The classical techniques include the use of Poisson integrals, interpolation theory and the Hardy–Littlewood maximal function.
The first term is bounded on the whole of [–π,π], so it suffices to show that the convolution operators Sε defined by
It also follows that, for a continuous function f on the circle, Hεf converges uniformly to Hf, so in particular pointwise.
Instead a direct comparison of Hεf with the Poisson integral of the Hilbert transform is used classically to prove this.
The convergence statement above follows by continuity from the result for trigonometric polynomials, where it is an immediate consequence of the formula for the Fourier coefficients of Kr.
The uniform boundedness of the operator norm of Hε follows because HPr − H1−r is given as convolution by the function ψr, where[7]
Define the Hardy space H2(R) to be the closed subspace of L2(R) consisting of functions for which the Fourier transform vanishes on the negative part of the real axis.
Its orthogonal complement is given by functions for which the Fourier transform vanishes on the positive part of the real axis.
carries the extended real line onto the circle, sending the point at ∞ to 1, and the upper halfplane onto the unit disk.
In Nikolski (1986), part of the L2 theory on the real line and the upper halfplane is developed by transferring the results from the circle and the unit disk.
In fact, identifying H2 with L2(0,∞) via the Fourier transform, for y > 0 multiplication by e−yt on L2(0,∞) induces a contraction semigroup Vy on H2.
In particular, the integral defining Hεf(x) can be computed by taking a standard semicircle contour centered on x.
This can also be deduced directly because, after passing to Fourier transforms, Hε and H become multiplication operators by uniformly bounded functions.
This relation has been used classically in Vekua (1962) and Ahlfors (1966) to establish the continuity properties of T on Lp spaces.
To see that their difference tends to 0 in the strong operator topology, it is enough to check this for f smooth of compact support in D. By Green's theorem[22]
Alternatively it can be proved directly from the result for the Hilbert transform on R using the expression of Rj as an integral over G.[25][26] The Poisson operators Ty on Rn are defined for y > 0 by[27]
It can be checked directly that the operators RjTε − Rj,ε are given by convolution with functions uniformly bounded in L1 norm.
In fact, write f = f+ + f− according to the ±i eigenspaces of H. Since f ± iHf extend to holomorphic functions in the upper and lower half plane, so too do their squares.
[30] The same identity of Cotlar is easily verified on trigonometric polynomials f by writing them as the sum of the terms with non-negative and negative exponents, i.e. the ±i eigenfunctions of H. The Lp bounds can therefore be established when p is a power of 2 and follow in general by interpolation and duality.
The method of rotation for Riesz transforms and their truncations applies equally well on Lp spaces for 1 < p < ∞.
The integration of the functions Φ from the group T or SO(n) into the space of operators on Lp is taken in the weak sense:
The continuity of the Lp norms of a fixed Riesz transform is a consequence of the Riesz–Thorin interpolation theorem.
The proofs of pointwise convergence for Hilbert and Riesz transforms rely on the Lebesgue differentiation theorem, which can be proved using the Hardy-Littlewood maximal function.
These finer estimates form an important part of the techniques involved in Lennart Carleson's solution in 1966 of Lusin's conjecture that the Fourier series of L2 functions converge almost everywhere.
A more refined argument can be given[47] to show that, as in case of Lp, the difference tends to zero at all Lebesgue points of f. In combination with the result for the conjugate Poisson integral, it follows that, if f is in L1(T), then Hεf converges to Hf almost everywhere, a theorem originally proved by Privalov in 1919.
Calderón & Zygmund (1952) introduced general techniques for studying singular integral operators of convolution type.
To prove boundedness on Lp spaces, Calderón and Zygmund introduced a method of decomposing L1 functions, generalising the rising sun lemma of F. Riesz.
The Marcinkiewicz interpolation theorem and duality then implies that the singular integral operator is bounded on all Lp for 1 < p < ∞.
For points x in the complement, they lie in a nested set of intervals with lengths decreasing to 0 and on each of which the average of f is bounded by α.
[50] In fact by the Marcinkiewicz interpolation argument and duality, it suffices to check that if f is smooth of compact support then