Dedicated to Professor M. S. Agranovich on the Occasion of ...into an orthogonal sum of countably many operators of Sturm{Liouville type, each acting on a semi-in nite interval. The

Russian Journal of Mathematical Physics, Vol. 8, No. 3, 2001, pp. 322–335.Copyright c© 2001 by MAIK “Nauka/Interperiodica” (Russia).

Dedicated to Professor M. S. Agranovich on the Occasion of His 70th Birthday

Geometry of Sobolev Spaces on RegularTrees and the Hardy Inequalities

K. Naimark and M. SolomyakDepartment of Physics of Complex Systems and Department of Mathematics,

The Weizmann Institute of Science, Rehovot 76100, IsraelE-mail: [email protected], [email protected]

Received May 15, 2001

Abstract. Regular metric trees (see Definition 2.1) possess a rich group of symmetries. As a

consequence, the Sobolev space H1(Γ) on such a tree Γ admits a direct decomposition thatis simultaneously orthogonal with respect to many inner products, including those definedby

RΓ uv dx and

RΓ u′v′ dx. This decomposition was discovered by the authors in [5]; here we

describe the construction in detail. Applications to spectral analysis of Schrodinger operatorsand to the Hardy inequalities on trees are given.

1. INTRODUCTION

Regular trees (see Definition 2.1) form an interesting class of general metric trees, i.e., treeswhose edges are regarded as nondegenerate line segments rather than pairs of vertices. The specialmetric and combinatorial structure of regular trees is reflected by the geometry of the correspondingL2-spaces and Sobolev spaces. Revealing this geometry is useful in the study of various analyticproblems on such trees. Roughly speaking, this enables one to reduce a problem concerning a treeto a family of more elementary problems on intervals. Certainly, this reduction is possible only ifthe problem under consideration is in a sense compatible with the geometry of the given tree.

In Sections 2 and 3 we present the fundamentals concerning the L2-spaces and Sobolev spaceson a regular tree Γ. We construct a special decomposition of these spaces that is simultaneouslyorthogonal with respect to the inner products in L2(Γ), in the Sobolev space H1(Γ), and in manyother Hilbert function spaces. This decomposition was discovered in our paper [5]. Here we give anew presentation of this material, which is more consistent and detailed, following the approachin [5] and also using some technical tools of the Carlson paper [1] (Carlson found a similar scheme,a bit later and in a different setting). The main difference between our approaches is that Carlsonis mainly concentrated on trees of finite total length or at least on those of finite radius (the lattermeans that the distance between two points is a bounded function on Γ× Γ). On the contrary, aregular tree of infinite radius is the main object of our investigation.

The basic material is illustrated by two applications. The first (Section 4) is related to the spectraltheory of a class of Schrodinger operators on regular trees of infinite radius. For compatibility withthe symmetries of the tree, we assume that the potential is symmetric, i.e., depends only on thedistance from a point x ∈ Γ to the root of Γ. We show that any such operator can be decomposedinto an orthogonal sum of countably many operators of Sturm–Liouville type, each acting on asemi-infinite interval. The domain of each component is described by means of specific matchingconditions on the functions and their derivatives at some points tn, tn →∞. Further applicationsof this result are given in [8].

The other application (Section 5) is related to Hardy inequalities of the form∫Γ

V (x)|u(x)|2dx 6 C∫

Γ

|u′(x)|2 dx, u(o) = 0. (1.1)

Here the point o ∈ Γ is the root of the tree. The problem is to describe the class of all functions V > 0(“Hardy weights”) for which inequality (1.1) holds. There are many other versions of the Hardy

The work of the second author was supported by the EPSRC grant no. GR/N 37193/01.

322

GEOMETRY OF SOBOLEV SPACES ON REGULAR TREES 323

inequality, namely, (Lp−Lq)-estimates, inequalities with two weight functions, etc. We concentrateon the simplest version (1.1) because of its close connection to the spectral theory of Laplacians ontrees.

An important question, which seems to have been ignored in the past, is as follows: what is thespace of functions u satisfying the inequality (1.1) for a given V ? It was agreed “by default”that the only natural space is the “homogeneous Sobolev space” H(Γ) consisting of all functionsu on Γ such that u(o) = 0 and the integral in the right-hand side of (1.1) is finite. For this space,the complete description of Hardy weights on arbitrary trees was found by Evans, Harris, and Pickin [2]. Their conditions imply that, in typical cases, such weights decay rather rapidly along thetree.

In our opinion, there is another natural function space for this problem, namely, the spaceH◦(Γ),i.e., the closure in H(Γ) of the set of all compactly supported functions. It is exactly this subspace,rather than the entire space H(Γ), that arises in connection with spectral analysis of the Laplacian.For some trees, these two spaces coincide. Such trees are said to be recurrent. For other trees, whichare said to be transient, H◦(Γ) is a proper subspace of H(Γ). The orthogonal complement H(Γ)of H◦(Γ) consists of the so-called harmonic functions on Γ, which rapidly grow along the tree.The functions u ∈ H◦(Γ) have slower growth because they can be approximated by functions withcompact support. Thus, there is a dichotomy between the growth properties of functions in H(Γ)and in H◦(Γ). For this reason, the decay restrictions for the Hardy weights on H◦(Γ) must beweaker than those for the entire space H(Γ).

We think that it is useful to have a result for the class H◦(Γ) similar to that in [2] for the entirespace. In this paper we make a step in this direction, namely, we solve the problem for the regulartrees and for symmetric weight functions V , i.e., for functions depending only on the distance froma point x ∈ Γ to the root o. As a consequence, in Corollary 5.3 we obtain a simple necessary andsufficient condition for the Laplacian on a regular tree to be positive definite.

2. REGULAR TREES

2.1. Geometry of a Tree

Let Γ be a rooted tree. Assume that o is the root, V = V(Γ) is the set of vertices, and E = E(Γ)is the set of edges of Γ. Suppose that #V = #E =∞. Write

V ′(Γ) = V(Γ) \ {o}.

Each edge e is regarded as a nondegenerate line segment. The distance ρ(y, z) between any twopoints y, z ∈ Γ (and thus the metric topology on Γ) is introduced in the natural way. Everywherebelow, |y| stands for ρ(y, o).

Write y � z if |z| = |y|+ρ(y, z). The relation y ≺ z means that y � z and z 6= y. The relation ≺defines a partial ordering on Γ. For y ≺ z we write 〈y, z〉 := {x ∈ Γ : y � x � z}. In particular,if e = 〈y, z〉 is an edge, then we say that y is its initial point and that e is emanating from y andterminating at z.

For any vertex z, its generation Gen(z) is defined by Gen(z) = #{x ∈ V : o ≺ x � z}. We assumethat Gen(z) <∞ for any vertex z. For an edge e, we define Gen(e) as the generation of the initialpoint of e.

The branching number b(z) of a vertex z is defined as the number of edges emanating from z.We assume that b(o) = 1 and that 1 < b(z) < ∞ for any z 6= o. Denote by e1

z, . . . , eb(z)z the edgesemanating from a vertex z ∈ V ′. The only edge that emanates from the root o is denoted by eo.

Definition 2.1. A tree Γ is said to be regular if all vertices of the same generation have equalbranching numbers and all edges of the same generation are of the same length.

In this paper we consider regular trees only. Obviously, any regular tree is completely determinedby specifying two number sequences, {bk} and {tk}, k = 0, 1, . . . , such that

b(z) = bGen(z), |z| = tGen(z) for each z ∈ V(Γ).

RUSSIAN JOURNAL OF MATHEMATICAL PHYSICS Vol. 8 No. 3 2001

324 K. NAIMARK, M. SOLOMYAK

According to our assumptions, one has b0 = 1 and bk > 2 for k > 1. It is clear that t0 = 0, and thesequence {tk} is strictly increasing. Write

r(Γ) = limk→∞

tk = supx∈Γ|x|.

It is natural to refer to r(Γ) as the radius of the tree.Let T ⊂ Γ be a subtree. Let oT be the root of T . The branching function gT (t) of T is defined by

gT (t) = #{x ∈ T : |x| = t}.

There are two types of subtrees, which are associated with vertices and edges of Γ, respectively.Namely, for any vertex z and for any edge e = 〈y, w〉, we set

Tz = {x ∈ Γ : x � z}, Te = e ∪ Tw. (2.1)

Obviously, To = Teo = Γ. If T = Te and Gen(e) = k, then gT (t) = gk(t), where

gk(t) =

0, t < tk,

1, tk 6 t 6 tk+1,

bk+1 . . . bn, tn < t 6 tn+1, n > k.

In particular, g0(0) = 1 and g0(t) = b0 . . . bn, tn < t 6 tn+1, n > 0. Thus,

gk(t) = (b0 · · · bk)−1g0(t), t > tk. (2.2)

Note thatgTz(t) = bkgk(t), Gen(z) = k. (2.3)

The natural measure dx on Γ is induced by the Lebesgue measure on the edges. Below we writeL2(Γ) = L2(Γ, dx) and denote by M(Γ) the linear space of all measurable functions that are finitealmost everywhere on Γ. The notation M(c, d), where (c, d) is an interval, has a similar meaning,and Mc(Γ) is defined as the space of all functions in M(Γ) supported by finitely many edges only.

2.2. Level-Wise Orthogonality and a Decomposition of M(Γ)

We say that two functions u, v ∈ M(Γ) are level-wise orthogonal if∑x∈Γ:|x|=t

u(x)v(x) = 0 for almost all t > 0. (2.4)

Two subspaces F,G ⊂M(Γ) are level-wise orthogonal if relation (2.4) holds for any u ∈ F , v ∈ G.Our technique is based on a suitable decomposition of the space M(Γ) into a family of mutually

level-wise orthogonal subspaces associated with subtrees T ⊂ Γ.For a subtree T ⊂ Γ, we say that a function u ∈ M(Γ) belongs to the classMT (which is a linear

subspace) if

u = 0 outside T, u(x) = u(y) if x, y ∈ T and |x| = |y|. (2.5)

Any function u ∈ MT can naturally be identified with the corresponding function f = JTu ∈M(|oT |, r(Γ)) such that u(x) = f(|x|) almost everywhere on T . In particular, the subspace MΓ

consists of all symmetric functions on the entire tree Γ (i.e., these functions depend on |x| only).The operator

(PTu)(x) =

{gT (|x|)−1

∑y∈T : |y|=|x| u(y) for x ∈ T,

0 for x /∈ T(2.6)



acts on M(Γ) and defines a projection ontoMT . It follows from (2.6) that the function u−PTu islevel-wise orthogonal to the subspace MT . It is natural to say that PT is the level-wise orthogonalprojection onto MT .

We need the subspaces MT associated with the subtrees Tz and Te (2.1). Let Gen(z) = k.For brevity, writeMz and Mj

z, j = 1, . . . , bk, instead ofMTz and MTejz

, respectively. For a given

function u ∈ M(Γ), we define the functions fz,u, f jz,u ∈ M(tk, r(Γ)) as follows:

fz,u = JTzPTzu, f jz,u = JTejz

PTejz

u, j = 1, . . . , bk.

According to (2.6) and (2.3), one has

fz,u(t) = (bkgk(t))−1∑

y∈Tz : |y|=tu(y) almost everywhere on (tk, r(Γ));

f jz,u(t) = gk(t)−1∑

y∈Tejz

: |y|=tu(y) almost everywhere on (tk, r(Γ)).

The subspaces M1z, . . . ,Mbk

z are mutually level-wise orthogonal, and their linear sum

Mz =M1z + · · ·+Mbk

z

contains Mz. Any function u ∈ Mz can be identified with the vector-valued function f = Jzu ∈(M(tk, r(Γ))

)bk given by

f = {f1, . . . , fbk}, fj = f jz,u, j = 1, . . . , bk. (2.7)

It is clear that the mapping Jz is one-to-one for any z ∈ V ′. For our further needs, it is useful toexpand f with respect to another basis by using the discrete Fourier transform; here we follow themethod in [1]. Write ω = ωk = e(2πi)/bk. The vectors

h〈s〉 = b−1/2k {ωs, . . . , ωs(bk−1), 1} ∈ Cbk, s = 1, . . . , bk,

form an orthogonal basis in Cbk . To any s and any function f ∈ M(tk, r(Γ)) we assign the vector-valued function

f 〈s〉 = h〈s〉f ∈(M(tk, r(Γ))

)bk .Note that

‖f 〈s〉(t)‖Cb = |f(t)| almost everywhere on (tk, r(Γ)). (2.8)

For a chosen s, the set of functions Jz−1

f 〈s〉 is a subspace of Mz, which we denote by M〈s〉z .The subspaces M〈s〉z are mutually level-wise orthogonal and span Mz. According to this construc-tion, the level-wise orthogonal projection in M(Γ) onto a subspace M〈s〉z is given by the formula

P〈s〉z u = Jz−1(

h〈s〉f 〈s〉z), f 〈s〉z = b

−1/2k

bk∑j=1

f jz,uω−js. (2.9)

The subspace M〈bk〉z coincides with Mz. Introduce the subspace

M′z = {u ∈ Mz :∑

x∈Tz :|x|=tu(x) = 0 for almost all t > 0}. (2.10)



It follows from the definition ofMz that the subspaces M′z andMz are level-wise orthogonal, andhence M′z =M〈1〉z + · · ·+M〈bk−1〉

z .The operator

P ′z =b∑j=1

Pjz − Pz = P〈1〉z + · · ·+P〈bk−1〉z

defines the level-wise orthogonal projection onto M′z.The next theorem is implicitly contained in the results of [5], Sections 5 and 6. Here we present

the result explicitly and in a somewhat sharper form.

Theorem 2.2. Let Γ be a regular tree.

1◦ The subspacesM〈s〉z , z ∈ V ′, s = 1, . . . , b(z)−1, and the subspaceMΓ are mutually level-wiseorthogonal.

2◦ One has

Mc(Γ) ⊂MΓ +∑z∈V′M′z =MΓ +

∞∑k=1

∑Gen(z)=k

bk−1∑s=1

M〈s〉z , (2.11)

and, for an arbitrary u ∈ Mc(Γ), one has∑x∈Γ:|x|=t

|u(x)|2 = |fΓ(t)|2 +∞∑k=1

∑Gen(z)=k

bk−1∑s=1

|f 〈s〉z (t)|2 (2.12)

almost everywhere on (0, r(Γ)), where fΓ = fΓ,u, and the functions f 〈s〉z are defined by (2.9).

Proof. 1◦ For a pair of subspaces formed byMΓ and by anyM′z, level-wise orthogonality followsfrom property (2.5) for T = Γ and from the definition of M′z in (2.10). Further, let z, w ∈ V ′ andz 6= w. If z 6� w and w 6� z, then the subtrees Tz and Tw are disjoint, which implies (2.4) for M′zand M′w. Assume that w � z, and let u ∈M′z and v ∈M′w. The function u satisfies relation (2.5)on each subtree Tejz . The function v vanishes outside Tw, which is completely contained in somesubtree Tejz . Therefore, one has ∑

x∈Tejz

:|x|=tv(x) =

∑x∈Tw:|x|=t

v(x) = 0

by (2.10). This implies (2.4) for any subspaces M′z and M′w with w 6= z. Finally, the level-wiseorthogonality of the subspaces M〈s〉z with the same value of z and different values of s is alreadyknown.

2◦ It suffices to show that any function u supported by a single edge e belongs to the sum in theright-hand side of (2.11). Let Gen(e) = k. For k = 0, the result is evident. Indeed, in this case onehas u ∈ MΓ. For k > 0, we consider the space (which we denote by B) generated by all functionsthat can be obtained from u by natural transplantations from e to any other edge of the samegeneration k. Clearly,

dim(B) = #{e ∈ Γ : Gen(e) = k} = b1b2 . . . bk.

On the other hand, consider the subspace B ∩MΓ (of dimension 1) and the subspaces of theform B ∩M′z. The latter intersection is trivial for Gen(z) > k, and otherwise the dimension ofthe intersection is equal to b(z)− 1. Therefore, the total dimension of the linear sum of all thesesubspaces is

1 + (b1 − 1) + b1(b2 − 1) + · · ·+ b1 . . . bk−1(bk − 1) = b1b2 . . .bk.Since the dimensions are the same, it follows that the function u belongs to the sum in (2.11).

Relation (2.12) is an immediate consequence of 1◦ and of the inclusion in (2.11).

Note that (2.11) is certainly a proper inclusion. Indeed, any subspace MΓ,M〈s〉z contains func-tions with unbounded support, and these functions cannot belong to Mc.



2.3. Spaces L2V (Γ) with Symmetric Weight and Their Orthogonal Decompositions

Let V (t) be a measurable, nonnegative function on (0, r(Γ)). Everywhere below, L2V (Γ) stands

for the Hilbert space whose metric form is

‖u‖2L2V

(Γ) =∫

Γ

|u(x)|2V (|x|) dx.

In this paper we consider symmetric weight functions only, i.e., belonging toMΓ. The sets

LΓ,V =MΓ ∩ L2V (Γ), L〈s〉z,V =M〈s〉z ∩ L2

V (Γ), z ∈ V ′, s = 1, . . . , b(z)− 1 (2.13)

are closed, mutually orthogonal subspaces of L2V (Γ). We omit the index V in this notation if V ≡ 1.

The following result is an immediate consequence of Theorem 2.2.

Theorem 2.3. Let Γ be a regular tree and let V (|x|) be a symmetric weight function. Then thesubspaces (2.13) are mutually orthogonal in the space L2

V (Γ) and define an orthogonal decompositionof this space. For any function u ∈ L2

V (Γ), one has the identity

∫Γ

|u(x)|2V (|x|) dx=∫ r(Γ)

0

|fΓ(t)|2V (t)gΓ(t) dt+∞∑k=1

∑Gen(z)=k

bk−1∑s=1

∫ r(Γ)

tk

|f 〈s〉z (t)|2V (t)gk(t) dt,

(2.14)where fΓ = fΓ,u, and the functions f 〈s〉z are given by (2.9). In particular,

L2(Γ) = LΓ ⊕∞∑k=1

∑Gen(z)=k

bk−1∑s=1

⊕L〈s〉z , (2.15)

∫Γ

|u(x)|2 dx =∫ r(Γ)

0

|fΓ(t)|2gΓ(t) dt+∞∑k=1

∑Gen(z)=k

bk−1∑s=1

∫ r(Γ)

tk

|f 〈s〉z (t)|2gk(t) dt (2.16)

for any function u ∈ L2(Γ).

3. THE SOBOLEV SPACE AND THE HOMOGENEOUSSOBOLEV SPACE ON A REGULAR TREE

Starting from this section, we always assume that

r(Γ) = supx∈Γ|x| =∞.

Many facts remain valid for r(Γ) <∞, usually with minor modifications in the statements; see [1]for details.

We say that a function u on Γ belongs to the homogeneous Sobolev space H = H(Γ) if u iscontinuous, u(o) = 0, u�e ∈ H1(e) for any edge e, and

‖u‖2H :=∫

Γ

|u′|2dx <∞.

Equivalently, the space H can be introduced as the set of all functions admitting a representationof the form

u(x) =∫〈o,x〉

v(y)dy, v ∈ L2(Γ). (3.1)



In this representation one has v = u′, and the mapping u 7→ u′ defines a natural isometry of Honto L2(Γ). A function u ∈ H need not belong to L2(Γ).

Along with H(Γ), we also introduce the “ordinary” (nonhomogeneous) Sobolev space H1 =H1(Γ) that consists of all continuous functions on Γ such that u�e ∈ H1(e) for any edge e and

‖u‖2H1 :=∫

Γ

(|u′(x)|2 + |u(x)|2

)dx <∞.

Let H1,0 = H1,0(Γ) be the subspace {u ∈ H1(Γ) : u(o) = 0}. An equivalent description is H1,0 =H ∩ L2(Γ). Clearly, H1,0 is a subspace of codimension one in H1, and the orthogonal complementof H1,0 belongs to LΓ. For this reason, the geometry of the spaces H1 and H1,0 is basically thesame, and we can restrict ourselves to the comparative study of the spaces H and H1,0. Note thatthe space Hc formed by all boundedly supported functions u ∈ H is always dense in H1,0. Indeed,for any number L > 0, let ϕL(t) be the continuous function on R+ that is equal to 1 for t 6 L andto 0 for t > L+ 1 and is linear on [L, L+ 1]. For a function u ∈ H1,0(Γ), set uL(x) = ϕL(|x|)u(x).Then uL ∈ Hc, and an elementary calculation shows that uL → u in H1,0(Γ). This reasoning canbe applied to arbitrary trees (not necessarily regular).

The sets

HΓ =MΓ ∩H, H〈s〉z =M〈s〉z ∩H, z ∈ V ′, s = 1, . . . , b(z)− 1 (3.2)

(cf. (2.13)) are closed subspaces of H. The operator PΓ (see (2.6)) acts on H and defines theorthogonal projection onto HΓ. The similar assertion fails for PT with T 6= Γ because the functiondefined by (2.6) is discontinuous at the point oT in general. This means that the operator PT doesnot act on H. However, the operators P〈s〉z defined by (2.9) do act onH provided that s < bk = b(z).Indeed, for any u ∈ H, the function v = P〈s〉z u has a derivative v′ such that v′ ∈ L2(Tz) and vanishesoutside Tz. Thus, the only obstacle for v to be in H is its possible discontinuity at the point oT .For any continuous function u on Γ and for s = 1, . . . , bk−1, it follows from (2.9) that f 〈s〉z (tk+) = 0,and therefore v is continuous on the entire Γ. This ensures the relation v = P〈s〉z u ∈ H. Further,differentiation preserves the classes M〈s〉z . Hence, for any two functions u1, u2 ∈ H belonging totwo different subspaces in (3.2), the derivatives u′1 and u′2 are level-wise orthogonal, and thus thefunctions u1 and u2 are orthogonal in H. This implies that the restriction of the operator P〈s〉z to Hdefines the orthogonal projection in H onto H〈s〉z .

We can now formulate an analog of Theorem 2.3 for the space H.

Theorem 3.1. Let Γ be a regular tree such that r(Γ) = ∞. Then the subspaces (3.2) aremutually orthogonal in H and define its orthogonal decomposition. For any function u ∈ H one hasthe identity∫

Γ

|u′(x)|2 dx =∫ ∞

0

∣∣∣∣dfΓ(t)dt

∣∣∣∣2gΓ(t) dt+∞∑k=1

∑Gen(z)=k

bk−1∑s=1

∫ ∞tk

∣∣∣∣df 〈s〉z (t)dt

∣∣∣∣2gk(t) dt, (3.3)

where fΓ = fΓ,u, and the functions f 〈s〉z are given by (2.9).A similar statement holds for the space H1,0.

Proof. ForH, the result follows from Theorem 2.3 in view of the description (3.1) of the spaceH.Since the subspaces HΓ ∩LΓ and H〈s〉z ∩L〈s〉z are orthogonal both in H(Γ) and in L2(Γ), the resultfor H1,0 is immediate.

3.1. Harmonic Functions in H

The class Hc can be not dense in H. Denote by H◦ = H◦(Γ) the closure of Hc in H (which is alsothe closure of H1,0(Γ)) and by H = H(Γ) the orthogonal complement of Hc. Thus, by definitionone has

H = H◦ ⊕H (3.4)



(orthogonal sum in H). The functions in H are said to be harmonic. One can readily give theirdirect description. For this purpose, we recall the notion of Kirchhoff matching condition.

Let u be a continuous function on Γ such that u � e ∈ C1(e) for any (closed) edge e ∈ E . For avertex z ∈ V ′, denote by u− the restriction of u to the edge terminating at z and by u1, . . . , ub(z)the restrictions of u to the edges emanating from z. One says that the Kirchhoff condition holds atthe vertex z for the function u if

u′−(z) =b(z)∑j=1

u′j(z). (3.5)

Since any harmonic function u ∈ H is orthogonal to Hc, it follows that u is linear on every edgeand satisfies the Kirchhoff condition (3.5) at any vertex z ∈ V ′. Moreover, u(o) = 0. Conversely,any such function belongs to H and is orthogonal to Hc.

A tree Γ is said to be recurrent if H(Γ) is trivial, and transient otherwise.

It follows from formulas (2.6) and (2.9) that the projections PΓ and P〈s〉z preserve the class Hc.Therefore, they also preserve the closure H◦ and the orthogonal complement H of this class. Hence,for any function u ∈ H, the component of this function in any subspace of the form (3.2) is alsoin H, and thus

H =(MΓ ∩H

)⊕∞∑k=1

∑gen(z)=k

bk−1∑s=1

⊕(M〈s〉z ∩H

).

The next theorem is a slight modification of Lemma 5.2 in [5]. It gives a complete description ofthe subspaces

MΓ ∩H, M〈s〉z ∩H, z ∈ V ′, s = 1, . . . , b(z)− 1. (3.6)

The functions

hk(t) :=∫ t

tk

ds

gk(s), t > tk, k = 0, 1, . . . , (3.7)

are involved in the statement. Note that the transiency criterion (3.8) for a regular tree is wellknown (see, e.g., [7, Example 8.3]).

Theorem 3.2. Let Γ be a regular tree such that r(Γ) =∞. The tree Γ is transient if and only if

L(Γ) :=∫ ∞

0

dt

g0(t)=∞∑n=0

tn+1 − tnb0 . . . bn

<∞. (3.8)

If (3.8) is satisfied, then any subspace of the form (3.6) is one-dimensional. The subspace MΓ ∩His spanned by the function u(x) = h0(|x|), and each subspaceM〈s〉z ∩H, where z ∈ V ′, Gen(z) = k,

and s = 1, . . . , bk − 1, is spanned by the function u = Jz−1hk.

Proof. Let u(x) = f(|x|) be a harmonic function in MΓ. Then the function f is continuouson R+ and linear on any segment [tn, tn+1]. Denote by cn the constant equal to the derivative of fon (tn, tn+1). The Kirchhoff conditions (3.5) become

cn+1 = bncn, n = 0, 1, . . . (3.9)

Relations (3.9), together with the boundary condition f(0) = 0, determine the function f up to aconstant factor. Choosing f(t) = t on [0, t1], we see that the only possible solution is f(t) = h0(t).For the corresponding function u(x) = h0(|x|) we have

∫Γ

|u′(x)|2 dx =∫ ∞

0

∣∣∣∣dh0(t)dt

∣∣∣∣2g0(t) dt =∫ ∞

0

dt

g0(t).



Thus, u ∈ H if (3.8) is satisfied, andMΓ ∩H = {0} otherwise.

Let now u ∈M〈s〉z ∩H for a vertex z ∈ V ′. Let Gen(z) = k and s 6 bk−1. Then u = 0 outside Tz,and the vector-valued function f = Jzu on Tz (see (2.7)) acquires the form f(x) = h〈s〉f(|x|).The function f(t) is continuous on [tk,∞) and linear on any segment [tn, tn+1]. The Kirchhoffconditions for f are the same as in (3.9); however, they hold for n > k only. Repeating the aboveargument for MΓ, we see that the only possible solution (up to a constant factor) is the function hkgiven by (3.7). The corresponding function u is in H if and only if∫ ∞

tk

gk(t)−1 dt <∞.

Considering the relation (2.3), we see that our entire collection of conditions (for any k) is equivalentto the sole condition (3.8). This completes the proof.

4. SCHRODINGER OPERATORS WITHSYMMETRIC POTENTIAL ON A REGULAR TREE

Let V (t) be a function on R+ that is measurable, real-valued, and bounded below and let V (|x|)be the corresponding symmetric potential on Γ. To simplify our presentation, we assume thatV ∈ L∞(R+). Then the quadratic form

aV [u] =∫

Γ

(|u′|2 + V (|x|)|u|2

)dx, u ∈ H1,0(Γ),

is bounded below and closed in L2(Γ). Let AV be the self-adjoint operator in L2(Γ) associatedwith this quadratic form. According to Friedrichs’ construction, the relations u ∈ Dom(AV ) andAV u = f mean that u ∈ H1,0(Γ) and that the relation∫

Γ

(u′v′ + V (|x|)uv

)dx =

∫Γ

fv dx

holds for an arbitrary function v ∈ H1,0(Γ). Integrating by parts, one can see that a functionu ∈ H1,0(Γ) belongs to Dom(AV ) if and only if u ∈ H2(e) on each edge e ∈ E , the Kirchhoffcondition (3.5) is satisfied at any vertex z ∈ V ′, and∫

Γ

|u′′|2 dx <∞.

On this domain, the operator AV acts according to the natural rule

(AV u)(x) = −u′′(x) + V (|x|)u(x), x /∈ V(Γ).

In particular, the operator A0 is referred to as the Laplacian on Γ. Our objective is to expand AV

in the orthogonal sum of a family of operators AV,k acting on the spaces L2(tk,∞). These auxiliaryoperators will also be described via their quadratic forms.

For any k = 0, 1, . . . , consider the function space

Dk ={y ∈

∑n>k⊕H1(tn, tn+1) : y(tk) = 0, y(tn+) = b1/2

n y(tn−), n > k}.

Obviously, Dk is a dense linear subspace of L2(tk,∞). The quadratic form

aV,k[y] =∫ ∞tk

(|y′|2 + V (t)|y|2

)dt, y ∈ Dk, (4.1)



is positive definite and closed in L2(tk,∞). The domain and the action of the corresponding self-adjoint operator AV,k can be found just like those for the operator AV . Namely, y ∈ Dom(AV,k) ifand only if y ∈ H2(tn, tn+1) for any n > k, y(tk) = 0, the matching conditions

y(tn+) = b1/2n y(tn−); y′(tn+) = b−1/2

n y′(tn−)

hold for n > k, and, finally, ∑n>k

∫ tn+1

tn

(|y′′|2 + |y|2

)dt <∞.

The next statement is the main result of this section. It should be compared with relation (7.1)in [5] and also with the results in [1]. Below we write A ∼ B if operators A and B are unitarilyequivalent, and A[m] stands for the direct orthogonal sum of m copies of a self-adjoint operator A.

Theorem 4.1. Let Γ be a regular tree, let r(Γ) =∞, let V = V (|x|) be a real-valued symmetricand bounded potential on Γ, and let AV be the corresponding Schrodinger operator. Then

AV ∼ AV,0 ⊕∞∑k=1

⊕AV,k [b1...bk−1(bk−1)]. (4.2)

Proof. Formulas (2.14), (2.16), and (3.3) show that the orthogonal decomposition (2.15) diag-onalizes the quadratic form aV , and therefore reduces the operator AV . It follows from (2.8) thatany subspace L〈s〉z with Gen(z) = k > 0 becomes the weighted space L2

gk(tk,∞). Accordingly, the

part of the quadratic form aV in the space L〈s〉z becomes∫ ∞tk

(|f ′|2 + V (t)|f |2

)gk(t)dt, f = f 〈s〉z .

The domain of this quadratic form is defined by the conditions∫ ∞tk

(|f ′|2 + |f |2

)gk(t)dt <∞, f(tk) = 0.

It is now convenient to make the substitution f(t) = gk(t)1/2y(t). This defines a unitary operatorfrom L2

gk(tk,∞) onto L2(tk,∞) that transforms the quadratic form aV � JzM〈s〉z into the quadratic

form (4.1) in L2(tk,∞). Here it is of importance that the weight function gk is constant on anyinterval (tn, tn+1), and hence one has f ′(t) = gk(t)1/2y′(t) on any such interval.

We see that any component AV �M〈s〉z with Gen(z) = k > 0 and s = 1, . . . , bk − 1 leads to thesame operator AV,k. The total number of these components is b1 . . .bk−1(bk− 1). In the same way,the component AV �LΓ leads to the operator AV,0. The proof is complete.

5. SYMMETRIC HARDY WEIGHTS5.1. Preliminaries

Let Γ be a regular tree and let r(Γ) =∞. We say that a measurable nonnegative function V (|x|)is a (symmetric) Hardy weight on the space H(Γ) if the inequality∫

Γ

V (|x|)|u(x)|2dx 6 C∫

Γ

|u′(x)|2 dx (5.1)

holds for all functions u ∈ H(Γ). We say that V (|x|) is a Hardy weight on H◦(Γ) (on H(Γ))if inequality (5.1) holds for any u ∈ H◦(Γ) (for any u ∈ H(Γ), respectively). In the case of H◦(Γ),



it suffices to verify inequality (5.1) for the functions u in the class Hc, which is dense in H◦. Denotethe least possible constant in the corresponding version of the inequality by CH(V ), CH◦(V ), orCH(V ), respectively. The notion of Hardy weight on H(Γ) becomes meaningless if the tree Γ isrecurrent. It follows from the decomposition (3.4) that a function V is a Hardy weight on H(Γ) ifand only if it is a Hardy weight on the subspaces H◦(Γ) and H(Γ) simultaneously. For this reason,we need not carry out a separate study of the Hardy weights on H, provided that this is donefor H◦ and H. Moreover, the result for H is simply a special case of [2, Theorem 3.1]. Nevertheless,we include the discussion of the H case because we think that this can help to compare technicaltools applied in each of these cases.

The following simple (but important) fact is a direct consequence of the results in Sections 2and 3.

Lemma 5.1. Let Γ be a regular transient tree, and let r(Γ) =∞. A function V (|x|) is a Hardyweight on H(Γ) if and only if the inequality∫ ∞

tk

V (t)|f(t)|2gk(t) dt 6 C∫ ∞tk

|f ′(t)|2gk(t) dt (5.2)

is satisfied for all k = 0, 1, . . . and all f ∈ H1,0loc (tk,∞), where the constant C does not depend on k.

A function V (|x|) is a Hardy weight on H◦(Γ) if and only if the same condition is satisfied forthe functions f ∈ H1,0(tk,∞) with bounded support.

A function V (|x|) is a Hardy weight on H(Γ) if and only if inequalities (5.2) hold for the functionsf = hk defined in (3.7).

In each of these cases, the best possible constant in inequality (5.1) is equal to the least possibleconstant C in (5.2).

5.2. Symmetric Hardy Weights on H and on H◦

Lemma 5.1 reduces the description of the Hardy weights on a tree to the similar “weighted”problem on the half-line. The latter problem was solved by Muckenhoupt [4]; see also the expositionof his results in [3] and [6].

To any measurable function V (t) > 0, t > 0, we associate the quantities

B0(V ) := supt>0

(∫ ∞t

V (s)g0(s) ds ·∫ t

0

ds

g0(s)

), B1(V ) := sup

t>t1

(∫ t

t1

V (s)g0(s) ds ·∫ ∞t

ds

g0(s)

),

B2(V ) := supt<t1

(t ·∫ t1

t

V (s) ds).

The possibilities Bi(V ) =∞, i = 0, 1, 2, are not excluded. The value of B2(V ) contains almost noinformation concerning the tree Γ. This reflects the fact that the structure of Γ at the root is asthat for a single segment.

Theorem 5.2. Let Γ be a regular transient tree, and let r(Γ) =∞.1◦ A function V (|x|) is a Hardy weight on H if and only if B0(V ) <∞. Moreover,

B0(V ) 6 CH(V ) 6 4B0(V ).

2◦ A function V (|x|) is a Hardy weight on H◦ if and only if B1(V ) < ∞ and B2(V ) < ∞.Moreover,

CH◦(V )/4 6 B1(V ) + B2(V ) (5.3)

B1(V ) 6 CH◦(V ), B2(V ) 6(

1 +b1t1t2 − t1

)CH◦(V ). (5.4)



Proof. We begin with some elementary remarks applicable to both statements.According to (2.2), for any k, the weight function gk in (5.2) can be replaced by g0. Further,

any function f ∈ H1,0loc (tk,∞) can be extended by zero to the entire semi-axis R+, which defines a

function in H1,0loc (R+). The integral on the right-hand side of (5.2) is preserved under this procedure.

Hence, under the assumptions of the theorem, the family of inequalities (5.2) can be reduced tothe single inequality ∫ ∞

0

V (t)|f(t)|2g0(t) dt 6 C∫ ∞

0

|f ′(t)|2g0(t) dt. (5.5)

Inequality (5.5) must hold for any function f ∈ H1,0loc (R+) under the assumptions of assertion 1◦

and for any function f ∈ H1,0(R+) with bounded support under the assumptions of assertion 2◦.1◦ Any function f ∈ H1,0

loc (R+) can be represented as

f(t) =∫ t

0

ϕ(s)ds, ϕ = f ′. (5.6)

Substituting this representation into (5.5), we obtain the inequality∫ ∞0

V (t)g0(t)∣∣∣∫ t

0

ϕ(s)ds∣∣∣2 dt 6 C ∫ ∞

0

|ϕ(t)|2g0(t) dt.

A necessary and sufficient condition for this inequality to hold is the classical Muckenhoupt con-dition (see, e.g., [3, Theorem 1.3.1/1]). Applying this theorem to the measures dµ = V g0 dt anddν = g0 dt with exponents p = q = 2, we obtain the desired result.

2◦ Along with the representation (5.6), any compactly supported function f ∈ H1,0(R+) admitsa representation of the form

f(t) =∫ ∞t

ψ(s)ds, ψ = −f ′. (5.7)

Substituting this relation into (5.5), we obtain the inequality∫ ∞0

V (t)g0(t)∣∣∣∫ ∞t

ψ(s) ds∣∣∣2dt 6 C ∫ ∞

0

|ψ(t)|2g0(t) dt.

A version of Muckenhoupt’s condition for this inequality is given in [3, Theorem 1.3.1/3].Let us split the function V into a sum V = V1 + V2, where V1(t) = V (t) for t > t1, V2(t) = V (t)

for t 6 t1, and V1(t) = 0 and V2(t) = 0 otherwise. Then the integral in the left-hand side of (5.5)splits into the sum∫ ∞

0

(V1(t) + V2(t)

)|f(t)|2g0(t) dt =

∫ ∞0

V1(t)|f(t)|2g0(t) dt+∫ ∞

0

V2(t)|f(t)|2 dt.

Now we use the representation (5.7) for the first term of the sum and the representation (5.6) forthe other. We obtain the estimate (5.3) by applying [3, Theorem 1.3.1/3 ] to the function V1 and[3, Theorem 1.3.1/1] to the function V2.

To prove inequalities (5.4), suppose that V is a Hardy weight on H◦. It follows from (5.5) that∫ ∞0

V2|f |2 dt =∫ t1

0

V |f |2g0dt 6 CH◦(V )∫ ∞

0

|f ′|2g0 dt, (5.8)

where the integral on the left-hand side does not depend on the values f(t) for t > t1. We applyinequality (5.8) to the function f1 that is continuous on R+, coincides with f on [0, t1], vanishesfor t > t2, and is linear on [t1, t2]. By the inequality

|f(t1)|2 6 t1∫ t1

0

|f ′|2 dt



and the relation g0(t) = b1 on [t1, t2], inequality (5.8) implies∫ ∞0

V2|f |2 dt 6(

1 +b1t1t2 − t1

)CH◦(V )

∫ t1

0

|f ′|2 dt.

Applying [3, Theorem 1.3.1/1] (lower estimate), we obtain the other inequality in (5.4).We can derive from (5.5) in the same way that∫ ∞

0

V1|f |2g0(t) dt 6 CH◦(V )∫ ∞

0

|f ′|2g0 dt.

This readily shows that V1 is a Hardy weight on the class of compactly supported functions inH1,0(R+), and the first inequality in (5.4) follows from [3, Theorem 1.3.1/3].

In the next statement we apply this result to Schrodinger operators on Γ.

Corollary 5.3. Let Γ be a regular transient tree, and let r(Γ) = ∞. The Laplacian on Γ ispositive definite in the space L2(Γ) if and only if

supt>0

(∫ t

0

g0(s)ds ·∫ ∞t

ds

g0(s)

)<∞ . (5.9)

Indeed, we are interested in the inequality∫Γ

|u′(x)|2 dx > c∫

Γ

|u(x)|2 dx, u ∈ H1,0(Γ),

for some c > 0. The above inequality can be extended by continuity to the space H◦, and thusmeans that the function V ≡ 1 is a Hardy weight on H◦. For this function V , the two conditionsB1(V ) <∞ and B2(V ) <∞ can be reduced to the single condition (5.9).

Example 5.4. Choose an integer b > 1 and consider a tree Γb all of whose vertices z ∈ V ′have the same branching number b and all edges are of the same length 1. In other words, tk = k,k = 0, 1, . . . , and bk = b, k = 1, 2, . . .

For the trees Γb, the behavior of the function gΓ(t) at 0 and at ∞ is as that of eβt, β = ln b.Therefore, condition (5.9) is satisfied, and the Laplacian on Γb is positive definite. For the completedescription of the spectrum of the Laplacian on the trees Γb, see [8].

It is also clear from Corollary 5.3 that the Laplacian on a recurrent regular tree is never positivedefinite.

5.3. Symmetric Hardy Weights on H

This problem is rather elementary because, for any k, inequality (5.2) must hold for a singlefunction f = fk (see (3.7)). For this function, inequality (5.2) becomes∫ ∞

tk

V (t)gk(t)(∫ t

tk

ds

gk(s)

)2

dt 6 C∫ ∞tk

dt

gk(t).

Considering relation (2.2), we can rewrite this formula as follows:

Jk(V ) :=∫ ∞tk

V (t)g0(t)(∫ t

tk

ds

g0(s)

)2

dt 6 C∫ ∞tk

dt

g0(t), k = 0, 1, . . . (5.10)



Theorem 5.5. Let Γ be a regular transient tree, and let r(Γ) =∞. A function V (|x|) > 0 is aHardy weight on the space H(Γ) if and only if the following two integrals are finite:

R1(V ) :=∫ ∞t1

V (t)g0(t) dt, R2(V ) :=∫ t1

0

V (t)t2 dt,

in which caset21R1(V ) + R2(V ) 6 L(Γ)CH(V ) 6 L(Γ)2R1(V ) +R2(V ). (5.11)

Proof. Upper estimate. Consider the integral J0(V ). Let us split it into the sum of integralsover the intervals (0, t1) and (t1,∞). On the first interval one has g0(t) = 1, and hence the integraldoes not exceed R2(V ). On the other interval we use the inequality∫ t

0

ds

g0(s)6 L(Γ),

which implies that the integral over (t1,∞) does not exceed L(Γ)2R1(V ). Thus, we have provedthat

L(Γ)J0(V ) 6 L(Γ)2R1(V ) +R2(V ).For k > 1, one has

Jk(V ) 6∫ ∞tk

dt

g0(t)·∫ ∞tk

V (t)g0(t)∫ t

tk

ds

g0(s)dt 6 L(Γ)R1(V )

∫ ∞tk

dt

g0(t).

This proves the upper estimate in (5.11).Lower estimate. It suffices to use the inequality (5.10) for k = 0 only:

CH(V )L(Γ) > J0(V ) >∫ t1

0

V (t)t2 dt+∫ ∞t1

V (t)g0(t)t21 dt > t21R1(V ) +R2(V ).

One can readily see that B0(V ) 6 L(Γ)R1(V )+B2(V ) for any function V . Therefore, a functionV is a Hardy weight on the entire space H(Γ) (where Γ is a transient regular tree) if and only ifR1(V ) < ∞ and B2(V ) < ∞. Thus, the behavior at infinity of the Hardy weights on the entirespace H is determined by the comparatively poor subspace H ⊂ H. At the qualitative level, thiseffect was discussed in the introduction.

ACKNOWLEDGMENTS

This paper was written during the stay of the second author in King’s College in London, andhe takes this opportunity to express his deep gratitude for the warm hospitality to the College,and especially to Professor Yu. Safarov. We are also grateful to W. D. Evans and D. J. Harris forfruitful discussions, and to I. Benjamini and B. Solomyak for useful consultations.

REFERENCES

1. Carlson, R., Nonclassical Sturm–Liouville Problems and Schrodinger Operators on Radial Trees, Elec-tron. J. Differential Equations, 2000, no. 71, 24 pp. (electronic).

2. Evans, W. D., Harris, D. J., and Pick, L., Weighted Hardy and Poincare Inequalities on Trees, J. LondonMath. Soc. (2), 1995, vol. 52, no. 1, pp. 121–136.

3. Maz’ya, V. G., Sobolev Spaces, Leningrad: Leningrad. Univ., 1985; English transl.: Maz’ja, V. G.,Sobolev Spaces, Berlin: Springer, 1985.

4. Muckenhoupt, B., Hardy’s Inequality with Weights, Studia Math., 1972, vol. 44, no. 1, pp. 31–38.5. Naimark, K. and Solomyak, M., Eigenvalue Estimates for the Weighted Laplacian on Metric Trees,

Proc. London Math. Soc. (3), 2000, vol. 80, no. 3, pp. 690–724.

6. Opic, B. and Kufner, A., Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series, 219,Longman: Harlow, 1990.

7. Peres, Y., Probability on Trees: an Introductory Climb, in Lectures on Probability Theory and Statistics(Saint-Flour, 1997), Lecture Notes in Math., vol. 1717, Berlin: Springer, 1999, pp. 193–280.

8. Sobolev, A. V. and Solomyak, M., Schrodinger Operator on Trees, Spectrum in Gaps (in preparation).


Documents

Dedicated to Professor M. S. Agranovich on the Occasion of ...into an orthogonal sum of countably many operators of Sturm{Liouville type, each acting on a semi-in nite interval. The