The NTV two-weight conjecture for the Hilbert transform

Let $\sigma$ and $\omega$ be positive locally finite Borel measures on $\mathbb{R}$. For the Hilbert transform

$$Hf(x)=\operatorname{pv}\int_{\mathbb{R}}\frac{f(y)}{y-x}\,dy,$$

write

$$\mathfrak{N}_{H}(\sigma,\omega)=\sup_{f\in L^{2}(\mathbb{R};\sigma)}\frac{\|H(f\sigma)\|_{L^{2}(\omega)}}{\|f\|_{L^{2}(\sigma)}}$$

for the operator norm, uniformly over appropriate truncations of $H$. For an interval $I$, let $c_I$ and $\ell(I)$ denote its center and length, and set

$$\mathbf{s}_{I}(x)=\frac{\ell(I)}{\ell(I)+|x-c_I|}.$$

Define the two-tailed Muckenhoupt characteristic

$$\mathcal{A}_{2}(\sigma,\omega)=\sup_{I\in\mathcal{I}}\left(\frac{1}{|I|}\int_I\mathbf{s}_{I}(x)^2\,d\omega(x)\right)\left(\frac{1}{|I|}\int_I\mathbf{s}_{I}(y)^2\,d\sigma(y)\right),$$

and the testing characteristic

$$\mathfrak{T}_{H}(\sigma,\omega)=\sup_{I\in\mathcal{I}}\frac{1}{\sqrt{|I|_{\sigma}}}\left\|H\mathbf{1}_{I}\sigma\right\|_{L^{2}(\omega)}.$$

The NTV conjecture. If $\mathcal{A}_{2}(\sigma,\omega)$ is finite and both testing characteristics $\mathfrak{T}_{H}(\sigma,\omega)$ and $\mathfrak{T}_{H}(\omega,\sigma)$ are finite, then $\mathfrak{N}_{H}(\sigma,\omega)$ is finite uniformly over appropriate truncations of $H$. This two-weight characterization would identify the precise conditions for boundedness of the Hilbert transform between the two weighted spaces. The conjecture was proved by Hytönen, Lacey, Sawyer, Shen and Uriarte-Tuero, building on work of Nazarov, Treil, Volberg and others; the source paper presents an alternate organization of that proof.