Sobolev and isoperimetric inequalities with monomial weights

We consider the monomial weight $\mathrm{\mid }{x}_1{\mathrm{\mid }}^{A_1}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{x}_n{\mathrm{\mid }}^{A_n}$ in ${\mathbb{R}}^n$, where $A_i\ge 0$ is a real number for each $i=1,\mathrm{.}\mathrm{.}\mathrm{.},n$, 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 $\mathrm{\mid }{x}_1{\mathrm{\mid }}^{A_1}\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{\mid }{x}_n{\mathrm{\mid }}^{A_n}dx$, 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 $\mathit{\text{integers}}$, 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 $\mathbb R^D=\mathbb R^{A_1+1}\times…\times\mathbb R^{A_n+1}$.