Necessary and sufficient conditions are given for the reflected Cauchy's operator (the reflected double layer potential operator) to be continuous as an operator from the space of all continuous functions on the boundary of the investigated domain to the space of all holomorphic functions on this domain (to the space of all harmonic functions on this domain) equipped with the topology of locally uniform convergence.
A mixed boundary value problem on a doubly connected domain in the complex plane is investigated. The solution is given in an integral form using reflection mapping. The reflection mapping makes it possible to reduce the problem to an integral equation considered only on a part of the boundary of the domain.
This note discusses to problem of the minimization of energy by the equilibrium measure obtained by the method of last exit in reference Ann. Inst. Fourier, 23-3 (1973), 313–322.
The main motivation for this work comes from the century-old Painlevé problem: try to characterize geometrically removable sets for bounded analytic functions in C.
We give a method for constructing functions and for which has a specified subharmonic minorant . By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class , for . In particular, the case yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in . We also apply the method to produce a new proof of the...