Sobolev and isoperimetric inequalities with monomial weights

Abstract

We consider the monomial weight |x1|A1⋯|xn|An in Rn, where Ai⩾0 is a real number for each i=1,…,n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by View the MathML source, and they contain the best or critical exponent (which depends on A1,…,An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions.

When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD=RA1+1×⋯×RAn+1.