Sobolev embedding for M1,p spaces is equivalent to a lower bound of the measure
Introduction
A metric-measure space is a metric space with a Borel measure μ such that for all and all . We will always assume that metric spaces have at least two points. Sobolev spaces on metric-measure spaces, denoted by , 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 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 and the growth properties of the measure μ.
Let be a metric-measure space. We say that , , if and there is a non-negative function such that More precisely, there is a set of measure zero, , such that inequality (1) holds for all . By we denote the class of all functions for which the above inequality is satisfied, and we set
The space is equipped with a ‘norm’ We put the word ‘norm’ in inverted commas, because it is a norm only when . In fact, if , the space is a Banach space, [9].
If is an open set, then is a metric-measure space and hence, the space is well defined. In other words, if and there is such that (1) holds for almost all .
The space is a natural generalization of the classical Sobolev space, , because if and is a bounded domain with the -extension property, then see [8]. Here we regard Ω as a metric-measure space with the Euclidean metric , and the Lebesgue measure . When , the space in the Euclidean setting is equivalent to the Hardy-Sobolev space [22]. While the spaces for 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 have different character when , , or . 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 measure With this condition one can prove Sobolev embedding theorems for spaces where the embedding has a different character if , or , [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 -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 is a uniformly perfect2 metric-measure space and fix parameters , and . Then the following statements are equivalent. There exists a finite constant such that There exist and such that for every ball with and finite , one has whenever and . Here, . There exist and such that for every ball with and finite , one has whenever and . Here, . There exist constants such that whenever is a ball with radius at most , and . There exist and such that Hence, every function has a Hölder continuous representative of order on X.
Remark 2 The expression ‘for every finite ’ is a concise way of saying ‘for every if and for every if ’.
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, , the measure μ is said to be doubling provided there exists a constant such that The smallest constant C for which (8) is satisfied will be denoted by . It follows from (8) that if X contains at least two elements then (see [1, Proposition 3.1, p. 72]). Moreover, as is well-known, the doubling property in (8) implies the following quantitative condition: for each , there exists satisfying whenever satisfy and (see, e.g., [8, Lemma 4.7]). Conversely, any measure satisfying (9) for some 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 is a uniformly perfect metric-measure space and fix parameters , and . Then the following statements are equivalent. There exists a constant satisfying whenever satisfy and . There exist and such that for every ball with and , one has whenever and . Here, . There exist and such that for every ball with and , one has whenever and . Here, . There exist and such that for each and , and each ball with and , finite, there holds Hence, every function has a Hölder continuous representative of order on .
Remark 5 Note that Theorem 4 does not cover the case . For that case see Theorem 27, Theorem 28 in the body of the paper.
Open balls in a metric space, , shall be denoted by while the notation will be used for closed balls. We allow the radius of a ball to equal zero. If , then , but . As a sign or warning, note that in general is not necessarily equal to the topological closure of . 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 we will mean that the constant depends on parameters s and p only. will stand for the set of positive integers, while . The characteristic function of a set E will be denoted by .
Section snippets
Sobolev embedding on metric-measure spaces
The next result from [8, Theorem 8.7] provides a general embedding theorem for Sobolev spaces 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 , inequality (11) easily follows from (12) (proven in [8]) we do not know how to conclude it from (12) when . 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 is a metric-measure space with the Euclidean metric and Lebesgue measure. If and , then we can always find a radius such that . However, in a general metric-measure space
The case
Theorem 17 Suppose that is a metric-measure space. Fix parameters , , , and let . Then the following statements are equivalent. There exists a constant such that There exists a constant such that for every ball with and finite , one has whenever and .
If, in addition,
The case
Theorem 25 Suppose that is a uniformly perfect metric-measure space. Then for each fixed and , the following two statements are equivalent. There exists a constant such that There exist constants such that whenever is a ball with radius at most , and .
Remark 26 The implication (a) ⇒ (b) holds in any metric-measure space, not necessarily uniformly
The case
Theorem 29 Suppose that is a uniformly perfect metric-measure space and fix satisfying . Then the following two statements are equivalent. There exists a finite constant such that There exists a constant with the property that for each and , there holds Hence, every function has a Hölder continuous representative of order on X.
Proof We begin
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P.H. was supported by NSF grant DMS-1800457.