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
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 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,
Thus
When r = 1, the integrand on the right-hand side has a singularity at θ = 0. The truncated Hilbert transform is defined by
where δ = |1 – eiε|. Since it is defined as convolution with a bounded function, it is a bounded operator on L2(T). Now
If f is a polynomial in z then
By Cauchy's theorem the right-hand side tends to 0 uniformly as ε, and hence δ, tends to 0. So
uniformly for polynomials. On the other hand, if u(z) = z it is immediate that
Thus if f is a polynomial in z−1 without constant term
uniformly.
Define the Hilbert transform on the circle by
Thus if f is a trigonometric polynomial
uniformly.
It follows that if f is any L2 function
in the L2 norm.
This is an immediate consequence of the result for trigonometric polynomials once it is established that the operators Hε are uniformly bounded in operator norm. But on [–π,π]
The first term is bounded on the whole of [–π,π], so it suffices to show that the convolution operators Sε defined by
are uniformly bounded. With respect to the orthonormal basis einθ convolution operators are diagonal and their operator norms are given by taking the supremum of the moduli of the Fourier coefficients. Direct computation shows that these all have the form
with 0 < a < b. These integrals are well known to be uniformly bounded.
It also follows that, for a continuous function f on the circle, Hεf converges uniformly to Hf, so in particular pointwise. The pointwise limit is a Cauchy principal value, written
If f is just in L2 then Hεf converges to Hf pointwise almost everywhere. In fact define the Poisson operators on L2 functions by
for r < 1. Since these operators are diagonal, it is easy to see that Trf tends to f in L2 as r increases to 1. Moreover, as Lebesgue proved, Trf also tends pointwise to f at each Lebesgue point of f. On the other hand, it is also known that TrHf − H1 − r f tends to zero at each Lebesgue point of f. Hence H1 – r f tends pointwise to f on the common Lebesgue points of f and Hf and therefore almost everywhere.
Results of this kind on pointwise convergence are proved more generally below for Lp functions using the Poisson operators and the Hardy–Littlewood maximal function of f.
The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.See: Thus if H is a diffeomorphism of the circle with
then the operators
are uniformly bounded and tend in the strong operator topology to H. Moreover, if Vf(z) = f(H(z)), then VHV−1 − H is an operator with smooth kernel, so a Hilbert–Schmidt operator.
In fact if G is the inverse of H with corresponding function g(θ), then
Since the kernel on the right hand side is smooth on T × T, it follows that the operators on the right hand side are uniformly bounded and hence so too are the operators Hεh. To see that they tend strongly to H, it suffices to check this on trigonometric polynomials. In that case
In the first integral the integrand is a trigonometric polynomial in z and ζ and so the integral is a trigonometric polynomial in ζ. It tends in L2 to the trigonometric polynomial
The integral in the second term can be calculated by the argument principle. It tends in L2 to the constant function 1, so that
where the limit is in L2. On the other hand, the right hand side is independent of the diffeomorphism. Since for the identity diffeomorphism, the left hand side equals Hf, it too equals Hf (this can also be checked directly if f is a trigonometric polynomial). Finally, letting ε → 0,
The direct method of evaluating Fourier coefficients to prove the uniform boundedness of the operator Hε does not generalize directly to Lp spaces with 1 < p < ∞. Instead a direct comparison of Hεf with the Poisson integral of the Hilbert transform is used classically to prove this. If f has Fourier series
its Poisson integral is defined by
where the Poisson kernel Kr is given by
In f is in Lp(T) then the operators Pr satisfy
In fact the Kr are positive so
Thus the operators Pr have operator norm bounded by 1 on Lp. 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
for 1 − r ≤ |θ| ≤ π, and, for |θ| < 1 − r,
These estimates show that the L1 norms ∫ |ψr| are uniformly bounded. Since H is a bounded operator, it follows that the operators Hε are uniformly bounded in operator norm on L2(T). The same argument can be used on Lp(T) once it is known that the Hilbert transform H is bounded in operator norm on Lp(T).
Hilbert transform on the real line
As in the case of the circle, the theory for L2 functions is particularly easy to develop. In fact, as observed by Rosenblum and Devinatz, the two Hilbert transforms can be related using the Cayley transform.See:
The Hilbert transform HR on L2(R) is defined by
where the Fourier transform is given by
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. It is the complex conjugate of H2(R). If PR is the orthogonal projection onto H2(R), then
The Cayley transform
carries the extended real line onto the circle, sending the point at ∞ to 1, and the upper halfplane onto the unit disk.
Define the unitary operator from L2(T) onto L2(R) by
This operator carries the Hardy space of the circle H2(T) onto H2(R). In fact for |w| < 1, the linear span of the functions
is dense in H2(T). Moreover,
where
On the other hand, for z ∈ H, the linear span of the functions
is dense in L2((0,∞)). By the Fourier inversion formula, they are the Fourier transforms of
so the linear span of these functions is dense in H2(R). Since U carries the fw's onto multiples of the hz's, it follows that U carries H2(T) onto H2(R). Thus
In , 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. The natural replacements for concentric circles in the disk are lines parallel to the real axis in H. Under the Cayley transform, these correspond to circles in the disk that are tangent to the unit circle at the point one. The behaviour of functions in H2(T) on these circles is part of the theory of Carleson measures. However, the theory of singular integrals can be developed more easily by working directly on R.
H2(R) consists exactly of L2 functions f that arise of boundary values of holomorphic functions on H in the following sense: f is in H2 provided that there is a holomorphic function F(z) on H such that the functions fy(x) = f(x + iy) for y > 0 are in L2 and fy tends to f in L2 as y → 0. In this case F is necessarily unique and given by Cauchy's integral formula:
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. Hence for f in L2
If f is in H2, F(z) is holomorphic for Im z > 0, since the family of L2 functions gz depends holomorphically on z. Moreover, fy = Vyf tends to f in H2 since this is true for the Fourier transforms. Conversely if such an F exists, by Cauchy's integral theorem and the above identity applied to fy
for t > 0. Letting t tend to 0, it follows that Pfy = fy, so that fy lies in H2. But then so too does the limit f. Since
uniqueness of F follows from
For f in L2, the truncated Hilbert transforms are defined by
The operators Hε,R are convolutions by bounded functions of compact support, so their operator norms are given by the uniform norm of their Fourier transforms. As before the absolute values have the form
with 0 < a < b, so the operators Hε,R are uniformly bounded in operator norm. Since Hε,Rf tends to Hεf in L2 for f with compact support, and hence for arbitrary f, the operators Hε are also uniformly bounded in operator norm.
To prove that Hε f tends to Hf as ε tends to zero, it suffices to check this on a dense set of functions. On the other hand,
so it suffices to prove that Hεf tends to if for a dense set of functions in H2(R), for example the Fourier transforms of smooth functions g with compact support in (0,∞). But the Fourier transform f extends to an entire function F on C, which is bounded on Im(z) ≥ 0. The same is true of the derivatives of g. Up to a scalar these correspond to multiplying F(z) by powers of z. Thus F satisfies a Paley-Wiener estimate for Im(z) ≥ 0:
for any m, N ≥ 0. In particular, the integral defining Hεf(x) can be computed by taking a standard semicircle contour centered on x. It consists of a large semicircle with radius R and a small circle radius ε with the two portions of the real axis between them. By Cauchy's theorem, the integral round the contour is zero. The integral round the large contour tends to zero by the Paley-Wiener estimate. The integral on the real axis is the limit sought. It is therefore given as minus the limit on the small semicircular contour. But this is the limit of
Where Γ is the small semicircular contour, oriented anticlockwise. By the usual techniques of contour integration, this limit equals if(x). In this case, it is easy to check that the convergence is dominated in L2 since
so that convergence is dominated by
which is in L2 by the Paley-Wiener estimate.
It follows that for f on L2(R)
This can also be deduced directly because, after passing to Fourier transforms, Hε and H become multiplication operators by uniformly bounded functions. The multipliers for Hε tend pointwise almost everywhere to the multiplier for H, so the statement above follows from the dominated convergence theorem applied to the Fourier transforms.
As for the Hilbert transform on the circle, Hεf tends to Hf pointwise almost everywhere if f is an L2 function. In fact, define the Poisson operators on L2 functions by