Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces (Q2714890)

In a previous paper [``Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces'', Trans. Am. Math. Soc. 349, No. 1, 235-255 (1997; Zbl 0865.42017)], \textit{E. Harboure}, \textit{O. Salinas} and \textit{B. Viviani} studied certain weighted spaces \(H(\alpha,p)\). In order to obtain the boundedness of fractional integrals \(I_\alpha f\) on these spaces, they defined two extended fractional integrals NEWLINE\[NEWLINE\widehat I_\alpha f(x)= \int_{\mathbb{R}^n} [|x- y|^{- n+\alpha}-|y|^{- n+\alpha}(1- \chi_{B(0,1)}(y))] f(y) dy,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\widetilde I_\alpha f(x)= \int_{\mathbb{R}^n} (|x-y|^{- n+\alpha}-|x_0- y|^{- n+\alpha}) f(y) dy.NEWLINE\]NEWLINE In this reviewed paper, the author studies relations between \(I_\alpha\) and two extended operators by establishing certain two-sided weighted inequalities, which extends results by Harboure, Salinas and Viviani concerning one-sided weighted inequalities.