Positive Toeplitz Operators from a Harmonic Bergman-Besov space into Another

TL;DR

This paper characterizes positive Toeplitz operators between harmonic Bergman-Besov spaces on the unit ball, providing conditions for boundedness, compactness, and Schatten class membership using Carleson measures, averaging functions, and Berezin transforms.

Contribution

It extends existing results for harmonic weighted Bergman spaces to the broader setting of harmonic Bergman-Besov spaces with full parameter ranges.

Findings

Characterization of bounded and compact Toeplitz operators via Carleson measures.

Criteria for Toeplitz operators to belong to Schatten classes using Berezin transforms.

Extension of known results from weighted Bergman spaces to harmonic Bergman-Besov spaces.

Abstract

We define positive Toeplitz operators between harmonic Bergman-Besov spaces $b^p_\alpha$ on the unit ball of $\mathbb{R}^n$ for the full ranges of parameters $0<p<\infty$, $\alpha\in\mathbb{R}$. We give characterizations of bounded and compact Toeplitz operators taking one harmonic Bergman-Besov space into another in terms of Carleson and vanishing Carleson measures. We also give characterizations for a positive Toeplitz operator on $b^{2}_{\alpha}$ to be a Schatten class operator $S_{p}$ in terms of averaging functions and Berezin transforms for $1\leq p<\infty$, $\alpha\in\mathbb{R}$. Our results extend those known for harmonic weighted Bergman spaces.