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

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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-Poincaré inequality and a weak Bakry–Émery 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–Émery 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.

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Acknowledgements

The authors thank Naotaka Kajino for many stimulating and helpful discussions. The authors also thank the anonymous referee for comments that helped improve the exposition of the paper. P.A-R. was partly supported by the Feodor Lynen Fellowship, Alexander von Humboldt Foundation (Germany) the grant DMS #1951577 and #1855349 of the NSF (U.S.A.). F.B. was partly supported by the grant DMS #1660031 of the NSF (U.S.A.) and a Simons Foundation Collaboration grant. L.R. was partly supported by the grant DMS #1659643 of the NSF (U.S.A.). N.S. was partly supported by the grants DMS #1800161 and #1500440 of the NSF (U.S.A.). A.T. was partly supported by the grant DMS #1613025 of the NSF (U.S.A.).

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, Texas, 77843, USA

    Patricia Alonso-Ruiz

  2. Department of Mathematics, University of Connecticut, Storrs, CT, 06269, USA

    Fabrice Baudoin, Luke Rogers & Alexander Teplyaev

  3. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Li Chen

  4. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, 45221, USA

    Nageswari Shanmugalingam

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  1. Patricia Alonso-Ruiz
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Correspondence to Fabrice Baudoin.

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Alonso-Ruiz, P., Baudoin, F., Chen, L. et al. Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates. Calc. Var. 59, 103 (2020). https://doi.org/10.1007/s00526-020-01750-4

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