A two weight weak type inequality for fractional integrals

For $1 < p \leqslant q < \infty ,0 < \alpha < n$ and $w(x),\upsilon (x)$ nonnegative weight functions on ${R^n}$ we show that the weak type inequality \[ \int _{\{ {T_\alpha }f > \lambda \} } w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left ( \int |f(x){|^p}\;\upsilon (x)\;dx \right )^{q/p}}\] holds for all $f \geqslant 0$ if and only if \[ \int _Q [{T_\alpha }({\chi _Q}w) (x)]^{p’}\upsilon (x)^{1 - p’} dx \leqslant B\left ( \int _Qw \right )^{p’/q’} < \infty \] for all cubes $Q$ in ${R^n}$. Here ${T_\alpha }$ denotes the fractional integral of order $\alpha ,{T_\alpha }f(x) = \int |x - y{|^{\alpha - n}}f(y) dy$. More generally we can replace ${T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $|x|$.