Control of the Bilinear Indicator Cube Testing property

Abstract

We show that the α-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure σ x ω is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding 2(n-α). 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^{α}} for α>0 yields a two weight T1 theorem for doubling weights with appropriate diagonal reverse doubling, i.e. the norm inequality for T^{α} is controlled by cube testing constants and the α-fractional one-tailed Muckenhoupt constants (without any energy assumptions), and also yields a corresponding cancellation condition theorem for the kernel of T^{α}, both of which hold for arbitrary α-fractional Calderón-Zygmund operators T^{α}. We do not know if the analogous result for BICT_{H}(σ,ω) holds for the Hilbert transform H in case α=0, but we show that BICT_{H^{dy}}(σ,ω) is not controlled by the Muckenhoupt condition for the dyadic Hilbert transform H^{dy} and doubling weights σ,ω.

15 pages