This paper studies Banach space valued Hausdorff-Young inequalities. The largest part considers ways of changing the underlying group. In particular the possibility to deduce the inequality for open subgroups as well as for quotient groups arising from compact subgroups is secured. A large body of results concerns the classical groups Tⁿ, Rⁿ and Z_k. Notions of Fourier type are introduced and they are shown to be equivalent to properties expressed by finite groups alone.

Letf and g be nonlinear entire functions. The relations between the dynamics of f⊗g and g⊗f are discussed. Denote byℐ (·) and F(·) the Julia and Fatou sets. It is proved that if z∈C, then z∈ℐ8464 (f⊗g) if and only if g(z)∈ℐ8464 (g⊗f); if U is a component of F(f○g) and V is the component of F(g○g) that contains g(U), then U is wandering if and only if V is wandering; if U is periodic, then so is V and moreover, V is of the same type according to the classification of periodic components as U. These results are used to show that certain new classes of entire functions do not have wandering domains.

We characterize in geometric terms the zero sets of holomorphic functions f in the bidisk such that log |f|∈L^p(D²) for 1< p<∞.

It is known how to obtain a uniform estimate of e.g. a polynomial in terms of its logarithmic sum over the integers provided that the sum is sufficiently small. This result is generalized here and we obtain estimates in terms of logarithmic sums taken over certain discrete subsets of the real axis.

First we define the dyadic Hardy space H_X(d) for an arbitrary rearrangement invariant space X on [0, 1]. We remark that previously only a definition of H_X(d) for X with the upper Boyd index q_x<∞ was available. Then we get a natural description of the dual space of H_x, in the case X having the property 1<-p_X<-q_X<2, imporoving an earlier result [P1].

In this paper, I study the group of analytic automorphisms of a bounded product domain in the space C(S,C) of continuous functions on a compact space S. I prove that its automorphism group is a Lie group and I am able to prove which are the bounded symmetric ones.