ON WEAK TYPE BOUNDS FOR A FRACTIONAL INTEGRAL ASSOCIATED WITH THE BESSEL DIFFERENTIAL OPERATOR

Abstract

In this study we show that $I_{\Omega ,\alpha ,v}$ and $M_{\Omega ,\alpha ,v},$ the fractional integral and maximal operators the generalized shift operator generated by Bessel differential operator respectively, are bounded operators from $L_{1,v}\left( \left\vert x\right\vert ^{\frac{\beta (n+2\left\vert v\right\vert -\alpha )}{n+2\left\vert v\right\vert }}\right. ,$ $\left.\mathbb{R}_{n}^{+}\right) $ to $L_{\frac{n+2\left\vert v\right\vert }{n+2\left\vert v\right\vert -\alpha },\infty }\left( \left\vert x\right\vert ^{\beta }, \mathbb{R}_{n}^{+}\right) $ where $0\lt \alpha 0,...,v_{n}\gt 0,\left\vert v\right\vert =v_{1}+...+v_{n}$and $-1\lt \beta \lt 0.$.