Hardy's inequality and embeddings in holomorphic Triebel-Lizorkin spaces

Abstract

In this work we study some properties of the holomorphic TriebeI-Lizorkin spaces $H F^{pq}_{s}$, $0 \lt p$, $q \leq \infty$, $s \in \mathbb{R}$, in the unit ball $B$ of $\mathbb{C}^{n}$, motivated by some well-known properties of the Hardy-Sobolev spaces $H^{p}_{s} = H F^{p^{2}}_{s}$, $0 \lt p \lt \infty$.

We show that $\sum_{n \geq 0}|a_{n}|/(n + 1) \lesssim ||\sum_{n \geq 0}a_{n}z^{n}||_{H F^{1 \infty}_{0}}$, which improves the classical Hardy's inequality for holomorphic functions in the Hardy space $H^{1}$ in the disc. Moreover, we give a characterization of the dual of $HF^{1q}_{s}$, which includes the classical result $(H^{1})^{\ast} = \mathrm{BMOA}$. Finally, we prove some embeddings between holomorphic Triebel-Lizorkin and Besov spaces, and we apply them to obtain some trace theorems.