  In this paper we continue to advance the theory regarding the Riesz
fractional gradient in the calculus of variations and fractional partial
differential equations begun in an earlier work of the same name. In particular
we here establish an $L^1$ Hardy inequality, obtain further regularity results
for solutions of certain fractional PDE, demonstrate the existence of
minimizers for integral functionals of the fractional gradient with non-linear
dependence in the field, and also establish the existence of solutions to
corresponding Euler-Lagrange equations obtained as conditions of minimality. In
addition we pose a number of open problems, the answers to which would fill in
some gaps in the theory as well as to establish connections with more classical
areas of study, including interpolation and the theory of Dirichlet forms.

Functional Analysis;Analysis of PDEs