# Revista Matemática Iberoamericana

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**Volume 32, Issue 1, 2016, pp. 79–174**

**DOI: 10.4171/RMI/882**

Published online: 2016-02-29

A two weight theorem for $\alpha$-fractional singular integrals with an energy side condition

Eric T. Sawyer^{[1]}, Chun-Yen Shen

^{[2]}and Ignacio Uriarte-Tuero

^{[3]}(1) McMaster University, Hamilton, Canada

(2) National Central University, Jhongli City, Taoyuan County, Taiwan

(3) Michigan State University, East Lansing, USA

Let $\sigma $ and $\omega $ be locally finite positive Borel measures on $\mathbb{R}^{n}$ with no common point masses, and let $T^{\alpha}$ be a standard $\alpha$-fractional Calderón–Zygmund operator on $\mathbb{R}^{n}$ with $0 \leq \alpha < n$. Furthermore, assume as side conditions the $\mathcal{A}_{2}^{\alpha}$ conditions and certain $\alpha$*-energy conditions*. Then we show that $T^{\alpha}$ is bounded from $L^{2}(\sigma ) $ to $L^{2}( \omega ) $ if the cube testing conditions hold for $T^{\alpha}$ and its dual, and if the weak boundedness property holds for $T^{\alpha}$.

Conversely, if $T^{\alpha}$ is bounded from $L^{2}( \sigma ) $to $L^{2}( \omega )$, then the testing conditions hold, and the weak boundedness condition holds. If the vector of $\alpha $-fractional Riesz transforms $\mathbf{R}_{\sigma }^{\alpha}$ (or more generally a strongly elliptic vector of transforms) is bounded from $L^{2}( \sigma) $ to $L^{2}( \omega ) $, then the $\mathcal{A}_{2}^{\alpha}$ conditions hold. We do not know if our energy conditions are necessary when $n \geq 2$.

The innovations in this higher dimensional setting are the control of functional energy by energy modulo $\mathcal{A}_{2}^{\alpha}$, the necessity of the $\mathcal{A}_{2}^{\alpha}$ conditions for elliptic vectors, the extension of certain one-dimensional arguments to higher dimensions in light of the differing Poisson integrals used in $\mathcal A_2$ and energy conditions, and the treatment of certain complications arising from the Lacey–Wick monotonicity lemma. The main obstacle in higher dimensions is thus identified as the pair of energy conditions.

*Keywords: *Weighted norm inequalities, singular integrals, T1 theorems

Sawyer Eric, Shen Chun-Yen, Uriarte-Tuero Ignacio: A two weight theorem for $\alpha$-fractional singular integrals with an energy side condition. *Rev. Mat. Iberoamericana* 32 (2016), 79-174. doi: 10.4171/RMI/882