Restricted Testing for Positive Operators

We prove that for certain positive operators T, such as the Hardy–Littlewood maximal function and fractional integrals, there is a constant D>1$$D>1$$, depending only on the dimension n, such that the two weight norm inequality ∫RnTfσ2dω≤C∫Rnf2dσ$$\begin{aligned} \int _{{\mathbb {R}}^{n}}T\left( f\sigma \right) ^{2}\mathrm{{d}}\omega \le C\int _{{\mathbb {R}}^{n}}f^{2}\mathrm{{d}}\sigma \end{aligned}$$holds for all f≥0$$f\ge 0$$if and only if …