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Journal of the London Mathematical Society 2000 62(2):599-612; doi:10.1112/S0024610700001393
© 2000 by London Mathematical Society
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© The London Mathematical Society

Off-Diagonal Bounds of Non-Gaussian Type for the Dirichlet Heat Kernel

Gabriele Grillo

Dipartimento di Matematica, Politecnico di Torino Corso Duca degli Abruzzi 24, 10129 Torino, Italy, grillo{at}calvino.polito.it

Received 10 May 1999.

The paper considers the heat kernel K{Omega}(t, x, y) of the operator –{Delta} on a proper Euclidean domain {Omega}, with Dirichlet boundary conditions. A general pointwise lower bound for K{Omega}, which is valid for t larger than a suitable t0(x,y), is proved (the short-time behaviour being well understood). The resulting non-Gaussian bounds describe simultaneously both the case of bounded domains and the case, modelled on the half-space example, of domains which satisfy a twisted infinite internal cone condition. Bounds for the Green's function are given as well.


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