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 $\tilde{T}$ of the Calderón--Zygmund operator $T$ on $J{N}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$, 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 $\tilde{T}$ is bounded on $J{N}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$ if and only if, for any $\gamma \in {\mathbb{Z}}_{+}^n$ with $|\gamma |\le s$, $T^{\ast }(x^{\gamma })=0$, 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 $H{K}_{(p,q,s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$, the predual space of $J{N}_{(p^{\prime },q^{\prime },s{)}_{\alpha }}^{\mathrm{con}}({\mathbb{R}}^n)$ with $\frac{1}{p}+\frac{1}{p^{\prime }}=1=\frac{1}{q}+\frac{1}{q^{\prime }}$, if and only if, for any $\gamma \in {\mathbb{Z}}_{+}^n$ with $|\gamma |\le s$, $T^{\ast }(x^{\gamma })=0$.

47 pages; Submitted

References in corpus (1)

• Integrable fractional mean functions on spaces of homogeneous type