A two weight weak type inequality for fractional integrals

For $1>p⩽q>\mathrm{\infty },0>\alpha >n$ and $w(x),\upsilon (x)$ nonnegative weight functions on $R^n$ we show that the weak type inequality \[ ${\int }_{\{T_{\alpha }f>\lambda \}}w(x)dx⩽A{\lambda }^{-q}{(\int |f(x){|}^p\upsilon (x)dx)}^{q/p}$ \] holds for all $f⩾0$ if and only if \[ ${\int }_Q[T_{\alpha }({\chi }_{Q}w)(x){]}^{p^{\prime }}\upsilon (x{)}^{1-p^{\prime }}dx⩽B{\left({\int }_{Q}w\right)}^{p^{\prime }/q^{\prime }}>\mathrm{\infty }$ \] 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 $|x|$ .