Let $\sigma$ and $w$ be locally finite positive Borel measures on $\mathbb{R}$ which do not share a common point mass. Assume that the pair of weights satisfy a Poisson $A_2$ condition, and satisfy the testing conditions below, for the Hilbert transform $H$,

$${\int }_{I}H\left(\sigma {\mathbf{1}}_I{)}^{2}dw\lesssim \sigma \right(I),{\int }_{I}H(w{\mathbf{1}}_I{)}^{2}d\sigma \lesssim w\left(I\right),$$

with constants independent of the choice of interval $I$. Then $H(\sigma \cdot )$ maps $L^2\left(\sigma \right)$ to $L^2\left(w\right)$, verifying a conjecture of Nazarov, Treil, and Volberg. The proof has two components, a global-to-local reduction, carried out in this article, and an analysis of the local problem, to be elaborated in a future Part II version of this article.

Let $\sigma$ and $w$ be locally finite positive Borel measures on $\mathbb{R}$ which do not share a common point mass. Assume that the pair of weights satisfy a Poisson $A_2$ condition, and satisfy the testing conditions below, for the Hilbert transform $H$,

$${\int }_{I}H\left(\sigma {\mathbf{1}}_I{)}^{2}dw\lesssim \sigma \right(I),{\int }_{I}H(w{\mathbf{1}}_I{)}^{2}d\sigma \lesssim w\left(I\right),$$

with constants independent of the choice of interval $I$. Then $H(\sigma \cdot )$ maps $L^2\left(\sigma \right)$ to $L^2\left(w\right)$, verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.

We study extensions and generalizations of the Schmidt subspace theorem in various settings. In particular, we prove results for algebraic points of bounded degree, giving a sharp version of Schmidt’s theorem for quadratic points in the projective plane and a more general result that resolves a conjecture of Schlickewei.

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline{\partial }$-cohomology of $(n,q)$-forms on $X$. The central tool is an $L^2$-resolution for the Grauert–Riemenschneider canonical sheaf ${\mathcal{K}}_X$. As an application, we obtain a Grauert–Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If $X$ is a Gorenstein space with canonical singularities, then we also get an $L^2$-representation of the flabby cohomology of the structure sheaf ${\mathcal{O}}_X$. To understand also the $L^2$-$\overline{\partial }$-cohomology of $(0,q)$-forms on $X$, we introduce a new kind of canonical sheaf, namely, the canonical sheaf of square-integrable holomorphic $n$-forms with some (Dirichlet) boundary condition at the singular set of $X$. If $X$ has only isolated singularities, then we use an $L^2$-resolution for that sheaf and a resolution of singularities to give a smooth representation of the $L^2$-$\overline{\partial }$-cohomology of $(0,q)$-forms.

We construct a geometric, real-analytic parameterization of the Hitchin component ${\mathrm{Hit}}_n\left(S\right)$ of the ${\mathrm{PSL}}_n\left(\mathbb{R}\right)$-character variety ${\mathcal{R}}_{{\mathrm{PSL}}_n\left(\mathbb{R}\right)}\left(S\right)$ of a closed surface $S$. The approach is explicit and constructive. In essence, our parameterization is an extension of Thurston’s shearing coordinates for the Teichmüller space of a closed surface, combined with Fock–Goncharov’s coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination $\lambda \subset S$ with finitely many leaves, we introduce two types of invariants for elements of the Hitchin component: shear invariants associated with each leaf of $\lambda$ and triangle invariants associated with each component of the complement $S-\lambda$. We describe identities and relations satisfied by these invariants, and we use the resulting coordinates to parameterize the Hitchin component.