LetCbe a regular cone inand in some instances a more general proper open connected subset of. We study holomorphic functions in tubeswhich satisfy a norm growth inLrwith the bound involving the associated functionM*(ρ) corresponding to sequencesMpp= 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 typeD′((MP),Lr) which are generalizations of the Schwartz distributionsD′Lr. TheLrnorm growth which we study here is motivated by the norm growth proved here for the Cauchy integral of elements inD′(*,Lr), where * is either (Mp) or {Mp}, ultradistributions of Beurling type or Roumieu type, respectively.