Weighted inequalities through factorization (Q756068)

The author proves a general factorization theorem for weights u, v associated to positive sublinear operators bounded from \(L^ p(X,v)\) to \(L^ q(Y,u)\), where X and Y are two possibly different measure spaces and p, q are any indices between 1 and \(\infty\). This result yields some theorems related to boundedness of the integral transform \(K: f\to \int k(x,y)f(y)dy\) generated by a nonnegative measurable kernel k on \(X\times Y\). To formulate one such theorem denote by M(X), M(Y) the classes of measurable functions defined on X and Y, respectively. Theorem 3. Let \(v\in M(X)\) and \(u\in M(Y)\) be non-negative. A necessary and sufficient condition for K to be bounded from \(L^ p(X,v)\) to \(L^ q(Y,u)\) is that there exist non-negative functions \(u_ 0\in M(X)\), \(v_ 0\in M(Y)\), \(v_ 1\in M(X)\), \(u_ 1\in M(Y)\) and finite constants \(C_ 0\), \(C_ 1\) such that \(v=u_ 0^{-p/p'}v_ 1,\) \(u=v_ 0^{-p/p'}u_ 1,\) \(K(u_ 0)\leq C_ 0v_ 0\) and \(K(u_ 1)\leq C_ 1v_ 1.\) Moreover \(\| K\| \leq C_ 0^{1/p'}C_ 1^{1/p}.\) Applications to Hardy operator, fractional integrals, Laplace transform and Riemann-Liouville operator are given as well.