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Proceedings of the London Mathematical Society Advance Access originally published online on November 24, 2007
Proceedings of the London Mathematical Society 2008 96(2):507-544; doi:10.1112/plms/pdm050
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© 2007 London Mathematical Society

Gaussian heat kernel upper bounds via the Phragmén–Lindelöf theorem

Thierry Coulhon

Département de Mathématiques
Université de Cergy-Pontoise
Site de Saint-Martin
2, rue Adolphe Chauvin
F 95302 Cergy-Pontoise Cedex
France

Adam Sikora

Department of Mathematical Sciences
New Mexico State University
Las Cruces
NM 88003-800
USA
asikora@nmsu.edu

Received 13 September 2006. Revision received 19 April 2007.

We prove that in the presence of L2 Gaussian estimates, the so-called Davies–Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces.

2000 Mathematics Subject Classification 35K05, 58J35.

Coulhon's research was partially supported by the European Commission (IHP Network ‘Harmonic Analysis and Related Problems’ 2002–2006, Contract HPRN-CT-2001-00273-HARP). Sikora's research was partially supported by an Australian Research Council (ARC) Discovery Grant DP 0451016 and New Mexico State University Summer Research Award.


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