Banach Journal of Mathematical Analysis

Boundedness of intrinsic Littlewood--Paley functions on Musielak--Orlicz Morrey and Campanato spaces

Yiyu Liang, Eiichi Nakai, Dachun Yang, and Junqiang Zhang

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $\varphi: {\mathbb R^n}\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is nondecreasing, $\varphi(x,0)=0$, $\varphi(x,t)>0$ when $t>0$, $\lim_{t\to\infty}\varphi(x,t)=\infty$ and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty({\mathbb R^n})$ weight uniformly in $t$. Let $\phi: [0,\infty)\to[0,\infty)$ be nondecreasing. In this article, the authors introduce the Musielak--Orlicz Morrey space $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ and obtain the boundedness on $\mathcal M^{\varphi,\phi}(\mathbb R^n)$ of the intrinsic Lusin area function $S_{\alpha}$, the intrinsic $g$-function $g_{\alpha}$, the intrinsic $g_{\lambda}^*$-function $g^\ast_{\lambda, \alpha}$ and their commutators with ${\rm BMO}(\mathbb{R}^n)$ functions, where $\alpha\in(0,1]$, $\lambda\in(\min\{\max\{3,\,p_1\},3+2\alpha/n\},\infty)$ and $p_1$ denotes the uniformly upper type index of $\varphi$. Let $\Phi: [0,\infty)\to[0,\infty)$ be nondecreasing, $\Phi(0)=0$, $\Phi(t)>0$ when $t>0$, and $\lim_{t\to\infty}\Phi(t)=\infty$, $w\in A_\infty(\mathbb R^n)$ and $\phi: (0,\infty)\to(0,\infty)$ be nonincreasing. The authors also introduce the weighted Orlicz--Morrey space $M_w^{\Phi,\phi}(\mathbb R^n)$ and obtain the boundedness on $M_w^{\Phi,\phi}(\mathbb R^n)$ of the aforementioned intrinsic Littlewood--Paley functions and their commutators with ${\rm BMO}(\mathbb{R}^n)$ functions. Finally, for $q\in[1,\infty)$, the boundedness of the aforementioned intrinsic Littlewood--Paley functions on the Musielak-Orlicz Campanato space $\mathcal L^{\varphi,q}(\mathbb R^n)$ is also established.

Article information

Source
Banach J. Math. Anal., Volume 8, Number 1 (2014), 221-268.

Dates
First available in Project Euclid: 14 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1381782098

Digital Object Identifier
doi:10.15352/bjma/1381782098

Mathematical Reviews number (MathSciNet)
MR3161693

Zentralblatt MATH identifier
1280.42016

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems

Keywords
intrinsic Littlewood--Paley function commutator Musielak--Orlicz space Morrey space Campanato space

Citation

Liang, Yiyu; Nakai, Eiichi; Yang, Dachun; Zhang, Junqiang. Boundedness of intrinsic Littlewood--Paley functions on Musielak--Orlicz Morrey and Campanato spaces. Banach J. Math. Anal. 8 (2014), no. 1, 221--268. doi:10.15352/bjma/1381782098. https://projecteuclid.org/euclid.bjma/1381782098


Export citation