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

Abstract

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

47pages, Banach J. Math. Anal. (to appear)

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• On sharp aperture-weighted estimates for square functions