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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.

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].

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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.