Mathematical Research Letters

Volume 22 (2015)

Number 1

A note on weak convergence of singular integrals in metric spaces

Pages: 11 – 21

DOI: https://dx.doi.org/10.4310/MRL.2015.v22.n1.a2

Authors

Vasilis Chousionis (Department of Mathematics, University of Connecticut, Storrs, Conn., U.S.A.; and Department of Mathematics and Statistics, University of Helsinki, Finland)

Mariusz Urbanski (Department of Mathematics, University of North Texas, Denton, Tx., U.S.A.)

Abstract

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

Keywords

singular integrals, metric spaces

2010 Mathematics Subject Classification

Primary 32A55. Secondary 30L99.

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Published 13 April 2015