## Tohoku Mathematical Journal 2010 June SECOND SERIES VOL. 62, NO. 2

 Tohoku Math. J. 62 (2010), 269-286

Title CARLESON INEQUALITIES ON PARABOLIC BERGMAN SPACES

Author Masaharu Nishio, Noriaki Suzuki and Masahiro Yamada

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.