Hardy operators and commutators on generalized central function spaces

Abstract

In this article, we study the boundedness of operators of Hardy type on generalized central function spaces, such as the generalized central Hardy space $\mathbf{HA}^{p,r}_\varphi(\mathbb{R}^n)$, the generalized central Morrey space $\dot{\mathbf{M}}^{p,r}_\varphi (\mathbb{R}^n)$, and the generalized central Campanato space $\dot{{\rm CMO}}^{p,r}_\varphi (\mathbb{R}^n)$, with $p\in(1,\infty)$, and $\varphi(t):(0,\infty)\to (0,\infty)$. We first show that $\mathbf{HA}^{p',r'}_\varphi (\mathbb{R}^n)$ is the predual of $\dot{{\rm CMO}}^{p,r}_\varphi (\mathbb{R}^n)$. After that, we investigate the boundedness of operators of Hardy type on those spaces. By duality, we obtain the boundedness characterization of function $b\in \dot{{\rm CMO}}^{p,r}_\varphi (\mathbb{R}^n)$ via the $\dot{\textbf{M}}^{p,r}_\varphi (\mathbb{R}^n)$-boundedness of commutator $[b,\mathcal{H}^*]$.

Keywords

• hardy operators
• commutator
• generalized central function space
• central atomic space