An ε-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 $|F|L^6(Q\lambda )\lesssim {\lambda }^{{\Sigma }_{\lambda }}{D}^{{\Sigma }_D}({\sum }_{\Theta }|F_{\Theta }|{L^6}^2{)}^{1/2}$ 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 $T{T}^{}$) yield $|K|L^2\to L^2\lesssim {\lambda }^{-9/2}{D}^{-3}$; (iii) robust Kakeya: a density threshold $>cD$ 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 ${\lambda }^{\epsilon }$- and $D^{\epsilon }$-losses are removed.

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