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 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
Hilbert transform on the circle
The theory for L2 functions is particularly simple on the circle. If f ∈ L2(T), then it has a Fourier series expansion
f ( θ ) = ∑ n ∈ Z a n e i n θ . {\displaystyle f(\theta )=\sum _{n\in \mathbf {Z} }a_{n}e^{in\theta }.}
Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 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
F ( z ) = ∑ n ≥ 0 a n z n , {\displaystyle F(z)=\sum _{n\geq 0}a_{n}z^{n},}
in the sense that the functions
f r ( θ ) = F ( r e i θ ) , {\displaystyle f_{r}(\theta )=F(re^{i\theta }),}
defined by the restriction of F to the concentric circles |z| = r, satisfy
‖ f r − f ‖ 2 → 0. {\displaystyle \|f_{r}-f\|_{2}\rightarrow 0.}
The orthogonal projection P of L2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on L2(T) with operator norm 1. By Cauchy's integral formula,
F ( z ) = 1 2 π i ∫ | ζ | = 1 f ( ζ ) ζ − z d ζ = 1 2 π ∫ − π π f ( θ ) 1 − e − i θ z d θ . {\displaystyle F(z)={1 \over 2\pi i}\int _{|\zeta |=1}{\frac {f(\zeta )}{\zeta -z}}\,d\zeta ={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\theta ) \over 1-e^{-i\theta }z}\,d\theta .}
Thus
F ( r e i φ ) = 1 2 π ∫ − π π f ( φ − θ ) 1 − r e i θ d θ . {\displaystyle F(re^{i\varphi })={1 \over 2\pi }\int _{-\pi }^{\pi }{f(\varphi -\theta ) \over 1-re^{i\theta }}\,d\theta .}
When r = 1, the integrand on the right-hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
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