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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Davies’ method for heat-kernel estimates: An extension to the semi-elliptic setting


Authors: Evan Randles and Laurent Saloff-Coste
Journal: Trans. Amer. Math. Soc. 373 (2020), 2525-2565
MSC (2010): Primary 35K08; Secondary 35K25, 35H30
DOI: https://doi.org/10.1090/tran/8050
Published electronically: January 23, 2020
MathSciNet review: 4069227
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Abstract: We consider a class of constant-coefficient partial differential operators on a finite-dimensional real vector space which exhibit a natural dilation invariance. Typically, these operators are anisotropic, allowing for different degrees in different directions. The “heat” kernels associated to these so-called positive-homogeneous operators are seen to arise naturally as the limits of convolution powers of complex-valued measures, just as the classical heat kernel appears in the central limit theorem. Building on the functional-analytic approach developed by E. B. Davies for higher-order uniformly elliptic operators with measurable coefficients, we formulate a general theory for (anisotropic) self-adjoint variable-coefficient operators, each comparable to a positive-homogeneous operator, and study their associated heat kernels. Specifically, under three abstract hypotheses, we show that the heat kernels satisfy off-diagonal (Gaussian-type) estimates involving the Legendre-Fenchel transform of the operator’s principle symbol. Our results extend those of E. B. Davies and G. Barbatis and partially extend results of A. F. M. ter Elst and D. Robinson.


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Additional Information

Evan Randles
Affiliation: Department of Mathematics & Statistics, Colby College, Waterville, Maine 04901
MR Author ID: 930680
Email: evan.randles@colby.edu

Laurent Saloff-Coste
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
MR Author ID: 153585
Email: lsc@math.cornell.edu

Received by editor(s): August 1, 2019
Published electronically: January 23, 2020
Additional Notes: This material was based upon work supported by the National Science Foundation under Grant No. DMS-1707589.
Article copyright: © Copyright 2020 American Mathematical Society