A Good-λ Lemma, Two Weight T1 Theorems Without Weak Boundedness, and a Two Weight Accretive Global Tb Theorem

Let σ and ω be locally finite positive Borel measures on $$\mathbb{R}^{n}$$, let Tα be a standard α-fractional Calderón-Zygmund operator on $$\mathbb{R}^{n}$$ with 0 ≤ α < n, and assume as side conditions the $$\mathcal{A}_{2}^{\alpha }$$ conditions, punctured A2α conditions, and certain α -energy conditions. Then the weak boundedness property associated with the operator Tα and the weight pair $$\left (\sigma,\omega \right )$$, is ‘good-λ’ controlled by the testing conditions and the Muckenhoupt and energy conditions. As a consequence, assuming the side conditions, we can eliminate the weak boundedness property from Theorem 1 of Sawyer et al. (A two weight fractional singular integral theorem with side conditions, energy and k-energy dispersed. arXiv:1603.04332v2) to obtain that Tα is bounded from $$L^{2}\left (\sigma \right )$$ to $$L^{2}\left (\omega \right )$$ if and only if the testing conditions hold for Tα and its dual. As a corollary we give a simple derivation of a two weight accretive global Tb theorem from a related T1 theorem. The role of two different parameterizations of the family of dyadic grids, by scale and by translation, is highlighted in simultaneously exploiting both goodness and NTV surgery with families of grids that are common to both measures.