  The study of what we now call Sobolev inequalities has been studied for
almost a century in various forms, while it has been eighty years since
Sobolev's seminal mathematical contributions. Yet there are still things we
don't understand about the action of integral operators on functions. This is
no more apparent than in the $L^1$ setting, where only recently have optimal
inequalities been obtained on the Lebesgue and Lorentz scale for scalar
functions, while the full resolution of similar estimates for vector-valued
functions is incomplete. The purpose of this paper is to discuss how some often
overlooked estimates for the classical Poisson equation give an entry into
these questions, to the present state of the art of what is known, and to
survey some open problems on the frontier of research in the area.
