Almost holomorphic extensions of ultradifferentiable functions
type: Item
instance of: scholarly article
title: Almost holomorphic extensions of ultradifferentiable functions
zbMATH Open document ID: 1061.30028
DOI: 10.1007/BF02893087
author: Mats Andersson
author: Bo Berndtsson
published in: Journal d'Analyse Mathématique
publication date: 2003-10-27T00:00:00Z
review text: For a function \(f\) on the real line (or the unit circle), the existence of extension to the complex \(F\) such that \(\overline\partial F\) tends to zero in a prescribed manner when approaching the real line (respectively the unit circle) (this \(F\) is called an almost holomorphic extensions) is related to regularity properties of the function \(f\). The authors extends this sort of result in the case of several variable. To achieve this, they review, in the proper setting, the results of \textit{A. Beurling} [Acta Math. 128, 153--182 (1972; Zbl 0235.30003) and Lecture Notes, Stanford 1961] and \textit{E. M. Dyn'kin} [Math. Sb., n. Ser. 89(131), 182--190 (1972; Zbl 0251.30033), J. Anal. Math. 60, 45--70 (1993; Zbl 0795.30034)]. More precisely, let \(H\) the class of continuous functions on \(\mathbb{R}\), nonnegative and increasing on \(\mathbb{R}^+\) and concave and decreasing on \(\mathbb{R}\) -- and tending to \(\infty\) for \(t\to\pm\infty\), \({\mathcal A}_h= \{f\in{\mathcal C}(\mathbb{R})(\int|\widehat f| e^h< \infty\}\) (here \(\widehat f\) is the Fourier transformation of \(f\)) \(\widetilde{\mathcal A}_h= \{f\in{\mathcal C}(\mathbb{R}\mid\sup|\widehat f| e^h< \infty\}\), and \(G\) the class of functions \(g\) nonnegative on \(\mathbb{R}\setminus\{0\}\), convex and decreasing on \(\mathbb{R}^+\), increasing on \(\mathbb{R}^-\), and \(\lim_{y\to\pm 0}g(y)= \infty\). The class of weights \(H\) and \(G\) are related by the Legendre transformation: Let \(h\) be nonnegative, increasing on \(\mathbb{R}^+\) decreasing in \(\mathbb{R}^-\). Define \(h^\# (y)= \sup_{\{t:yt>0\}}(h(t)- ty)\) and for \(g\) nonnegative decreasing on \(\mathbb{R}^+\) and increasing on \(\mathbb{R}^-\), \(g^b(t)= \inf_{\{y,yt>0\}}(g(y) +ty)\). One has that \(f^{\#b}\) is the smallest concave majorant of \(h\) and \(g^{b,\#}\), is the largest convex minorant of \(g\). If \(h\in H\) then \(h= h^{\# b}\); if \(g\in G\), \(g= g^{b\#}\). A precise version of Dyn'kin result is proved: Let \(h\…
reviewed by: Gheorghe Gussi
Mathematics Subject Classification ID: 30D60
Mathematics Subject Classification ID: 32-XX
Mathematics Subject Classification ID: 26E10
zbMATH DE Number: 1997294
zbMATH Keywords: almost holomorphic extension
zbMATH Keywords: Legendre transform
zbMATH Keywords: 0251.30033
MaRDI profile type: Publication
cites work: Q5698983
cites work: Weighted estimates for \(\overline\partial\) in domains in \(\mathbb{C}\)
cites work: Analytic continuation across a linear boundary
cites work: Q4264393
cites work: Holomorphic approximation of ultradifferentiable functions
cites work: Q4097773
cites work: Q5664175
cites work: The pseudoanalytic extension
cites work: Q5584797
cites work: Q3999037
cites work: Q5613105
cites work: Q5613348
full work available at URL: https://doi.org/10.1007/bf02893087
OpenAlex ID: W2060945504
description: scientific article; zbMATH DE number 1997294