Bergman Projections, Kernel $p$-Norm Estimates, and Toeplitz Operators with Békollé and Bonami weights

Abstract

In this paper, we establish entirely new $p$-norm estimates for reproducing kernels to characterize the bounded and compact Toeplitz operators $T_{\hat{I}}¼$ acting between weighted Békollé--Bonami Bergman spaces $A_u^p(\mathbb{D})$ and $A_u^q(\mathbb{D})$ for all positive exponents $0<p,q<\infty$. These operator-theoretic properties are completely described in terms of generalized Berezin transforms, averaging functions, and Carleson measures. We introduce two explicit conditions on the weights to ensure the boundedness of the weighted Bergman projection $P_u$, generalizing results from Hilbert spaces to Banach spaces.Our work generalizes the main results of Tong, Li, and Arroussi \cite{TLA} from Hilbert spaces to the more general setting of Banach spaces.