Uniformly local spaces and refinements of the classical Sobolev embedding theorems

We prove that if f is a distribution on R-N with N>1 and if partial derivative(j)f is an element of L-pj,L-sigma j boolean AND L-uloc(N,1) with 1 <= p(j) <= N and sigma(j) = 1 when p(j) = 1 or N, then f is bounded, continuous and has a finite constant radial limit at infinity. Here, L-p,L-sigma is the classical Lorentz space and L-uloc(p,sigma) is a "uniformly local" subspace of L-loc(p,sigma) larger than L-p,L-sigma when p<infinity. We also show that f is an element of BUC if, in addition, partial derivative(j) f is an element of L-pj,L-sigma j boolean AND L-uloc(q) with q>N whenever p(j)<N and that, if so, the limit of f at infinity is uniform if the p(j) 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 (p(j) = N and f vanishing at infinity, or partial derivative(j)f is an element of L-p boolean AND L-q 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 is an element of L-1 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.