Localized special John–Nirenberg–Campanato spaces via congruent cubes with applications to boundedness of local Calderón–Zygmund singular integrals and fractional integrals

Let \(p,q\in [1,\infty )\) $p,q\in [1,\infty )$ , s be a nonnegative integer, \(\alpha \in \mathbb {R}\) $\alpha \in \mathbb{R}$ , and \(\mathcal {X}\) $\mathcal{X}$ be \(\mathbb {R}^n\) ${\mathbb{R}}^n$ or a cube \(Q_0\subsetneqq \mathbb {R}^n\) $Q_0⫋{\mathbb{R}}^n$ . In this article, the authors introduce the localized special John–Nirenberg–Campanato spaces via congruent cubes, \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathcal {X})\) $j{n}_{{(p,q,s)}_{\alpha }}^{\text{con}}(\mathcal{X})$ , and show that, when \(p\in (1,\infty )\) $p\in (1,\infty )$ , the predual of \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathcal {X})\) $j{n}_{{(p,q,s)}_{\alpha }}^{\text{con}}(\mathcal{X})$ is a Hardy-kind space \(hk_{(p',q',s)_{\alpha }}^{\textrm{con}}(\mathcal {X})\) $h{k}_{{(p^{\prime },q^{\prime },s)}_{\alpha }}^{\text{con}}(\mathcal{X})$ , where \(\frac{1}{p}+\frac{1}{p'}=1=\frac{1}{q}+\frac{1}{q'}\) $\frac{1}{p}+\frac{1}{p^{\prime }}=1=\frac{1}{q}+\frac{1}{q^{\prime }}$ . As applications, in the case \(\mathcal {X}=\mathbb {R}^n\) $\mathcal{X}={\mathbb{R}}^n$ , the authors obtain the boundedness of local Calderón–Zygmund singular integrals and local fractional integrals on both \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)\) $j{n}_{{(p,q,s)}_{\alpha }}^{\text{con}}({\mathbb{R}}^n)$ and \(hk_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)\) $h{k}_{{(p,q,s)}_{\alpha }}^{\text{con}}({\mathbb{R}}^n)$ . One novelty of this article is to find the appropriate expression of local Calderón–Zygmund singular integrals on \(jn_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)\) $j{n}_{{(p,q,s)}_{\alpha }}^{\text{con}}({\mathbb{R}}^n)$ and the other novelty is that, for the boundedness on \(hk_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)\) $h{k}_{{(p,q,s)}_{\alpha }}^{\text{con}}({\mathbb{R}}^n)$ , the authors use the duality theorem to overcome the essential difficulties caused by the deficiency of both the molecular and the maximal function characterizations of \(hk_{(p,q,s)_{\alpha }}^{\textrm{con}}(\mathbb {R}^n)\) $h{k}_{{(p,q,s)}_{\alpha }}^{\text{con}}({\mathbb{R}}^n)$ .