Control of the bilinear indicator cube testing property

We show that the \(\alpha\)-fractional bilinear indicator/cube testing constant \(\mathcal{BICT}_{T^{\alpha }}\left( \sigma ,\omega \right) \equiv \sup_{Q\in \mathcal{P}^{n}}\sup_{E,F\subset Q}\frac{1}{\sqrt{\left\vert Q\right\vert_{\sigma }\left\vert Q\right\vert _{\omega }}}\left\vert \int_{F}T_{\sigma}^{\alpha }\left( \mathbf{1}_{E}\right) \omega \right\vert ,\) defined for any singular integral \(T^{\alpha }\) on \(\mathbf{R}^{n}\) with \(0<\alpha 0\) yields a \(T1\) theorem weights appropriate doubling, i.e. norm \(T^{\alpha}\) cube constants one-tailed \(\mathcal{A}_{2}^{\alpha (without energy assumptions), also corresponding cancellation condition kernel }\), both hold arbitrary Calderon-Zygmund operators }\). We do not know analogous result \(\mathcal{BICT}_{H}\left(\sigma,\omega holds Hilbert transform \(H\) case \(\alpha=0\), but we \(\mathcal{BICT}_{H^{\operatorname{dy}}}\left(\sigma ,\omega\right)\) ,\sigma dyadic \(H^{\operatorname{dy}}\) $\).