Musielak-Orlicz Campanato Spaces and Applications

Let $\varphi: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce the Musielak-Orlicz Campanato space ${\mathcal L}_{\varphi,q,s}({\mathbb R}^n)$ and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space $H^{\varphi}(\mathbb R^n)$, which in the case when $q=1$ and $s=0$ was obtained by L. D. Ky [arXiv: 1105.0486]. The authors also establish a John-Nirenberg inequality for functions in ${\mathcal L}_{\varphi,1,s}({\mathbb R}^n)$ and, as an application, the authors also obtain several equivalent characterizations of ${\mathcal L}_{\varphi,q,s}({\mathbb R}^n)$, which, in return, further induce the $\varphi$-Carleson measure characterization of ${\mathcal L}_{\varphi,1,s}({\mathbb R}^n)$.