Carleson inequalities on parabolic Bergman spaces

Abstract

We study Carleson inequalities on parabolic Bergman spaces on the upper half space of the Euclidean space. We say that a positive Borel measure satisfies a $(p,q)$-Carleson inequality if the Carleson inclusion mapping is bounded, that is, $q$-th order parabolic Bergman space is embedded in $p$-th order Lebesgue space with respect to the measure under considering. In a recent paper [6], we estimated the operator norm of the Carleson inclusion mapping for the case $q$ is greater than or equal to $p$. In this paper we deal with the opposite case. When $p$ is greater than $q$, then a measure satisfies a $(p,q)$-Carleson inequality if and only if its averaging function is $\sigma$-th integrable, where $\sigma$ is the exponent conjugate to $p/q$. An application to Toeplitz operators is also included.