Bounded and almost periodic ultradistributions as boundary values of holomorphic functions
For a nonquasianalytic weight \( \omega \), the space \(\mathcal H_{\omega^*} \) is defined as the smallest algebra of holomorphic functions on \( \mathbb C \setminus \mathbb R \) that contains all bounded holomorphic functions and all exponentials \( \exp(i\lambda z) \) for \( \lambda \in \mathbb R \) and is closed under the application of all \(\omega\)-ultradifferentiable operators of Beurling type. The main result states that the boundary values of elements in \(\mathcal H_{\omega^*} \) are bounded \(\omega\)-ultra\-dis\-tri\-bu\-tions of Beurling type and that, conversely, each bounded \(\omega\)-ultradistribution of Beurling type is the boundary value of an element in \( \mathcal H_{\omega^*} \). The corresponding result for Roumieu type is also stated. This completes results by \textit{S. Pilipović} [J. Math. Anal. Appl. 203, 719--737 (1996; Zbl 0891.46019)]. An important class of examples of bounded \(\omega\)-ultradistributions is given by the class of almost periodic \(\omega\)-ultradistributions. In this special case, those \(\omega\)-ultradistributions which are boundary values of holomorphic functions vanishing on \( \{ \text{Im} z < 0 \} \) are characterized.
scientific article; zbMATH DE number 2104670