On the class of almost subadditive weight functions (Q6614379)

A non-quasianalytic weight (in the sense of Braun-Meise-Taylor) is a continuous increasing function \(\omega:[0,\infty)\to [0,\infty)\) satisfying the following conditions:\N\begin{itemize}\N\item[(\(\alpha\))] \(\omega(2t) = O(\omega(t))\) as \(t\to \infty\);\N\item[(\(\beta\))] \(\int_1^\infty\frac{\omega(t)}{t^2}\, dt < \infty\);\N\item[(\(\gamma\))] \(\log(t) = o(\omega(t))\) as \(t\to \infty\);\N\item[(\(\delta\))] \(\varphi_\omega(x):=\omega(e^x)\) is convex.\N\end{itemize}\NA function \(f:[0,\infty)\to [0,\infty)\) is called almost subadditive if for each \(p > 1\) there exists \(C > 0\) such that \N\[\Nf(x+y) \leq p\left(f(x) + f(y)\right) + C\ \ \mbox{for all}\ x,y\geq 0.\N\]\NIt was proved by \textit{A. V. Abanin} and \textit{P. T. Tien} in [J. Math. Anal. Appl. 366, 296--301 (2010; Zbl 1194.46058)] that for every non-quasianalytic weight \(\omega\) there exists an almost subadditive (non-quasianalytic) weight \(\sigma\) such that \(\omega(t) \leq \sigma(t)\) for all \(t\geq 0.\) This paper answers a question posed by Abanin and Tien by showing that there exist weights \(\omega\) that are not equivalent to any almost subadditive weight. In the proof of the main theorem, the growth rate \(\gamma(\omega)\) plays an essential role. See [\textit{J. Jiménez-Garrido} et al., Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 4, 3659--3697 (2019; Zbl 1436.46029)]. The result obtained also applies to quasianalytic weights and to weights that meet weaker conditions than those usually required. The case where \(\omega = \omega_M\) is the weight associated with a weight sequence \(M\) is discussed in detail, allowing explicit counterexamples to be found.