Localization and Toeplitz operators on doubling Fock spaces

Chunxu Xu and Hong Zou

Vol. 344 (2026), No. 2, 449–474

DOI: 10.2140/pjm.2026.344.449

Abstract

Let $F_{\phi }^p$ be the weighted Fock space over the complex plane, where $\phi$ is subharmonic and $△\phi dA$ is a doubling measure. Firstly we introduce the concept of weakly localized operators from $F_{\phi }^p$ to $F_{\phi }^q$ for $1\le p,q<\infty$. Then we show that weakly localized operators form an algebra and characterize the compactness of weakly localized operators from one doubling Fock space to another. As applications, the boundedness and compactness of Toeplitz operators with ${\mathrm{BMO}<!--FUNCTION\; APPLICATION-->}^s$ symbols from $F_{\phi }^p$ to $F_{\phi }^q$ for $1<p\le q<\infty$ and $p=q=1$, and with ${\mathrm{IMO}<!--FUNCTION\; APPLICATION-->}^s$ symbols from $F_{\phi }^p$ to $F_{\phi }^q$ for $1\le q<p<\infty$, are considered.

Keywords

doubling Fock space, Toeplitz operator, $\mathrm{BMO}^s$, weakly localized operator, $\mathrm{IMO}^s$

Mathematical Subject Classification

Primary: 47B35, 47B47

Secondary: 30H20

Milestones

Received: 20 January 2025

Revised: 16 June 2026

Accepted: 23 June 2026

Published: 8 September 2026