On weighted inequalities for singular integrals

Abstract:

In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in $(0,\infty )$ then the one-sided $A_{p}$ condition, $A_{p}^{-}$, is a sufficient condition for the singular integral to be bounded in $L^{p}(w)$, $1<p<\infty$, or from $L^{1}(wdx)$ into weak-$L^{1}(wdx)$ if $p=1$. This one-sided $A_{p}$ condition becomes also necessary when we require the uniform boundedness of the singular integrals associated to the dilations of a kernel which is not identically zero in $(0,\infty )$. The two-sided version of this result is also obtained: Muckenhoupt’s $A_{p}$ condition is necessary for the uniform boundedness of the singular integrals associated to the dilations of a general Calderón-Zygmund kernel which is not the function zero either in $(-\infty ,0)$ or in $(0,\infty )$.