## Tokyo Journal of Mathematics

### Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function and Its Commutator on Non-homogeneous Generalized Morrey Spaces

#### Abstract

Let $\mu$ be a non-negative Radon measure on $\mathbb{R}^{d}$ which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function is bounded on the generalized Morrey space $\mathcal{L}^{p,\phi}(\mu)$, and also obtain that the commutator $g^{\ast}_{\lambda,\mu,b}$ generated by the Littlewood-Paley function $g^{\ast}_{\lambda,\mu}$ and the regular bounded mean oscillation space $(=$RBMO$)$, which is due to X. Tolsa, is bounded on $\mathcal{L}^{p,\phi}(\mu)$. As a corollary, the authors prove that the commutator $g^{\ast}_{\lambda,\mu,b}$ is bounded on the Morrey space $\mathcal{M}^{p}_{q}(\mu)$ defined by Sawano and Tanaka when we take $\phi(t)=t^{1-\frac{p}{q}}$ with $1<p<q<\infty$.

#### Article information

Source
Tokyo J. Math., Volume 41, Number 2 (2018), 617-626.

Dates
First available in Project Euclid: 20 November 2017

https://projecteuclid.org/euclid.tjm/1511221573

Mathematical Reviews number (MathSciNet)
MR3908814

Zentralblatt MATH identifier
07053496

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis

#### Citation

WANG, Miaomiao; MA, Shaoxian; LU, Guanghui. Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function and Its Commutator on Non-homogeneous Generalized Morrey Spaces. Tokyo J. Math. 41 (2018), no. 2, 617--626. https://projecteuclid.org/euclid.tjm/1511221573

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