$L^p$ measure of growth and higher order Hardy-Sobolev-Morrey inequalities

Abstract

When the growth at infinity of a function $u$ on ${\mathbb{R}}^N$ is compared with the growth of $|x{|}^s$ for some $s\in \mathbb{R},$ this comparison is invariably made pointwise. This paper argues that the comparison can also be made in a suitably defined $L^p$ sense for every $1\le p<\infty$ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub $|x{|}^{-N/p}$ growth of $\nabla u$ in the sense implies sub $|x{|}^{1-N/p}$ growth of $u$ in the $L^q$ sense for well chosen values of $q.$ By investigating how sub $|x{|}^s$ growth of ${\nabla }^{k}u$ in the $L^p$ sense implies sub $|x{|}^{s+j}$ growth of ${\nabla }^{k-j}u$ in the $L^q$ sense for (almost) arbitrary $s\in \mathbb{R}$ and for $q$ in a $p$-dependent range of values, a family of higher order Hardy/Sobolev/Morrey type inequalities is obtained, under optimal integrability assumptions. These optimal inequalities take the form of estimates for ${\nabla }^{k-j}(u-{\pi }_u),1\le j\le k,$ where ${\pi }_u$ is a suitable polynomial of degree at most $k-1,$ which is unique if and only if $s<-k.$ More generally, it can be chosen independent of $(s,p)$ when $s$ remains in the same connected component of $\mathbb{R}\backslash \{-k,...,-1\}.$