**
J. Rákosník (ed.), **

Function spaces, differential operators and nonlinear analysis.

Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.

Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996

p. 141 - 158

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Banach function spaces and the two-weight problem for maximal functions

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Carlos Pérez

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain cperezmo@ccuam3.sdi.uam.es

**Abstract:** The purpose of this talk is to discuss sharp sufficient conditions on the couple of weights $(w,v)$ such that $$ \int\limits_{{\Bbb R}^n}\big(w(y)Mf(y)\big)^pdy\le c\int\limits_{{\Bbb R}^n}\big(v(y)|f(y)|\big)^pdy, $$ where $M$ denotes the Hardy-Littlewood maximal function. The conditions that we find are ``close'' in form to the classical $A_p$ condition for two weights which is necessary but not sufficient in general. Then we shall apply these results to deduce sharp non standard estimates for singular integrals and for the vector-valued maximal function.

**Full text of the article:**

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