Fractional Laplacians on the sphere, the Minakshisundaram zeta function and semigroups

TL;DR

This paper explores the connections between fractional Laplacians on the sphere, special functions from number theory, and geometric analysis, providing new formulas, characterizations, and inequalities for these operators.

Contribution

It introduces novel pointwise formulas and a local extension characterization for fractional Laplacians on the sphere, linking them to zeta functions and semigroup methods.

Findings

Derived a pointwise description of fractional Laplacians using Dirichlet-to-Neumann maps.

Established formulas involving the heat semigroup for fractional powers.

Proved a local extension problem and Harnack inequality for these operators.

Abstract

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 Minakshisundaram's ideas, we find a precise pointwise description of $(-\Delta_{\mathbb{S}^{n-1}})^s u(x)$ 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.