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 Calderón–Zygmund operator T on \(JN_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\), the special John–Nirenberg–Campanato space via congruent cubes, which when \(p=\infty \) coincides with the Campanato space \({\mathcal {C}}_{\alpha ,q,s}({\mathbb {R}}^n)\). 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 \(|\gamma |\le s\), \(T^*(x^{\gamma })=0\), 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 \(|\gamma |\le s\), \(T^*(x^{\gamma })=0\). The main interesting integrands in the latter boundedness are that, to overcome the difficulty caused by that \(\Vert \cdot \Vert _{HK_{(p,q,s)_{\alpha }}^{\mathrm {con}}({\mathbb {R}}^n)}\) is no longer concave, the authors first find an equivalent norm of \(\Vert \cdot \Vert _{HK_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)}\), and then establish a criterion for the boundedness of linear operators on \(HK_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\) via introducing molecules of \(HK_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\), using the boundedness of \({\widetilde{T}}\) on \(JN_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\), and skillfully applying the dual relation \((HK_{(p,q,s)_{\alpha }}^{\mathrm {con}}({\mathbb {R}}^n))^* =JN_{(p',q',s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\).

设\(p\in [1,\infty ]\) , \(q\in (1,\infty )\) , \(s\in {\mathbb {Z}}_+:={\mathbb {N }}\cup \{0\}\)和\(\alpha \in {\mathbb {R}}\)。在这篇文章中，作者引入合理版本\（{\ widetilde【T}} \）的卡尔德-Zygmund算子的Ť上\（JN _ {（P，Q，S）_ \阿尔法} ^ {\ mathrm {CON }}({\mathbb {R}}^n)\)，通过全等立方体的特殊约翰-尼伦伯格-坎帕纳托空间，当\(p=\infty \)与坎帕纳托空间\({\mathcal {C }}_{\alpha ,q,s}({\mathbb {R}}^n)\)。然后作者证明\({\widetilde{T}}\)有界于\(JN_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\)当且仅当，对于任何\(\gamma \in { \mathbb {Z}}_+^n\)与\(|\gamma |\le s\)，\(T^*(x^{\gamma })=0\)，这是一个众所周知的假设. 为此，作者找到了这个假设的等效版本。此外，作者表明T可以扩展到 Hardy 类空间上的唯一连续线性算子\(HK_{(p,q,s)_{\alpha }}^{\mathrm {con}}({\ mathbb {R}}^n)\)，\(JN_{(p',q',s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n )\)与\(\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}\)，如果和仅当，对于任何\(\gamma \in {\mathbb {Z}}_+^n\)与\(|\gamma |\le s\)，\(T^*(x^{\gamma })=0\)。后有界中主要有趣的被积函数是，为了克服由\(\Vert \cdot \Vert _{HK_{(p,q,s)_{\alpha }}^{\mathrm {con} }({\mathbb {R}}^n)}\)不再是凹的，作者首先找到了\(\Vert \cdot \Vert _{HK_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)}\)，然后在\(HK_{(p,q,s)_\alpha上建立线性算子有界的判据}^{\mathrm {con}}({\mathbb {R}}^n)\)通过引入\(HK_{(p,q,s)_\alpha }^{\mathrm {con}}( {\mathbb {R}}^n)\)，使用有界\({\widetilde{T}}\)在\(JN_{(p,q,s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\) 上，和巧妙地应用对偶关系\((HK_{(p,q,s)_{\alpha }}^{\mathrm {con}}({\mathbb {R}}^n))^* =JN_{(p ',q',s)_\alpha }^{\mathrm {con}}({\mathbb {R}}^n)\)。