This problem can be reduced to that of finding univalent holomorphic maps f, g of the unit disk and its complement into the extended complex plane, both admitting continuous extensions to the closure of their domains, such that the images are complementary Jordan domains and such that on the unit circle they differ by a given quasisymmetric homeomorphism.
[3] Sharon & Mumford (2006) describe the first two methods of conformal welding as well as providing numerical computations and applications to the analysis of shapes in the plane.
They extend continuously to homeomorphisms fi of the unit circle onto the Jordan curve on the boundary.
In fact an element in the kernel would consist of a pair of holomorphic functions on D and Dc which have smooth boundary values on the circle related by f. Since the holomorphic function on Dc vanishes at ∞, the positive powers of this pair also provide solutions, which are linearly independent, contradicting the fact that I − Kf is a Fredholm operator.
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.