Periodic Reporting for period 1 - HARHCS (Harmonic Analysis on Real Hypersurfaces in Complex Space) Reporting period: 2019-07-29 to 2021-07-28 Summary of the context and overall objectives of the project The project addressed various open problems at the interface of Complex and Harmonic Analysis, two branches of mathematics characterized by a rich tradition and a vibrant present. More precisely, the main goal has been the study of two mathematical objects naturally attached to real hypersurfaces embedded in complex manifolds: Cauchy-Szegö projections and spectral multipliers of Kohn Laplacians. The former are higher dimensional incarnations of the classical Cauchy integral, and as such they are of central importance in modern complex analysis in several variables. The latter are the natural Laplacians in this context and the study of their spectral multipliers fits into a wider set of problems lying at the heart of contemporary harmonic analysis. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In the thirteen months covered by this report, the Experienced Researcher obtained several novel results, both in the expected directions indicated in the grant proposal, and in unexpected ones. The most notable outcomes of the project are: the discovery of a new extremality property of Lp operator norms of Szegő projections on abstract CR manifolds, in collaboration with B. Lamel (University of Vienna); a proof of L1 unboundedness of Bergman and Cauchy-Szegő projections in great generality; new pointwise estimates for Bergman kernels under appropriate geometric conditions, in collaboration with D. N. Son (University of Vienna); the development of a method to study Bergman projections on domains containing complex manifolds in their boundary; the introduction of a new transfinite construction in several complex variables related to the Diederich-Fornæss index, in collaboration with S. Mongodi (Politecnico di Milano); a new sharp multiplier theorem of Mihlin-Hörmander type for two-dimensional Grushin operators, in collaboration with the Supervisor A. Martini. Moreover, the researcher visited several European research institutions (University of Vienna, Centre de Recerca Matematica in Barcelona, Istituto Nazionale di Alta Matematica in Rome, Isaac Newton Institute in Cambridge, Politecnico di Milano, Università degli Studi di Milano Statale, Università di Milano-Bicocca), gave in-presence and virtual research seminars, and helped organize seminars at the Host institution. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The work performed allowed to expand our understanding of the interaction between analysis and geometry in several complex variables. Fundamental properties like L1 unboundedness of Bergman projections or the existence of off-diagonal singularities of Cauchy-Szegő kernels have been investigated in greater generality than has been done before. New estimates and constructions in several complex variables have been introduced in collaboration with S. Mongodi and D. N. Son, and they are expected to generate new interesting mathematics in the future. Finally, in collaboration with the Supervisor A. Martini a new method to study spectral multipliers of Grushin operators has been introduced, which already proved to be powerful enough to obtain new and very satisfactory results in two real dimensions.