(A₂)-conditions and Carleson inequalities in Bergman spaces

Takahiko Nakazi and Masahiro Yamada

Vol. 173 (1996), No. 1, 151–171

DOI: 10.2140/pjm.1996.173.151

Abstract

Let ν and μ be finite positive measures on the open unit disk D. We say that ν and μ satisfy the (ν,μ)-Carleson inequality, if there is a constant C > 0 such that

for all analytic polynomials f. In this paper, we study the necessary and sufficient condition for the (ν,μ)-Carleson inequality. We establish it when ν or μ is an absolutely continuous measure with respect to the Lebesgue area mesure which satisfy the (A₂)-condition. Moreover, many concrete examples of such measures are given.

Mathematical Subject Classification 2000

Primary: 46E20

Secondary: 30D99, 30E20

Milestones

Received: 1 July 1993

Revised: 1 June 1994

Published: 1 March 1996