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Maximal and singular integral operators via Fourier transform estimates. (English) Zbl 0568.42012
In this paper \(L^ p\) inequalities are proved for singular integral operators written as \(Tf=\sum_{k}\sigma_ k*f\) and maximal operators defined by \(Mf=\sup_{k}| \mu_ k*f|,\) where \(\sigma_ k\) and \(\mu_ k\) are Borel measures with uniformly bounded total variation, the \(\sigma_ k's\) have zero integral and the \(\mu_ k's\) are positive. The inequalities follow from the regularity at zero and the decay at infinity of the Fourier transforms of \(\sigma_ k\) and \(\mu_ k\). Several applications are given: lacunary maximal functions (which generalize lacunary spherical means), homogeneous singular integrals with size conditions on the kernel and some of their variants, including weighted inequalities for kernels bounded on the unit sphere, maximal functions and Hilbert transforms along different types of curves (homogeneous, approximately homogeneous, convex plane curves). Some of them are old resuls with easier proofs, others are new. For all the singular integral operators the almost everywhere convergence of the truncated kernels is also obtained.

42B25 Maximal functions, Littlewood-Paley theory
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