Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups. (English) Zbl 1517.35101
Danielli, Donatella (ed.) et al., New developments in the analysis of nonlocal operators. AMS special session, University of St. Thomas, Minneapolis, MN, USA, October 28–30, 2016. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 723, 167-189 (2019).
Summary: In this paper we show novel underlying connections between fractional powers of the Laplacian on the unit sphere and functions from analytic number theory and differential geometry, like the Hurwitz zeta function and the Minakshisundaram zeta function. Inspired by S. Minakshisundaram’s ideas [J. Indian Math. Soc., New Ser. 13, 41–48 (1949; Zbl 0033.11605)], we find a precise pointwise description of \((-\Delta _{\mathbb{S}^{n-1}})^su(x)\) in terms of fractional powers of the Dirichlet-to-Neumann map on the sphere. The Poisson kernel for the unit ball will be essential for this part of the analysis. On the other hand, by using the heat semigroup on the sphere, additional pointwise integro-differential formulas are obtained. Finally, we prove a characterization with a local extension problem and the interior Harnack inequality.
For the entire collection see [Zbl 1416.35007].
For the entire collection see [Zbl 1416.35007].
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 47D06 | One-parameter semigroups and linear evolution equations |
| 11M35 | Hurwitz and Lerch zeta functions |
| 11M41 | Other Dirichlet series and zeta functions |
| 26A33 | Fractional derivatives and integrals |
| 35K08 | Heat kernel |
| 35R11 | Fractional partial differential equations |
Keywords:
fractional Laplacian on the sphere; zeta function; method of semigroups; spherical harmonics; Harnack inqualityCitations:
Zbl 0033.11605Software:
DLMFReferences:
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