Carleson inequalities on parabolic bergman spaces

摘要

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 σ-th integrable, where σ is the exponent conjugate to p/q. An application to Toeplitz operators is also included.