Calderón-Zygmund operators on the predual of a Morrey space
Let \(1<p<\infty\), \(1/p+1/p'=1\), and \(\varphi\) be a non-increasing function on \([0, \infty)\) such that \(t^{1/p'}\varphi(t)\) is non-decreasing. A function \(a\) on \({\mathbb R}^n\) is called an \((H^p, \varphi)\)-atom if \(\text{ supp} (a)\subset B(x_0, r)\), a ball of radius \(r\) centered at \(x_0\), \(\int_{{\mathbb R}^n}a(x) dx=0\) and \(\|a\|_{L^p({\mathbb R}^n)}\leq r^{n(1/p-1)}\varphi(r^n)^{-1}.\) Using \((H^p, \varphi)\)-atoms, \textit{J. Alvarez} in [Ryan, John (ed.), Clifford algebras in analysis and related topics. Based on a conference, Fayetteville, AR, USA, April 8-10, 1993. Boca Raton, FL: CRC Press. Studies in Advanced Mathematics. 309-319 (1996; Zbl 0842.42008)] introduced the atomic Hardy space \(H^{p, \varphi}({\mathbb R}^n)\) and proved that the dual space of \(H^{p, \varphi}({\mathbb R}^n)\) is the Morrey space \(L^{p', \varphi}({\mathbb R}^n)\). In this paper, the author introduces the local version, \(h^{p, \varphi}({\mathbb R}^n)\), of the atomic Hardy space \(H^{p, \varphi}({\mathbb R}^n)\) in the sense of Goldberg. Let \(T\) be a \(\delta\)-Calderón-Zygmund operator, \(0<\delta\leq 1\), such that \(T^{\ast}1\in \text{ Lip} _{\varepsilon}({\mathbb R}^n)\), \(0<\varepsilon<1\), and \(1<p\leq n/(n-\varepsilon)\). The author proves that \(T\) is bounded from \(H^{p, \varphi}({\mathbb R}^n)\) to \(h^{p, \varphi}({\mathbb R}^n)\). As an application, he verifies that Calderón's commutator is also bounded from \(H^{p, \varphi}({\mathbb R}^n)\) to \(h^{p, \varphi}({\mathbb R}^n)\), \(1<p\leq 1/(1-\varepsilon)\).
scientific article; zbMATH DE number 1941603