Heat Kernels on Metric Spaces with Doubling Measure

2009 ◽  
pp. 3-44 ◽  
Author(s):  
Alexander Grigor’yan ◽  
Jiaxin Hu ◽  
Ka-Sing Lau
Keyword(s):  
Metric Spaces ◽  
Heat Kernels ◽  
Doubling Measure

scholarly journals Duality of Moduli and Quasiconformal Mappings in Metric Spaces

2020 ◽  
Vol 8 (1) ◽  
pp. 166-181
Author(s):  
Rebekah Jones ◽  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Complete Metric Space ◽  
Poincaré Inequality ◽  
Quasiconformal Mappings ◽  
Duality Relation ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Family Of Curves ◽  
The Family

AbstractWe prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families of surfaces.

scholarly journals Dichotomy of global capacity density in metric measure spaces

2018 ◽  
Vol 11 (4) ◽  
pp. 387-404 ◽  
Author(s):  
Hiroaki Aikawa ◽  
Anders Björn ◽  
Jana Björn ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Exact Formula ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Geodesic Metric ◽  
Measure Spaces ◽  
Variational Capacity ◽  
John Domains ◽  
Superlevel Sets ◽  
Sobolev Capacity

AbstractThe variational capacity {\operatorname{cap}_{p}} in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every {E\subset{\mathbb{R}}^{n}},\inf_{x\in{\mathbb{R}}^{n}}\frac{\operatorname{cap}_{p}(E\cap B(x,r),B(x,2r))}% {\operatorname{cap}_{p}(B(x,r),B(x,2r))}is either zero or tends to 1 as {r\to\infty}. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincaré inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in {{\mathbb{R}}^{n}}. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.

Heat kernels on metric spaces and a characterisation of constant functions

2004 ◽  
Vol 115 (3) ◽  
pp. 389-399 ◽  
Author(s):  
Katarzyna Pietruska-Pałuba
Keyword(s):  
Metric Spaces ◽  
Heat Kernels

scholarly journals Diffusion processes and heat kernels on metric spaces

1998 ◽  
Vol 26 (1) ◽  
pp. 1-55 ◽  
Author(s):  
K. T. Sturm
Keyword(s):  
Metric Spaces ◽  
Diffusion Processes ◽  
Heat Kernels

scholarly journals Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

2016 ◽  
Vol 45 (4) ◽  
pp. 609-633 ◽  
Author(s):  
Niko Marola ◽  
Michele Miranda ◽  
Nageswari Shanmugalingam
Keyword(s):  
Metric Spaces ◽  
Heat Kernels ◽  
Sets Of Finite Perimeter ◽  
Finite Perimeter

The variational 1-capacity and BV functions with zero boundary values on doubling metric spaces

2018 ◽  
Vol 0 (0) ◽  
Author(s):  
Panu Lahti
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Boundary Values ◽  
Poincare Inequality ◽  
Compactly Supported ◽  
Bv Functions

AbstractIn the setting of a metric space that is equipped with a doubling measure and supports a Poincaré inequality, we define and study a class of{\mathrm{BV}}functions with zero boundary values. In particular, we show that the class is the closure of compactly supported{\mathrm{BV}}functions in the{\mathrm{BV}}norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and{\mathrm{BV}}analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.

scholarly journals Semiregular and strongly irregular boundary points for p-harmonic functions on unbounded sets in metric spaces

2021 ◽  
Author(s):  
Anders Björn ◽  
Daniel Hansevi
Keyword(s):  
Harmonic Functions ◽  
Metric Spaces ◽  
Poincaré Inequality ◽  
Doubling Measure ◽  
Poincare Inequality ◽  
Irregular Boundary ◽  
Complete Metric Spaces ◽  
Boundary Points ◽  
Local Properties ◽  
Harmonic Measures

AbstractThe trichotomy between regular, semiregular, and strongly irregular boundary points for $$p$$ p -harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $$p$$ p -Poincaré inequality, $$1<p<\infty $$ 1 < p < ∞ . We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $$p$$ p -harmonic measures, removability, and semibarriers.

scholarly journals AN ANALYTICAL APPROACH TO HEAT KERNEL ESTIMATES ON STRONGLY RECURRENT METRIC SPACES

2008 ◽  
Vol 51 (1) ◽  
pp. 171-199 ◽  
Author(s):  
Jiaxin Hu
Keyword(s):  
Dirichlet Boundary ◽  
Metric Spaces ◽  
Dirichlet Forms ◽  
Upper Bounds ◽  
Heat Kernels ◽  
Dirichlet Boundary Conditions ◽  
Green Functions ◽  
Compact Embedding ◽  
Kernel Estimates ◽  
Compact Metric Spaces

AbstractIn this paper we prove that sub-Gaussian estimates of heat kernels of regular Dirichlet forms are equivalent to the regularity of measures, two-sided bounds of effective resistances and the locality of semigroups, on strongly recurrent compact metric spaces. Upper bounds of effective resistances imply the compact embedding theorem for domains of Dirichlet forms, and give rise to the existence of Green functions with zero Dirichlet boundary conditions. Green functions play an important role in our analysis. Our emphasis in this paper is on the analytic aspects of deriving two-sided sub-Gaussian bounds of heat kernels. We also give the probabilistic interpretation for each of the main analytic steps.

scholarly journals Uniformization with Infinitesimally Metric Measures

2021 ◽  
Author(s):  
Kai Rajala ◽  
Martti Rasimus ◽  
Matthew Romney
Keyword(s):  
Metric Spaces ◽  
Doubling Measure ◽  
Quasiconformal Maps ◽  
Uniformization Theorem ◽  
Quasiconformal Map

AbstractWe consider extensions of quasiconformal maps and the uniformization theorem to the setting of metric spaces X homeomorphic to $${{\mathbb {R}}}^2$$ R 2 . Given a measure $$\mu $$ μ on such a space, we introduce $$\mu $$ μ -quasiconformal maps$$f:X \rightarrow {{\mathbb {R}}}^2$$ f : X → R 2 , whose definition involves deforming lengths of curves by $$\mu $$ μ . We show that if $$\mu $$ μ is an infinitesimally metric measure, i.e., it satisfies an infinitesimal version of the metric doubling measure condition of David and Semmes, then such a $$\mu $$ μ -quasiconformal map exists. We apply this result to give a characterization of the metric spaces admitting an infinitesimally quasisymmetric parametrization.

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