Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space: Its sharp constants and extremal functions

We investigate the sharp constants in a Brézis-Gallouët-Wainger type inequality with a double logarithmic term in the Hölder space in a bounded domain in Rⁿ. Ibrahim, Majdoub and Masmoudi gave the sharp constant in the two-dimensional case. We make precise estimates to give the sharp constants, and pass to the case of higher dimensions n≥2. We can also show that the inequality with fixed constants including the sharp ones admits an extremal function under a suitable condition when the domain is a ball.