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

Abstract

Let σ and ω be locally finite positive Borel measures on $\mathbb{R}^{n}$, let T ^α 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 A ₂ ^α conditions, and certain α -energy conditions. Then the weak boundedness property associated with the operator T ^α 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 T ^α is bounded from $L^{2}\left (\sigma \right )$ to $L^{2}\left (\omega \right )$ if and only if the testing conditions hold for T ^α 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.

This paper is dedicated to Dick Wheeden on the occasion of his retirement from Rutgers University, and for all of his fundamental contributions to the theory of weighted inequalities, in particular for the beautiful paper of Hunt, Muckenhoupt and Wheeden that started it all back in 1973.

Appendix

We assume notation as above. Define the bilinear form

$$\displaystyle{ \mathcal{B}\left (\,f,g\right ) \equiv \left \langle T_{\sigma }^{\alpha }f,g\right \rangle _{\omega },\ \ \ \ \ f \in L^{2}\left (\sigma \right ),g \in L^{2}\left (\omega \right ), }$$

restricted to functions f and g of compact support and mean zero. For each dyadic grid $\mathcal{D}$ we then have

$$\displaystyle{ \mathcal{B}\left (\,f,g\right ) =\sum _{I,J\in \mathcal{D}\text{ }}\left \langle T_{\sigma }^{\alpha } \bigtriangleup _{I}^{\sigma }f,\bigtriangleup _{ J}^{\omega }g\right \rangle _{\omega }. }$$

Now define the bilinear forms

$$\displaystyle{ \mathcal{C}_{\mathcal{D}}\left (\,f,g\right ) =\sum _{I,J\in \mathcal{D}:\ I\text{ and }J\text{ are }\mathbf{r}\text{-close }}\left \langle T_{\sigma }^{\alpha } \bigtriangleup _{I}^{\sigma }f,\bigtriangleup _{ J}^{\omega }g\right \rangle _{\omega },\ \ \ \ \ f \in L^{2}\left (\sigma \right ),g \in L^{2}\left (\omega \right ). }$$

Thus the form $\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )$ sums over those pairs of cubes in the grid $\mathcal{D}$ that are close in both scale and position, these being the only pairs where the need for a weak boundedness property traditionally arises. We also consider the subbilinear form

$$\displaystyle{ \mathcal{S}_{\mathcal{D}}\left (\,f,g\right ) =\sum _{I,J\in \mathcal{D}:\ I\text{ and }J\text{ are }\mathbf{r}\text{-close }}\left \vert \left \langle T_{\sigma }^{\alpha } \bigtriangleup _{I}^{\sigma }f,\bigtriangleup _{ J}^{\omega }g\right \rangle _{\omega }\right \vert,\ \ \ \ \ f \in L^{2}\left (\sigma \right ),g \in L^{2}\left (\omega \right ), }$$

which dominates $\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )$, i.e. $\left \vert \mathcal{C}_{\mathcal{D}}\left (\,f,g\right )\right \vert \leq \mathcal{S}_{\mathcal{D}}\left (\,f,g\right )$ for all $f \in L^{2}\left (\sigma \right ),g \in L^{2}\left (\omega \right )$. The main results above can be organized into the following two part theorem.

Theorem 5

With notation as above, we have:

• (1) For f and g of compact support and mean zero,

  $$\displaystyle\begin{array}{rcl} & & \mathbb{E}_{\Omega }\left \vert \mathcal{B}\left (\,f,g\right ) -\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )\right \vert {}\\ & \leq & C_{\alpha }\left (\sqrt{\mathfrak{A}_{2 }^{\alpha }} + \mathfrak{T}_{T^{\alpha }} + \mathfrak{T}_{T^{\alpha }}^{{\ast}} + \mathcal{E}_{\alpha }^{\mathop{\mathrm{strong}}} + \mathcal{E}_{\alpha }^{\mathop{\mathrm{strong}},{\ast}} + 2^{-\varepsilon \mathbf{r}}\mathfrak{N}_{ T^{\alpha }}\right )\left \Vert \,f\right \Vert _{L^{2}\left (\sigma \right )}\left \Vert g\right \Vert _{L^{2}\left (\omega \right )} {}\\ & & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +C_{\alpha }\mathbb{E}_{\Omega }\mathcal{S}_{\mathcal{D}}\left (\,f,g\right ). {}\\ \end{array}$$

• (2) For f and g of compact support and mean zero, and for $0 <\lambda <\frac{1} {2}$ ,

  $$\displaystyle{ \mathbb{E}_{\Omega }\mathcal{S}_{\mathcal{D}}\left (\,f,g\right ) \leq C_{\alpha }\left (\frac{1} {\lambda } \sqrt{\mathfrak{A}_{2 }^{\alpha }} + \mathfrak{T}_{T^{\alpha }} + \mathfrak{T}_{T^{\alpha }}^{{\ast}} + \root{4}\of{\lambda }\mathfrak{N}_{ T^{\alpha }}\right )\left \Vert \,f\right \Vert _{L^{2}\left (\sigma \right )}\left \Vert g\right \Vert _{L^{2}\left (\omega \right )}. }$$

The reason for emphasizing the two estimates in this way, is that a different proof strategy might produce a different bound for $\mathbb{E}_{\Omega }\left \vert \mathcal{B}\left (\,f,g\right ) -\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )\right \vert$, which can then be combined with the bound for $\mathbb{E}_{\Omega }\mathcal{S}_{\mathcal{D}}\left (\,f,g\right )$ to control $\left \vert \mathcal{B}\left (\,f,g\right )\right \vert$. Note also that the term $C_{\alpha }\mathbb{E}_{\Omega }\mathcal{S}_{\mathcal{D}}\left (\,f,g\right )$ is included in part (1) of the theorem, to allow for some of the inner products in the definition of $\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )$ to be added back into the form $\mathcal{B}\left (\,f,g\right ) -\mathcal{C}_{\mathcal{D}}\left (\,f,g\right )$ during the course of the proof of estimate (1). Indeed, this was done when controlling the form $\mathsf{T}_{\mathop{\mathrm{far}}\mathop{ \mathrm{below}}}^{2}\left (\,f,g\right )$ above.