Resumen de Stability properties and Borel-Ritt theorems in ultraholomorphic classes. An application to a generalized moment problem
The first aim of this dissertation is to characterize several stability properties, such as inverse or composition closedness, for ultraholomorphic function classes in unbounded sectors of the Riemann surface of the logarithm of both Roumieu and Beurling type defined in terms of a weight matrix. In the Roumieu case, we transfer and extend known results from J. Siddiqi and M. Ider, from the weight sequence setting and in sectors not wider than a half-plane, to the weight matrix framework and for sectors in the Riemann surface of the logarithm with arbitrary opening. The key argument rests on the construction, under suitable hypotheses, of characteristic functions in these classes for unrestricted sectors. As a by-product, we obtain new stability results when the growth control in these classes is expressed in terms of a weight sequence, or of a weight function in the sense of Braun-Meise-Taylor. In the Beurling case, we only deal with sectors not wider than a half-plane, due to the lack of characteristic functions, and the technique, completely different, rests on the theory of multiplicatively convex Fréchet algebras.
The second objective consists of improving known results of Borel-Ritt-type, dealing with the surjectivity of the asymptotic Borel mapping in Carleman ultraholomorphic classes, defined by the control of the size of the derivatives of their elements in terms of a weight sequence. We construct optimal flat functions in Carleman-Roumieu ultraholomorphic classes associated to general strongly nonquasianalytic weight sequences, and defined on sectors of suitably restricted opening. These functions provide kernel functions and their corresponding sequences of Stieltjes moments, in terms of which suitable formal Borel-like transforms, and truncated Laplace-like transforms, can be defined. These tools allow for the design of a general constructive procedure in order to obtain linear continuous extension operators, right inverses of the Borel mapping, for the case of regular weight sequences in the sense of Dyn'kin, i. e., those satisfying derivation closedness. The critical opening for which such results cease to be available is determined, and is given by an index of O-regular variation for the defining weight sequence. Some examples are discussed where such optimal flat functions can be obtained in a more explicit way.
Furthermore, a much weaker condition for the weight sequence, that of having shifted moments, is shown to be sufficient to obtain these extension results, although its necessity is not clear. Regarding Beurling classes, we are able to slightly improve a classical result of J. Schmets and M. Valdivia and reprove a result of A. Debrouwere, both under derivation closedness. Our new condition also allows us to obtain surjectivity results for Beurling classes in suitably small sectors, but the technique is now adapted from a classical procedure already appearing in the work of V. Thilliez, in its turn inspired by that of J. Chaumat and A.-M. Chollet.
Finally, the condition of shifted moments allows for a new framework when considering the Stieltjes moment problem within the general Gelfand-Shilov spaces defined via weight sequences. The novelty consists of allowing for a naturally larger target space for the moment mapping, which sends a function to its sequence of Stieltjes moments. The injectivity and surjectivity of the moment mapping in this new setting is studied and, in some cases, characterized.
