Abstract. We extend the well-known Paley and Paley-Kahane-Khintchine inequalities on lacunary series to the unit polydisk of ‫ރ‬ n . Then we apply them to obtain sharp estimates for the mean growth in weighted spaces h( p, α), h( p, log(α)) of Hardy-Bloch type, consisting of functions n-harmonic in the polydisk. These spaces are closely related to the Bloch and mixed norm spaces and naturally arise as images under some fractional operators.2000 Mathematics Subject Classification. 32A37, 32A05.

In continuation of recent studies, we discuss two constructive approaches for the generation of harmonic conjugates to find null solutions to the Riesz system in double-struckR3. This class of solutions coincides with the subclass of monogenic functions with values in the reduced quaternions. Our first algorithm for harmonic conjugates is based on special systems of homogeneous harmonic and monogenic polynomials, whereas the second one is presented by means of an integral representation. Some examples of function spaces illustrating the techniques involved are given. More specifically, we discuss the (monogenic) Hardy and weighted Bergman spaces on the unit ball in double-struckR3 consisting of functions with values in the reduced quaternions. We end up proving the boundedness of the underlying harmonic conjugation operators in certain weighted spaces. Copyright © 2013 John Wiley & Sons, Ltd.

No abstract

We study anisotropic mixed norm spaces of n-harmonic functions in the unit polydisc of C n . Bergman type reproducing integral formulas are established by means of fractional derivatives and some continuous inclusions. It gives us a tool to construct corresponding projections and related operators and prove their boundedness on the mixed norm and Besov spaces.

No abstract