A differentiable structure for metric measure spaces

In this note, we give several characterizations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy's original inequality. We give examples obtaining new weighted Hardy inequalities on R n , on homogeneous groups, on hyperbolic spaces and on Cartan–Hadamard manifolds. We note that doubling conditions are not required for our analysis.

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On Whitney-type Characterization of Approximate Differentiability on Metric Measure Spaces

Canadian Journal of Mathematics ◽

10.4153/cjm-2012-064-x ◽

2014 ◽

Vol 66 (4) ◽

pp. 721-742 ◽

Cited By ~ 1

Author(s):

E. Durand-Cartagena ◽

L. Ihnatsyeva ◽

R. Korte ◽

M. Szumańska

Keyword(s):

Type Theorem ◽

Maximal Functions ◽

Metric Measure Spaces ◽

Differentiable Structure ◽

Differentiable Functions ◽

Measure Spaces ◽

Approximate Differentiability

AbstractWe study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, BV, and maximal functions.

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Density and Extension of Differentiable Functions on Metric Measure Spaces

Analysis and Geometry in Metric Spaces ◽

10.1515/agms-2020-0130 ◽

2021 ◽

Vol 9 (1) ◽

pp. 254-268

Author(s):

Rafael Espínola García ◽

Luis Sánchez González

Keyword(s):

Banach Spaces ◽

Metric Measure Spaces ◽

Differentiable Structure ◽

First Order ◽

Differentiable Functions ◽

Measure Spaces ◽

The Given ◽

Vector Valued

Abstract We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions de-fined on these metric measure spaces.

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On the relaxation of variational integrals in metric Sobolev spaces

Advances in Calculus of Variations ◽

10.1515/acv-2013-0207 ◽

2015 ◽

Vol 8 (1) ◽

Cited By ~ 1

Author(s):

Omar Anza Hafsa ◽

Jean-Philippe Mandallena

Keyword(s):

Integral Representation ◽

Calculus Of Variations ◽

Sobolev Spaces ◽

General Framework ◽

Metric Measure Spaces ◽

Representation Theorems ◽

Differentiable Structure ◽

Convex Case ◽

Measure Spaces ◽

Variational Integrals

AbstractWe give an extension of the theory of relaxation of variational integrals in classical Sobolev spaces to the setting of metric Sobolev spaces. More precisely, we establish a general framework to deal with the problem of finding an integral representation for “relaxed” variational functionals of variational integrals of the calculus of variations in the setting of metric measure spaces. We prove integral representation theorems, both in the convex and non-convex case, which extend and complete previous results in the setting of euclidean measure spaces to the setting of metric measure spaces. We also show that these integral representation theorems can be applied in the setting of Cheeger–Keith's differentiable structure.

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Hardy inequalities on metric measure spaces, II: the case p > q

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2021.0136 ◽

2021 ◽

Vol 477 (2250) ◽

pp. 20210136

Author(s):

Michael Ruzhansky ◽

Daulti Verma

Keyword(s):

Integral Form ◽

Hyperbolic Spaces ◽

Hadamard Manifolds ◽

Metric Measure Spaces ◽

Differentiable Structure ◽

Hardy Inequalities ◽

Homogeneous Groups ◽

Link Type ◽

Measure Spaces

In this paper, we continue our investigations giving the characterization of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the inequalities are given in the integral form in the spirit of Hardy’s original inequality. This is a continuation of our paper (Ruzhansky & Verma 2018. Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 )) where we treated the case p ≤ q . Here the remaining range p > q is considered, namely, 0 < q < p , 1 < p < ∞. We give several examples of the obtained results, finding conditions on the weights for integral Hardy inequalities on homogeneous groups, as well as on hyperbolic spaces and on more general Cartan–Hadamard manifolds. As in the first part of this paper, we do not need to impose doubling conditions on the metric.

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