Weighted estimates of singular integrals and their applications (Q1060376)

This paper is a survey of some recent developments in the theory of weighted estimates of the form \[ \int_{{\mathbb{R}}^ n}| Tf|^ pw\leq C\int_{{\mathbb{R}}^ n}| f|^ pv \] and corresponding inequalities of the weak type \[ \int_{\{| T(f)| >\lambda \}}w\leq C1/| \lambda |^ p\int_{{\mathbb{R}}^ n}| f|^ pv,\quad \lambda >0, \] where T is a singular integral or maximal operator and v, w are weighted functions for which known conditions \((A_ p)\) are fulfilled. The following problems determine the content of the paper: Hardy inequalities, Stieltjes transform, weighted estimates of maximal function, fractional integrals, singular operators and integrals of Cauchy type.