Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Patricia Alonso-Ruiz; Fabrice Baudoin; Li Chen; Luke Rogers; Nageswari Shanmugalingam; Alexander Teplyaev

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Metric Geometry 2020-09-11 v4 Analysis of PDEs Functional Analysis Probability

Authors: Patricia Alonso-Ruiz , Fabrice Baudoin , Li Chen , Luke Rogers , Nageswari Shanmugalingam , Alexander Teplyaev

Abstract

We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincar\'e inequality and a weak Bakry-\'Emery curvature type condition, this BV class is identified with the heat semigroup based Besov class $\mathbf{B}^{1,1/2}(X)$ that was introduced in our previous paper. Assuming furthermore a quasi Bakry-\'Emery curvature type condition, we identify the Sobolev class $W^{1,p}(X)$ with $\mathbf{B}^{p,1/2}(X)$ for $p>1$ . Consequences of those identifications in terms of isoperimetric and Sobolev inequalities with sharp exponents are given.

Keywords

besov class dirichlet spaces bv functions heat semigroup sobolev class gaussian heat kernel

Cite

@article{arxiv.1811.11010,
  title  = {Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates},
  author = {Patricia Alonso-Ruiz and Fabrice Baudoin and Li Chen and Luke Rogers and Nageswari Shanmugalingam and Alexander Teplyaev},
  journal= {arXiv preprint arXiv:1811.11010},
  year   = {2020}
}

Comments

The notes arXiv:1806.03428 will be divided in a series of papers. This is the second paper dealing with strictly local Dirichlet forms. v2 corrects typos and changes some terminology. To appear in Cal. Var & PDE

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

Abstract

Keywords

Cite

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