Toeplitz operators and Carleson measures in strongly pseudoconvex domains

TL;DR

This paper investigates the conditions under which Toeplitz operators map between Bergman spaces on strongly pseudoconvex domains, providing geometric characterizations of Carleson measures using Kobayashi geometry.

Contribution

It offers sharp, generalized criteria for Toeplitz operator boundedness and introduces geometric characterizations of Carleson measures in complex domains.

Findings

Sharp conditions for Toeplitz operator mapping properties

Geometric characterization of Carleson measures

Extension of results to strongly pseudoconvex domains

Abstract

We study mapping properties of Toeplitz operators associated to a finite positive Borel measure on a bounded strongly pseudoconvex domain D in n complex variables. In particular, we give sharp conditions on the measure ensuring that the associated Toeplitz operator maps the Bergman space A^p(D) into A^r(D) with r>p, generalizing and making more precise results by Cuckovic and McNeal. To do so, we give a geometric characterization of Carleson measures and of vanishing Carleson measures of weighted Bergman spaces in terms of the intrinsic Kobayashi geometry of the domain, generalizing to this setting results obtained by Kaptanoglu for the unit ball.