Adams-type inequalities with logarithmic weights in fractional dimensions and the existence of extremals

Abstract

In this paper, we proved a sharp Adams-type inequality with logarithmic weights ${\omega }_{\beta }\left(r\right)={(\log \frac{1}{r})}^{\beta (p-1)}$ or ${\omega }_{\beta }\left(r\right)={(\log \frac{e}{r})}^{\beta (p-1)}$, $\beta \in (0,1)$ in the fractional dimensions. Furthermore, we show the existence of extremals for this kind of inequalities.