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Conjugate hyperharmonic functions and cauchy type integrals in douglis analysis

In the recent years the so-called hyperanalytic functions theory, i.e., null solutions of the Douglis operator in , has emerged as an increasingly important area of activity for mathematicians. The definition of conjugate hyperharmonic Douglis algebra-valued functions presented in this work is shown to be a generalization of the classical conjugate harmonic functions in the Complex analysis case. An approach to the problem of constructing a conjugate hyperharmonic function to a given hyperharmonic function is considered. The most interesting feature of the article is contained in the following results: • It is proved that singular Cauchy integral operator in the sense of Douglis analysis defines a bounded operator on generalized Hölder spaces. • A complete characterization of those classes of Douglis algebra-valued functions such that for any regular curve the Cauchy transform (in the Douglis analysis sense) has continuous limit values everywhere on the curve is also obtained.