Sobolev and isoperimetric inequalities with monomial weights

TL;DR

This paper establishes weighted Sobolev, isoperimetric, Morrey, and Trudinger inequalities in ^n with monomial weights, providing optimal constants and extremal functions, extending classical inequalities to weighted settings.

Contribution

It derives new weighted inequalities with best constants and extremals, generalizing classical results to monomial-weighted spaces and identifying their geometric and functional properties.

Findings

Derived weighted Sobolev and isoperimetric inequalities with optimal constants.

Identified extremal functions for the inequalities.

Extended classical inequalities to monomial-weighted ^n spaces.

Abstract

We consider the monomial weight $|x_1|^{A_1}...|x_n|^{A_n}$ in $\mathbb R^n$, where $A_i\geq0$ 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 $|x_1|^{A_1}...|x_n|^{A_n}dx$, and they contain the best or critical exponent (which depends on $A_1$, ..., $A_n$). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When $A_i$ are nonnegative \textit{integers}, these inequalities are exactly the classical ones in the Euclidean space $\mathbb R^D$ (with no weight) when written for axially symmetric functions and domains in $\mathbb R^D=\mathbb R^{A_1+1}\times...\times\mathbb R^{A_n+1}$.