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

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.

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