L^p measure of growth and higher order Hardy–Sobolev–Morrey inequalities on ℝ^N

Abstract

When the growth at infinity of a function u on ℝ^N is compared with the growth of |x|^s for some s ∈ 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 ≤ p < ∞ and that, in this perspective, inequalities of Hardy, Sobolev or Morrey type account for the fact that sub |x|^−N/p growth of ∇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 ∇^ku in the L^p sense implies sub |x|^s+j growth of ∇^k−ju in the L^q sense for (almost) arbitrary s ∈ 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 ∇^k−j(u − π_u), 1 ≤ j ≤ k, where π_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 ℝ\{−k,…,−1}.