On extension problems and the ultraholomorphic setting

This recent project ist the direct continuation of the FWF-project 10.55776/P33417 Ultraholomorphic and ultradifferentiable extension problems and thus the research questions are based on the progress made during project 10.55776/P33417. We investigate problems concerning the surjectivity of the (asymptotic) Borel map defined on classes of ultradifferentiable and ultraholomorphic functions. Mo- reover, we study basic properties of these classes, the defining weights and connections to further weighted spaces which are investigated in the field of Functional Analysis. Spaces of ultradifferentiable functions are certain sub-classes of the set of all infinitely differentiable functions. All derivatives of the functions belonging to such classes are required to satisfy particular growth control expressed typically by weight sequences or weight functions. In each setting one dis- tinguishes between the Roumieu- and the Beurling-type. Using weight matrices one can treat both classical settings in a unified way but also new spaces. A further advantage of weight matrices ist that one obtains automatically mixed weight sequence results (controlled loss of regularity). Ultraholomor- phic function classes are the complex differentiable counter-parts of the aforementioned spaces. In the field of Functional Analysis also further weighted spaces are studied (for example weighted spaces of integrable functions) and the weights are required to satisfy standard growth- and regularity assumpti- ons. Within this project we study the surjectivity of the Borel map in the ultraholomorphic case and inves- tigate the expected analogue behavior of the ultraholomorphic classes compared with the ultradifferen- tiable spaces. A further goal is to find new applications for ultraholomorphic classes defined by weight matrices. In the ultradifferentiable case we focus on the more general Whitney-jet-mapping, on the Beurling-type and we are looking for the existence of a continuous linear extension operator. We in- vestigate the image of the Borel mapping in the case of non-surjectivity. Another idea is the study of classes defined via anisotropic weights and to search for concrete applications in this case. Finally, we have the goal to study weights and their growth properties from an abstract point of view and to give applications and connections to other weighted spaces appearing in the field of Functional Analysis. The principal investigator Gerhard Schindl works together with his collaboration partners Chiara Boiti (Università di Ferrara), Javier Sanz Gil (Universidad de Valladolid), Céline Esser (Université de Liège) and Armin Rainer (Universität Wien).