Singular integral operators of convolution type

In mathematics, singular integral operators of convolution type are the singular integral operators that arise on Rⁿ and Tⁿ 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 L² is evident because the Fourier transform converts them into multiplication operators. Continuity on L^p 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 L^p spaces. This article explains the theory for the classical operators and sketches the subsequent general theory.

The theory for L² functions is particularly simple on the circle. If f ∈ L²(T), then it has a Fourier series expansion

Hardy space H²(T) consists of the functions for which the negative coefficients vanish, a_n = 0 for n < 0. These are precisely the square-integrable functions that arise as boundary values of holomorphic functions in the open unit disk. Indeed, f is the boundary value of the function

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