A generalized setting for ultradifferentiable vectors

In the mathematical subject Analysis smooth functions are functions which have derivatives of arbitrary order. Ultradifferentiable functions are smooth functions, whose derivatives satisfy certain uniform estimates. The most famous class of ultradifferentiable functions is the space of real- analytic functions. Ultradifferentiable functions play a significant role in the study of partial differential equations. A classical result states that all solutions of the Laplace equations are real- analytic. On the other hand the solutions of the heat equation are elements of an ultradifferentiable class, which is strictly larger than the class of real-analytic functions. The problem, which is in the focus of this project, can be formulated in the following way: Assume that a smooth function satisfies the defining estimates of a certain ultradifferentiable class for some derivatives. Can we conclude that all derivatives satisfy these estimates, i.e. is the function an element of the same class? A function as described above is usually called an ultradifferentiable vector in the literature. The study of ultradifferentiable vectors has several connections with and applications to the theory of certain partial differential equations. During the research stay at the Universidade de Sao Paulo it is planned to apply the general theory of ultradifferentiable classes, which has made a lot of progress in recent years, to the study of ultradifferentiable vectors. This allows to unify and generalise older results at the same time. On the other hand, this gives also rise to new interesting questions and problems, which demand further investigation. The research work on these problems will take place in cooperation with the research group of the host Paulo Cordaro at the Universidade de Sao Paulo.

In this project a new framework for the discussion of ultradifferentiable vectors has been devolped using the modern theory of ultradifferentiable classes. Ultradifferentiable classes are algebras of smooth functions defined by estimates on their derivatives. The primary examples of ud classes are the Gevrey classes which play an important role in the theory of PDEs. Gevrey classes have been generalized in different ways, most commonly using weight sequences and weight functions. Although Ultradifferentiable classes defined by weight sequences and weight functions, respectively, are usually different, the theory for both is quite similar. This project is dealing with ultradifferentiable regularity in the theory of PDEs, primarily with the regularity of ultradifferentiable vectors of linear differential operators. Ultradifferentiable vectors associated to a given operator are functions which satisfy the defining estimates of the given ud class a priori only for the iterates of the operator. If for an operator P the ultradifferentiable vectors are all ultradifferentiable functions of the same class U then we say the Theorem of Iterates holds for P and the ud class U. It is well known that the Theorem of Iterates holds for elliptic operators with analytic coefficients and all reasonable ultradifferentiable classes. In the case of ultradifferentiable coefficients one of the results of this project allowed to significantly loosen the conditions on the class necessary for the Theorem of Iterates holds especially in the case of classes given by weight sequences. For non-elliptic case the situation is more complicated. The Theorem of Iterates might or might not hold in the analytic class, e.g. any analytic vector of hypoelliptic operators of principal type is analytic, whereas the Theorem of Iterates can only hold for non-analytic Gevrey classes if the operator is elliptic. In this case one is interested what minimal Gevrey regularity it is possible to show for Gevrey vectors. The new framework for ultradifferentiable vectors developed during this project gives a method to systematically study the loss of regularity for ultradifferentiable vectors in the case of general ultradifferentiable classes. Generalizing theorems of Baouendi and Metivier the main results of the project show a distinct difference between classes given by weight sequences and weight functions. For a certain family of weigth functions it is shown that the Theorem of Iterates holds for hypoelliptic differential operators of principal type with respect to classes associated to weight functions from this family. On the other hand it is proven that a similar statement cannot hold in the case of classes given by weight sequences by extending the methods of Metivier used in the Gevrey case. Further results achieved during this project extend some results on Gevrey hypoellipticity to the ultradifferentiable category.