Cauchy and Bergman projection, sharp gradient estimates and certain operator norm equalities

Abstract

We get sharp pointwise estimates for the gradient of Pf, where P is Bergman projection in terms of Lp -norm of function f defined in Bn. Using limiting argument we transfer this result to Cauchy projection on Sn and hence, the optimal gradient estimates of solution of (Formula presented.) -problem, thus extending results from Kalaj, Vujadinović [Norm of the Bergman projection onto the Bloch space. J Oper Theory. 2015;73(1):113–126], Kalaj, Marković [Optimal estimates for the gradient of harmonic functions in the unit disk. Complex Anal Oper Theory. 2013;7:1167–1183], Melentijević [Norm of the Bergman projection onto the Bloch space with M -invariant gradient norm. arXiv 1711.08719[math.CV]]. As corollaries we get the sharp gradient estimate of a function in L2 Hardy and Bergman spaces and exact norms of Cauchy projection acting into the Bloch space equipped with several (quasi)-norms.