Sobolev embedding for M1,p spaces is equivalent to a lower bound of the measure

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Abstract

It has been known since 1996 that a lower bound for the measure, μ(B(x,r))brs, implies Sobolev embedding theorems for Sobolev spaces M1,p defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for M1,p spaces are equivalent to the lower bound of the measure.

Introduction

A metric-measure space (X,d,μ) is a metric space (X,d) with a Borel measure μ such that 0<μ(B(x,r))< for all xX and all r(0,). We will always assume that metric spaces have at least two points. Sobolev spaces on metric-measure spaces, denoted by M1,p, have been introduced in [9], and they play an important role in the so called area of analysis on metric spaces [3], [4], [15], [16]. Later, many other definitions have been introduced in [6], [10], [11], [24], but in the important case when the underlying metric-measure space supports the Poincaré inequality, all the definitions are equivalent [8], [19]. One of the features of the theory of M1,p spaces is that, unlike most of other approaches, they do not require the underlying measure to be doubling in order to have a rich theory. In this paper we will focus on understanding the relation between the Sobolev embedding theorems for spaces M1,p and the growth properties of the measure μ.

Let (X,d,μ) be a metric-measure space. We say that uM1,p(X,d,μ), 0<p<, if uLp(X,μ) and there is a non-negative function 0gLp(X,μ) such that|u(x)u(y)|d(x,y)(g(x)+g(y))for μ-almost all x,yX. More precisely, there is a set NX of measure zero, μ(N)=0, such that inequality (1) holds for all x,yXN. By D(u) we denote the class of all functions 0gLp(X,μ) for which the above inequality is satisfied, and we setD+(u):={gD(u):gLp(X,μ)>0}.

The space M1,p is equipped with a ‘norm’uM1,p(X,d,μ)=uLp(X,μ)+infgD(u)gLp(X,μ). We put the word ‘norm’ in inverted commas, because it is a norm only when p1. In fact, if p1, the space M1,p is a Banach space, [9].

If ΩX is an open set, then (Ω,d,μ) is a metric-measure space and hence, the space M1,p(Ω,d,μ) is well defined. In other words, uM1,p(Ω,d,μ) if uLp(Ω,μ) and there is 0gLp(Ω,μ) such that (1) holds for almost all x,yΩ.

The space M1,p is a natural generalization of the classical Sobolev space, W1,p, because if p>1 and ΩRn is a bounded domain with the W1,p-extension property, thenM1,p(Ω,dRn,Ln)=W1,p(Ω)and the norms are equivalent, see [8]. Here we regard Ω as a metric-measure space with the Euclidean metric dRn, and the Lebesgue measure Ln. When p=1, the space M1,1 in the Euclidean setting is equivalent to the Hardy-Sobolev space [22]. While the spaces M1,p for 0<p<1 do not have an obvious interpretation in terms of classical Sobolev spaces, they found applications to Hardy-Sobolev spaces as well as Besov spaces and Triebel-Lizorkin spaces (see, e.g., [22], [23]).

The classical Sobolev embedding theorems for W1,p(Rn) have different character when p<n, p=n, or p>n. Therefore, in the metric-measure context, in order to prove embedding theorems, we need a condition that would be the counterpart of the dimension of the space. It turns out that such a condition is provided by the lower bound for the growth of the measureμ(B(x,r))brs. With this condition one can prove Sobolev embedding theorems for M1,p spaces where the embedding has a different character if 0<p<s, p=s or p>s, [8], [9]. For a precise statement see Theorem 6, below. The purpose of this paper is to prove that condition (2) is actually equivalent to the existence of the embeddings listed in Theorem 6. Precise statements are given in Theorem 1. Partial or related results have been obtained in [5], [7], [12], [13], [14], [17], [18], [20], [21]. An extension of the results in this work to certain classes of Triebel-Lizorkin and Besov spaces is given in a forthcoming paper [2].

The first main result of this paper highlights the fact that the lower measure condition in (2) characterizes certain M1,p-Sobolev embeddings. See Theorem 17, Theorem 25, and Theorem 29 in the body of the paper for a more detailed account of the following theorem.

Theorem 1

Suppose that (X,d,μ) is a uniformly perfect2 metric-measure space and fix parameters σ(1,), and s(0,). Then the following statements are equivalent.

  • (a)

    There exists a finite constant κ>0 such thatμ(B(x,r))κrsfor every xX and every finite r(0,diam(X)].

  • (b)

    There exist p(0,s) and C(0,) such that for every ball B0:=B(x0,R0) with x0X and finite R0(0,diam(X)], one has(B0|u|pdμ)1/pC(μ(σB0)R0s)1/p[R0(σB0gpdμ)1/p+(σB0|u|pdμ)1/p], whenever uM1,p(σB0,d,μ) and gD(u). Here, p:=sp/(sp).

  • (c)

    There exist p(0,s) and C(0,) such that for every ball B0:=B(x0,R0) with x0X and finite R0(0,diam(X)], one hasinfγR(B0|uγ|pdμ)1/pC(μ(σB0)R0s)1/pR0(σB0gpdμ)1/p, whenever uM1,p(σB0,d,μ) and gD(u). Here, p:=sp/(sp).

  • (d)

    There exist constants C1,C2,γ(0,) such thatB0exp(C1|uuB0|gLs(σB0))γdμC2, whenever B0X is a ball with radius at most diam(X), uM1,s(σB0,d,μ) and gD+(u).

  • (e)

    There exist p(s,) and C(0,) such that|u(x)u(y)|Cd(x,y)1s/pgLp(X,μ),x,yX. Hence, every function uM1,p(X,d,μ) has a Hölder continuous representative of order (1s/p) on X.

Here and in what follows the integral average is denoted byuE=Eudμ=1μ(E)Eudμ where EX is a μ-measurable set of positive measure. Also, τB denotes the dilation of a ball B by a factor τ(0,), i.e., τB:=B(x,τr).

Remark 2

The expression ‘for every finite r(0,diam(X)] is a concise way of saying ‘for every r(0,diam(X)] if diam(X)< and for every r(0,) if diam(X)=’.

Remark 3

Theorem 1 asserts that in particular, if just one of the Sobolev embeddings (b) - (e) holds for some p, then all of the embeddings (b) - (e) hold for all p. This is a new self-improvement phenomena.

Given a metric-measure space, (X,d,μ), the measure μ is said to be doubling provided there exists a constant C(0,) such thatμ(2B)Cμ(B) for all ballsBX. The smallest constant C for which (8) is satisfied will be denoted by Cμ. It follows from (8) that if X contains at least two elements then Cμ>1 (see [1, Proposition 3.1, p. 72]). Moreover, as is well-known, the doubling property in (8) implies the following quantitative condition: for each s[log2(Cμ),), there exists κ(0,) satisfyingμ(B(x,r))μ(B(y,R))κ(rR)s, whenever x,yX satisfy B(x,r)B(y,R) and 0<rR< (see, e.g., [8, Lemma 4.7]). Conversely, any measure satisfying (9) for some s(0,) is necessarily doubling. Note that if the space X is bounded then the above quantitative doubling property implies the lower measure bound in (3).

The following theorem, which constitutes the second main result of our paper, is an analogue of Theorem 1 for doubling measures. The reader is referred to Theorem 20 and Theorem 30.

Theorem 4

Suppose that (X,d,μ) is a uniformly perfect metric-measure space and fix parameters σ(1,), and s(0,). Then the following statements are equivalent.

  • (a)

    There exists a constant κ(0,) satisfyingμ(B(x,r))μ(B(y,R))κ(rR)s, whenever x,yX satisfy B(x,r)B(y,R) and 0<rR<.

  • (b)

    There exist p(0,s) and C(0,) such that for every ball B0:=B(x0,R0) with x0X and R0(0,), one has(B0|u|pdμ)1/pCR0(σB0gpdμ)1/p+C(σB0|u|pdμ)1/p, whenever uM1,p(σB0,d,μ) and gD(u). Here, p:=sp/(sp).

  • (c)

    There exist p(0,s) and C(0,) such that for every ball B0:=B(x0,R0) with x0X and R0(0,), one hasinfγR(B0|uγ|pdμ)1/pCR0(σB0gpdμ)1/p, whenever uM1,p(σB0,d,μ) and gD(u). Here, p:=sp/(sp).

  • (d)

    There exist p(s,) and C(0,) such that for each uM1,p(X,d,μ) and gD(u), and each ball B0:=B(x0,R0) with x0X and R0(0,diam(X)], finite, there holds|u(x)u(y)|Cd(x,y)1s/pR0s/p(σB0gpdμ)1/pfor everyx,yB0. Hence, every function uM1,p(σB0,d,μ) has a Hölder continuous representative of order (1s/p) on B0.

Remark 5

Note that Theorem 4 does not cover the case p=s. For that case see Theorem 27, Theorem 28 in the body of the paper.

Open balls in a metric space, (X,d), shall be denoted by B(x,r)={y:d(x,y)<r} while the notation B(x,r)={y:d(x,y)r} will be used for closed balls. We allow the radius of a ball to equal zero. If r=0, then B(x,r)=, but B(x,r)={x}. As a sign or warning, note that in general B(x,r) is not necessarily equal to the topological closure of B(x,r). By C we will denote a general constant whose value may change within a single string of estimates. While the center and the radius of a ball in a metric space is not necessarily uniquely determined, our balls will always have specified centers and radii so formally a ball is a triplet: a set, a center and a radius. By writing C(s,p) we will mean that the constant depends on parameters s and p only. N will stand for the set of positive integers, while N0:=N{0}. The characteristic function of a set E will be denoted by χE.

Section snippets

Sobolev embedding on metric-measure spaces

The next result from [8, Theorem 8.7] provides a general embedding theorem for Sobolev spaces M1,p defined on balls in a metric-measure space X. While this result has been proven in [8] we decided to include a proof for the following reasons. The paper [8] does not include inequality (11). While in the case p1, inequality (11) easily follows from (12) (proven in [8]) we do not know how to conclude it from (12) when p<1. Also some of the arguments given in [8] are somewhat sketchy and hard to

Auxiliary results

In this section we will collect some lemmata of a purely technical nature that will be needed in the proofs of the main results. Since the results collected here are not interesting on its own, the reader may skip this section for now and return to it when needed.

An open set ΩRn is a metric-measure space with the Euclidean metric and Lebesgue measure. If xΩ and r(0,), then we can always find a radius rx<r such that |B(x,rx)Ω|=12|B(x,r)Ω|. However, in a general metric-measure space (X,d,μ)

The case p<s

Theorem 17

Suppose that (X,d,μ) is a metric-measure space. Fix parameters σ(1,), s(0,), p(0,s), and let p:=sp/(sp). Then the following statements are equivalent.

  • (a)

    There exists a constant κ(0,) such thatμ(B(x,r))κrsfor every xX and every finite r(0,diam(X)].

  • (b)

    There exists a constant CS(0,) such that for every ball B0:=B(x0,R0) with x0X and finite R0(0,diam(X)], one has(B0|u|pdμ)1/pCS(μ(σB0)R0s)1/p[R0(σB0gpdμ)1/p+(σB0|u|pdμ)1/p], whenever uM1,p(σB0,d,μ) and gD(u).

If, in addition, (X,d)

The case p=s

Theorem 25

Suppose that (X,d,μ) is a uniformly perfect metric-measure space. Then for each fixed s(0,) and σ(1,), the following two statements are equivalent.

  • (a)

    There exists a constant κ(0,) such thatμ(B(x,r))κrsfor every xX and every finite r(0,diam(X)].

  • (b)

    There exist constants C1,C2,γ(0,) such thatB0exp(C1|uuB0|gLs(σB0,μ))γdμC2, whenever B0X is a ball with radius at most diam(X), uM1,s(σB0,d,μ) and gD+(u).

Remark 26

The implication (a) ⇒ (b) holds in any metric-measure space, not necessarily uniformly

The case p>s

Theorem 29

Suppose that (X,d,μ) is a uniformly perfect metric-measure space and fix s,p(0,) satisfying p>s. Then the following two statements are equivalent.

  • (a)

    There exists a finite constant κ>0 such thatμ(B(x,r))κrsfor everyxXand every finiter(0,diam(X)].

  • (b)

    There exists a constant CH(0,) with the property that for each uM1,p(X,d,μ) and gD(u), there holds|u(x)u(y)|CHd(x,y)1s/pgLp(X,μ),x,yX. Hence, every function uM1,p(X,d,μ) has a Hölder continuous representative of order (1s/p) on X.

Proof

We begin

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    P.H. was supported by NSF grant DMS-1800457.

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