Kernel Smoothing Operators on Thick Open Domains

We define the notion of a thick open set $\mathrm{\Omega }$ in a Euclidean space and show that a local Hardy-Littlewood inequality holds in $L^p(\mathrm{\Omega })$, $p\in (1,\mathrm{\infty }]$. We then establish pointwise and $L^p(\mathrm{\Omega })$ convergence for families of convolution operators with a Markov normalization on $\mathrm{\Omega }$. We demonstrate application of such smoothing operators to piecewise-continuous density, velocity, and stress fields from discrete element models of sea ice dynamics.