Operators of type 1,1

This major project concerns a kind of pseudo-differential operators known in the literature as type 1,1-operators. They have been known in modern mathematical analysis since around 1980, when it was shown by Y. Meyer that they play a central role for the treatment of general non-linear partial differential equations. The peculiar properties of type 1,1-operators was further elucidated in important contributions of G. Bourdaud (1983{1988) and L. Hörmander(1985-1989).
However, the principal purpose of the project is to clarify the notion of a type 1,1-operator; i.e. to make the first general definition of these operators (when theory of type 1,1-operators, by consistent use of the definition (from 2008).
On this basis a 30 year old conjecture of C. Parenti and L. Rodino, that no type 1,1-operator can create singularities, has been confirmed. In addition the fact that they sometimes change existing singularites has been shown using Weierstrass' nowhere differentiable function. How they change the supports and spectra of functions has also been described in details (2008). The question which functions these operators may act on has also been amply discussed. As a result of the definition, they are always defined on all temperate smooth functions, i.e. the maximal space (2010). In general such operators are shown to be defined only on functions belonging to specific Lizorkin-Triebel spaces, which are optimal within the framework of Fourier analysis (2005). However, they are always bounded on the functions in Lp with compact spectra (2010). For operators in the self-adjoint subclass (in particular those that fulfil the twisted diagonal condition of Hörmander) it is shown that they always are continuous on the entire space of temperate distributions (2010). The paradifferential 3-term decomposition is generalised (including its representation by infinite series) to operators of type 1,1, with two terms belonging to the self-adjoint subclass; while the `symmetric' term is shown explicitly to cause the domain limitations (2011).
An extensive continuity analysis has also been worked out via Fourier analysis, leading to results within the scales of Hölder-Zygmund spaces and Lp-theory in the scales of Sobolev (Bessel potential) spaces; and more generally in Besov and Lizorkin-Triebel spaces (2011).
Type 1,1-operators have also been shown to enter as a main ingredient in a parametrix construction for boundary value problems for semilinear partial differential equations. In particular their non-creation of singularities have been utilised to carry local regularity improvements over from the data to the solutions (2008).
Eight manuscripts have been worked out, including an approved dissertation for the dr.scient.-degree (DSc); one is a technical report, five are published, two submitted.