A two weight inequality for the fractional integral when $p=n/\alpha$

Abstract:

Let ${I_\alpha }$ be the fractional integral operator defined as \[ {I_\alpha }f(x) = \int {f(y){{\left | {x - y} \right |}^{\alpha - n}}dy.} \] Given a weight $w$ (resp. $\upsilon$), necessary and sufficient conditions are given for the existence of a nontrivial weight $\upsilon$ (resp. $w$) such that \[ {\left \| {\upsilon {\chi _B}} \right \|_\infty }\frac {1}{{\left | B \right |}}\int _B {\left | {{I_\alpha }f(x) - {m_B}({I_\alpha }f)} \right |} dx \leqslant C{\left ( {\int {{{\left | f \right |}^{n/\alpha }}w} } \right )^{\alpha /n}}\] holds for any ball $B$ such that ${\left \| {v{\chi _B}} \right \|_\infty } > 0$.