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On weighted inequalities for singular integrals. (English) Zbl 0868.42007
Summary: In this note we consider singular integrals associated to Calderón-Zygmund kernels. We prove that if the kernel is supported in (0,) 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<, 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,). The two-sided version of this result is also obtained: Muckenhoupts 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 (-,0) or in (0,).

MSC:
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory