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Heat kernel asymptotics for real powers of Laplacians
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 2 / April 2024
- Published online by Cambridge University Press:
- 23 January 2023, pp. 367-393
- Print publication:
- April 2024
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- Article
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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 
$[0,\infty )\times M \times M$.
