Projektdetails
Abstract
Theoretical framework
Spaces of ultradifferentiable functions are subclasses of smooth functions with certain growth conditions on all their derivatives. By considering weight matrices one is able to treat both classical settings, using weight sequences or functions, in a unified approach but also more spaces. Ultraholomorphic classes are their complex counterparts; mostly defined in terms of weight sequences but we have also introduced new spaces defined by weight functions and matrices. Based on the knowledge of growth properties of the weights in these settings we establish connections to different fields dealing with weighted spaces.
Research questions
This project is mainly devoted to the study of the injectivity and surjectivity of the (asymptotic) Borel map considered on ultradifferentiable and ultraholomorphic function classes defined in terms of general weight matrices. When the Borel map is not surjective then we study the image. In the ultradifferentiable setting we are interested in the more general Whitney jet mapping admitting a controlled loss of regularity. We also investigate the structure of ultraholomorphic classes; e.g. characterize inclusion relations, dual spaces and stability properties. A further focus are applications for new spaces defined in terms of weight matrices in order to underline their importance. Finally, the aim is to investigate other topics dealing with weighted spaces in Functional Analysis and hence to connect different areas of research.
Methods
We are working within the general weight matrix setting and treat the sequence and function case simultaneously. We apply knowledge of growth and regularity properties for sequences and functions, appearing in the ultradifferentiable setting, to other weighted spaces.
Innovation
The research will complete and extend the information for classes defined by weight sequences and also provide completely new insights for classes defined by weight functions. By working with matrices we automatically obtain results for mixed weight sequence situations, i.e. a controlled loss of regularity, and dealing with weaker conditions on sequences than usually assumed; especially when studying the Borel and Whitney jet mapping. New results about the intrinsic structure of ultraholomorphic classes are expected and applications for such spaces defined by weight functions will be given to emphasize the relevance of this recent setting. Finally, we will obtain information about the importance and meaning of known growth and regularity conditions for weight sequences and functions in new directions when studying relations to other weighted spaces.
Primary researchers involved
On this project will work the applicant Gerhard Schindl at the University of Vienna and his collaboration partners Chiara Boiti (Univ. di Ferrara), Céline Esser (Univ. de Liège), Armin Rainer (Univ. of Vienna), and Javier Sanz (Univ. de Valladolid).
Spaces of ultradifferentiable functions are subclasses of smooth functions with certain growth conditions on all their derivatives. By considering weight matrices one is able to treat both classical settings, using weight sequences or functions, in a unified approach but also more spaces. Ultraholomorphic classes are their complex counterparts; mostly defined in terms of weight sequences but we have also introduced new spaces defined by weight functions and matrices. Based on the knowledge of growth properties of the weights in these settings we establish connections to different fields dealing with weighted spaces.
Research questions
This project is mainly devoted to the study of the injectivity and surjectivity of the (asymptotic) Borel map considered on ultradifferentiable and ultraholomorphic function classes defined in terms of general weight matrices. When the Borel map is not surjective then we study the image. In the ultradifferentiable setting we are interested in the more general Whitney jet mapping admitting a controlled loss of regularity. We also investigate the structure of ultraholomorphic classes; e.g. characterize inclusion relations, dual spaces and stability properties. A further focus are applications for new spaces defined in terms of weight matrices in order to underline their importance. Finally, the aim is to investigate other topics dealing with weighted spaces in Functional Analysis and hence to connect different areas of research.
Methods
We are working within the general weight matrix setting and treat the sequence and function case simultaneously. We apply knowledge of growth and regularity properties for sequences and functions, appearing in the ultradifferentiable setting, to other weighted spaces.
Innovation
The research will complete and extend the information for classes defined by weight sequences and also provide completely new insights for classes defined by weight functions. By working with matrices we automatically obtain results for mixed weight sequence situations, i.e. a controlled loss of regularity, and dealing with weaker conditions on sequences than usually assumed; especially when studying the Borel and Whitney jet mapping. New results about the intrinsic structure of ultraholomorphic classes are expected and applications for such spaces defined by weight functions will be given to emphasize the relevance of this recent setting. Finally, we will obtain information about the importance and meaning of known growth and regularity conditions for weight sequences and functions in new directions when studying relations to other weighted spaces.
Primary researchers involved
On this project will work the applicant Gerhard Schindl at the University of Vienna and his collaboration partners Chiara Boiti (Univ. di Ferrara), Céline Esser (Univ. de Liège), Armin Rainer (Univ. of Vienna), and Javier Sanz (Univ. de Valladolid).
| Kurztitel | Extensionsprobleme |
|---|---|
| Status | Laufend |
| Tatsächlicher Beginn/ -es Ende | 1/09/24 → 31/08/28 |