## Tohoku Mathematical Journal

### 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.

#### Article information

Source
Tohoku Math. J. (2), Volume 62, Number 2 (2010), 269-286.

Dates
First available in Project Euclid: 23 June 2010

https://projecteuclid.org/euclid.tmj/1277298649

Digital Object Identifier
doi:10.2748/tmj/1277298649

Mathematical Reviews number (MathSciNet)
MR2663457

Zentralblatt MATH identifier
1204.35013

#### Citation

Nishio, Masaharu; Suzuki, Noriaki; Yamada, Masahiro. Carleson inequalities on parabolic Bergman spaces. Tohoku Math. J. (2) 62 (2010), no. 2, 269--286. doi:10.2748/tmj/1277298649. https://projecteuclid.org/euclid.tmj/1277298649

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