Bergman and Cauchy–Szegő projections

The Bergman and Cauchy–Szegő projections, being orthogonal projections onto the Bergman and Hardy spaces of holomorphic functions, respectively, are two fundamental operators in complex analysis. It is of interest to determine the mapping properties of these operators on Lebesgue spaces. This talk discusses results pertaining to these mapping properties, beginning with the unit disc/ball and progressing to more arbitrary domains in several variables, including weighted \(L^p\) spaces and the connection to Calderón–Zygmund theory, plus some open questions.

Further reading: Forelli–Rudin, “Projections on spaces of holomorphic functions in balls” (Indiana Univ. Math. J., 1974); Bekollé, “Inégalités à poids pour le projecteur de Bergman dans la boule unité de \(\mathbb{C}^n\)” (Studia Math., 1982); Zhu, Spaces of Holomorphic Functions in the Unit Ball (Springer, 2005).