L~p measure of growth and higher order Hardy-Sobolev-Morrey inequalities on R~N

摘要

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-j)u 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}.