Uniformly local spaces and refinements of the classical Sobolev embedding theorems

Abstract

We prove that if $f$ is a distribution on $\mathbb{R}^N$ with $N \gt 1$ and if $\partial_j f \in L^{p_j},\sigma_j \cap L^{N,1}_{uloc}$ with $1 \leq p_j \leq N$ 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, $L^{p,\sigma}$ is the classical Lorentz space and $L^{p,\sigma}_{uloc}$ is a “uniformly local” subspace of $L^{p,\sigma}_{loc}$ larger than $L^{p,\sigma}$ when $p \lt \infty$.

We also show that $f \in \mathcal{BUC}$ if, in addition, $\partial_j f \in L^{p_j ,\sigma_j} \cap L^q_{uloc}$ with $q \gt N$ whenever $p_j \lt N$ 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 $\partial_j f \in L^p \cap L^q$ with $p \lt N \lt q$).

Various similar results hold under integrability conditions on the higher order derivatives of $f$. All of them are applicable to $g \ast f$ with $g \in L^1$ 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.