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

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.

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