Abstract
We consider analytic functions in tubes Rn+iB⊂Cn with values in Banach space or Hilbert space. The base of the tube B will be a proper open connected subset of Rn, an open connected cone in Rn, an open convex cone in Rn, and a regular cone in Rn, with this latter cone being an open convex cone which does not contain any entire straight lines. The analytic functions satisfy several different growth conditions in Lp norm, and all of the resulting spaces of analytic functions generalize the vector valued Hardy space Hp in Cn. The analytic functions are represented as the Fourier–Laplace transform of certain vector valued Lp functions which are characterized in the analysis. We give a characterization of the spaces of analytic functions in which the spaces are in fact subsets of the Hardy functions Hp. We obtain boundary value results on the distinguished boundary Rn+i{0¯} and on the topological boundary Rn+i∂B of the tube for the analytic functions in the Lp and vector valued tempered distribution topologies. Suggestions for associated future research are given.
Highlights
In [1] and related work, we defined and analyzed vector-valued Hardy Hp(TB, X ) functions on tubes TB = Rn + iB ⊂ Cn with values in Banach space X
We showed that any Banach space X vector-valued analytic function on TB which obtained a X vector-valued distributional boundary value was a Hp(TB, X ), 1 ≤ p ≤ ∞, function with values in Banach space X if the X vector-valued boundary value was a Lp(Rn, X ), 1 ≤ p ≤ ∞, function
We showed that the Hp(TB, X ), 1 ≤ p ≤ ∞, functions admitted a representation by the Poisson integral of Lp(Rn, X ), 1 ≤ p ≤ ∞, functions if the values of the analytic functions were in a certain type of Banach space and obtained a pointwise growth estimate for the Hp(TB, X ) functions for this Banach space
Summary
We showed that the Hp(TB, X ), 1 ≤ p ≤ ∞, functions admitted a representation by the Poisson integral of Lp(Rn, X ), 1 ≤ p ≤ ∞, functions if the values of the analytic functions were in a certain type of Banach space and obtained a pointwise growth estimate for the Hp(TB, X ) functions for this Banach space. We have obtained many general results concerning Hp(TB, X ) functions with values in Banach space including representations as Fourier–Laplace, Cauchy, and Poisson integrals and the existence of boundary values. The generalizations of the vector-valued analytic functions in Hp(TB, X ), X being a Banach space, which we consider here are defined in Section 4 of this paper.
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