  This work addresses the regularity of solutions for a nonlocal diffusion
equation over the space of periodic distributions. The spatial operator for the
nonlocal diffusion equation is given by a nonlocal Laplace operator with a
compactly supported integral kernel. We follow a unified approach based on the
Fourier multipliers of the nonlocal Laplace operator, which allows the study of
regular as well as distributional solutions of the nonlocal diffusion equation,
integrable as well as singular kernels, in any spatial dimension. In addition,
the results extend beyond operators with singular kernels to nonlocal
super-diffusion operators. We present results on the spatial and temporal
regularity of solutions in terms of regularity of the initial data or the
diffusion source term. Moreover, solutions of the nonlocal diffusion equation
are shown to converge to the solution of the classical diffusion equation for
two types of limits: as the spatial nonlocality vanishes or as the singularity
of the integral kernel approaches a certain critical singularity that depends
on the spatial dimension. Furthermore, we show that, for the case of integrable
kernels, discontinuities in the initial data propagate and persist in the
solution of the nonlocal diffusion equation. The magnitude of a jump
discontinuity is shown to decay overtime.
