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

On a complete doubling metric measure space ( X , d , μ ) supporting the weak Poincaré inequality, by establishing some capacitary strong-type inequalities for the Hardy–Littlewood maximal operator, we characterize such a measure ν on X × R + that f ↦ ∫ X p t 2 ( · , y ) f ( y ) d μ ( y ) is bounded...

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Bibliographic Details
Published in:Calculus of variations and partial differential equations 2019-10, Vol.58 (5), p.1-40, Article 165
Main Authors: Liu, Liguang, Xiao, Jie, Yang, Dachun, Yuan, Wen
Format: Article
Language:English
Subjects:
Analysis
Calculus of Variations and Optimal Control
Optimization
Control
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Sobolev space
Systems Theory
Theoretical
Thermodynamics
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Summary:On a complete doubling metric measure space ( X , d , μ ) supporting the weak Poincaré inequality, by establishing some capacitary strong-type inequalities for the Hardy–Littlewood maximal operator, we characterize such a measure ν on X × R + that f ↦ ∫ X p t 2 ( · , y ) f ( y ) d μ ( y ) is bounded from Newton–Sobolev space N 1 , p ( X ) under p ∈ [ 1 , ∞ ) into the Lebesgue space L q ( X × R + , ν ) with q ∈ 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 X . Via taking t → 0 , a characterization of ν on X ensuring the continuity of N 1 , p ( X ) ⊂ L q ( X , ν ) is also obtained.
ISSN:0944-2669
1432-0835
DOI:10.1007/s00526-019-1611-3