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

Abstract

When the growth at infinity of a function $u$ on $\Bbb{R}^{N}$ is compared with the growth of $|x|^{s}$ for some $s\in \Bbb{R},$ 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\leq 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 $% L^{p} $ 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 $s\in \Bbb{R}$ 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 $\nabla ^{k-j}(u-\pi _{u}),1\leq j\leq k,$ 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 $\Bbb{R}\backslash \{-k,...,-1\}.$