Dirichlet problem for holomorphic functions in generalized Hölder spaces (Q610577)

The authors study the following Dirichlet problem: Let \(f\) be a function on the boundary of the unit disc in the complex plane that belongs to a certain regularity class of functions. Can one find a function \(g\) of the same regularity class on the closed disc which is holomorphic in the open disc and satisfies the conditions \(\text{Re\,} g=f\) on the boundary and \(\text{Im\,}g|_{z=z_{0}}=c\)? This is known to hold for the class of Hölder continuous functions of any order \(\lambda\in(0,1)\). The authors sketch the proof of a more general statement, that is, the result remains true for functions from the so-called generalized Hölder space, that is the space of functions for which \[ \max\left\{\sup_{K}|f(t)|,\sup_{z_{1}\neq z_{2}}\frac{|f(z_{1})-f(z_{2})|}{\mu(|z_{1}-z_{2}|)}\right\}<\infty, \] where \(K\) is a given compact subset of the complex plane and \(\mu\) is a function (not necessarily continuous) with positive values that satisfies some additional assumptions (for \(\mu(t)=t^{\lambda}\) one obtains the classical Hölder functions). Complete proofs are not provided.