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Singular integral operators of convolution type
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on R''n'' and T''n'' through convolution by distributions; equivalently they are the singular integral operators that commute with translations. The classical examples in harmonic analysis are the harmonic conjugation operator on the circle, the Hilbert transform on the circle and the real line, the Beurling transform in the complex plane and the Riesz transforms in Euclidean space. The continuity of these operators on ''L''2 is evident because the Fourier transform converts them into multiplication operators. Continuity on ''Lp'' spaces was first established by Marcel Riesz. The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function. For more general operators, fundamental new techniques, introduced by Alberto Calderón and Antoni Zygmund in 1952, were developed by a number of authors to give general criteria for continuity on ''Lp'' spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.
==L2 theory==
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