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Musielak-Orlicz Campanato Spaces and Applications

Let $φ: \mathbb R^n\times [0,\infty)\to[0,\infty)$ be such that $φ(x,\cdot)$ is an Orlicz function and $φ(\cdot,t)$ is a Muckenhoupt $A_\infty(\mathbb R^n)$ weight uniformly in $t$. In this article, the authors introduce the Musielak-Orlicz Campanato space ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$ and, as an application, prove that some of them is the dual space of the Musielak-Orlicz Hardy space $H^φ(\mathbb R^n)$, 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 ${\mathcal L}_{φ,1,s}({\mathbb R}^n)$ and, as an application, the authors also obtain several equivalent characterizations of ${\mathcal L}_{φ,q,s}({\mathbb R}^n)$, which, in return, further induce the $φ$-Carleson measure characterization of ${\mathcal L}_{φ,1,s}({\mathbb R}^n)$.