Local weighted inequalities for the fractional integral operator (Q2730857)

The author studies a local version of the Riesz potential, given by NEWLINE\[NEWLINE I_{\alpha,B(x_0,R_0)}f(x)=\int_{y\in B(x_0,R_0)}|x-y|^{\alpha-n}f(y) dy, NEWLINE\]NEWLINE where \(x_0\in\mathbb R^n\), \(R_0>0\), \(n\geq 1\) and \(0<\alpha<n\), with \(R_0=\infty\) as a possibility. The main purpose of the paper is to derive criteria for boundedness of this operator from a weighted Lebesgue space \(L^p(v)\) into another such space, \(L^q(u)\), where \(1\leq p\leq q<\infty\). The criteria for this boundedness are formulated in terms of two Hardy-type conditions and one Muckenhoupt-type condition. The sufficient conditions are not quite the same as the necessary ones as they involve a stronger version of the Muckenhoupt-type condition. On the other hand the conditions seem to be more manageable than the known necessary and sufficient ones.