The Borel map in ultraholomorphic function classes

Spaces of ultradifferentiable functions are subclasses of infinitely differentiable functions with certain growth conditions on all their derivatives. In the literature two different ap- proaches are considered, either using a weight sequence or using a weight function. In each setting one distinguishes between the Beurling type and the Roumieu type and one can divide each type into the quasi- and the non-quasianalytic case. Recently we have introduced and considered classes defined by (one-parameter) weight ma- trices. The spaces defined by weight sequences and weight functions were identified as par- ticular cases of weight matrix classes but one is able to describe more classes. So using this new method one is able to transfer results from one setting into the other one and to prove results for both cases simultaneously. Analogously to the ultradifferentiable case also classes of ultraholomorphic functions were introduced in the literature (defined by weight sequences). Also these classes can be divided into quasi- and non-quasianalytic ones. To study and characterize the (non)-quasianalyticity, i.e. the (non)-injectivity, and the surjectivity of the Borel map several growth indices were introduced and investigated. The general setting for working with ultraholomorphic classes is the notion of strongly regular weight sequences. This research project has two main topics. On the one hand several problems concerning the injectivity and surjectivity of the Borel map and the relations between different growth indices are remaining open even for the single weight sequence Roumieu case. The Beurling case has been studied far less and the question is if resp. which results can be transferred from the Roumieu to the Beurling case. Another question is if we can weaken the strong assumptions on the weight sequence. On the other hand the aim is to transfer the results from the weight sequence to the more gen- eral weight matrix case, in particular we are interested in the weight function case. Such ul- traholomorphic classes have not been introduced and studied so far and it turns out that in general the notion of strongly regular sequences cannot be applied to this approach. Moreover important theorems and techniques which were used and necessary for the (strongly regular) weight sequence case are not valid respectively cannot be applied any more. The applicant will work with Prof. Javier Sanz and his research team at Departamento de lgebra, Anlisis Matemtico, Geometra y Topologa, Facultad de Ciencias, Paseo de Belén 7, 47011 Valladolid (Spain).

During the two years phase abroad, staying at the Universidad de Valladolid (ESP) and working with the host Prof. Javier Sanz Gil and his research team, this research project has been devoted to questions concerning the injectivity and surjectivity of the (asymptotic) Borel map considered on classes of ultraholomorphic functions. First, study this mapping on classes defined by a single weight sequence and second introduce new ultraholomorphic classes defined in terms of weight functions. Finally, during the one year return phase in Vienna, the plan has been to prove for ultraholomorphic classes the "exponential law" analogously to the ultradifferentiable case and to provide applications of this result. We have been able to put forward the knowledge of the injectivity and surjectivity of the (asymptotic) Borel map in the ultraholomorphic weight sequence setting. In order to do so we have also studied growth indices and we have been able to detect a connection between them and the concept of O-regular variation. In this context we have been able to see strong connections to similar growth conditions for different weighted spaces arising in Functional Analysis. We have introduced ultraholomorphic classes defined in terms of weight functions. For these new classes we have shown extension results, introduced new growth indices on the weight and also given a connection to O-regular variation. We have studied in detail the, in general big, difference between the different notions of strong non-quasianalyticity in the weight sequence and weight function setting. In the weight function setting one is able to prove as a by-product mixed weight sequence extension results as well. Concerning the surjectivity we have also obtained results with controlled loss of regularity (mixed settings); more precisely we have treated both the ultraholomorphic weight sequence and weight function setting for the Borel map and the ultradifferentiable weight function setting even for the more general Whitney jet mapping. In order to do so we have introduced new mixed growth indices. One can expect that these mixed growth conditions will have consequences in other fields of Functional Analysis dealing with weighted structures. Concerning the injectivity in the ultradifferentiable setting (quasianalyticity) we have seen in a more precise and quantitative way "how large the failure" of being surjective is in this case. Our developed results give some idea how to tackle the problem to give a precise characterization of the image of the Borel map in the quasianalytic setting which is still an open question in this context. Currently we are working on the exponential law for ultraholomorphic classes and a concrete application of this theorem by proving an extension result for ultraholomorphic classes of several variables in polysectors defined in terms of a weight function.