# Control of the bilinear indicator cube testing property

## Authors

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

## Keywords:

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

### Abstract

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 $$\sigma ,\omega$$.
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Articles

2021-09-09

## 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 https://afm.journal.fi/article/view/111181