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.
Hilbert transform on the circle.
The theory for "L"2 functions is particularly simple on the circle. If "f" ∈ "L"2(T), then it has a Fourier series expansion
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
in the sense that the functions
defined by the restriction of "F" to the concentric circles |"z"| = "r", satisfy
The orthogonal projection "P" of "L"2(T) onto H2(T) is called the Szegő projection. It is a bounded operator on "L"2(T) with operator norm 1. By Cauchy's theorem
then "T" defines a bounded operator on L"p" for 1 < "p" < ∞ and a continuous operator from L1 into functions of weak type L1.
In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if "f" is smooth of compact support then
Take a Calderón−Zygmund decomposition of "f" as above
with intervals "J""n" and with α = λμ, where μ > 0. Then
The term for "g" can be estimated using Chebychev's inequality:
If "J"* is defined to be the interval with the same centre as "J" but twice the length, the term for "b" can be broken up into two parts:
The second term is easy to estimate:
To estimate the first term note that
Thus by Chebychev's inequality:
By construction the integral of "bn" over "Jn" is zero. Thus, if "yn" is the midpoint of "Jn", then by Hörmander's condition:
Hence
Combining the three estimates gives
The constant is minimized by taking
The Markinciewicz interpolation argument extends the bounds to any L"p" with 1 < "p" < 2 as follows. Given "a" > 0, write
where "f""a" = "f" if |"f"| < "a" and 0 otherwise and "f""a" = "f" if |"f"| ≥ "a" and 0 otherwise. Then by Chebychev's inequality and the weak type L1 inequality above
Hence
Continuity of the norms can be shown by a more refined argument or follows from the Riesz–Thorin interpolation theorem.