A Sobolev type embedding theorem for Besov spaces defined on doubling metric spaces

Martín, Joaquim; Ortiz, Walter A.

doi:10.1016/j.jmaa.2019.07.032

Cited by 8 publications

(2 citation statements)

References 26 publications

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“…Unfortunately, given R > 0, in many cases there is a 'gap' between these parameters, that is, q µ (R) can be much smaller than Q µ (R). The reader can find interesting examples illustrating this phenomenon in [34].…”

Section: 25)mentioning

confidence: 95%

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Tyulenev

2023

Sb. Math.

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Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},\mu)$ be a metric measure space with uniformly locally doubling measure $\mu$ supporting a weak local $(1,p)$-Poincaré inequality. For each $\theta \in [0,p)$ we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower $\theta$-codimensional content regular closed sets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},\mu)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we obtain an intrinsic description of the trace-space of the Sobolev space $W^{1}_{p}(\operatorname{X})$ to arbitrary closed nonempty sets $S \subset \operatorname{X}$. Bibliography: 43 titles.

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Section: 25)mentioning

confidence: 95%

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Tyulenev

2023

Sb. Math.

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“…Finally we consider the case in which ν additionally satisfies a reverse-doubling property (11.1), which holds in particular when Z is uniformly perfect [25,Lemma 4.1].…”

Section: Introductionmentioning

confidence: 99%

Extension and trace theorems for noncompact doubling spaces

Butler

2020

Preprint

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We generalize the extension and trace results of Björn-Björn-Shanmugalingam [3] to the setting of complete noncompact doubling metric measure spaces and their uniformized hyperbolic fillings. This is done through a uniformization procedure introduced by the author that uniformizes a Gromov hyperbolic space using a Busemann function instead of the distance functions considered in the work of Bonk-Heinonen-Koskela [7]. We deduce several new corollaries for the Besov spaces that arise as trace spaces in this fashion, including density of Lipschitz functions with compact support, embeddings into Hölder spaces for appropriate exponents, and the existence of Lebesgue points quasieverywhere with respect to the Besov capacity under an additional reverse doubling hypothesis on the measure. Under this same reverse doubling hypothesis we also obtain a hyperbolic Poincaré inequality for Besov functions that controls the mean oscillation of a Besov function over a ball through an integral over an extended ball of an upper gradient for an extension of this function to a uniformized hyperbolic filling of the space.

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Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted L spaces on metric spaces

Björn

Kałamajska

2022

Journal of Functional Analysis

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A Sobolev type embedding theorem for Besov spaces defined on doubling metric spaces

Cited by 8 publications

References 26 publications

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Traces of Sobolev spaces to irregular subsets of metric measure spaces

Extension and trace theorems for noncompact doubling spaces

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted L spaces on metric spaces

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