Invertible Toeplitz products, weighted norm inequalities, and A${}_p$ weights

TL;DR

This paper characterizes invertible Toeplitz products on various analytic function spaces using weighted norm inequalities and A_p weights, extending known inequalities to a broader range of p values.

Contribution

It provides new characterizations of invertible Toeplitz products on multiple spaces and extends the reverse Hölder inequality to all p > 1.

Findings

Characterization of invertible Toeplitz products on weighted Bergman, Hardy, and Fock spaces.

Analysis and comparison of A_p weight conditions in these characterizations.

Extension of the reverse Hölder inequality to p > 1.

Abstract

In this paper, we characterize invertible Toeplitz products on a number of Banach spaces of analytic functions, including weighted Bergman space $L^p_a (\mathbb{B}_n, dv_\gamma)$, the Hardy space $H^p(\partial \mathbb{D})$, and the weighted Fock space F${}_\alpha ^p$ for $p > 1$. The common tool in the proofs of our characterizations will be the theory of weighted norm inequalities and A${}_p$ type weights. Moreover, we analyze and compare the various A${}_p$ type conditions that arise in our characterizations. Finally, we extend the "reverse H\"older inequality" of Zheng and Stroethoff \cite{SZ1, SZ2} for $p = 2$ to the general case of $p > 1$.