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 $α\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calderón--Zygmund operator $T$ on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, the special John--Nirenberg--Campanato space via congruent cubes, which coincides with the Campanato space $\mathcal{C}_{α,q,s}(\mathbb{R}^n)$ when $p=\infty$. Then the authors prove that $\widetilde T$ is bounded on $JN_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$ if and only if, for any $γ\in\mathbb{Z}_+^n$ with $|γ|\leq s$, $T^*(x^γ)=0$, 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 $HK_{(p,q,s)_α}^{\mathrm{con}}(\mathbb{R}^n)$, the predual space of $JN_{(p',q',s)_α}^{\mathrm{con}}(\mathbb{R}^n)$ with $\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}$, if and only if, for any $γ\in\mathbb{Z}_+^n$ with $|γ|\leq s$, $T^*(x^γ)=0$.