An optimal limiting 2𝐷 Sobolev inequality

Abstract

The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Hölder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality\[‖u‖L∞⩽‖∇u‖L22παln⁡(1+62πα‖u‖C˙α‖∇u‖L2ln⁡(1+2πα‖u‖C˙α‖∇u‖L2))\|u\|_{L^{\infty }} \leqslant \frac {\|\nabla u\|_{L^2}}{\sqrt {2\pi \alpha }} \sqrt {\ln \Bigl (1+6\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}}{\|\nabla u\|_{L^{2}}} \sqrt {\ln (1+\sqrt {2\pi \alpha } \tfrac {\|u\|_{{\left .\rm \! \dot C\right .^{\!\alpha }}}} {\|\nabla u\|_{L^{2}}} )\!}\, \Bigr )}\]for functionsu∈W01,2(B1)u\in W^{1,2}_0(B_1)on the unit diskB1B_1inR2\mathbb R^2,α∈(0,1].\alpha \in (0,1].