Uniform short Hilbert-transform bounds for Lipschitz directions

There are absolute constants \(a_*\in(0,1/2)\) and \(C_*<\infty\)
such that every Lipschitz field \(v:\R^2\to S^1\) with
\(\Lip(v)\le1\) satisfies
\[
 \sup_{0<\epsilon<a_*}
 \norm{H_{v,a_*}^{\epsilon}f}_{L^2(\R^2)}
 \le C_*\norm{f}_{L^2(\R^2)}
 \qquad(f\in\mathcal S(\R^2)).
\]
Its symmetric principal value therefore satisfies
\[
 \norm{H_{v,a_*}f}_2\le C_*\norm f_2,
 \qquad
 \big|\{x:|H_{v,a_*}f(x)|>\lambda\}\big|
 \le C_*^2\lambda^{-2}\norm f_2^2
 \quad(\lambda>0).
\]

Let \(v:\R^2\to S^1\) be Lipschitz and \(0<\delta<1\). Restrict to
rectangles satisfying
\[
 L(R)\le \frac1{100\Lip(v)},\qquad
 \abs{R\cap v^{-1}(\operatorname{EX}(R))}\ge\delta\abs R,
\]
where the first restriction is void when \(\Lip(v)=0\). Then
\[
 \mathcal M_{v,\delta}h(z)
   =\sup_R\frac{\one_R(z)}{\abs R}\int_R\abs h
 \quad\text{satisfies}\quad
 \norm{\mathcal M_{v,\delta}h}_{L^{2,\infty}}
       \le C\delta^{-1/2}\norm h_2.
\]
Here the supremum is tested at every point of \(R\), and the constant
is independent of \(v\) and \(\delta\).

For the fixed absolute constants in \eqref{red:local-operator}, there
is an absolute \(C\) such that, for every field \(u\) with
\(\abs u\le1\) and \(\Lip(u)\le C_{\rm Lip}\), every
\(0<\epsilon<a\), every dyadic \(w\), and every vertical band
cutoff \(P_w\),
\begin{equation}
 \abs{\langle H_{u,a}^{\epsilon}P_w f,g\rangle}
 \le C\bigl(1+\abs{\log(|F|/|E|)}\bigr)^{-3}
       \sqrt{|F||E|}
\end{equation}
whenever \(F,E\subset I\times\R\) have positive finite measure,
\(\abs f\le\one_F\), \(\abs g\le\one_E\), and
\(\abs I\le b_0\).

For \(\abs t\le b_0\) and \(T\) as in \Cref{red:commutator},
\begin{equation}
 \norm{TC_t h-C_tTh}_2
      \le C A(\mathfrak m)\abs t\,\norm h_2.
\end{equation}
The constant is uniform over \(\abs u\le1\),
\(\Lip(u)\le C_{\rm Lip}\).

The uniform restricted estimate in \Cref{thm:band-restricted} implies
\Cref{thm:main}.

The diagonal form
\[
 \sum_j\langle\mathcal A_{u,\mathcal L}
                         \widetilde Q_j f_j,g_j\rangle
\]
is a sum, with absolutely summable scalar coefficients, of averages
of forms \eqref{eq:src-3}. Each form satisfies
\Cref{red:packets}, uniformly in \(w,m,\mathcal L\), and has one
common angular grid and common spatial lattices across components.
The same assertion holds for \(m=1\) with the original
positive-frequency band cutoff in place of \(\widetilde Q_1\).
The sums can be obtained as limits of finite packet collections.

Suppose that the following bounds hold, uniformly for every admissible
finite form \(\Lambda_{\mathcal S}\) in
\Cref{red:packets}, with the fixed packet constants there.
The field \(u:\R^2\to\R\) is bounded and measurable, the sets
\(F,E\subset\R^2\) are measurable with positive finite measure,
and the input and test satisfy
\(\norm{f(z)}_{\mathcal H}\le\one_F(z)\),
\(\norm{g(z)}_{\mathcal H}\le\one_E(z)\).
\begin{enumerate}[label=\textup{(\roman*)}]
\item For \(m\ge1\), there is a fixed exponent \(C_1\) such that
\begin{equation}
 |\Lambda_{\mathcal S}(f,g)|
     \le C(1+\log(2m))^{C_1}\sqrt{|F||E|}.
\end{equation}
\item For \(L\ge L_0\), with \(\tau,m,W\) as in
\eqref{red:small-time-parameters}, assume in addition that
\(\abs u\le1\), \(\Lip(u)\le C_{\rm Lip}\), and
\(F,E\subset I\times\R\) for an interval \(I\) with
\(\abs I\le b_0\). Whenever
\(L/2\le\log\sqrt{|E|/|F|}\le2L\),
\begin{equation}
 |\Lambda_{\mathcal S}(f,g)|
       \le C L^{-20}\sqrt{|F||E|}.
\end{equation}
\item For \(m=1\), one scalar component, and \(|E|\le|F|\),
there is a fixed \(c>0\) such that
\begin{equation}
 |\Lambda_{\mathcal S}(f,g)|
       \le C\sqrt{|F||E|}(|E|/|F|)^c.
\end{equation}
\end{enumerate}
Then \Cref{thm:band-restricted} holds. The constants remain uniform
in \(w\), \(\epsilon\), and the number of retained scales.