On quasianalytic local rings

This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasianalytic classes are well-known in harmonic analysis since several decades, their study from the viewpoint of differential analysis and analytic geometry has begun much more recently and, to some extent, has earned them a new interest. Therefore, we focus on contemporary questions closely related to topics in local algebra. We study, in particular, Weierstrass division problems and the role of hyperbolicity, together with properties of ideals of quasianalytic germs. Incidentally, we also present a simplified proof of Carleman's theorem on the non-surjectivity of the Borel map in the quasianalytic case.Comment: Final Manuscrip

A Phragm\'en-Lindel\"of theorem via proximate orders, and the propagation of asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragm\'en-Lindel\"of theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindel\"of and G. Valiron.Comment: 20 page

El impacto del análisis funcional en algunos problemas del análisis

Discurso leído en el acto de recepción como académico del número de la Real Academia de Ciencias Exactas, Físicas y Naturales, por D. José Bonet Solves y contestación de D. Manuel Valdivia Ureña el 23 de abril de 2008. El propósito general de esta presentación es discutir el impacto del análisis funcional en algunos problemas del análisis matemático y su relevancia actual.This is the text of the entrance lecture as full member of the Real Academia de Ciencias Exactas, Físicas y Naturales by José Bonet Solves and the reply by Manuel Valdivia Ureña on April 28, 2008. The purpose of this talk is to discuss the impact of functional analysis in several problems of mathematical analysis and its relevance in present research.Bonet Solves, JA. (2008). El impacto del análisis funcional en algunos problemas del análisis. Editorial Universitat Politècnica de València. http://hdl.handle.net/10251/342

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.Comment: 48 pages; minor changes, accepted for publication in Journal of Mathematical Analysis and Applications; some typos correcte

On the singular locus of sets definable in a quasianalytic structure

Abstract Given a quasianalytic structure, we prove that the singular locus of a quasi-subanalytic set E is a closed quasi-subanalytic subset of E