Function spaces of coercivity for the fractional Laplacian in spaces of homogeneous type

TL;DR

This paper establishes the existence of Green's functions for fractional Laplacians within certain function spaces on spaces of homogeneous type, using dyadic analysis and the Lax-Milgram theorem.

Contribution

It introduces a novel approach combining Haar wavelet analysis and the Lax-Milgram theorem to prove existence results for fractional Laplacians in new function spaces.

Findings

Existence of Green's functions for fractional Laplacians proven.

Application of dyadic analysis with Haar wavelets in this context.

Extension of fractional Laplacian theory to spaces of homogeneous type.

Abstract

We combine dyadic analysis through Haar type wavelets defined on Christ's families of generalized cubes, and Lax-Milgram theorem, in order to prove existence of Green's functions for fractional Laplacians on some function spaces of vanishing small resolution in spaces of homogeneous type.