Control of the Bilinear Indicator Cube Testing property

Abstract

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}.