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Heat kernels and Besov spaces on metric measure spaces

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Abstract

Let (M, ρ, μ) be a metric measure space satisfying the volume doubling condition. Assume also that (M, ρ, μ) supports a heat kernel satisfying the upper and lower Gaussian bounds. We study the problem of identity of two families of Besov spaces B sp, q and \(B_{p,q}^{s,{\cal L}}\), where the former one is defined using purely the metric measure structure of M, while the latter one is defined by means of the heat semigroup associated with the heat kernel. We prove that the identity \(B_{p,q}^s = B_{p,q}^{s,{\cal L}}\) holds for a range of parameters p, q, s given by some Hardy—Littlewood—Sobolev—Kato diagram.

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Acknowledgement

This work was done during a stay of the first author at the University of Bielefeld in 2017–2019. He would like to thank this university for the hospitality. The authors would like to thank the referee for his/her comments and suggestions to improve the manuscript.

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Authors and Affiliations

  1. Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou, 310023, People’s Republic of China

    Jun Cao

  2. Department of Mathematics, University of Bielefeld, 33501, Bielefeld, Germany

    Alexander Grigor’yan

Authors
  1. Jun Cao
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  2. Alexander Grigor’yan
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Corresponding author

Correspondence to Jun Cao.

Additional information

Jun Cao was supported by the NNSF of China (Grant No. 12071431) and Zhejiang Provincial Natural Science Foundation of China (Grant No. LR22A010006)

Alexander Grigor’yan was supported by SFB1283 of the German Research Foundation.

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Cao, J., Grigor’yan, A. Heat kernels and Besov spaces on metric measure spaces. JAMA 148, 637–680 (2022). https://doi.org/10.1007/s11854-022-0239-y

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  • DOI: https://doi.org/10.1007/s11854-022-0239-y

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