Control of the bilinear indicator cube testing property


  • Eric T. Sawyer McMaster University, Department of Mathematics and Statistics
  • Ignacio Uriarte-Tuero Michigan State University, Department of Mathematics


Hilbert transform, T1 theorem, two weights, Muckenhoupt conditions, doubling weights, reverse doubling weights, energy conditions, bilinear indicator testing, Bellman function



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 \(\alpha\)-fractional singular integral \(T^{\alpha }\) on \(\mathbf{R}^{n}\) with \(0<\alpha <n\), is controlled by the classical \(\alpha\)-fractional Muckenhoupt constant \(A_{2}^{\alpha }\left( \sigma ,\omega\right)\), provided the product measure \(\sigma \times \omega\) is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding \(2\left(n-\alpha \right)\).

Moreover, this control is sharp within the class of diagonally reverse doubling product measures. In fact, every product measure \(\mu \times \mu\), where \(\mu\) is an Ahlfors-David regular measure \(\mu\) with exponent \(n-\alpha\), has diagonal exponent \(2\left( n-\alpha \right)\) and satisfies \(A_{2}^{\alpha }\left( \mu ,\mu \right)<\infty\) and \(\mathcal{BICT}_{I^{\alpha }}\left( \mu ,\mu \right)=\infty\), which has implications for the \(L^{2}\) trace inequality of the fractional integral \(I^{\alpha}\) on domains with fractional boundary.
When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of \(\mathcal{BICT}_{T^{\alpha }}\) for \(\alpha>0\) yields a \(T1\) 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 \(\mathcal{A}_{2}^{\alpha }\) (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 Calderón-Zygmund operators \(T^{\alpha }\).
We do not know if the analogous result for \(\mathcal{BICT}_{H}\left(\sigma,\omega \right)\) holds for the Hilbert transform \(H\) in case \(\alpha=0\), but we show that \(\mathcal{BICT}_{H^{\operatorname{dy}}}\left(\sigma ,\omega\right)\) is not controlled by the Muckenhoupt condition \(\mathcal{A}_{2}^{\alpha }\left( \omega ,\sigma \right)\) for the dyadic Hilbert transform \(H^{\operatorname{dy}}\) and doubling weights \(\sigma ,\omega $\).



How to Cite

Sawyer, E. T., & Uriarte-Tuero, I. (2021). Control of the bilinear indicator cube testing property. Annales Fennici Mathematici, 46(2), 1105–1122. Retrieved from