Ultraholomorphic and ultradifferentiable extension problems

This project is devoted to questions and problems concerning the injectivity and surjectivity of the (asymptotic) Borel map considered on classes of so-called ultradifferentiable and ultraholomorphic functions. Spaces of ultradifferentiable functions are subclasses of the set of all infinitely differentiable func- tions with certain growth conditions on all their derivatives; classically defined in terms of weight sequences or functions by assuming standard growth and regularity assumptions on the weights. Ultraholomorphic classes are their complex differentiable counterparts; in the literature mostly defi- ned in terms of weight sequences. Within this project we consider both the Roumieu- and Beurling-type and classes defined in terms of weight sequences, functions and most generally weight matrices. By using a weight matrix one is able to treat both settings in a unified approach but also more classes. Moreover, one is able to get auto- matically results for mixed weight sequence situations (controlled loss of regularity) and dealing with weaker assumptions on the sequences than usually assumed. In the case when the Borel map is not surjective the aim is to get some information or even a full characterization of the image of this mapping. In the ultradifferentiable setting, for the more general Whitney jet mapping, and for Beurling-type classes defined by weight functions and matrices, we are treating the problem when the Borel map is not surjective but a controlled loss of regularity is possible (mixed setting). We will also study the analogous behaviour of ultraholomorphic classes compared with the ultradif- ferentiable setting and we want to find applications for ultraholomorphic spaces defined in terms of weight functions and weight matrices. This will underline the autonomy and importance of these new classes. The project leader Gerhard Schindl will work together with a Postdoc at the University of Vienna and his collaboration partners Javier Sanz Gil from the Universidad de Valldadolid, Céline Esser from the Université de Liège and Armin Rainer from the University of Vienna.

Within this research project classes of ultraholomorphic and ultradifferentiable functions and their defining weights were studied. Those weights control the growth of the functions and of all their complex resp. real derivatives, and technical regularity properties on the weights are required. Classically, two, in general mutually distinct, settings are considered: weight sequences or weight functions. A generalized method, namely weight matrices, was introduced in the project leader's PhD-thesis. We focused on the investigation of the size of the image of the Borel-mappping which is naturally defined on such spaces. We were able to prove a precise characterization of the controlled loss of regularity expressed in terms of the defining weight matrices. During this research we recognized a known condition which is optimal in order to show new ultraholomorphic extension results when assuming very general assumptions on the weight sequence. Moreover, we started to investigate important stability properties of ultraholomorphic classes; e.g. closedness under composition with functions. Another aim was the study of growth properties of weights and to establish connections to other weighted spaces appearing in the field of Functional Analysis. The goal is to transfer known information to other structures. We focused on Orlicz-classes and weighted spaces of entire functions. In addition, we studied ultradifferentiable classes defined via non-standard (small) weight sequences. Finally, intrinsic properties of classes of ultradifferentiable functions were analyzed. We were able to show that new generalized classes can equivalently be described by using weight matrices. In addition, for ultradifferentiable classes new versions of the Joris-Theorem and generalized interpolation results were shown. Also applications of weight matrices to the theory of partial differential operators were given. During this project we characterized the inclusion relations of new weighted spaces of globally defined functions of Gelfand-Shilov-type. The project leader has worked with his collaboration partners Céline Esser from the Université de Liège, Armin Rainer from the University of Vienna, and Javier Sanz Gil from the Universidad de Valladolid. Moreover, within this project David Nicolas Nenning has worked as a postdoc from September 1st, 2020 until May 31st, 2023 at the University of Vienna and he crucially participated in solving some of the research problems. With Chiara Boiti (Ferrara), David Jornet (Valencia), and Alessandro Oliaro (Torino) we have worked on Gelfand-Shilov-type spaces. With Stefan Fürdös (University of Vienna) we have worked on applications in the theory of partial differential operators. Finally, Javier Jiménez-Garrido (Univ. Cantabria, Santander) and Ignacio Miguel-Cantero (Univ. de Valladolid) made contributions. Jiménez-Garrido enjoyed a research stay at the University of Vienna from April 1st to June 30th, 2022 and Miguel-Cantero from February 15th to May 15th, 2023. During their stays both were supervised by the project leader.