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

$$\int_{\{ {T_\alpha }f > \lambda \} }\,w(x)\;dx \leqslant A{\lambda ^{ - q}}{\left( \int \vert f(x){\vert^p}\;\upsilon (x)\;dx \right)^{q/p}}$$

holds for all $f \geqslant 0$ if and only if

$$\int_Q\,[{T_\alpha }({\chi_Q}w)\,(x)]^{p'}\upsilon (x)^{1 - p'}\,dx \leqslant B\left( \int_Qw \right)^{p^{\prime}/q^{\prime}} < \infty$$

for all cubes $Q$ in ${R^n}$. Here ${T_\alpha }$ denotes the fractional integral of order $\alpha ,{T_\alpha }f(x) = \int \vert x - y{\vert^{\alpha - n}}f(y)\,dy$. More generally we can replace ${T_\alpha }$ by any suitable convolution operator with radial kernel decreasing in $\vert x\vert$.