# On A Finite Range Decomposition of the Resolvent of a Fractional Power of the Laplacian II. The Torus

## TL;DR

This paper extends previous work on finite range decompositions of the resolvent of fractional Laplacians to lattice tori, establishing existence, regularity, differentiability, and uniform continuity properties for the resolvent on these compact structures.

## Contribution

It proves the existence and regularity of finite range decompositions for the fractional Laplacian resolvent on lattice tori, including differentiability and uniform continuity with respect to the resolvent parameter.

## Key findings

- Established finite range decomposition on lattice tori for fractional Laplacians.
- Proved regularity, differentiability, and uniform continuity of the decompositions.
- Extended previous lattice results to compact torus settings.

## Abstract

In previous papers, [M1, M2], [M3], we proved the existence as well as regularity of a finite range decomposition for the resolvent $G_{\alpha} (x-y,m^2) = ((-\Delta)^{\alpha\over 2} + m^{2})^{-1} (x-y) $, for $0<\alpha <2$ and all real $m$, in the lattice ${\bf Z}^{d}$ for dimension $d\ge 2$. In this paper, which is a continuation of the previous one, we extend those results by proving the existence as well as regularity of a finite range decomposition for the same resolvent but now on the lattice torus ${\bf Z}^{d}/L^{N+1}{\bf Z}^{d} $ for $d\ge 2$ provided $m\neq 0$ and $0<\alpha <2$. We also prove differentiability and uniform continuity properties with respect to the resolvent parameter $m^{2}$. Here $L$ is any odd positive integer and $N\ge 2$ is any positive integer.