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 $\mathbb C^n$ with smooth boundary. In this paper, we prove the following inequality: \[ \left (\int _0^{\delta _0}\mathcal M_q^\lambda (f;t)~ t^{\lambda n(1/p-1/q)-1}dt\right )^{1/\lambda }\leq C_{p,q}\|f\|_{p,0}, \] where $1<p<q<\infty , f\in H^p(D)$, and $p\leq \lambda <\infty$. 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.