Best constants for two families of higher order critical Sobolev embeddings

Abstract

In this paper we obtain the best constants in some higher order Sobolev inequalities in the critical exponent. These inequalities can be separated into two types: those that embed into L-infinity(R-N) and those that embed into slightly larger target spaces. Concerning the former, we show that for k is an element of{1,..., N - 1}, N - k even, one has an optimal constant c(k) > 0 such that parallel to u parallel to L-infinity <= c(k) integral vertical bar del(k)(-Delta)((N- k)/2)u vertical bar for all u is an element of C-c(infinity) (R-N) (the case k = N was handled in Shafrir, 2018). Meanwhile the most significant of the latter is a variation of D. Adams' higher order inequality of J. Moser: For Omega subset of R-N, m is an element of N and p = N/m, there exists A > 0 and optimal constant beta(0) > 0 such that integral(Omega) exp(beta(0)vertical bar u vertical bar(p')) <= A vertical bar Omega vertical bar for all u such that parallel to del(m)u parallel to(Lp(Omega)) <= 1, where parallel to del(m)u parallel to(Lp(Omega)) is the traditional seminorm on the space W-m,W-p(Omega). (C) 2018 Elsevier Ltd. All rights reserved.