We show that the { \alpha}-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure { \sigma} x { \omega} is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding 2(n-{ \alpha}). Moreover, this control is sharp within the class of diagonally reverse doubling product measures. When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of BICT_{ T^{ { \alpha}}} for { \alpha}>0 yields a two weight T1 theorem for doubling weights with appropriate diagonal reverse doubling, i.e. the norm inequality for T^{ { \alpha}} is controlled by cube testing constants and the { \alpha}-fractional one-tailed Muckenhoupt constants (without any energy assumptions), and also yields a corresponding cancellation condition theorem for the kernel of T^{ { \alpha}}, both of which hold for arbitrary { \alpha}-fractional Calder\'on-Zygmund operators T^{ { \alpha}}. We do not know if the analogous result for BICT_{ H}({ \sigma},{ \omega}) holds for the Hilbert transform H in case { \alpha}=0, but we show that BICT_{ H^{ dy}}({ \sigma},{ \omega}) is not controlled by the Muckenhoupt condition for the dyadic Hilbert transform H^{ dy} and doubling weights { \sigma},{ \omega}.
Categories: math.CA