Weighted Local Estimates for Singular Integral Operators

Abstract

A local median decomposition is used to prove that a weighted local mean of a function is controlled by a weighted local mean of its local sharp maximal function. Together with (a local version of) the estimate $M_{0,s}^{♯}(Tf)(x)\le cMf(x)$ for Calderón-Zygmund singular integral operators, this allows us to express the local weighted integral control of $Tf$ by $Mf$. Similar estimates hold for $T$ replaced by singular integrals with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $M$ replaced by an appropriate maximal function $M_T$. Using sharper bounds in the local median decomposition we prove two-weight, $L_v^p$-$L_w^q$ estimates for singular integral operators for $1<p\le q<\infty$. In all cases, the results include weights that are not necessarily $A_{\infty }$. The local nature of these estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.

53 pages; Theorem 6.6 now gives two-weight, $L_v^p$-$L_w^q$ estimates with $1<p\le q<\infty$ for Calderón-Zygmund operators based on "bump" conditions introduced by Pérez