In the special case of Fourier series for the unit circle, the operators become the classical Cauchy transform, the orthogonal projection onto Hardy space, and the Hilbert transform a real orthogonal linear complex structure.
The range of the Cauchy transform is the Hardy space of the bounded region enclosed by the Jordan curve.
If f is in L2(T), then it has a Fourier series expansion[1][2] Hardy space H2(T) consists of the functions for which the negative coefficients vanish, an = 0 for n < 0.
The pointwise limit is a Cauchy principal value, written The Hilbert transform has a natural compatibility with orientation-preserving diffeomorphisms of the circle.
The Hardy space on the unit circle can be generalized to any multiply connected bounded domain Ω with smooth boundary ∂Ω.
Moreover[6] Hence differentiating the integral means with respect to s, the derivative in the direction of the inward pointing normal, gives using Green's theorem.
Furthermore, the restrictions fs of f to ∂Ωs, which can be naturally identified with ∂Ω, tend in L2 to the original function in Hardy space.
Moreover, in this case the functions tend uniformly to the boundary value and hence also in L2, using the natural identification of the spaces L2(∂Ωs) with L2(∂Ω).
Since h is a limit in L2 of rational functions g, the same results hold for h and Ch, with the same inequalities for the integral means.
[8] The Hardy space H2(Ω) has a natural partner, namely the closure in L2(∂Ω) of boundary values of rational functions vanishing at ∞ with poles only in Ω. Denoting this subspace by H2+(∂Ω) to distinguish it from the original Hardy space, which will also denoted by H2−(∂Ω), the same reasoning as above can be applied.
When applied to a function h in H2+(∂Ω), the Cauchy integral operator defines a holomorphic function F in Ωc vanishing at ∞ such that near the boundary the restriction of F to the level curves, each identified with the boundary, tend in L2 to h. Unlike the case of the circle, H2−(∂Ω) and H2+(∂Ω) are not orthogonal spaces.
The kernel can also be computed explicitly using the truncated Hilbert transforms for ∂Ω: and it can be verified directly that this is a smooth function on T × T.[10] Let C− and C+ be the Cauchy integral operators for Ω and Ωc.
Then Since the operators C−, C+ and H are bounded, it suffices to check this on rational functions F with poles off ∂Ω and vanishing at ∞ by the Hartogs–Rosenthal theorem.
Since E is an idempotent with range H2(∂Ω), P is given by the Kerzman–Stein formula: Indeed, since E − E* is skew-adjoint its spectrum is purely imaginary, so the operator I + E − E* is invertible.
An argument based on the Carathéodory kernel theorem shows that this condition is satisfied whenever there is a family of Jordan curves in Ω, eventually containing any compact subset in their interior, on which the integral means of F are bounded.
In fact, if Cn is the Cauchy integral operator corresponding to Ωsn, then[19] Since the first term on the right hand side is defined by pairing h − hn with a fixed L2 function, it tends to zero.
Singular integral operators on spaces of Hölder continuous functions are discussed in Gakhov (1990).