Boundedness of Calderón--Zygmund Operators on Special John--Nirenberg--Campanato and Hardy-Type Spaces via Congruent Cubes

Abstract: Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}+:=\mathbb{N}\cup{0}$, and $\alpha\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calder\'on--Zygmund operator $T$ on $JN{(p,q,s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$, the special John--Nirenberg--Campanato space via congruent cubes, which coincides with the Campanato space $\mathcal{C}{\alpha,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then the authors prove that $\widetilde T$ is bounded on $JN_{(p,q,s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any $\gamma\in\mathbb{Z}+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$, which is a well-known assumption. To this end, the authors find an equivalent version of this assumption. Moreover, the authors show that $T$ can be extended to a unique continuous linear operator on the Hardy-kind space $HK_{(p,q,s){\alpha}}^{\mathrm{con}}(\mathbb{R}^n)$, the predual space of $JN{(p',q',s)\alpha}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, if and only if, for any $\gamma\in\mathbb{Z}+^n$ with $|\gamma|\leq s$, $T^*(x^{\gamma})=0$.