AN OPTIMAL LIMITING 2D SOBOLEV INEQUALITY

The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Holder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality parallel to u parallel to(L)infinity <= parallel to del u parallel to(L)2/root 2 pi alpha root ln(1+6 root 2 pi alpha parallel to u parallel to((C) over dot)alpha/parallel to del u parallel to(L)2 root ln(1+root 2 pi alpha parallel to u parallel to((C) over dot)alpha/parallel to del u parallel to(L)2)) for functions u is an element of W-0(1,2)(B-1) on the unit disk B-1 in R-2, alpha is an element of (0, 1].