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Restriction of heat equation with Newton–Sobolev data on metric measure space

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Abstract

On a complete doubling metric measure space \(({\mathcal {X}},d,\mu )\) supporting the weak Poincaré inequality, by establishing some capacitary strong-type inequalities for the Hardy–Littlewood maximal operator, we characterize such a measure \(\nu \) on \({\mathcal {X}}\times {\mathbb {R}}_+\) that \(f\mapsto \int _{{\mathcal {X}}} p_{t^2}(\cdot ,y)f(y)\,d\mu (y)\) is bounded from Newton–Sobolev space \(N^{1,p}({\mathcal {X}})\) under \(p\in [1,\infty )\) into the Lebesgue space \(L^q({\mathcal {X}}\times {\mathbb {R}}_+,\nu )\) with \(q\in {\mathbb {R}}_+\), where the kernel \(p_t\) satisfies certain two-sided estimate. This offers a priori estimate for the solution to the heat equation with a Newton–Sobolev data on the given metric measure space \({\mathcal {X}}\). Via taking \(t\rightarrow 0\), a characterization of \(\nu \) on \({\mathcal {X}}\) ensuring the continuity of \(N^{1,p}({\mathcal {X}})\subset L^q({\mathcal {X}},\nu )\) is also obtained.

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Acknowledgements

The authors would like to thank the anonymous referees for several helpful comments on the original version of this paper.

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Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, People’s Republic of China

    Liguang Liu

  2. Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada

    Jie Xiao

  3. Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, People’s Republic of China

    Dachun Yang & Wen Yuan

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  1. Liguang Liu
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  2. Jie Xiao
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  3. Dachun Yang
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  4. Wen Yuan
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Correspondence to Wen Yuan.

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Communicated by R. Schoen.

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L. Liu was supported by NNSF of China Grant # 11771446; J. Xiao was supported by NSERC of Canada Grant # 202979463102000; D. Yang was supported by NNSF of China Grant # 11571039 and #11671185; W. Yuan (the corresponding author) was supported by NNSF of China Grant # 11871100; L. Liu, D. Yang and W. Yuan were also supported by NNSF of China Grant #11761131002.

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Liu, L., Xiao, J., Yang, D. et al. Restriction of heat equation with Newton–Sobolev data on metric measure space. Calc. Var. 58, 165 (2019). https://doi.org/10.1007/s00526-019-1611-3

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