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

(Received April 1, 2009)
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.

2000 Mathematics Subject Classification. Primary 35K05; Secondary 26D10, 31B10.
Key words and phrases. Carleson measure, Toeplitz operator, parabolic operator of fractional order, Bergman space.

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