A Singular Moser-Trudinger Inequality on Metric Measure Space

Let(X,d,μ)be a metric space with a Borel-measureμ,supposeμsatisfies the Ahlfors-regular condition,i.e.birs≤μu(Br(x))≤b2rs,VBr(x)CX,r>0,where bi,b2 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.