Orlicz Sobolev Inequalities and the Doubling Condition

Korobenko, Lyudmila

doi:10.48550/arxiv.1906.04088

Cited by 2 publications

(3 citation statements)

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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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confidence: 99%

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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confidence: 99%

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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“…The idea of the proof in the case 0 < p < s is to estimate the series at (22) by the series in (20). Similar ideas are also used in other cases p = s and p > s.…”

Section: Sobolev Embedding On Metric-measure Spacesmentioning

confidence: 99%

“…Precise statements are given in Theorem 1. Partial or related results have been obtained in [7,9,14,15,16,19,20,22,23]. An extension of the results in this work to certain classes of Triebel-Lizorkin and Besov spaces is given in a forthcoming paper [4].…”

Section: Introductionmentioning

confidence: 99%

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

Alvarado,

Górka,

Hajłasz

2019

Preprint

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Orlicz Sobolev Inequalities and the Doubling Condition

Cited by 2 publications

References 10 publications

A note on metric-measure spaces supporting Poincaré inequalities

A note on metric-measure spaces supporting Poincaré inequalities

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

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