Bergman and Szego projections, Extremal Problems, and Square Functions

2019/09/20 by Timothy Ferguson, Ferguson, Timothy
Mathematics · #30H10 #30H20 #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.1909.09666

openalex publication_date 2019/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study estimates for Hardy space norms of analytic projections. We first find a sufficient condition for the Bergman projection of a function in the unit disc to belong to the Hardy space H^p for 1 < p < ∞. We apply the result to prove a converse to an extension of Ryabykh's theorem about Hardy space regularity for Bergman space extremal functions. We also prove that the H^q norm of the Szegö projection of F^p/2 F^(p/2)-1 cannot be too small if F is analytic, for certain values of p and q. We apply this to show that the best analytic approximation in L^p of a function in both L^p and L^q will also lie in L^q, for certain values of p and q.

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