Factorization and Extrapolation of Pairs of Weights in Two Different Measure Spaces

✍ Scribed by Francisco J. Ruiz; José L. Torrea

• Publisher: John Wiley and Sons
• Year: 1991
• Tongue: English
• Weight: 408 KB
• Volume: 150
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.19911500114

✦ Synopsis

The technique introduced by RUBIO D E FRANCIA ([13], [14]) in order to obtain factorization and extrapolation results for Ap weights is used here in a more general setting in which can appear two different measure spaces and two different operators. "Let S be a sublinear operator which is bounded in Lp(p) (where p is an arbitrary masure), and assume that Sf 2 0 for every nonnegalive f E Lp(p). Then, to every g > 0 In Lp(p) we can associate a function G satisfying: (i) g(z) 5 G(z) a.e. (iii) SG(z) 5 CG(z) a.e. (C independent of 9)". This result has been used by several authors ([l], [2], [5], [12], [18]) always in a unique appropriate measure space.

The main observation in this paper is that the RDFA can be used also for two operators acting between two different measure spaces and therefore as a consequence we obtain results about factorization of weights in different contexts : POISSON integral, maximal operators related with tent spaces, trace inequalities for potential operators, maximal ergodic operators or one side HARDY-LITTLEWOOD maximal operators.

If the operators are linear and one is the adjoint of the other then the S. BLOOM'S result ([ 13) is also extended obtaining necessary and sufficient conditions. Moreover, in this case it is possible to obtain an extrapolation result as in [14] (see also [6]).

The organization of this paper is as follows: in 2 we introduce the notations and we *) The authors have been supported by a grant of CAICYT PB 85-0338 (ii) IlQllp 2 2 llgllp