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

Abstract

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 $0\le \alpha <n$. 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 $n\ge 2$.

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.