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Heat kernel asymptotics for real powers of Laplacians

Part of: Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  23 January 2023

Cipriana Anghel Open the ORCID record for Cipriana Anghel [Opens in a new window]
Cipriana Anghel*
Affiliation:
Institute of Mathematics of the Romanian Academy, Bucharest, Romania
*
e-mail: cianghel@imar.ro
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Abstract

We describe the small-time heat kernel asymptotics of real powers $\operatorname {\Delta }^r$, $r \in (0,1)$ of a non-negative self-adjoint generalized Laplacian $\operatorname {\Delta }$ acting on the sections of a Hermitian vector bundle $\mathcal {E}$ over a closed oriented manifold M. First, we treat separately the asymptotic on the diagonal of $M \times M$ and in a compact set away from it. Logarithmic terms appear only if n is odd and r is rational with even denominator. We prove the non-triviality of the coefficients appearing in the diagonal asymptotics, and also the non-locality of some of the coefficients. In the special case $r=1/2$, we give a simultaneous formula by proving that the heat kernel of $\operatorname {\Delta }^{1/2}$ is a polyhomogeneous conormal section in $\mathcal {E} \boxtimes \mathcal {E}^* $ on the standard blow-up space $\operatorname {M_{heat}}$ of the diagonal at time $t=0$ inside $[0,\infty )\times M \times M$.

Keywords

Heat kernel asymptoticsfractional powers of Laplaciansblow-up heat spacepolyhomogeneous expansions

MSC classification

Primary: 58J37: Perturbations; asymptotics 58J35: Heat and other parabolic equation methods
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 27
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This work was partially supported from the project PN-III-P4-ID-PCE-2020-0794 funded by UEFSCDI.

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