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Positive-Homogeneous Operators, Heat Kernel Estimates and the Legendre-Fenchel Transform

Conference paper
Part of the Progress in Probability book series (PRPR, volume 72)

Abstract

We consider a class of homogeneous partial differential operators on a finite-dimensional vector space and study their associated heat kernels. The heat kernels for this general class of operators are seen to arise naturally as the limiting objects of the convolution powers of complex-valued functions on the square lattice in the way that the classical heat kernel arises in the (local) central limit theorem. These so-called positive-homogeneous operators generalize the class of semi-elliptic operators in the sense that the definition is coordinate-free. More generally, we introduce a class of variable-coefficient operators, each of which is uniformly comparable to a positive-homogeneous operator, and we study the corresponding Cauchy problem for the heat equation. Under the assumption that such an operator has Hölder continuous coefficients, we construct a fundamental solution to its heat equation by the method of Levi, adapted to parabolic systems by Friedman and Eidelman. Though our results in this direction are implied by the long-known results of Eidelman for \(2\mathbf{b}\)-parabolic systems, our focus is to highlight the role played by the Legendre-Fenchel transform in heat kernel estimates. Specifically, we show that the fundamental solution satisfies an off-diagonal estimate, i.e., a heat kernel estimate, written in terms of the Legendre-Fenchel transform of the operator’s principal symbol—an estimate which is seen to be sharp in many cases.

Keywords

\(2\mathbf{b}\)-parabolic operators Heat kernel estimates Legendre-Fenchel transform Quasi-elliptic operators Semi-elliptic operators 

Mathematics Subject Classification

Primary 35H30; Secondary 35K25 

Notes

Acknowledgements

We thank the anonymous referee for many helpful comments. This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144153 (Evan Randles) and the National Science Foundation under Grant No. DMS-1404435 (Laurent Saloff-Coste).

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California, Los AngelesLos AngelesUSA
  2. 2.Department of MathematicsCornell UniversityIthacaUSA

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