Boundedness of Intrinsic Littlewood-Paley Functions on Musielak-Orlicz Morrey and Campanato Spaces

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