Regularity and comparison principles in complex analysis and locally integrable structures - Silva, Vinícius Novelli da (2024)
This thesis deals with three topics, related to complex analysis in several variables, CR geometry and systems of partial differential equations. Firstly, we study a comparison principle between Levi-flat CR structures and systems of real vector fields. In a family of models given by fibre bundles with complex fiber, we compute completely the cohomology of the associated differential complex. We also introduce some compact models (parametrized by a certain number of forms of type (0, 1)) and, in this case, study questions of global hypoellipticity and prove, in several cases, a comparison isomorphism between the cohomology spaces. In the second part, we study (in the one dimensional case) the Fréchet algebra A^\\infty(K) (introduced recently by Cordaro, Della Sala and Lamel, to treat questions about the Borel map). We prove some function-theoretic properties about this algebra (like a localization property, in analogy with the uniform case P(K)) and study the question of determining when this algebra coincide with the full algebra of formal power series with coefficients in C(K). To do this, we introduce as the main tool a generalization of the Cauchy transform. Finally, we study a problem of regularizability of locally integrable structures. Inspired by a recent result of Kossovskiy and Zaitsev, we present a family of structures (of non-CR type) and a sufficent condition that guarantees when such structures are equivalent to real-analytic ones.