Iterates of differential operators and vector valued functions on non quasi analytic classes

Resumen

Classes of smooth functions defined by estimates on the growth of the successive iterates of a partial differential operator were introduced by Komatsu in order to study regularity properties of certain partial differential equations. This research was continued until recent days by several authors like Bolley, Camus, Kotake, Langenbruch, Métivier, Narasimhan, Newberger, Rodino, Zanghirati, Zielezny and others. All this work is related to the problem of iterates which consists, roughly speaking, in characterizing the functions in a given class of functions in terms of the behavior of the iterates of a fixed differential operator. In the first part of this thesis we continue the research described above in a more general setting: non quasi analytic classes of ultradifferentiable functions as defined by Braun, Meise and Taylor. The study of these non quasianalytic classes of ultradifferentiable functions has received much attention in recent years due to its applications to the theory of partial differential operators: we refer to the work of Bonet, Braun, Doma' nski, Fernández, Frerick, Galbis, Taylor and Vogt. In Chapter 1 we introduce these classes and give the properties that will be used throughout this memoir. In the second chapter we define non quasianalytic classes with respect to the iterates of a partial differential operator P(D) and study their locally convex properties like completeness and nuclearity. In fact, we prove that theses classes are a complete locally convex space if and only if the operator P(D) is hypoelliptic and that, in this case, such spaces are nuclear. After that, it is proved that these classes verify a Paley-Wiener type theorem. The aim of Chapter 3 is to obtain results concerning the problem of iterates on non quasianalytic classes. We extend results by Newberger, Zielezny, Métivier and Komatsu and give characterizations in order that a non quasianalytic class defined with respect to the iterates of a partial differential operator coincides with a class of ultradifferentiable functions in the sense of Braun, Meise and Taylor. We want to emphasize that the previous literature on iterates of diffe- rential operators only deals with Roumieu classes. However, all the results presented in Chapters 2 and 3 remain true in the Beurling case. In 1990, Langenbruch and Voigt proved that a Fréchet space of distributions which is stable under a single hypoelliptic differential operator is continuously included in the space of smooth functions. In Chapter 4 we introduce ultradifferential operators and give extensions of the result of Langenbruch and Voigt to the ultradifferential setting. The new notion of (w, P(D))-stable Fréchet space imposes an equicontinuity condition on the successive iterates of P(D) and allows us to show the connections of this topic with the problem of iterates. In the second part of this thesis we investigate vector valued functions in a locally convex space. Motivated by previous work by Bonet, Doma ' nski, Komatsu, Kriegl, Michor, Rainer and Schwartz, we introduce in Chapter 5 several notions of vector valued ultradifferentiable function and we prove that every weakly non-quasianalytic ultradifferentiable function with values in a Fréchet space E is topologically (or strongly) ultradifferentiable if and only if the space E satisfies the topological invariant (DN) of Vogt. Thus, we solve a problem posed by Kriegl and Michor in a meeting celebrated in Paderborn (Germany) in November 2008