Equivalent conditions for bergman space and littlewood-paley type inequalities

Abstract

In this paper we show that the following integrals ∫B |f(z)p(1 - |z|)αdV(z), ∫B |f(z)| p-q|∇ f(z)|q(1 - |z|)α+qdV(z), and ∫B |f(z)|p-q|R f(z)|q(1 -|z|) α+qdV(z), where p > 0, q ∈ [0, p], α ∈ (-1, ∞), and where f is a holomorphic function on the unit ball B in ℂn are comparable. This result confirms a conjecture proposed by the second author at several meetings, for example, at the International twoday meeting on complex, harmonic, and functional analysis and applications, Thessaloniki, December 12 and 13, 2003. Also we generalize the well-known inequality of Littlewood-Paley in the unit ball.