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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Davies’ method for heat-kernel estimates: An extension to the semi-elliptic setting
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by Evan Randles and Laurent Saloff-Coste PDF
Trans. Amer. Math. Soc. 373 (2020), 2525-2565 Request permission

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.
  • © Copyright 2020 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 373 (2020), 2525-2565
  • MSC (2010): Primary 35K08; Secondary 35K25, 35H30
  • DOI: https://doi.org/10.1090/tran/8050
  • MathSciNet review: 4069227