Stein's conjecture on the Hilbert transform along Lipschitz vector fields

Theorem 1.1: there are absolute $a\in(0,1/2)$ and $C$ such that every Lipschitz unit field $v$ on $\mathbb R^2$ with $\mathrm{Lip}(v)\le1$ satisfies $\sup_{0<\epsilon<a}\|H^{\epsilon}_{v,a}f\|_2\le C\|f\|_2$; the principal value is $L^2$ bounded and weak-(2,2). By scaling this holds at outer length $a/\mathrm{Lip}(v)$. Fields may depend on both coordinates. It does not give $L^p$ bounds for $p\neq2$, does not address the maximal (Zygmund) averaging operator, and does not give estimates at arbitrary outer lengths for varying fields.

Theorem 1.1: there are absolute $a\in(0,1/2)$ and $C$ such that every Lipschitz unit field $v$ on $\mathbb R^2$ with $\mathrm{Lip}(v)\le1$ satisfies $\sup_{0<\epsilon<a}\|H^{\epsilon}_{v,a}f\|_2\le C\|f\|_2$; the principal value is $L^2$ bounded and weak-(2,2). By scaling this holds at outer length $a/\mathrm{Lip}(v)$. Fields may depend on both coordinates. It does not give $L^p$ bounds for $p\neq2$, does not address the maximal (Zygmund) averaging operator, and does not give estimates at arbitrary outer lengths for varying fields.