Transference of fractional Laplacian regularity

TL;DR

This paper establishes a method to transfer regularity estimates for the fractional Laplacian from Euclidean space to the torus, enabling the derivation of inequalities and regularity results in the periodic setting.

Contribution

It provides a precise formula for transferring fractional Laplacian regularity estimates from ^n to \u211d^n, addressing the subtlety of function space differences.

Findings

Transferred Harnack inequalities to the torus setting

Related extension problems between ^n and \u211d^n

Derived pointwise formulas and Hölder regularity estimates

Abstract

In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus $\mathbb{T}^n$ from the fractional Laplacian on $\mathbb{R}^n$. Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that $L^2$ functions on the torus can not be identified with $L^2$ functions on $\mathbb{R}^n$. The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise formulas and H\"older regularity estimates.