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

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.

scholarly journals Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

2009 ◽  
Vol 7 (1) ◽  
pp. 61-89 ◽  
Author(s):  
Natasha Samko
Keyword(s):  
Operator Theory ◽  
Sufficient Conditions ◽  
Metric Measure Spaces ◽  
Lower And Upper Bounds ◽  
Index Numbers ◽  
Weights And Measures ◽  
Monotonic Functions ◽  
Measure Spaces ◽  
Necessary And Sufficient

In connection with application to various problems of operator theory, we study almost monotonic functionsw(x, r) depending on a parameterxwhich runs a metric measure spaceX, and the so called index numbersm(w, x),M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indicesm(w, x) andM(w, x) , for the uniform belongness of functionsw(·,r) to Zygmund-Bary-Stechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.

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.

The Riesz–Herz equivalence for capacitary maximal functions on metric measure spaces

2016 ◽  
Vol 7 (2) ◽  
Author(s):  
Pilar Silvestre
Keyword(s):  
Maximal Function ◽  
Maximal Functions ◽  
Metric Measure Spaces ◽  
Measure Spaces

AbstractWe prove a Riesz–Herz estimate for the maximal function

N-Lusin property in metric measure spaces: A new sufficient condition

2018 ◽  
Vol 30 (6) ◽  
pp. 1475-1486
Author(s):  
Marcela Garriga ◽  
Pablo Ochoa
Keyword(s):  
Measure Zero ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Metric Measure Spaces ◽  
Type Estimate ◽  
Measure Spaces ◽  
Approximate Differentiability

Abstract In this work, we are concerned with the study of the N-Lusin property in metric measure spaces. A map satisfies that property if sets of measure zero are mapped to sets of measure zero. We prove a new sufficient condition for the N-Lusin property using a weak and pointwise Lipschitz-type estimate. Relations with approximate differentiability in metric measure spaces and other sufficient conditions for the N-Lusin property will be provided.

The distance of L∞ from BMO on metric measure spaces

2014 ◽  
Vol 5 (2) ◽  
Author(s):  
Juha Kinnunen ◽  
Parantap Shukla
Keyword(s):  
Metric Measure Spaces ◽  
Measure Spaces

Abstract.This work has two principal aims. The first is to obtain a characterization of the closure of

scholarly journals SMOOTH METRIC MEASURE SPACES AND QUASI-EINSTEIN METRICS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250110 ◽  
Author(s):  
JEFFREY S. CASE
Keyword(s):  
Einstein Metrics ◽  
Conformal Geometry ◽  
Point Of View ◽  
Metric Measure Spaces ◽  
Unified Framework ◽  
Variational Characterization ◽  
Gradient Ricci Solitons ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

Smooth metric measure spaces have been studied from the two different perspectives of Bakry–Émery and Chang–Gursky–Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.

Some Notes on Function Spaces and Dirichlet Forms on Self-Similar Sets

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Yuan Liu ◽  
Zhi-Ying Wen
Keyword(s):  
Dirichlet Forms ◽  
Function Spaces ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Self Similarity ◽  
Equivalent Characterization ◽  
Measure Spaces ◽  
Self Similar ◽  
Sierpinski Gaskets

AbstractGrigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.

scholarly journals Characterization of Low Dimensional RCD*(K, N) Spaces

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Hausdorff Dimension ◽  
Isoperimetric Inequality ◽  
Type Inequality ◽  
Metric Measure Spaces ◽  
One Dimensional ◽  
Curvature Bounds ◽  
Measure Spaces ◽  
Low Dimensional ◽  
Regular Sets

AbstractIn this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov’s isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

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