A note on weak convergence of singular integrals in metric spaces

TL;DR

This paper demonstrates that singular integral operators in metric spaces converge weakly in dense subspaces of L^2 under minimal regularity assumptions, extending classical results to more general metric settings.

Contribution

It establishes weak convergence of singular integrals in metric spaces with minimal regularity, broadening the scope of classical harmonic analysis results.

Findings

Weak convergence in metric spaces

Minimal regularity assumptions used

Extension of classical results to general metric spaces

Abstract

We prove that in any metric space $(X,d)$ the singular integral operators {equation*} T^k_{\mu,\ve}(f)(x)=\int_{X\setminus B(x,\varepsilon)}k(x,y)f(y)d\mu (y).{equation*} converge weakly in some dense subspaces of $L^2(\mu)$ under minimal regularity assumptions for the measures and the kernels.