A Singular Moser-Trudinger Inequality on Metric Measure Space

Abstract

Let $(X,d,\mu)$ be a metric space with a Borel-measure $\mu$, suppose $\mu$ satisfies the Ahlfors-regular condition, i.e. \begin{equation*} b_1r^s\leq\mu(B_r(x))\leq b_2r^s,\qquad \forall B_r(x)\subset X, \;\; \  r>0, \end{equation*} where $b_1$, $b_2$ are two positive constants and $s$ is the volume growth exponent. In this paper, we mainly study two things, one is to consider the best constant of the Moser-Trudinger inequality on such metric space under the condition that $s$ is not less than 2. The other is to study the generalized Moser-Trudinger inequality with a singular weight.