Research proof index

Let $1 < q \le 2$ and let $q'$ be its Hölder conjugate. If every phase-linearized nontangential operator has the required uniform $L^2$ bound, then for measurable $F,G$ and measurable $f$ with $\lVert f(x)\rVert \le \mathbf{1}_F(x)$, the linearized Carleson operator satisfies

$\int_G^+ \mathcal{C}^{\mathrm{lin}}_{Q,K}f(x)\,dx \le C(a,q)\,\mu(G)^{1/q'}\mu(F)^{1/q}.$

For measurable $f$ with $\lVert f(x)\rVert\le1$, the scale-$n$ quantity lcoConvergent, which records a supremum of truncated linearized Carleson integrals over rational radii, is a measurable function of $x$.