Weighted inequalities of weak type for the fractional integral operator (Q1381127)

The Riesz potential \(I_\alpha\), \(0<\alpha<n\), is given at \(f\in L^1_{\text{loc}} (\mathbb{R}^n)\) by \(I_\alpha f(x)=\int_{\mathbb{R}^n}| x-y| ^{\alpha-n} dy\). In the paper, the boundedness of \(I_\alpha\) from the weighted Lebesgue space \(L^p(v)\) into the weighted Marcinkiewicz space \(L^{p,\infty}(u)\), that is, the inequality \(\lambda^p u(\{I_\alpha f>\lambda\})\leq C\int_{\mathbb{R}^n}f^pv\) is studied. A characterization of this inequality is known [\textit{E. T. Sawyer}, Trans. Am. Math. Soc. 281, 339-345 (1984; Zbl 0539.42008)], but the criterion is expressed through the operator \(I_\alpha\) itself and thus might be difficult to verify in practice. Further, a criterion for \(I_\alpha: L^p(v)\to L^{q,\infty}(u)\), \(1<p<q<\infty\) is known [\textit{M. Gabidzashvili} and \textit{V. Kokilashvili}, Czech. Akad. Sci. 45, 1-11 (1989), per bibl.], but the method cannot be extended to the case \(p=q\). In the paper under review, a new characterization of the boundedness is given in terms of a modified Riesz potential operator. As a corollary, a relatively simple sufficient condition is obtained.