(untitled) (Journal of Partial Differential Equations, 2022)
Let (X,d,μ) be a metric space with a Borel-measure μ, suppose μ satisfies the Ahlfors-regular condition, i.e. equation* b_1r^s≤μ(B_r(x))≤ b_2r^s,B_r(x)X, r>0, 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.