The limiting case in the Sobolev embedding theorem and radial-symmetric functions

Abstract

Denoting by B<inf>r0</inf> the open ball with radius r<inf>0</inf>, centered at the origin, we consider the so called “limiting case” in the Sobolev embedding theorem, W^j+m,p (B<inf>r0</inf>) → W^j,q (B<inf>r0</inf>), namely the case mp = n, 1 < p ≤ q, where the embedding for q = ∞ does not hold. We show that in the case j = 1, contrary to the case j = 0, radial-symmetric counterex-amples, that is radial-symmetric functions in W^m+1,p (B<inf>r0</inf>) \ W^1,∞ (B<inf>r0</inf>) do not exist, if one assumes C²-regularity away from the origin. Moreover, we characterize in dimension n = 2 the set W^m+1,p (B<inf>r0</inf>) \ W^1,∞ (B<inf>r0</inf>), i.e. W^2,2 (B<inf>r0</inf>) \ W^1,∞ (B<inf>r0</inf>) within a reasonable large class of functions.