## 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.## References

- Pascal Auscher, Alan McIntosh, and Philippe Tchamitchian,
*Heat kernels of second order complex elliptic operators and applications*, J. Funct. Anal.**152**(1998), no. 1, 22–73. MR**1600066**, DOI 10.1006/jfan.1997.3156 - G. Barbatis and E. B. Davies,
*Sharp bounds on heat kernels of higher order uniformly elliptic operators*, J. Operator Theory**36**(1996), no. 1, 179–198. MR**1417193** - S. Blunck and P. C. Kunstmann,
*Generalized Gaussian estimates and the Legendre transform*, J. Operator Theory**53**(2005), no. 2, 351–365. MR**2153153** - Felix E. Browder,
*The asymptotic distribution of eigenfunctions and eigenvalues for semi-elliptic differential operators*, Proc. Nat. Acad. Sci. U.S.A.**43**(1957), 270–273. MR**90741**, DOI 10.1073/pnas.43.3.270 - Thierry Coulhon,
*Ultracontractivity and Nash type inequalities*, J. Funct. Anal.**141**(1996), no. 2, 510–539. MR**1418518**, DOI 10.1006/jfan.1996.0140 - Edward Brian Davies,
*One-parameter semigroups*, London Mathematical Society Monographs, vol. 15, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London-New York, 1980. MR**591851** - E. B. Davies,
*$L^p$ spectral theory of higher-order elliptic differential operators*, Bull. London Math. Soc.**29**(1997), no. 5, 513–546. MR**1458713**, DOI 10.1112/S002460939700324X - E. B. Davies,
*Limits on $L^p$ regularity of self-adjoint elliptic operators*, J. Differential Equations**135**(1997), no. 1, 83–102. MR**1434916**, DOI 10.1006/jdeq.1996.3219 - E. B. Davies,
*Uniformly elliptic operators with measurable coefficients*, J. Funct. Anal.**132**(1995), no. 1, 141–169. MR**1346221**, DOI 10.1006/jfan.1995.1103 - Ennio De Giorgi,
*Un esempio di estremali discontinue per un problema variazionale di tipo ellittico*, Boll. Un. Mat. Ital. (4)**1**(1968), 135–137 (Italian). MR**0227827** - G. V. Demidenko,
*Integral operators defined by quasi-elliptic equations. I*, Sibirsk. Mat. Zh.**34**(1993), no. 6, 52–67, ii, vii (Russian, with English and Russian summaries); English transl., Siberian Math. J.**34**(1993), no. 6, 1044–1058. MR**1268157**, DOI 10.1007/BF00973468 - S. D. Èĭdel′man,
*Parabolic systems*, North-Holland Publishing Co., Amsterdam-London; Wolters-Noordhoff Publishing, Groningen, 1969. Translated from the Russian by Scripta Technica, London. MR**0252806** - S. D. Èĭdel′man,
*On a class of parabolic systems*, Soviet Math. Dokl.**1**(1960), 815–818. MR**0123845** - G. B. Folland and Elias M. Stein,
*Hardy spaces on homogeneous groups*, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR**657581** - Avner Friedman,
*Partial differential equations of parabolic type*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964. MR**0181836** - Leonard Gross,
*Logarithmic Sobolev inequalities and contractivity properties of semigroups*, Dirichlet forms (Varenna, 1992) Lecture Notes in Math., vol. 1563, Springer, Berlin, 1993, pp. 54–88. MR**1292277**, DOI 10.1007/BFb0074091 - Lars Hörmander,
*The analysis of linear partial differential operators. II*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 257, Springer-Verlag, Berlin, 1983. Differential operators with constant coefficients. MR**705278**, DOI 10.1007/978-3-642-96750-4 - Yakar Kannai,
*On the asymptotic behavior of resolvent kernels, spectral functions and eigenvalues of semi-elliptic systems*, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)**23**(1969), 563–634. MR**415092** - Elliott H. Lieb and Michael Loss,
*Analysis*, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR**1817225**, DOI 10.1090/gsm/014 - V. G. Maz’ya,
*Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients*, Funct. Anal. Appl., 2(3):230-235, 1968. - El Maati Ouhabaz,
*Analysis of heat equations on domains*, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. MR**2124040** - Evan Randles and Laurent Saloff-Coste,
*Convolution powers of complex functions on $\Bbb Z^d$*, Rev. Mat. Iberoam.**33**(2017), no. 3, 1045–1121. MR**3713040**, DOI 10.4171/RMI/964 - Evan Randles and Laurent Saloff-Coste,
*Positive-homogeneous operators, heat kernel estimates and the Legendre-Fenchel transform*, Stochastic analysis and related topics, Progr. Probab., vol. 72, Birkhäuser/Springer, Cham, 2017, pp. 1–55. MR**3737622**, DOI 10.1007/978-3-319-59671-6_{1} - Laurent Saloff-Coste,
*The heat kernel and its estimates*, Probabilistic approach to geometry, Adv. Stud. Pure Math., vol. 57, Math. Soc. Japan, Tokyo, 2010, pp. 405–436. MR**2648271**, DOI 10.2969/aspm/05710405 - Martin Schechter,
*Spectra of partial differential operators*, 2nd ed., North-Holland Series in Applied Mathematics and Mechanics, vol. 14, North-Holland Publishing Co., Amsterdam, 1986. MR**869254** - A. F. M. ter Elst and Derek W. Robinson,
*High order divergence-form elliptic operators on Lie groups*, Bull. Austral. Math. Soc.**55**(1997), no. 2, 335–348. MR**1438852**, DOI 10.1017/S0004972700034006

## 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