Singularity removal in nonlocal operator inequalities on domains and applications

We prove that certain classes of nonlocal integral estimates on domains, involving operators, may be reduced to the local analogues without the singular weight on cubes. The theory applies to two-sided inverse doubling and quasi-monotone kernels with some moment conditions, and with a left-hand-side operator with average domain-bi-Lipschitz properties. Direct applications to fractional semi-norm inequalities in Peridynamics, such as fractional Korn's inequalities and Geometric Rigidity Estimates are presented.