We obtain a two weight local $Tb$ theorem for any elliptic and gradient elliptic fractional singular integral operator $T^{\alpha}$ on the real line $\mathbb{R}$, and any pair of locally finite positive Borel measures $(\sigma,\omega)$ on $\mathbb{R}$. The Hilbert transform is included in the case $\alpha = 0$, and is bounded from $L^{2}(\sigma)$ to $L^{2}(\omega)$ if and only if the Muckenhoupt and energy conditions hold, as well as $b_{Q}$ and $b_{Q}^{\ast}$ testing conditions over intervals $Q$, where the families $\{b_{Q}\}$ and $\{b_{Q}^{\ast}\}$ are $p$-weakly accretive for some $p > 2$. A number of new ideas are needed to accommodate weak goodness, including a new method for handling the stubborn nearby form, and an additional corona construction to deal with the stopping form. In a sense, this theorem improves the $T1$ theorem obtained by the authors and M. Lacey.

中文翻译：

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Hilbert变换的两个权重的局部$ Tb $定理

对于实线$ \ mathbb {R} $上的任何椭圆和梯度椭圆分数奇异积分算子$ T ^ {\ alpha} $，以及任意一对局部有限的正Borel度量$，我们获得一个两个权重的局部$ Tb $定理。 （\ sigma，\ omega）$在$ \ mathbb {R} $上。希尔伯特变换包含在$ \ alpha = 0 $的情况下，并且当且仅当Muckenhoupt和能量满足时，才从$ L ^ {2}（\ sigma）$到$ L ^ {2}（\ omega）$条件以及间隔为$ Q $的$ b_ {Q} $和$ b_ {Q} ^ {\ ast} $测试条件成立，家庭$ \ {b_ {Q} \} $和$ \ {b_ {Q} ^ {\ ast} \} $是$ p $-对于$ p> 2 $来说是微不足道的。需要许多新的方法来适应弱点，包括处理附近顽固形式的新方法，以及用于处理停止形式的其他电晕构造。从某种意义上说