Weighted Sobolev inequality
The authors consider the two-weight Sobolev inequality. They show that if \(1<p \leq q < \infty\) and \(v\) and \(w\) are any two weights such that the fractional integral operator \(I_ 1\) is of weak type \((L^ p(w), L^ q(v))\), then for every \(f \in C^ 1_ * (\mathbb{R}^ n)\) \[ \Bigl( \int_{\mathbb{R}^ n} | f |^ q vdx \Bigr)^{1/q} \leq C \Bigl( \int_{\mathbb{R}^ n} | \nabla f |^ p wdx \Bigr)^{1/p}, \] where \(C\) only depends on the weak norm \(I_ 1\) and dimension \(n\).
scientific article; zbMATH DE number 96635