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

Abstract

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