Sobolev inequalities for products of powers

Abstract:

We derive weighted Sobolev inequalities of the form ${\left \| f \right \|_{L_u^q}} \leq C{\left \| {\nabla f} \right \|_{L_v^p}}$, $f \in C_0^\infty ({{\mathbf {R}}^n})$, $1 < p \leq q < \infty$, for classes of weight functions $u$, $v$ which include $v$’s that are a finite product of certain power weights times an ${A_p}$ function.