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

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.