Hardy–bloch Type Spaces and Lacunary Series on the Polydisk

Abstract

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.

Weighted Integrals and Bloch Spaces of n-Harmonic Functions on the Polydisc

“…The first space B corresponds to that introduced by Timoney [24] for holomorphic functions in U n (see also [5]), while the second space Bh agrees with Definition 1 for p = q = ∞ (see also [2,4,5,25]). …”

Lacunary Series in Weighted HyperholomorphicBp,q(G) Spaces

“…the Green's function in 3 (see [6]). Then, we define the hyperholomorphic B p,q (G ) spaces in the three dimensional case as follows: One of our tasks in this article is to study these B p,q (G ) spaces and their relations to the above mentioned hyperholomorphic -Bloch space.…”

“…In the past few decades both Taylor and Fourier series expansions were studied by the help of lacunary series (see [4,5,25,27,31,33,34,36] and others). On the other hand there are some characterizations in higher dimensions using several complex variables and quaternion sense (see [3,11,13,16,20,23,29,30]). …”

Products of integral-type operators and composition operators between Bloch-type spaces

“…It makes B α into a Banach space. When α = 1, B 1 = B is the well-known Bloch space (see, for example, [2,7,23,34,35]). Let B α 0 denote the subspace of B α consisting of those f ∈ B α for which…”