A Singular Moser-Trudinger Inequality on Metric Measure Space

Let (X,d, mu) be a metric space with a Borel-measure mu, suppose mu satisfies the Ahlfors-regular condition, i.e. b(1)r(s) <= mu( B-r(x))= b(2)r(s), for all B-r(x) subset of X, r> 0, 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.