THE A_{∞} PROPERTY FOR YOUNG FUNCTIONS AND WEIGHTED NORM INEQUALITIES

D. CRUZ-URIBE, SFO AND A. FIORENZA

Abstract. Using Orlicz space norms we define a set function property analogous to the A_{∞} condition for weights, and we characterize the Young functions which produce set functions with this property. As an application, we improve two-weight norm inequalities due to Cruz-Uribe and Pérez [4] for fractional integrals and commutators.

1. Introduction

A weight w satisfies the A_{∞} condition if there exist positive constants C and δ such that, given any cube Q and a measurable set E ⊂ Q,

w(E) ≤ C (\frac{|E|}{|Q|})^{δ} w(Q).

(Cf. García-Cuerva and Rubio de Francia [8, p. 401].) The A_{∞} condition is central to proving weighted good-λ inequalities which in turn are used to prove one-weight, weighted norm inequalities for a variety of operators. (See, for example, Coifman and Fefferman [2], Journé [9] and Muckenhoupt and Wheeden [13].) Given a weight u and r > 1, define the set function A_{u,r} by

A_{u,r}(E) = |E| (\frac{1}{|E|} ∫_{E} u^{r} dx)^{1/r} . (1)

It is immediate that A_{u,r} has the following property analogous to the A_{∞} condition: given E ⊂ Q,

A_{u,r}(E) ≤ (\frac{|E|}{|Q|})^{1/r′} A_{u,r}(Q). (2)

Date: 12/26/00. 1991 Mathematics Subject Classification. 42B20,42B25. Key words and phrases. weights, Young functions, Orlicz spaces, singular integral operators,

Calderón-Zygmund operators, commutators, fractional integral operators.

The authors would like to thank Carlo Sbordone for his support and encouragement which made this paper possible. The first author was partially supported by a Ford Foundation post-doctoral fellowship.