Two-weight norm inequalities for the rough fractional integrals
Suppose that \(0<\alpha<n\), \(\Omega(x)\) is a function homogeneous of degree zero in \({\mathbb R}^n\) and its restriction to the unit sphere \(S^{n-1}\) is in \(L^s(S^{n-1})\) for some \(s>1\). Define the fractional integral operator \(T_{\Omega ,\alpha }\) and the fractional maximal operator \(M_{\Omega ,\alpha}\), respectively, by \[ T_{\Omega ,\alpha }f(x)=\int_{{\mathbb R}^n} \frac {\Omega (x-y)}{|x-y|^{n-\alpha}}f(y) dy \] and \[ M_{\Omega ,\alpha }f(x)=\sup_{r>0} \frac 1{r^{n-\alpha }} \int_{|x-y|<r} |\Omega (x-y)||f(y)|dy. \] In this paper, the authors establish the weighted \((L^p,L^q)\)-boundedness of \(T_{\Omega ,\alpha }\) and \(M_{\Omega ,\alpha}\) with two different weights.
scientific article; zbMATH DE number 1751242