scholarly journals Density and Extension of Differentiable Functions on Metric Measure Spaces

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.

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

2014 ◽  
Vol 66 (4) ◽  
pp. 721-742 ◽  
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.

scholarly journals Hardy inequalities on metric measure spaces

2019 ◽  
Vol 475 (2223) ◽  
pp. 20180310
Author(s):  
Michael Ruzhansky ◽  
Daulti Verma
Keyword(s):  
Integral Form ◽  
Hyperbolic Spaces ◽  
Hadamard Manifolds ◽  
Metric Measure Spaces ◽  
Differentiable Structure ◽  
Hardy Inequalities ◽  
Homogeneous Groups ◽  
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.

scholarly journals First order Poincaré inequalities in metric measure spaces

2013 ◽  
Vol 38 ◽  
pp. 287-308 ◽  
Author(s):  
Estibalitz Durand-Cartagena ◽  
Jesús A. Jaramillo ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Measure Spaces ◽  
Poincaré Inequalities ◽  
First Order ◽  
Poincare Inequalities ◽  
Measure Spaces

On the Basis Problem for Vector Valued Function Spaces

1956 ◽  
Vol 8 ◽  
pp. 417-422 ◽  
Author(s):  
H. W. Ellis
Keyword(s):  
Banach Spaces ◽  
Function Spaces ◽  
Length Function ◽  
General Measure ◽  
Basis Problem ◽  
Measure Spaces ◽  
Vector Valued Function ◽  
Vector Valued

1. Introduction. In a recent paper (2) Halperin and the author considered separable Banach spaces Lλ of real valued functions on general measure spaces and proved the existence of 1-regular (§2) Haar or σ-Haar bases when λ was the classical p-norm or any levelling length function (3) and, more generally, of K-regular Haar or σ-Haar bases when λ was a continuous length function satisfying certain additional conditions (2, Theorem 3.2).

scholarly journals On the relaxation of variational integrals in metric Sobolev spaces

2015 ◽  
Vol 8 (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.

scholarly journals Newtonian Spaces Based on Quasi-Banach Function Lattices

2016 ◽  
Vol 119 (1) ◽  
pp. 133 ◽  
Author(s):  
Lukáš Malý
Keyword(s):  
Absolute Continuity ◽  
General Setting ◽  
Sobolev Type ◽  
Metric Measure Spaces ◽  
First Order ◽  
Upper Gradient ◽  
Measure Spaces ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Sobolev Capacity

In this paper, first-order Sobolev-type spaces on abstract metric measure spaces are defined using the notion of (weak) upper gradients, where the summability of a function and its upper gradient is measured by the "norm" of a quasi-Banach function lattice. This approach gives rise to so-called Newtonian spaces. Tools such as moduli of curve families and Sobolev capacity are developed, which allows us to study basic properties of these spaces. The absolute continuity of Newtonian functions along curves and the completeness of Newtonian spaces in this general setting are established.

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