On the $\mathrm{Lip}(\omega)$-continuity of the operator of harmonic reflection over boundaries of simple Carathéodory domains

We study continuity conditions for the operator of harmonic reflection of functions over boundaries of simple Carathéodory domains. This operator is viewed as one acting from a space of functions of Lipschitz–Hölder type, defined by a general modulus of continuity, into another space of such kind. The results obtained are based on the continuity criterion for the Poisson operator (acting in the same spaces of functions) in the domains in question, which are also obtained in the paper; they generalize and refine the results of the recent work by the second author and P. Paramonov (Analysis and Mathematical Physics, 2019).