Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

Stevan Pilipović,Filip Tomić,Nenad Teofanov

doi:10.3390/math9010007

Stevan Pilipović, Filip Tomić + Show 1 more

Open Access

https://doi.org/10.3390/math9010007

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Journal: Mathematics	Publication Date: Dec 22, 2020
Citations: 4	License type: CC BY 4.0

Affiliation: University of Novi Sad

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.

Highlights

In this paper we describe certain intermediate spaces between the spaces of Schwartz distributions and any space of Gevrey ultradistributions as boundary values of analytic functions
Another approach is given through their representations as finite sums of boundary values of analytic functions
We give a characterization of analytic functions whose boundary values are elements of intermediate spaces between the spaces of Schwartz distributions and any space of Gevrey ultradistributions

Summary

Introduction

In this paper we describe certain intermediate spaces between the spaces of Schwartz distributions and any space of Gevrey ultradistributions as boundary values of analytic functions. Consequences of the composition of ultradifferentiable functions determined by different classes of such sequences are discussed They consider weighted matrices, that is, a family of sequences of the form k!Mkλ, k ∈ N, λ ∈ Λ (partially ordered and directed set), and make the unions, again considering various properties such as compositions and boundary values. Their analysis follows the approach of [7,23]. Throughout the paper we always assume τ > 0 and σ > 1

Test Spaces and Duals

Main Result

Wave Front Sets

Conclusions