Holomorphlc functions in tubes associated with ultradistributions

Abstract

Let C be a regular cone in and in some instances a more general proper open connected subset of . We study holomorphic functions in tubes which satisfy a norm growth in L r with the bound involving the associated function M∗(ρ) corresponding to sequences M p p = 0,1,2,…, which are used to define ultradistributions. We show that these holomorphic functions haveFourier–Laplace integral representations and obtain boundary values in the ultradistribution spaces of Beurling type D′((M P ),L r ) which are generalizations of the Schwartz distributions D′ L r. The L r norm growth which we study here is motivated by the norm growth proved here for the Cauchy integral of elements in D′(∗,L r ), where ∗ is either (M p ) or {M p }, ultradistributions of Beurling type or Roumieu type, respectively.