Nonlocal Poincar\'{e} Inequalities for Integral Operators with Integrable Nonhomogeneous Kernels

The paper provides two versions of nonlocal Poincar'e-type inequalities for integral operators with a convolution-type structure and functions satisfying a zero-Dirichlet like condition. The inequalities extend existing results to a large class of nonhomogeneous kernels with supports that can vary discontinuously and need not contain a common set throughout the domain. The measure of the supports may even vanish allowing the zero-Dirichlet condition to be imposed on only a lower-dimensional manifold, with or without boundary. The conditions may be imposed on sets with co-dimension larger than one or even at just a single point. This appears to currently be the first such results in a nonlocal setting with integrable kernels. The arguments used are direct, and examples are provided demonstrating the explicit dependence of bounds for the Poincar'{e} constant upon structural parameters of the kernel and domain.