Almost subadditive weight functions form Braun-Meise-Taylor theory of ultradistributions
The study of several classes of ultra-distributions has been a very active area of research during the last two decades. These are enlargements of the class of Schwartz distributions. Each ultradistribution theory is based on a family \(({\mathcal D}_{\alpha})_{\alpha \in A} \) of spaces of smooth and compactly supported functions satisfying some assumptions. Given two theories of ultradistributions \(({\mathcal D}_{\alpha})_{\alpha \in A} \) and \(({\mathcal D}_{\beta})_{\beta \in B} \), we say that \(({\mathcal D}_{\alpha})_{\alpha \in A} \) is larger than \(({\mathcal D}_{\beta})_{\beta \in B} \) if, for each \(\beta \in B\), there is \(\alpha \in A\) such that \({\mathcal D}_{\alpha} \subset {\mathcal D}_{\beta}\). Two theories of ultradistributions are said to be equivalent if each one is larger than the other. There are essentially two ways to introduce ultradistributions, the theory of \textit{H.\,Komatsu} [J.~Fac.\ Sci., Univ.\ Tokyo, Sect.\ I A 20, 25--105 (1973; Zbl 0258.46039)], in which test functions are defined by looking at the growth of the derivatives on compact sets, and the theory developed by \textit{G.\,Björk} [Ark.\ Math.\ 6, 351--407 (1965; Zbl 0166.36501)], following the ideas previously announced by Beurling, in which one pays attention at the growth of the Fourier transforms. In 1990, \textit{R.\,W.\thinspace Braun, R.\,Meise} and \textit{B.\,A.\thinspace Taylor} [Result.\ Math.\ 17, No.\,3--4, 206--237 (1990; Zbl 0735.46022)] developed a theory of ultradistributions, defined as in the Beurling-Björck case by weight functions. Their point of view permits a unified treatment of both theories, it is equivalent to Komatsu's theory and, as was shown by \textit{U.\,Franken} [Result.\ Math.\ 25, No.\,1--2, 50--53 (1994; Zbl 0822.46037)], it is strictly larger than Beurling-Björck's one, which is based on subadditive weights. In the paper under review, the author considers weight functions in the sense of Braun, Meise and Taylor that are almost subadditive. More precisely, a weight \(\omega\) is called almost subadditive if, for each \(K>1\), there is \(C>0\) such that \(\omega(x+y) \leq K(\omega(x)+\omega(y))+C\) for all \(x,y \geq 0\). If \(AS\) denotes the class of almost subadditive weight functions, the author shows that \(({\mathcal D}_{\alpha})_{\alpha \in AS}\) is equivalent, as a theory of ultradistributions, to the one of Braun, Meise and Taylor. Finally, recall that two weight functions \(\omega, \sigma\) are called equivalent if there are constants \(C, D\) such that \(C \sigma(t) \leq \omega(t) \leq D \sigma(t)\) for \(t\) big enough. The question whether each weight function in the sense of Braun, Meise and Taylor is equivalent to an almost subadditive weight function remains open.
scientific article; zbMATH DE number 5644929
scientific article