Transference of fractional Laplacian regularity

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 Hölder regularity estimates.

7 pages. To appear in Special Functions, Partial Differential Equations and Harmonic Analysis. Proceedings of the conference in honor of Calixto P. Calderón, Roosevelt University at Chicago, November 16-18, 2012. C. Georgakis, A. Stokolos and W. Urbina (eds)