A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

Salahaddine Boutayeb, Thierry Coulhon*, Adam Sikora

*Corresponding author for this work

    Research output: Contribution to journalArticle

    20 Citations (Scopus)

    Abstract

    On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies-Gaffney estimates.

    Original languageEnglish
    Pages (from-to)302-374
    Number of pages73
    JournalAdvances in Mathematics
    Volume270
    DOIs
    Publication statusPublished - 2 Jan 2015

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