Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions

Abstract

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.

arXiv comment36 pages

References in corpus (8)

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• Flat functions in Carleman ultraholomorphic classes via proximate orders
• Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes
• Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions
• Extension of Whitney jets of controlled growth
• Smooth solutions of quasianalytic or ultraholomorphic equations
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Cited by in corpus (18)

• Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis
• Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions
• Indices of O-regular variation for weight functions and weight sequences
• On subadditivity-like conditions for associated weight functions
• On generalized definitions of ultradifferentiable classes
• The Theorem of Iterates for elliptic and non-elliptic Operators
• The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces
• On the regularization of sequences and associated weight functions
• On inclusion relations between weighted spaces of entire functions
• Equality of ultradifferentiable classes by means of indices of mixed O-regular variation
• On the construction of large Algebras not contained in the image of the Borel map
• The Borel map in the mixed Beurling setting
• Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting
• Ultraholomorphic sectorial extensions of Beurling type
• Ellipticity and the problem of iterates in Denjoy-Carleman classes
• On the class of almost subadditive weight functions
• On inclusion relations of weighted $L^p$-type spaces defined in terms of weight function matrices
• On the conjugate weight function and ultradifferentiable classes of entire functions