Estimates on the mean growth of $H^p$ functions in convex domains of finite type

Abstract

Let D be a bounded convex domain of finite type in C n with smooth boundary. In this paper, we prove the following inequality: (∫ δ0 0 M λ q (f;t)t λn(1/p-1/q-1 dt) 1/λ ≤ C p,q ∥f∥ p,0, where 1 < p < q < ∞, f ∈ H P (D), and p < A < ∞. This is a generalization of some classical result of Hardy-Littlewood for the case of the unit disc. Using this inequality, we can embed the H P space into a weighted Bergman space in a convex domain of finite type.