Title: An ε-free rank-6 decoupling estimate for the paraboloid surface Show Chinese title

Title: 一个无ε的六阶解耦估计对于抛物面表面

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. Show Chinese abstract

Abstract: 对于抛物面分解 $F=\sum_{\Theta} F_{\Theta}$ ，带有 $\Theta\subset{|\xi|\sim\lambda}$ 和半径 $r=\lambda^{-2/3}$，我们证明了一个无对数估计 $|F|{L^{6}(Q{\lambda})}\lesssim \lambda^{\Sigma_{\lambda}} D^{\Sigma_{D}} \big(\sum_{\Theta}|F_{\Theta}|{L^{6}}^{2}\big)^{1/2}$ ，如 $\lambda\to\infty$所示，其中 $D=\lambda^{1/12}$。 关键组成部分：(i) 3阶广义几何：法线的双利普希茨行为给出$\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim \lambda^{-5/4}$，通过三线性Kakeya-BCT插入在$\lambda$中贡献$+5/36$；(ii) 核估计：十二次积分（6次在$t$，6次在$x^{\prime}$）和测度分析（Schur和$TT^{}$）产生$|K|{L^2\to L^2}\lesssim \lambda^{-9/2} D^{-3}$；(iii) 鲁棒Kakeya：密度阈值$> c{} D$带来因子$D$（在$\lambda$中$+1/12$，在$D$中$+1$）；(iv) 代数壳：排除一个邻域$N_{\beta}(P)$在$\lambda$中贡献$-1/12$并在$D$中贡献$-1$；(v) 管道包装：仅作解释；(vi) 窄级联：一个双重$7/8$缩放退出窄区域并在$\lambda$中贡献$-5/64$（在$D$中为零）。 指数相加：$\Sigma_{\lambda}=5/36-9/2-5/64=-2557/576\approx -4.44<0$和$\Sigma_{D}=-3+1-1=-3<0$，因此两者$\lambda^{\varepsilon}$- 和$D^{\varepsilon}$-损失都被移除。