Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces
Let \(0<p,\theta \leq \infty \) and \(w\) be non-negative measurable function on \(\left( 0,\infty \right) .\) The local and global Morrey-type spaces \(LM_{p\theta ,w}\) and \(GM_{p\theta ,w}\) are defined to be the space of \(f\in L_{p}^{loc}\left(\mathbb{R}^{n}\right) \) with finite quasinorms \[ \begin{multlined} \left\| f\right\| _{LM_{p\theta ,w}}=\left\| w\left( r\right) \left\| f\right\| _{L_{p}\left( B\left( 0,r\right) \right)}\right\| _{L_{\theta }\left( 0,\infty \right) }\text{ and \;}\\ \left\| f\right\| _{GM_{p\theta ,w}}=\underset{x\in\mathbb{R}^{n}}{\sup }\left\| f\left( x+.\right) \right\| _{LM_{p\theta ,w}}\end{multlined} \] respectively. The local and global weak Morrey-type spaces \(LWM_{p\theta ,w}\) and \(GWM_{p\theta ,w}\) are defined similarly. Note that \(LM_{p\infty,1}=GM_{p\infty,1}=L_{p}\left(\mathbb{R}^{n}\right)\). Furthermore, \(GM_{p\infty ,r^{\frac{-\lambda}{p}}}=M_{P,\lambda }\) for \(1\leq p\leq \infty \) and \(0<\lambda <n,\) where \(M_{P,\lambda}\) is the Morrey space. Let \(T\) \ be a Calderon-Zygmund operator. We say that \(T\) \ is genuine Calderon-Zygmund operator if for \(n\geq 2\) there exists \(C_{1},C_{2}>0\) and a rotation \(R\) \ such that \[ K\left( x,y\right) \geq \frac{C_{1}}{\left| x-y\right| ^{n}} \] for all \(x\in\mathbb{R}^{n}\) and \(y\in C_{x},\) where \(K\left( x,y\right) \) is kernel of \(T\), \( C_{x}=x+R\left( C\right) \) and \[ C=\left\{ y=\left( y_{1},y_{2},...,y_{n}\right) =\left( \overset{-}{y},y_{n}\right) \in\mathbb{R}^{n}:y_{n}>C_{2}\left| \overset{-}{y}\right| ,\overset{-}{y}\in\mathbb{R}^{n-1}\right\}. \] In this work, the authors give results on the boundedness of Calderon-Zygmund and genuine Calderon-Zygmund operator on\ \(\;\)the spaces \( LM_{p\theta ,w}\), \(GM_{p\theta ,w}\) and \ \(LWM_{p\theta ,w}\), \( GWM_{p\theta ,w}\). Theorem1, Theorem 2 and Theorem 3 of this paper state that the Calderon-Zygmund operator \(T\) is bounded from \(LM_{p\theta_{1},w_{1}}\) to \(LM_{p\theta _{2},w_{2}}\), from \(GM_{p\theta _{1},w_{1}}\) to \(GM_{p\theta _{2},w_{2}}\) and from \(LM_{p\infty ,w_{1}}\) to \(LM_{p\theta _{2},w_{2}}\), from \(GM_{p\infty ,w_{1}}\) to \(GM_{p\theta _{2},w_{2}}\) under some conditions. In Theorem 3, it is also discussed the necessary and sufficient condition for the boundedness of genuine Calderon-Zygmund operator \(T\) from \(LM_{p\theta _{1},w_{1}}\) to \(LM_{p\theta _{2},w_{2}}\). At the end of this paper the authors work on necessary and sufficient conditions for the boundedness of the Calderon-Zygmund and genuine Calderon-Zygmund operator from \(LM_{p\theta _{1},w_{1}}\) to \(LWM_{p\theta_{2},w_{2}}.\)
scientific article; zbMATH DE number 5656067
scientific article