Elements of script D′^(Mp)<inf>L</inf>^s and script D′^{Mp}<inf>L</inf>^s as boundary values of holomorphic functions

Abstract

We give representations of elements belonging to script D′(Mp)Lsand script D′{Mp}Ls s ∈ (1, infin;), where Mp is a non-quasianalytic sequence, as boundary values of holomorphic functions which satisfy appropriate estimates on the boundary of their domains. Conversely, we prove that the same estimates on holomorphic functions imply their script D′(Mp)Ls - and script D′{Mp}Ls -boundary values. With the stronger assumptions on Mp we obtain that appropriate L∞, resp. L1, estimates on holomorphic functions imply their boundary values in script D′(Mp)L∞ and script D′{Mp}L∞, resp. script D′(Mp)L1 and script D′{Mp}L1. © 1996 Academic Press, Inc.