An optimal limiting $2D$ Sobolev inequality

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

$$\Vert u\Vert _{L^{\infty}} \leqslant \frac{\Vert\nabla u\Vert _{L... ...{{\left.\rm\dot C\right.^{\alpha}}}} {\Vert\nabla u\Vert _{L^{2}}} )} \Bigr)}$$

for functions $u\in W^{1,2}_0(B_1)$ on the unit disk $B_1$ in $\mathbb{R}^2$, $\alpha\in(0,1].$