Singular integral operators of convolution type
In mathematics, singular integral operators of convolution type are the singular integral operators that arise on R and T 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" is evident because the Fourier transform converts them into multiplication operators. Continuity on "L" 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" spaces. This article explains the theory for the classical operators and sketches the subsequent general theory. Section::::"L" theory. Hilbert transform on the circle. The theory for "L" functions is particularly simple on the circle. If "f" ∈ "L"(T), then it has a Fourier series expansion formula_1  Hardy space "H"(T) consists of the functions for which the negative coefficients vanish, "a" = 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 formula_2 in the sense that the functions formula_3 defined by the restriction of "F" to the concentric circles |"z"| = "r", satisfy formula_4 The orthogonal projection "P" of "L"(T) onto H(T) is called the Szegő projection. It is a bounded operator on "L"(T) with operator norm 1. By Cauchy's theorem  formula_5 then "T" defines a bounded operator on L for 1 < "p" < ∞ and a continuous operator from L into functions of weak type L. In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if "f" is smooth of compact support then formula_6 Take a Calderón−Zygmund decomposition of "f" as above formula_7 with intervals "J" and with "α" = "λμ", where "μ" > 0. Then formula_8 The term for "g" can be estimated using Chebychev's inequality: formula_9 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: formula_10 The second term is easy to estimate: formula_11 To estimate the first term note that formula_12 Thus by Chebychev's inequality: formula_13 By construction the integral of "b" over "J" is zero. Thus, if "y" is the midpoint of "J", then by Hörmander's condition: formula_14 Hence formula_15 Combining the three estimates gives formula_16 The constant is minimized by taking formula_17 The Markinciewicz interpolation argument extends the bounds to any "L" with 1 < "p" < 2 as follows. Given "a" > 0, write formula_18  where "f" = "f" if |"f"| < "a" and 0 otherwise and "f" = "f" if |"f"| ≥ "a" and 0 otherwise. Then by Chebychev's inequality and the weak type "L" inequality above formula_19 Hence formula_20 Continuity of the norms can be shown by a more refined argument or follows from the Riesz–Thorin interpolation theorem.