Two weight norm inequalities for fractional one-sided maximal operators

Abstract

In this paper we introduce a new maximal function, the dyadic one-sided maximal function. We prove that this maximal function is equivalent to the one-sided maximal function studied by the authors and Ortega in Weighted inequalities for one-sided maximal functions (Trans. Amer. Math. Soc. 319 (1990)) and by Sawyer in Weighted inequalities for the one-sided Hardy-Littlewood maximal functions (Trans. Amer. Math. Soc. 297 (1986)), but our function, being dyadic, is much easier to deal with, and it allows us to study fractional maximal operators. In this way we obtain a geometric proof of the characterization of the good weights for fractional maximal operators, answering a question raised by Andersen and Sawyer in Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators (Trans. Amer. Math. Soc. 308 (1988)). Our methods, avoiding complex interpolation, give also the case of different weights for the fractional maximal operator, which was an open problem.