Nonlocal Differential Operators with Integrable, Nonsymmetric Kernels: Part I--Operator Theoretic Properties

Abstract: Recent decades have provided a host of examples and applications motivating the study of nonlocal differential operators. We discuss a class of such operators acting on bounded domains, focusing on those with integrable kernels having compact support. Notably, we make no explicit symmetry assumptions on the kernel and discuss some implications of this decision. We establish a nonlocal-to-local convergence result, showing that these operators coincide with the classical derivative as the nonlocality vanishes. We also provide a new integration by parts result, a characterization of the compactness of these nonlocal operators, an implication for the nonlocal Poincar\'e inequality, and a variety of examples. This work establishes several key results needed to analyze nonlocal variational problems, given in Part II. We hope that this paper can serve as a relatively gentle introduction to the field of nonlocal modeling.