An {\epsilon}-free rank-6 decoupling estimate for the paraboloid surface

Abstract

For the paraboloid decomposition $F=\sum_{\Theta} F_{\Theta}$ with $\Theta\subset{|\xi|\sim\lambda}$ and radius $r=\lambda^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}} \big(\sum_{\Theta}|F_{\Theta}|{L^{6}}^{2}\big)^{1/2}$ as $\lambda\to\infty$, where $D=\lambda^{1/12}$. Key components: (i) broad geometry of rank 3: bilipschitz behavior of normals gives $\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim \lambda^{-5/4}$, which via a trilinear Kakeya-BCT insertion contributes $+5/36$ in $\lambda$; (ii) kernel estimate: twelve integrations (6 in $t$, 6 in $x^{\prime}$) and measure analysis (Schur and $TT^{}$) yield $|K|{L^2\to L^2}\lesssim \lambda^{-9/2} D^{-3}$; (iii) robust Kakeya: a density threshold $> c{} D$ brings a factor $D$ ($+1/12$ in $\lambda$, $+1$ in $D$); (iv) algebraic shell: excluding a neighborhood $N_{\beta}(P)$ contributes $-1/12$ in $\lambda$ and $-1$ in $D$; (v) tube packing: explanatory only; (vi) narrow cascade: a double $7/8$ rescaling exits the narrow regime and contributes $-5/64$ in $\lambda$ (zero in $D$). Summing exponents: $\Sigma_{\lambda}=5/36-9/2-5/64=-2557/576\approx -4.44<0$ and $\Sigma_{D}=-3+1-1=-3<0$, hence both $\lambda^{\varepsilon}$- and $D^{\varepsilon}$-losses are removed.