Ultradifferentiable regularity of PDEs

One of the fundamental problems in the theory of partial differential equations is the question under which conditions the solutions of a certain differential equation are well-behaved or, to formulate in a more mathematical language, how regular are the solutions. The focus of this project is the investigation of the regularity of solutions of partial differential equations, in particular the hypoellipticity of differential operators, in the ultradifferentiable setting. Here ultradifferentiable classes are families of smooth functions which contain all real-analytic functions and satisfy certain invariant properties. One well-known example of such classes are the Gevrey classes which in particular play an important role in the theory of differential operators with constant coefficients. The problem of ultradifferentiable hypoellipticty can now be formulated as follows: A differential operator P is ultradifferentiable hypoelliptic with respect to a given ultradifferentiable class if every function u is in this class in the case that Pu is an element of that same class. We note that in the smooth case the problem of hypoellipticity goes back to Laurent Schwarz. The main goal of the project is to systematically develop and extend, in the ultradifferentiable setting, tools for the investigation of regularity problems, in particular from microlocal analysis, like pseudodifferential operators or Fourier integral operators. For such tools there is an extensive and well developed theory in the smooth and real-analytic setting and partially also in the Gevrey case. Moreover it is planned to apply these tools to different regularity problems in the theory of partial differential equations and CR geometry and on the way to extend several known smooth regularity results to the ultradifferentiable category, like the famous theorem of Francois Treves on the characterization of the hypoellipticity of differential operators of principal type or the results of Lamel and Mir on the regularity of CR mappings (from the point of view of partial differential equations this problem is equivalent to the regularity of solutions of a system of linear partial differential equations of first order with a nonlinear side condition).