Conformal welding

Welding using the Hilbert transform on the circle

The use of the Hilbert transform to establish conformal welding was first suggested by the Georgian mathematicians D.G. Mandzhavidze and B.V. Khvedelidze in 1958. A detailed account was given at the same time by F.D. Gakhov and presented in his classic monograph (Gakhov (1990)).

Let e_n(θ) = e^inθ be the standard orthonormal basis of L²(T). Let H²(T) be Hardy space, the closed subspace spanned by the e_n with n ≥ 0. Let P be the orthogonal projection onto Hardy space and set T = 2P - I. The operator H = iT is the Hilbert transform on the circle and can be written as a singular integral operator.

Given a diffeomorphism f of the unit circle, the task is to determine two univalent holomorphic functions

${\displaystyle {\displaystyle f_{-}(z)=a_0+a_{1}z+a_2{z}^2+\cdots ,f_{+}(z)=z+b_1{z}^{-1}+b_2{z}^{-2}+\cdots ,}}$

defined in |z| < 1 and |z| > 1 and both extending smoothly to the unit circle, mapping onto a Jordan domain and its complement, such that

${\displaystyle {\displaystyle f_{-}(e^{i\theta })=f_{+}(f(e^{i\theta })).}}$

Let F be the restriction of f₊ to the unit circle. Then

${\displaystyle {\displaystyle TF=-F+2e^{i\theta }}}$

and

${\displaystyle {\displaystyle TF\circ f=F\circ f.}}$

Hence

${\displaystyle {\displaystyle T(F\circ f)\circ f^{-1}-T(F)=2F-2e^{i\theta }.}}$

If V(f) denotes the bounded invertible operator on L² induced by the diffeomorphism f, then the operator

${\displaystyle {\displaystyle K_f=V(f)PV(f{)}^{-1}-P}}$

is compact, indeed it is given by an operator with smooth kernel because P and T are given by singular integral operators. The equation above then reduces to

${\displaystyle {\displaystyle (I-K_f)F=e^{i\theta }.}}$

The operator I − K_f is a Fredholm operator of index zero. It has zero kernel and is therefore invertible. In fact an element in the kernel would consist of a pair of holomorphic functions on D and D^c which have smooth boundary values on the circle related by f. Since the holomorphic function on D^c vanishes at ∞, the positive powers of this pair also provide solutions, which are linearly independent, contradicting the fact that I − K_f is a Fredholm operator. The above equation therefore has a unique solution F which is smooth and from which f_± can be reconstructed by reversing the steps above. Indeed, by looking at the equation satisfied by the logarithm of the derivative of F, it follows that F has nowhere vanishing derivative on the unit circle. Moreover F is one-to-one on the circle since if it assumes the value a at different points z₁ and z₂ then the logarithm of R(z) = (F(z) − a)/(z - z₁)(z − z₂) would satisfy an integral equation known to have no non-zero solutions. Given these properties on the unit circle, the required properties of f_± then follow from the argument principle.