Ft(x,w) = t−d/2F

(w−xt1/2

, tV (x), . . . , t(|β|+2)/2DβV (x), . . .).

82 J. Dziubanski

For a function ϕ ∈ Sm0 ([0,∞)) we denote by F[ϕ]t the convolution kernel of

the operator ϕ(tP ) on G, and by Φt(x,w) the integral kernel of the operatorϕ(tA) = ΠF[ϕ]

t. By (3.12), (3.13), Theorem 3.6, Corollary 3.7, and the

homogeneity of P , we have the following:

Corollary 3.14. For every integer k > 0 there exists m > 0 such that if

ϕ ∈ Sm0 ([0,∞)), then the kernels Φt and F[ϕ] = F[ϕ]1 are related by

(3.15) Φt(x,w) = t−d/2Φ(w−xt1/2

, tV (x), . . . , t(|β|+2)/2DβV (x), . . .),

where

(3.16) Φ(x,ξ) = (FYF[ϕ])(x,ξ) ∈W k(Rd×RD).

If, moreover, ϕ ∈ S∞0 ([0,∞)), then

Φ(x,ξ) ∈ S(Rd×RD).

Here and subsequently, S(Rd×RD) denotes the Schwartz class of functions onRd×RD.

Let us denote by Tt(x,w) the integral kernels of the semigroup Tt = Πqt .

Proposition 3.17. For every b > 0 there is a constant Cb such that

0 ≤ Tt(x,w)(3.18)

≤ Cbt−d/2(1 + t−1/2|x−w|)−b∏β≤α

(1 + t(|β|+2)/2|DβV (x)|)−b.

For every multiindex β ∈ Nd there is a constant C > 0 such that

(3.19)∣∣∣∣ ∂|β|∂xβ

Tt(x,w)∣∣∣∣ ≤ Ct−(d+|β|)/2(1 + t−1/2|x−w|)−d−2−|β|.

Proof. The estimate (3.18) was proved in [D2, Proposition 3.17]. To prove(3.19) we use (3.5), (3.12), and (3.4).

It was pointed out by the referee that the estimate (3.18) can be written as

(3.18′) 0 ≤ Tt(x,w) ≤ Cbt−d/2(1 + t−1/2|x−w|)−b(1 + t1/2m(x,V ))−b′.

Hp Spaces Associated with Some Schrodinger Operators 83

Assume that ψ ∈ S([0,∞)) be such that 0 /∈ suppψ. Similarly, we denote by

F[ψ]t the convolution kernel of the operator ψ(tP ) and by Ψt(x,w) the integral

kernel of the operator ψ(tA). Obviously, by Corollary 3.14, we have

(3.15′) Ψt(x,w) = t−d/2Ψ(w−xt1/2

, tV (x), . . . , t(|β|+2)/2DβV (x), . . .),

where

(3.16′) Ψ(x,ξ) = (FYF[ψ])(x,ξ) ∈ S(Rd×RD).

The corollary below follows from Proposition 3.8.

Corollary 3.20. For every multi-index γ ∈ ND and β ∈ Nd we have

(3.21)∫

Rd

∂|γ|

∂ξγΨ(u−w,ξ)

∣∣∣ξ=0

wβ dw = 0.

4. Tangential maximal functions associated to A.

Lemma 4.1. Assume that m ∈ {1,2,3, . . . } and θ, ϕ ∈ Sm0 ([0,∞)),ϕ(0) 6= 0. Then there are ψ(0) ∈ Sm0 ([0,∞)) and ψ ∈ C∞c (0,1) such that

(4.2) 1 = ψ(0)(λ)ϕ(λ) +∞∑j=1

ψ(2−jλ)ϕ(2−jλ) for λ ≥ 0,

and, consequently

θ(λ) = ψ(0)(λ)θ(λ)ϕ(λ) +∞∑j=1

ψ(2−jλ)θ(λ)ϕ(2−jλ) for λ ≥ 0.

Proof. Since ϕ(0) 6= 0 there exists a positive integer k and a constantc > 0 such that |ϕ(λ)| > c for λ ∈ [0,2−k]. Therefore there is a functionψ ∈ C∞c (2−k−7,2−k−5) such that

∑∞j=−∞ψ(2−jλ)ϕ(2−jλ) = 1 for λ > 0. Set

η(λ) =∞∑j=1

ψ(2−jλ)ϕ(2−jλ).

84 J. Dziubanski

Obviously η(λ) = 1 for λ > 2−k−1. Put

ψ(0)(λ) =

1− η(λ)ϕ(λ)

for λ ∈ [0,2−k]

0 for λ > 2−k .

It is easy to verify that ψ(0) ∈ Sm0 ([0,∞)) and (4.2) holds.

For functions ψ(0), ψ, and θ from Lemma 4.1 let us denote by K(0)t (x,y) and

K2−jt(x,y) the integral kernels of the operators ψ(0)(tA)θ(tA) and ψ(2−jtA)θ(tA)respectively. The following lemma can be deduced from (3.4), (3.13), Theorem3.6 and Corollary 3.14.

Lemma 4.3. For every M, M ′′, k > 0 there there exists m > 0 such thatif θ, ϕ, ψ(0), ψ ∈ Sm0 ([0,∞)) are from Lemma 4.1. Then there exists a constantC > 0 such that∣∣∣∣ ∂|β|∂xβ

K(0)t (x,y)

∣∣∣∣ ≤ Ct−(d+|β|)/2(

1 +|x− y|t1/2

)−M,(4.4)

∣∣∣∣ ∂|β|∂xβK2−jt(x,y)

∣∣∣∣ ≤ C(2−jt)−(d+|β|)/22−jM′′(

1 +|x− y|

(2−jt)1/2

)−M,(4.5)

for every β, |β| ≤ k.

Remark. One should not confuse the functions ψ(0), ψ that are defined on[0,∞) with the functions ψ(n,k) (from Lemma 2.3) that are defined on Rd.

For a positive real number N and ϕ ∈ S([0,∞)) we define an analogue ofPeetre tangential maximal operator M∗ϕ,N by

(4.6) M∗ϕ,Nf(x) = supt>0, y∈Rd

{|ϕ(tA)f(y)|(1 + t−1/2|y−x|)−N}.

The operators M∗ϕ,N have many of the properties of classical maximaloperators. We present some of them which we shall need later. For theconvenience of the reader we provide the proofs.

Lemma 4.7. For every N > 0 there exists m > 0 such that if θ, ϕ ∈Sm0 ([0,∞)) such that ϕ(0) 6= 0, then there exists a constant C such that

(4.8) Mθf(x) ≤ CM∗ϕ,Nf(x) .

Hp Spaces Associated with Some Schrodinger Operators 85

Proof. Lemmas 4.1 and 4.3 imply

Mθf(x) = supt>0

∣∣∣ψ(0)(tA)θ(tA)ϕ(tA)f(x) +∞∑j=1

ψ(2−jtA)θ(tA)ϕ(2−jtA)f(x)∣∣∣

≤ supt>0

∫|K(0)

t (x,y)ϕ(tA)f(y)|dy

+ supt>0

∞∑j=1

∫|K2−jt(x,y)ϕ(2−jtA)f(y)|dy

≤ supt>0

C∞∑j=0

2−jM′′∫

(2−jt)−d/2(

1 +|x− y|

(2−jt)1/2

)−M

×(

1 +|x− y|

(2−jt)1/2

)NM∗ϕ,Nf(x)dy,

which gives (4.8).

Lemma 4.9. For every N > 0 there exists m > 0 such that if ϕ ∈Sm0 ([0,∞)), ϕ(0) 6= 0, then there is a constant C > 0 such that

(4.10)∣∣∣∣ ∂∂xiϕ(tA)f(x)

∣∣∣∣ ≤ Ct−1/2(

1 +|x−w|t1/2

)NM∗ϕ,Nf(w).

Proof. Applying Lemmas 4.1 and 4.3 to θ = ϕ, we obtain

∣∣∣∣ ∂∂xiϕ(tA)f(x)∣∣∣∣

≤∫ ∣∣∣∣ ∂∂xiK(0)

t (x,y)ϕ(tA)f(y)∣∣∣∣ dy

+∞∑j=1

∫ ∣∣∣∣ ∂∂xiK2−jt(x,y)ϕ(2−jtA)f(y)∣∣∣∣ dy

≤ C∞∑j=0

∫(2−jt)−(d+1)/22−jM

′′(

1 +|x− y|

(2−jt)1/2

)−M

×(

1 +|w− y|

(2−jt)1/2

)N (1 +

|w− y|(2−jt)1/2

)−N|ϕ(2−jtA)f(y)|dy ≤

86 J. Dziubanski

≤ C∞∑j=0

∫t−(d+1)/22−jM

′′2j(d+N+1)/2

(1 +|x− y|t1/2

)−M

×(

1 +|x− y|t1/2

)N (1 +|x−w|t1/2

)NM∗ϕ,Nf(w)dy

≤ Ct−1/2(

1 +|x−w|t1/2

)NM∗ϕ,Nf(w).

Proposition 4.11. For every N > 0 there is m > 0 such that for everyfunction ϕ ∈ Sm0 ([0,∞)), ϕ(0) 6= 0, there is a constant C > 0 such that

(4.12) M∗ϕ,Nf(x) ≤ C[M[H−L](Mϕf)r(x)]1/r,

where r = d/N and M[H−L] is the classical Hardy-Littlewood maximal operator.

Proof. Fix 0 < δ1 < 1 and set δ = t1/2δ1. By the Mean Value Theorem weget

|ϕ(tA)f(x−u)|

≤ Cδ−d/r

(∫|x−u−y|≤δ

|ϕ(tA)f(y)|r dy)1/r

+Cδ sup|x−y−u|≤δ

|∇ϕ(tA)f(y)|

≤ Cδ−d/r(δ+ |u|)d/r[M[H−L]|ϕ(tA)f |r(x)]1/r

+ Cδ sup|x−u−y|≤δ

|∇ϕ(tA)f(y)|(

1 +|x− y|t1/2

)−N (1 +|x− y|t1/2

)N.

Lemma 4.9 leads to

|ϕ(tA)f(x−u)|

≤ Cδ−d/r(δ+ |u|)d/r[M[H−L]|ϕ(tA)f |r(x)]1/r +Ct−1/2δM∗ϕ,Nf(x)

(1 +

δ

t1/2+|u|t1/2

)N≤ Cδ

−d/r1

(1 +|u|t1/2

)d/r[M[H−L]|ϕ(tA)f |r(x)]1/r +Cδ1M

∗ϕ,Nf(x)

(1 + δ1 +

|u|t1/2

)N.

Hp Spaces Associated with Some Schrodinger Operators 87

Therefore,

|ϕ(tA)f(x−u)|(

1 +|u|t1/2

)−N≤ Cδ−d/r1 [M[H−L]|ϕ(tA)f |r(x)]1/r +Cδ1M

∗ϕ,Nf(x).

Taking δ1 sufficiently small, we get (4.12).

Proposition 4.13. For every p ∈ (0,1] there exists m > 0 such that ifθ, ϕ ∈ Sm0 ([0,∞)), ϕ(0) 6= 0, then there exists a constant C > 0 such that

(4.14) ‖Mθf‖Lp ≤ C‖Mϕf‖Lp .

Proof. Fix N > 0 such that r = d/N < p. Let m > 0 be such that Lemma4.7 and Proposition 4.11 hold. By Lemma 4.7 it suffices to show that

(4.15) ‖M∗ϕ,Nf‖Lp ≤ C‖Mϕf‖Lp .

By virtue of Proposition 4.11, we obtain

‖M∗ϕ,Nf‖Lp ≤ C‖[M[H−L](Mϕf)r]1/r‖Lp

= C‖M[H−L](Mϕf)r‖1/rLp/r

≤ C‖(Mϕf)r‖1/rLp/r

= C‖Mϕf‖Lp .

5. Proof of (1.13). For p ∈ (0,1] let m > 0 be large enough such thatProposition 4.13 holds. If ϕ ∈ S([0,∞)), then there are constants c1, c2, . . . , cm+1

such that θ(λ) = ϕ(λ)−∑m+1n=1 cne

−nλ ∈ Sm0 ([0,∞)). Therefore

‖Mϕf‖Lp ≤ C(‖Mθf‖Lp + ‖Mf‖Lp).

By Proposition 4.13 it suffices to show that there exists a function ϕ ∈Sm0 ([0,∞)), ϕ(0) 6= 0, such that

(5.1) ‖Mϕf‖Lp ≤ C‖Mf‖Lp .

Such a function can be constructed as a linear combination of functions e−nλ,

88 J. Dziubanski

that is, there are constants d1,d2, . . . ,dm+1 such that

(5.2) ϕ(λ) =m+1∑n=1

dne−nλ ∈ Sm0 ([0,∞)) and ϕ(0) = 1.

Obviously for ϕ of the form (5.2) the inequality (5.1) is satisfied.

6. Proof of (1.14). It suffices to show that there is a constant C > 0 suchthat

(6.1) ‖Ma‖pLp ≤ C

for every HpA atom a. Let a be an Hp

A atom associated to a ball B(x0,r). Assumethat x0 ∈ Bn. By the definition of the atoms r ≤ 21−n/2. It follows from (1.8)and the estimates (3.18) that

(6.2) Ma(x) ≤ Cr−d/p for x ∈ B∗(x0,r) = B(x0,2r),

and, consequently,

(6.3)∫B(x0,2r)

(Ma(x))p dx ≤ C.

In order to show that

(6.4)∫B(x0,2r)c

(Ma(x))p dx ≤ C ,

we consider two cases:

o Case 1: r ≤ 2−1−n/2

Then, by definition, a satisfies the moment conditions (1.10). Let τ be thesmallest integer > d(1/p− 1). For fixed x ∈ B(x0,2r)c let us consider the Taylorexpansion of the function w 7→ Tt(x,w) at x0. According to (3.19) and theequality Tt(x,w) = Tt(w,x), we get

Hp Spaces Associated with Some Schrodinger Operators 89

|Tta(x)| =∣∣∣∣∫ a(w)Tt(x,w)dw

∣∣∣∣=

∣∣∣∣∣∣∫a(w)[Tt(x,w)−

∑|γ|<τ

1γ!

(∂|γ|

∂wγTt

)(x,w)

∣∣∣w=x0

(w−x0)γ ]dw

∣∣∣∣∣∣≤ Ct−τ/2t−d/2

∫|a(w)|

(1 +|x−x0|t1/2

)−d−τ|w−x0|τ dw

≤ C|x−x0|−d−τrτ+dr−d/p.

Therefore

(6.5)∫B(x0,2r)c

(Ma(x))p dx ≤ Crpτ+pdr−d∫B(x0,2r)c

|x−x0|−(d+τ)pdx ≤ C.

o Case 2: 2−1−n/2 < r ≤ 2−n/2

Note that in this case no moment condition on a is required. For 0 < t ≤ 2−n

and x ∈ B(x0,2r)c, we have

|Tta(x)| ≤ C∫|a(w)|Tt(x,w)dw

≤ Ct−d/2∫B(x0,r)

r−d/p(

1 +|x−w|t1/2

)−Mdw

≤ Ct−d/2r−d/p∫B(x0,r)

(1 +|x−x0|t1/2

)−Mdw

≤ C2nd/(2p)(

1 +|x−x0|2−n/2

)−M,

with M > 0 being large, cf. (3.18). This gives

(6.6)∫B(x0,2r)c

(sup

0<t≤2−n|Tta(x)|

)pdx ≤ C

∫2nd/2

(1 +|x−x0|2−n/2

)−Mp

dx ≤ C.

It remains to estimate∫B(x0,2r)c

(supt>2−n |Tta(x)|)p dx.

90 J. Dziubanski

Let P tm be the operator defined by

P tmf(x) =∫|f(w)|t−d/2χB(0,m)(t−1/2(w−x))

×( ∏β≤α

χ[−m,m](t(|β|+2)/2DβV (x)

))dw.

As a consequence of (3.18), we obtain

(6.7) |Tta(x)| ≤∑m≥2

bmPtma(x),

where bm ≤ Cq(1 +m)−q for every q > 0. We shall use the following lemma.

Lemma 6.8. There is a constant C1 ≥ 1 independent of n such that forevery bounded function f such that suppf ⊂ B(xn,21−n/2), xn ∈ Bn, and everym ≥ 2, we have

(6.9) P tmf ≡ 0 for t > mC12−n.

Proof. Let f be a bounded function such that suppf ⊂ B(xn,21−n/2),

xn ∈ Bn. Then, by Lemma 2.1, suppf ⊂⋃n+C2k=n−C2

Bk. Assume that P tmf 6≡ 0.

Then there is x ∈ Rd and w ∈suppf such that

|t(|β|+2)/2DβV (x)| ≤ m for all β, β ≤ α, and |t−1/2(w−x)| ≤ m.

Since DγV (w) =∑β≤α(1/β!)Dβ+γV (x)(w−x)β for every γ ≤ α,

|DγV (w)| ≤ Cm|α|+1 t−(|γ|+2)/2.

On the other hand, there is γ such that

|DγV (w)|1/(|γ|+2) ≥ 12D

2(n−C2)/2.

Thus

2(n−C2)(|γ|+2))/2 ≤ Cm(|α|+1)t−(|γ|+2)/2.

This implies t ≤ mC12−n, which completes the proof of Lemma 6.8.

Hp Spaces Associated with Some Schrodinger Operators 91

By Lemma 6.8

∫B(x0,2r)c

(supt>2−n

|Tta(x)|)pdx ≤

∥∥∥ supt>2−n

∣∣∣ ∞∑m=2

bmPtma(x)

∣∣∣ ∥∥∥pLp

≤∥∥∥ ∞∑m=2

bm sup2−n<t≤2−nmC1

|P tma(x)|∥∥∥pLp(dx)

.

Note that sup2−n<t≤2−nmC1 |P tma(x)| ≤ C2nd/(2p)χB(x0,2−n/2mC3)(x). Therefore

∫B(x0,2r)c

(supt>2−n

|Tta(x)|)pdx ≤ C

∞∑m=2

∫bpm2nd/2χB(x0,2−n/2mC3)(x)dx ≤ C.

This ends the proof of (6.4).

7. Local maximal functions. For a function ϕ ∈ S([0,∞)) a real numberN > 0, and for a nonnegative integer n we define the local maximal functions

M∗(n)ϕ,N and M (n)

ϕ by

M∗(n)ϕ,N f(x) = sup

0<t≤2−n, w∈Rd

{|ϕ(tA)f(w)|(1 + t−1/2|w−x|)−N} , and(7.1)

M (n)ϕ f(x) = sup

0<t≤2−n

∣∣∣∣∫Rd

Φt(w−x,0)f(w)dw∣∣∣∣ ,(7.2)

where Φt(x,0) = t−d/2Φ(x/t1/2, 0), and Φ(x,0) is defined by (3.15) and (3.16),here (x,0) ∈ Rd×RD.

Let us observe, cf. [D2], that the function x 7→ Φ(−x,0) is the convolutionkernel of the operator ϕ(−∆), where ∆ is the Laplacian on Rd. ThereforeΦ(−x,0) (as a function of x) belongs to the Schwartz class S(Rd) and∫

Φ(−x,0)dx = ϕ(0). Our aim in this section is to prove the following result:

Theorem 7.3. For every l, N > 0 and every p ∈ (0,1] there exists m > 0such that if ϕ ∈ Sm0 ([0,∞)), ϕ(0) = 1, then there is a constant C > 0 such that

‖M (n)ϕ (ψ(n,k)f)‖Lp ≤ C‖(1 + 2n/2|x−x(n,k)|)−lM∗(n)

ϕ,N f(x))‖Lp(dx),

see Section 2 for the definitions of ψ(n,k) and x(n,k).

92 J. Dziubanski

For ϕ ∈ Sm0 ([0,∞)), ϕ(0) = 1 let ψ(0) ∈ Sm0 ([0,∞)), and ψ ∈ C∞c ((0,∞))be such that ψ(0)(λ) = 0 for λ > 1 and

(7.4) 1 = ϕ(λ)ψ(0)(λ) +∞∑j=1

ϕ(2−jλ)ψ(2−jλ) for λ ≥ 0.

Proposition 7.5. For every M , ` > 0 there is a constant C > 0 such thatfor every 0 < t ≤ 2−n we have∣∣∣∣∫ ψ(n,k)(w)Φt(w−x, 0)Ψ2−jt(w,u)dw

∣∣∣∣ ≤ C2−jM t−d/2(1 + 2j/2t−1/2|x−u|)−`.

Proof. For γ ∈ ND we shall denote by Dγξ the differential operator ∂|γ|/∂ξγ .

From (3.15′) it follows that∣∣∣∣∫ ψ(n,k)(w)Φt(w−x,0)Ψ2−jt(w,u)dw∣∣∣∣

≤∣∣∣∣∫ ψ(n,k)(w)Φt(w−x,0)

×[Ψ2−jt(w,u)−

∑|γ|<M ′

1γ!

2jd/2t−d/2Dγξ Ψ(

u−w(2−jt)1/2 , 0

)ξγ]dw

∣∣∣∣∣∣+

∣∣∣∣∣∣∫ψ(n,k)(w)Φt(w−x,0)

∑|γ|<M ′

1γ!

2jd/2t−d/2Dγξ Ψ(

u−w(2−jt)1/2 , 0

)ξγ dw

∣∣∣∣∣∣= I1 + I2,

where ξ = (2−jtV (w), . . . ,(2−jt)(|β|+2)/2DβV (w), . . .).In order to estimate I1 we observe that ‖ξ‖ ≤ C2−j, whenever w ∈ B(n,k)

and 0 < t ≤ 2−n. By virtue of the Taylor formula and Corollary 3.14, we obtain

I1 ≤ C∫B(n,k)

2−jM′t−d/2

(1 +|x−w|t1/2

)−b2jd/2t−d/2

(1 +

|w−u|(2−jt)1/2

)−`dw

≤ C∫B(n,k)

2−jM′t−d/2

(1 +|x−w|t1/2

)−b2jd/2t−d/2

(1 +

|w−u|(2−jt)1/2

)−`×(

1 +|x−u|

(2−jt)1/2

)`(1 +

|x−u|(2−jt)1/2

)−`dw ≤

Hp Spaces Associated with Some Schrodinger Operators 93

≤ C2−jM′t−d/2

(1 +

|x−u|(2−jt)1/2

)−`∫B(n,k)

2jd/2t−d/2(

1 +|x−w|t1/2

)−b×(

1 +|x−w|

(2−jt)1/2

)`dw

≤ C2−jM′t−d/2

(1 +

|x−u|(2−jt)1/2

)−`∫B(n,k)

2jd/2t−d/2(

1 +|x−w|t1/2

)−b× 2j`/2

(1 +|x−w|t1/2

)`dw

≤ C2−jM′t−d/22j(`+d)/2

(1 +

|x−u|(2−jt)1/2

)−`.

To estimate I2 we study each summand separately. Let

H(x,w) =1γ!ψ(n,k)(w)Φt(w−x,0)ξγ ,

where ξ = (2−jtV (w), . . . ,(2−jt)(|β|+2)/2DβV (w) . . .). We denote by DκwH(x,w)

the differentiation of the function H(x,w) with respect to the second variable.For fixed x and u we consider Taylor expansion of the function w 7→ H(x,w) atthe point u. By Corollary 3.20,

I2,γ =∣∣∣∣∫ 1

γ!ψ(n,k)(w)Φt(w−x,0)2jd/2t−d/2Dγ

ξ Ψ(

u−w(2−jt)1/2 , 0

)ξγ dw

∣∣∣∣=∣∣∣∣∫ H(x,w)2jd/2t−d/2Dγ

ξ Ψ(

u−w(2−jt)1/2 , 0

)dw

∣∣∣∣=

∣∣∣∣∣∣∫ [

H(x,w)−∑|κ|<M ′′

1κ!DκwH(x,u)(w−u)κ

]

× 2jd/2t−d/2Dγξ Ψ(

u−w(2−jt)1/2 , 0

)dw

∣∣∣∣∣∣≤∣∣∣∣∣∫|w−u|≤|x−u|/2

∣∣∣∣∣+∣∣∣∣∣∫|w−u|≥|x−u|/2

∣∣∣∣∣= I ′2,γ + I ′′2,γ .

94 J. Dziubanski

Since 0 < t ≤ 2−n and M ′′ is large, using Taylor’s formula, (2.6), and Corollary3.14, we have

I ′2,γ ≤ C∫|w−u|≤|x−u|/2

|w−u|M ′′t−d/2t−M ′′/2(

1 +|x−u|t1/2

)−`× (2−jt)−d/2

(1 +

|w−u|(2−jt)1/2

)−bdw

≤ C2−jM′′/2t−d/2

(1 +|x−u|t1/2

)−`

≤ C2−jM t−d/2(

1 +|x−u|

(2−jt)1/2

)−`.

Likewise,

I ′′2,γ ≤ C∫|w−u|≥|x−u|/2

|w−u|M ′′t−d/2t−M ′′/2(2−jt)−d/2(

1 +|w−u|

(2−jt)1/2

)−bdw

≤ C2−jM′′/2t−d/2

(1 +

|x−u|(2−jt)1/2

)−`,

which completes the proof of Proposition 7.5.

Corollary 7.6. For every M,l > 0 there is a constant C such that forevery 0 < t ≤ 2−n we have∣∣∣∣∫ ψ(n,k)(w)Φt(w−x,0)Ψ2−jt(w,u)dw

∣∣∣∣(7.7)

≤ C2−jM t−d/2(

1 +|x−u|

(2−jt)1/2

)−`(1 + 2n/2|x−x(n,k)|)−`.

Proof. Corollary 3.14 implies∣∣∣∣∫ ψ(n,k)(w)Φt(w−x,0)Ψ2−jt(w,u)dw∣∣∣∣(7.8)

≤ C

∫ψ(n,k)(w)t−d/2

(1 +|x−w|t1/2

)−`2dj/2t−d/2

(1 +

|w−u|(2−jt)1/2

)−`dw

≤ Ct−d/2(1 + 2n/2|x−x(n,k)|)−`∫

2dj/2t−d/2(

1 +|w−u|

(2−jt)1/2

)−`dw ≤

Hp Spaces Associated with Some Schrodinger Operators 95

≤ Ct−d/2(1 + 2n/2|x−x(n,k)|)−` .

Now (7.7) follows from (7.8) and Proposition 7.5.

Proof of Theorem 7.3. Let 0 < t ≤ 2−n. From (7.4) we conclude that∣∣∣∣∫ ψ(n,k)(w)f(w)Φt(w−x,0)dw∣∣∣∣

=∣∣∣∣∫ ψ(n,k)(w)Φt(w−x,0)

[ψ(0)(tA)ϕ(tA)f(w)

+∞∑j=1

ψ(2−jtA)ϕ(2−jtA)f(w)]dw

∣∣∣∣∣∣≤∣∣∣∣∣∫ ∫

ψ(n,k)(w)Φt(w−x,0)Ψ(0)t (w,u)

(1 +|x−u|t1/2

)N×(

1 +|x−u|t1/2

)−Nϕ(tA)f(u)dudw

∣∣∣∣∣+

∣∣∣∣∣∣∞∑j=1

∫ ∫ψ(n,k)(w)Φt(w−x,0)Ψ2−jt(w,u)

(1 +

|x−u|2−j/2t1/2

)N

×(

1 +|x−u|

2−j/2t1/2

)−Nϕ(2−jtA)f(u)dudw

∣∣∣∣∣ .Applying Corollaries 7.6 and 3.14 with m sufficiently large, we get∣∣∣∣∫ ψ(n,k)(w)f(w)Φt(w−x,0)dw

∣∣∣∣≤ C

∫ ∫ψ(n,k)(w)t−d/2

(1 +|x−w|t1/2

)−`−Nt−d/2

(1 +|w−u|t1/2

)−`−N×(

1 +|x−u|t1/2

)NM∗(n)ϕ,N f(x)dwdu

+ C∞∑j=1

2−jM t−d/2(1 + 2n/2|x−x(n,k)|)−`

×∫ (

1 +|x−u|

(2−jt)1/2

)−`−N (1 +

|x−u|(2−jt)1/2

)NM∗(n)ϕ,N f(x)du ≤

96 J. Dziubanski

≤ C(1 + 2n/2|x−x(n,k)|)−`M∗(n)ϕ,N f(x).

8. Proof of (1.15). Atomic decomposition. Before we turn to ourproof of atomic decomposition we state some results from the theory of localHardy spaces, cf. [Go].

Let Φ be a function from the Schwartz class on Rd such that∫

Φ = 1. For

every nonnegative integer n we define the local maximal function M(n)Φ

by

(8.1) M(n)Φf(x) = sup

0<t≤2−n|f ∗ Φt(x)|,

where Φt(x) = t−d/2Φ(x/t1/2).We say that f is in the local Hardy space hpn if

(8.2) ‖f‖phpn = ‖M (n)Φf‖pLp <∞.

A function a is an atom for the local Hardy space hpn associated to a ball B(x0,r),r ≤ 2−n/2 if

(8.3) supp a ⊂ B(x0,r), r ≤ 2−n/2,

(8.4) ‖a‖L∞ ≤ (volB(x0,r))−1/p,

(8.5) if r ≤ 2−1−n/2, then∫a(x)xβ dx = 0 for |β| ≤ d

(1p− 1).

The atomic quasi-norm in hpn is defined by

(8.6) ‖f‖phpa,n = inf{∑

j

|cj |p},

where the infimum is taken over all decompositions f =∑cj aj , where aj are hpn

atoms and cj are scalars.

Theorem 8.7 [Go]. The quasi-norms ‖ ‖hpn and ‖ ‖hpa,n are equivalent with

constants independent of n ∈ Z, that is, there exists a constant Cp > 0 such thatfor every n we have

C−1p ‖f‖hpn ≤ ‖f‖hpa,n ≤ Cp‖f‖hpn .

Hp Spaces Associated with Some Schrodinger Operators 97

Moreover, if f ∈ hpn, suppf ⊂ B(x,21−n/2), then there are hpn atoms ajsuch that supp aj ∈ B(x,22−n/2) and

(8.8) f =∑j

cj aj ,∑j

|cj |p ≤ C‖f‖phpn

with a constant C independent of n.

Proof of (1.15). We first prove that for every p ∈ (0,1] there exists m > 0such that (1.15) holds for every function ϕ such that ϕ ∈ Sm0 ([0,∞)), ϕ(0) = 1.

Indeed, it follows from Theorems 7.3, 8.7, Lemma 2.1, and the definition ofHpA atoms that

ψ(n,k)f =∑j

cj,(n,k)aj,(n,k),

where aj,(n,k) are HpA atoms and

∑j

|cj,(n,k)|p ≤ C‖(1 + 2n/2|x−x(n,k)|)−`M∗(n)ϕ,N f(x)‖pLp(dx),

provided ϕ ∈ Sm0 ([0,∞)) (with m large enough) and ϕ(0) = 1. Therefore

f =∑(n,k)

ψ(n,k)f =∑(n,k)

∑j

cj,(n,k)aj,(n,k)

and ∑(n,k)

∑j

|cj,(n,k)|p ≤ C∑(n,k)

‖(1 + 2n/2|x−x(n,k)|)−`M∗ϕ,Nf(x)‖pLp(dx).

Since∑

(n,k)(1 + 2n/2|x−x(n,k)|)−lp is a bounded function for sufficiently largel (cf. Lemmas 2.1 and 2.2), we have

∑(n,k)

∑j

|cj,(n,k)|p ≤ C‖M∗ϕ,Nf‖pLp ,

which, by (4.15), gives

∑(n,k)

∑j

|cj,(n,k)|p ≤ C‖Mϕf‖pLp .

98 J. Dziubanski

Now let ϕ ∈ S([0,∞)) be such that ϕ(0) 6= 0. Then there are constantsc1, c2, ...., cm+1 such that ϕ(1)(λ) =

∑m+1n=1 cnϕ(nλ) ∈ Sm0 ([0,∞)) and ϕ(1)(0) = 1.

By the above

‖f‖pHpA atom ≤ C‖Mϕ(1)f‖

pLp ≤ C1‖Mϕf‖pLp ,

which completes the proof of (1.15).

References

[CGGP] M. Christ, D. Geller, P. G lowacki and L. Polin, Pseudodifferentialoperators on groups with dilations, Duke Math. J. 68 (1992), 31–65.

[D1] J. Dziuba�nski, Schwartz spaces associated with some non-differential convolutionoperators on homogeneous groups, Colloq. Math. 63 (1992), 153–161.

[D2] J. Dziuba�nski, A note on Schrodinger operators with polynomial potentials,Colloq. Math. (to appear).

[DHJ] J. Dziuba�nski, A. Hulanicki, and J.W. Jenkins, A nilpotent Lie algebra andeigenvalue estimates, Coll. Math. 68 (1995), 7–16.

[DZ] J. Dziuba�nski and J. Zienkiewicz, Hardy space H1 associated to Schrodingeroperator with potential satisfying reverse Holder inequality, (Preprint).

[DZ1] , Hardy spaces associated with some Schrodinger operators, (Preprint).[Fe] C. Fefferman, The uncertainty principle, Bull. Amer. Marth. Soc. 9 (1983),

129–206.[FeS] C. Fefferman and E. Stein, Hp spaces of several variables, Acta Math. 129

(1972), 137–193.[FS] G. Folland and E. Stein, Hardy Spaces on Homogeneous Groups, Princeton

University Press, Princeton, New Jersey, 1982.[G] P. G lowacki, Stable semigroups of measures as commutative approximate

identities on nongraded homogeneous groups, Invent. Math. 83 (1986), 557–582.[Go] D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979),

27–42.[He] W. Hebisch, On operators satisfying Rockland condition, (Preprint of the

University of Wroc law).[HN] B. Helffer and J. Nourrigat, Une Inegalite L2, (Preprint).

[H] A. Hulanicki, A functional calculus for Rockland operators on nilpotent Liegroups, Studia Math. 78 (1984), 253–266.

[HJ] A. Hulanicki and J.W. Jenkins, Almost everywhere summability on nilmani-folds, Trans. Amer. Math. Soc. 278 (1983), 703–715.

[S] E. Stein, Harmonic Analysis, Princeton University Press, Princeton, 1993.[Z] J. Zhong, The Sobolev estimates for some Schrodinger type operators, Ph.D.

Thesis, Princeton University, Princeton, 1993.

Institute of MathematicsUniversity of Wroc lawPlac Grunwaldzki 2/450-384 Wroc law, POLANDE{mail: jdziuban@math.uni.wroc.pl

Received: July 3rd, 1997; revised: November 2nd, 1997.