Generalized upper and lower Legendre conjugates for Braun-Meise-Taylor weight functions
Gerhard Schindl
TL;DR
This work investigates how generalized Legendre conjugates $\sigma\check{\star}\tau$ and $\sigma\widehat{\star}\tau$ interact with Braun-Meise-Taylor weight functions and analyzes the induced transformations on associated weight matrices $\mathcal{M}_{\sigma}$ and $\mathcal{M}_{\tau}$. It develops a framework that expresses the conjugates via pointwise products and quotients of weight matrices, provides conditions for the well-definedness of $\sigma\widehat{\star}\tau$, and derives consequences for globally defined ultradifferentiable and Gelfand-Shilov spaces. An explicit application generalizes a recent result on the continuity and range of the resolvent operator on weighted Gelfand-Shilov spaces. Overall, the paper broadens the applicability of weight-matrix methods to Braun-Meise-Taylor weights and offers new tools for analyzing regularity transformations across weighted function classes.
Abstract
We apply recent knowledge and techniques of the new generalized upper and lower Legendre conjugates to the theory of weight functions in the sense of Braun-Meise-Taylor and study in detail the effects on the corresponding associated weight matrices. An immediate and concrete application of the main statements is also provided. More precisely, we generalize a very recent result concerning the continuity and the range of the resolvent operator when being considered on weighted spaces of globally defined functions of Gelfand-Shilov type.