## Weighted inequalities through factorization.(English)Zbl 0722.42012

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.

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory 44A10 Laplace transform 44A05 General integral transforms 44A15 Special integral transforms (Legendre, Hilbert, etc.)
Full Text: