We prove that if f is a distribution on RN with N>1 and if ∂jf∈Lpj,σj∩LN,1uloc with 1≤pj≤N and σj=1 when pj=1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, Lp,σ is the classical Lorentz space and Lp,σuloc is a “uniformly local” subspace of Lp,σloc larger than Lp,σ when p<∞. We also show that f∈BUC if, in addition, ∂jf∈Lpj,σj∩Lquloc with q>N whenever pj<N and that, if so, the limit of f at infinity is uniform if the pj are suitably distributed. Only a few special cases have been considered in the literature, under much more restrictive assumptions that do not involve uniformly local spaces (pj=N and f vanishing at infinity, or ∂jf∈Lp∩Lq with p<N<q). Various similar results hold under integrability conditions on the higher order derivatives of f. All of them are applicable to g∗f with g∈L1 and f as above, or under weaker assumptions on f and stronger ones on g. When g is a Bessel kernel, the results are provably optimal in some cases.

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 L1 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.

New spaces of Lipschitz type on metric-measure spaces are introduced and they are shown to be just the well-known Besov spaces or Triebel–Lizorkin spaces when the smooth index is less than 1. These theorems also hold in the setting of spaces of homogeneous type, which include Euclidean spaces, Riemannian manifolds and some self-similar fractals. Moreover, the relationships amongst these Lipschitz-type spaces, Hajłasz–Sobolev spaces, Korevaar–Schoen–Sobolev spaces, Newtonian Sobolev space and Cheeger–Sobolev spaces on metric-measure spaces are clarified, showing that they are the same space with equivalence of norms. Furthermore, a Sobolev embedding theorem, namely that the Lipschitz-type spaces with large orders of smoothness can be embedded in Lipschitz spaces, is proved. For metric-measure spaces with heat kernels, a Hardy–Littlewood–Sobolev theorem is establish, and hence it is proved that Lipschitz-type spaces with small orders of smoothness can be embedded in Lebesgue spaces.

Given a fixed$p$≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $ , provided that the dimension$d$is sufficiently large. The method also yields new$L$^{$q$}space-time regularity results for solutions of the wave equation in high dimensions.

We obtain characterizations of positive Borel measures$μ$on$B$^{$n$}so that some weighted holomorphic Besov spaces$B$_{$s$}^{$p$}($ω$,$B$^{$n$}) are embedded in$L$^{$p$}($d$$μ$).