An ε-free rank-6 decoupling estimate for the paraboloid surface
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arXiv
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| Format: | Preprint |
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2025
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| author | Cherevan, Pylyp |
| author_facet | Cherevan, Pylyp |
| contents | For the paraboloid decomposition $F=\sum_Θ F_Θ$ with $Θ\subset{|ξ|\simλ}$ and radius $r=λ^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Qλ)}\lesssim λ^{Σ_λ} D^{Σ_{D}} \big(\sum_Θ|F_Θ|{L^{6}}^{2}\big)^{1/2}$ as $λ\to\infty$, where $D=λ^{1/12}$. Key components: (i) broad geometry of rank 3: bilipschitz behavior of normals gives $\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim λ^{-5/4}$, which via a trilinear Kakeya-BCT insertion contributes $+5/36$ in $λ$; (ii) kernel estimate: twelve integrations (6 in $t$, 6 in $x^{\prime}$) and measure analysis (Schur and $TT^{}$) yield $|K|{L^2\to L^2}\lesssim λ^{-9/2} D^{-3}$; (iii) robust Kakeya: a density threshold $> c{} D$ brings a factor $D$ ($+1/12$ in $λ$, $+1$ in $D$); (iv) algebraic shell: excluding a neighborhood $N_β(P)$ contributes $-1/12$ in $λ$ 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 $λ$ (zero in $D$). Summing exponents: $Σ_λ=5/36-9/2-5/64=-2557/576\approx -4.44<0$ and $Σ_{D}=-3+1-1=-3<0$, hence both $λ^{\varepsilon}$- and $D^{\varepsilon}$-losses are removed. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_20834 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An ε-free rank-6 decoupling estimate for the paraboloid surface Cherevan, Pylyp General Mathematics 5Q30, 76D05, 42B20 For the paraboloid decomposition $F=\sum_Θ F_Θ$ with $Θ\subset{|ξ|\simλ}$ and radius $r=λ^{-2/3}$, we prove a log-free estimate $|F|{L^{6}(Qλ)}\lesssim λ^{Σ_λ} D^{Σ_{D}} \big(\sum_Θ|F_Θ|{L^{6}}^{2}\big)^{1/2}$ as $λ\to\infty$, where $D=λ^{1/12}$. Key components: (i) broad geometry of rank 3: bilipschitz behavior of normals gives $\max{i<j<k}|n_i\wedge n_j\wedge n_k|\gtrsim λ^{-5/4}$, which via a trilinear Kakeya-BCT insertion contributes $+5/36$ in $λ$; (ii) kernel estimate: twelve integrations (6 in $t$, 6 in $x^{\prime}$) and measure analysis (Schur and $TT^{}$) yield $|K|{L^2\to L^2}\lesssim λ^{-9/2} D^{-3}$; (iii) robust Kakeya: a density threshold $> c{} D$ brings a factor $D$ ($+1/12$ in $λ$, $+1$ in $D$); (iv) algebraic shell: excluding a neighborhood $N_β(P)$ contributes $-1/12$ in $λ$ 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 $λ$ (zero in $D$). Summing exponents: $Σ_λ=5/36-9/2-5/64=-2557/576\approx -4.44<0$ and $Σ_{D}=-3+1-1=-3<0$, hence both $λ^{\varepsilon}$- and $D^{\varepsilon}$-losses are removed. |
| title | An ε-free rank-6 decoupling estimate for the paraboloid surface |
| topic | General Mathematics 5Q30, 76D05, 42B20 |
| url | https://arxiv.org/abs/2510.20834 |