Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 82, pp. 1-17.

Title: Hardy operators and commutators on generalized central function spaces

Author: Le Trung Nghia (Ton Duc Thang Univ., Ho Chi Minh City, Vietnam)

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
$[b,\mathcal{H}^*]$.


Submitted July 2, 2025. Published  August 08, 2025.
Math Subject Classifications: 42B20, 42B35, 42B30, 46A20.
Key Words: Hardy operators; commutator; generalized central function space; central atomic space.