Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

Abstract

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, known as stability under ultradifferential operators in the classical ultradistribution theory, is replaced by a weaker condition, in which the growth properties are controlled by an additional parameter. For that reason, new techniques were used in the proofs. As an application, we discuss the corresponding wave front sets.

Extension operators for some ultraholomorphic classes defined by sequences of rapid growth

“…We note that the notation (M.2) , resp. (M.2), comes from [19], while in [17,18,20] (M.2) , resp. (M.2), is used.…”

“…These conditions (M.2) and (M.2) are different from the classical ones, namely (dc) and (mg), appearing in the literature when dealing with Carleman-like classes (see [13]). They play a prominent role in the study of the corresponding ultradifferentiable and ultradistributional classes carried out by S. Pilipović, N. Teofanov and F. Tomić in [17,18,19,20], as they allow for a precise control of the flexibility obtained by introducing the two parameter dependence.…”

“…The main aim of this paper is to give a constructive proof of the surjectivity of the Borel map in sectors of the complex plane for the ultraholomorphic class associated with some specific sequences, M τ,σ = (p τ p σ ) ∞ p=0 , which are not strongly regular, but for which the sequence M is regular as long as 1 < σ < 2. These sequences have been considered in a series of papers by S. Pilipović, N. Teofanov and F. Tomić [17,18,19,20], inducing ultradifferentiable spaces of so-called extended Gevrey regularity. However, it is important to note that the estimates they impose on the elements of those classes are different, as their derivatives are controlled by expressions of the form CA p σ p τ p σ instead.…”

Optimal flat functions in Carleman-Roumieu ultraholomorphic classes in sectors

“…On each such interval of indices the mapping q → b q is now clearly strictly increasing since 1 + δ j > 1 for all j. Moreover, by the first half in (20), we have b p j+1 −1 = (1 + δ j ) p j+1 −p j −1 a p j < b p j+1 . Hence the sequence q → b q is strictly increasing.…”

“…We conclude if we show that b q ≤ Aa q for all q with 1 + p j ≤ q ≤ p j+1 − 1, j ≥ 0. For this, since q → b q is strictly increasing, it suffices to observe that, thanks to the second half in (20), we have b…”