Sectorial Extensions, via Laplace Transforms, in Ultraholomorphic Classes Defined by Weight Functions
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We prove several extension theorems for Roumieu ultraholomorphic classes of functions in sectors of the Riemann surface of the logarithm which are defined by means of a weight function or weight matrix. Our main aim is to transfer the results of V. Thilliez from the weight sequence case to these different, or more general, frameworks. The technique rests on the construction of suitable kernels for a truncated Laplace-like integral transform, which provides the solution without resorting to Whitney-type extension results for ultradifferentiable classes. As a byproduct, we obtain an extension in a mixed weight-sequence setting in which assumptions on the sequence are minimal.
KeywordsUltraholomorphic classes weight sequences functions and matrices Legendre conjugates Laplace transform extension operators indices of O-regular variation
Mathematics Subject Classification46E10 30E05 26A12 44A05
The first two authors are partially supported by the Spanish Ministry of Economy, Industry and Competitiveness under the Project MTM2016-77642-C2-1-P. The first author is partially supported by the University of Valladolid through a Predoctoral Fellowship (2013 call) co-sponsored by the Banco de Santander. The third author is supported by FWF-Project J 3948-N35, as a part of which he is an external researcher at the Universidad de Valladolid (Spain) for the period October 2016-September 2018.
The authors wish to express their gratitude to Prof. Óscar Blasco, from the Universidad de Valencia (Spain), for his helpful comments regarding Proposition 7.3.
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