Weak type bounds for a class of rough operators with power weights

Abstract:

In this note we show that $T_{\Omega ,\alpha }$ and $M_{\Omega ,\alpha },$ the fractional integral and maximal operators with rough kernel respectively, are bounded operators from $L^{1}(|x|^{\beta (n-\alpha )/n},\mathbb {R}^{n})$ to $L^{n/(n-\alpha ),\infty }(|x|^{\beta },\mathbb {R}^{n}),$ where $0<\alpha <n$ and $-1<\beta <0.$