Regularity of the solutions for PDE and perturbation problem

A natural question in the theory of partial differential equation of second order with analytic coefficients is the analytic regularity of the solutions in local as well as in global setting. A strongly related problem concerns the conditions in order that such an operator preserves its regularity properties if it is perturbed by adding an operator of lower order. The project is devoted to investigate such problems. In 1967 Hörmander characterized the smoothness in the case of real coefficients. Later on it was shown how the Hörmander condition is not enough to ensure that the solutions are analytic functions. Although of considerable prominence, the progresses done to date are not yet able to provide a clear framework on the possible sufficient conditions in order that the solutions may have analytic regularity. Our research project fits in this context. The first goal will be to introduce, study and compare two new models, rich of properties. The purpose is to highlighting the major differences that distinguish them. In the same way we want to approach the problem of the global regularity for operators with real analytic coefficients defined on the torus with the main purpose to test the Treves` conjecture in this setting. One of the main tool will be the use of suitable a priori estimates. To obtain the results we also want to develop the knowledge concerning the so called Green operator associated to the operator. The subject is already studied in the literature but we need a fresh point of view more suitable for the degenerate case. A comprehension of the properties of the Green operator will allow us to solve the perturbation problem. As it was shown in a recent work by Cordaro and the applicant, the Green operator plays an important role to understand the regularity and the perturbation problem. In the case when the coefficients of the operators are complex-valued function the situation changes radically. The Hörmander theory cannot be extended to this setting as was recently shown. To understand the sufficient conditions that ensure the smoothness of the solutions we will start, following the ideas of Kohn, focusing on a particular class of operators with the purpose to develop strategies able to investigate the general case. To do this we will take advantage of the spectral property of the operator to construct a sort of ``inverse of the operator.

The aim of the project was to investigate, for an operator P, which is a sum of squares of real analytic vector fields, two main problems. The first is the study of the singularities of the solutions of P in the local/global setting. The second concerns the following perturbation property: assume that P is (global-)analytic-hypoelliptic ((G-)AH), and Q is a pseudodifferential operator of order less than the subelliptic index of P, is P+Q still ((G-)AH)? The hypoellipticity was characterised by Hörmander. The Hörmander condition is not enough to ensure the property (AH), as shown by the Baouendi-Goulaouic. In 1999 Treves introduced the Poisson-stratification, and formulated a conjecture on the (global-)analytic hypoellipticity. In 2016 Albano, Bove and Mughetti (ABM) produced the first model which is not consistent with the local Treves conjecture reopening the question concerning the identification of the sufficient conditions so that P is (AH). Our research project fits in this context. We analysed a class of operators that generalise the (ABM) model studying an ``intermediate" situation: we proved an analyticity result that is local in some variables and global in others. This showed that the (ABM) model is (GAH) and that some aspects of the global Treves conjecture are solid. We focused on the possibility to construct a model not consistent with the global Treves conjecture. We was not able to do it but several progress were done. We produced a precise study of the regularity of the solutions for a generalisation of a significant model introduced by Metivier in 1982 in two variable. The optimal regularity obtained is in accordance with that predicted by Bove and Tartakoff. The result obtained allowed to begin to investigate the optimal regularity for some class of operator in three or more variables, work in progress with Bove. Concerning the perturbation problem in a work with Bove a couple of results was obtained: the first on the minimal regularity related to the subelliptic index, the second concerning the analytic hypoellipticity for a class of operator satisfies to some conditions that ensure the (AH). These results show the link of the perturbation problem with the first main problem and with the characterisation of the regularity of a function via power of the operator P. In this direction with M. Derridj we gain via the FBI-technique a result concerning the minimal microlocal regularity of analytic vectors for P thus providing a microlocal version, of a result due to Derridj. The work with Derridj carry on in order to extend these results to other interesting operators. The possibility to interact with Prof. B. Lamel was a great opportunity to develop and advances knowledge, and I have contributed much of my expertise to the group.