Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure

Alvarado, Ryan; Górka, Przemysław; Hajłasz, Piotr

doi:10.48550/arxiv.1903.05793

2019

DOI: 10.48550/arxiv.1903.05793

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Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure

Ryan Alvarado,

Przemysław Górka,

Piotr Hajłasz

Abstract: It has been known since 1996 that a lower bound for the measure, µ(B(x, r)) ≥ br s , implies Sobolev embedding theorems for Sobolev spaces M 1,p defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for M 1,p spaces are equivalent to the lower bound of the measure.

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2020

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“…Different, but related iterative arguments to the one presented below were used in [7,11,15,16,17] in the proofs that a Sobolev inequality implies a measure density condition. Other applications of a method developed in [22] are given in [1,21].…”

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A note on metric-measure spaces supporting Poincaré inequalities

Alvarado

Hajłasz

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Dedicated to Professor Vladimir Maz'ya on the occasion of his 80th birthday.Abstract. Using a method of Korobenko, Maldonado and Rios we show a new characterization of doubling metric-measure spaces supporting Poincaré inequalities without assuming a priori that the measure is doubling.Non-smooth functions have played a key role in analysis since the nineteenth century. One fundamental development in this vein came with the introduction of Sobolev spaces, which turned out to be a key tool in studying nonlinear partial differential equations and calculus of variations. Although classically Sobolev functions themselves were not smooth, they were defined on smooth objects such as domains in the Euclidean space or, more generally, Riemannian manifolds. By the late 1970s it became well recognized that several results in real analysis required little structure from the underlying ambient space, and could be generalized to non-smooth settings, such as to the so-called spaces of homogeneous type. The latter spaces are (quasi)metric spaces equipped with a doubling Borel measure (see [9,10]). In fact, maximal functions, Hardy spaces, functions of bounded mean oscillation, and singular integrals of Calderón-Zygmund-type all continue to have a fruitful theory in the context of spaces of homogeneous type. However, this rich theory was, in a sense, only zeroth-order analysis given that no derivatives were involved. The study of first-order analysis with suitable generalizations of derivatives, a fundamental theorem of calculus, and Sobolev spaces, in the setting of spaces of homogeneous type, was initiated in the 1990s. This area, known as analysis on metric spaces, has since grown into a multifaceted theory which continues to play an important role in many areas of contemporary mathematics. For an introduction to the subject we recommend [2,3,4,5,8,12,13,19,20,23].One of the main objects of study in analysis on metric spaces are so called spaces supporting Poincaré inequalities introduced in [19]. To define this notion, recall that 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 x ∈ X and all r ∈ (0, ∞), where B(x, r) denotes the (open) metric ball with center x and radius r, i.e., B(x, r) := {y ∈ X : d(x, y) < r}. If the measure µ is doubling, that is, if there exists a finite constant C > 0 such that µ(2B) ≤ Cµ(B) for all balls B ⊆ X, then we call (X, d, µ) a doubling metric-measure space. The notation τ B denotes the dilation of a ball B by a factor τ ∈ (0, ∞), i.e., τ B := B(x, τ r). A Borel 2010 Mathematics Subject Classification. 30L99, 46E35.

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A note on metric-measure spaces supporting Poincaré inequalities

Alvarado

Hajłasz

2020

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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Sobolev embedding for $M^{1,p}$ spaces is equivalent to a lower bound of the measure

Abstract: It has been known since 1996 that a lower bound for the measure, µ(B(x, r)) ≥ br s , implies Sobolev embedding theorems for Sobolev spaces M 1,p defined on metric-measure spaces. We prove that, in fact Sobolev embeddings for M 1,p spaces are equivalent to the lower bound of the measure.

Cited by 1 publication

References 20 publications

A note on metric-measure spaces supporting Poincaré inequalities

A note on metric-measure spaces supporting Poincaré inequalities

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