A note on weak convergence of singular integrals in metric spaces

Abstract

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