On the comparison of ultradifferentiable weight function settings

TL;DR

The paper addresses the comparison of ultradifferentiable spaces defined via weight functions in the Beurling-Björck (Fourier-growth) and Braun-Meise-Taylor (derivative-growth) frameworks, emphasizing that the convexity of $\varphi_{\omega}$ is crucial for their coincidence. It develops a matrix-weight approach $\mathcal{M}_{\omega}$ to characterize inclusion and equality of Beurling-type spaces through growth relations such as $\omega(2t)=O(\omega(t))$, and extends these results to general weight matrices and Fourier-weighted subspaces. A central contribution is the construction of a Beurling-Björck weight that violates convexity and has no equivalent convex $\omega$, revealing uncountably many non-equivalent classes and illustrating limits of convexity-based equivalence. The work also engages with the growth index $\gamma(\omega)$ and discusses implications for Paley-Wiener-type results, quasianalyticity, and the limitations of the weight-sequence viewpoint in this context.

Abstract

We summarize, revisit and compare different notions of ultradifferentiable settings defined in terms of weight functions. More precisely, we emphasize the difference between the original Beurling-Björck-setting, when measuring the growth of the Fourier-transform, and the more recent approach by Braun, Meise and Taylor, when controlling the growth of the derivatives similarly like in the weight sequence framework. For the equality of both settings the convexity of the defining weight function is required and, somehow, has become a standard assumption. We study the meaning of this convexity condition and its failure in detail. First, we characterize the inclusion relations and hence the equality for Beurling-Björck-type classes in terms of weight functions analogously as it has already been done for the Braun-Meise-Taylor setting and work within the general weight matrix setting. Finally, we construct a technical (counter-)example which is a weight in the sense of Beurling-Björck but violates the convexity condition and which illustrates the different flavor of both weight function settings.