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Cauchy and Poisson integrals of ultradistributions. (English) Zbl 0712.46018

The authors define and study the generalized Cauchy and Poisson integrals of ultradistributions of Beurling type 𝒟((M p ),L s ) and of Roumieu type 𝒟({M p },L s ) both of which generalize Schwartz distributions 𝒟 L ' for appropriate values of s. Thirteen theorems have been proved in this convection.

The authors claim that the work contained in the paper under review, may form a foundation for future research concerning a study of holomorphic functions in tubes which are characterised by either pointwise or norm growths, their boundary values, their recovery in terms of generalized integrals including some related properties. A detailed account of the lines on which this can be accomplished, is also given at the end of the paper.

Reviewer: G.L.N.Rao
MSC:
46F12Integral transforms in distribution spaces
46F15Hyperfunctions, analytic functionals
46F20Distributions and ultradistributions as boundary values of analytic functions
32A07Special domains in n (Reinhardt, Hartogs, circular, tube)
32A10Holomorphic functions (several variables)
32A40Boundary behavior of holomorphic functions (several variables)