Kernel smoothing operators on thick open domains

Abstract

We study the convergence properties of a general family of normalized Markovian kernels on a domain $$\Omega$$ in the Euclidean space. Based on the notion of a thick set we show that a local Hardy-Littlewood inequality holds in $$L^p(\Omega)$$ , $$p \in (1, \infty]$$ when $$\Omega$$ is thick. We then establish pointwise and $$L^p(\Omega)$$ convergence for families of convolution operators with a Markov normalization on $$\Omega$$ . We further discuss convergence properties of bistochastic kernels where the corresponding integral operators preserve constant functions and integrals.