From Ultra-differentiable to Quasi-Analytic Analysis, Lecture 2

Abstract: The mini-course is intended as an introduction to ultradifferential analysis with special emphasis on ultra-differentiable extension theorems. The development of differential analysis in the last century was decisively influenced by Whitney’s work on the extension of differentiable functions from closed sets. We shall be interested in quantitative versions of Whitney’s extension theorem. The quantitative aspect is implemented by uniform growth properties of the multisequence of partial derivatives which measure the deviation from the Cauchy estimates and hence from analyticity. These growth conditions determine ultradifferentiable function classes which form a scale of regularity classes between the real analytic and the smooth class.

Lecture 2 (Tue, May 25, 13:00). Next we will study the image of the Borel map on Denjoy–Carleman classes which naturally sits in the sequence space defined by the characteristic bounds. The quasianalytic and the non-quasianalytic case are fundamentally different. While in the non-quasianalytic case we will give necessary and sufficient conditions for the Borel map being onto the sequence space, we shall see that on quasianalytic classes, strictly containing the real analytic class, the Borel map is never onto. This will be complemented by some results on the description of the Borel image and a discussion of further intricacies of the quasianalytic setting.

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Ultra-differentiable Analysis concerns sub-algebras of smooth functions with constrained growth of the Taylor series coefficients. Besides its importance in the analysis of partial differential equations the development of this theory was influenced by the classical Whitney extension problem and the composition problem. Both problems followed a path that often meets sub-analytic geometry, and later o-minimal geometry. Of special interest among the ultra-differentiable classes are the quasi-analytic classes which possess a (quasi-)analytic continuation property similar to the real analytic class. This property make these classes interesting from an analytic viewpoint and also suitable for questions of tame geometry. In relation with asymptotic expansions of solutions of analytic ODEs they also naturally appear when dealing with non-oscillation problems of the solutions. Armin Rainer will introduce us to ultra-differentiable classes, in particular to quasi-analytic ones.

algebraic geometrydifferential geometrymetric geometry

Audience: advanced learners

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School of Real Geometry in Fortaleza - ScReGeFor

Series comments: Main aim of School of Real Geometry in Fortaleza is to present current topics in Real Algebraic and Tame Geometries. There will be five courses on topics in Real Geometry. The target audience is primarily Phd Students, Post-docs and young researchers.

All courses and talks will be available online in google meet. The links will be sent to registered participants some days before the event.

More information available on the website sites.google.com/view/scregefor2020/home

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