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

By Salahaddine Boutayeb, T. Coulhon, Adam S. Sikora. 2013

47 citations recorded by the source corpus.

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 localized 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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6ee23bebc2c560eb30690a2973f486e410ce9ed0

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Gradient estimates for heat kernels and harmonic functions (2017, methodology): “for the time derivative of the heat kernel for all t > 0; see [20, 97, 105, 106].”
Heat semigroup and singular PDEs (2015, background): “See also [9] for many reformulations of local Sobolev inequalities, an alternative proof of the equivalence with (UE), and more references.”