Weighted norm inequalities for maximal operators and Pisier's theorem on factorization through L^p∞

Abstract

It is proved that the inequality

$$\int {(M_\alpha f)} ^q d\omega \leqslant C(\int {\left| f \right|^p dx} )^{q/p}$$

for the fractional maximal operator

$$M_\alpha f(x) \approx \begin{array}{*{20}c} {\sup } \\ {x \in Q} \\ \end{array} \left| Q \right|^{\alpha /n - 1} \int_Q {\left| f \right|dx'} (x \in R^n )$$

holds for all f∈L^p if and only if$\begin{array}{*{20}c} {\sup } \\ {x \in Q} \\ \end{array} (\left| Q \right|_\omega /\left| Q \right|^{1 - \alpha p/n} ) \in L^{q/ (p - q)} (\omega )$ provided 0<q<p<∞, 1<p<∞, 0<α<n/p. Similar two-weight inequalities, as well as some extensions, including Riesz potentials, are also given. (The case p≤q has been treated by E.T. Sawyer [18], [19].) The proofs make use of a result of G. Pisier [17] related to the theory of factorization of operators through (weak) L^p-spaces.