Geometric classification of the $\ell^2(L^6)$ decoupling for nondegenerate surfaces in $\mathbb{R}^4$

Bourgain and Demeter proved the sharp $\ell^p(L^p)$ decoupling for every $C^3$ surface $\Psi:[0,1]^2\to\mathbb{R}^4$ satisfying their nondegeneracy condition (rank of $[\Psi_t,\Psi_s,\Psi_{tt},\Psi_{ss},\Psi_{ts}]$ equal to 4), and observed that the stronger critical-index $\ell^2(L^6)$ decoupling at the canonical cap scale $N^{-1/2}$ holds for some such surfaces (e.g. $(t,s,t^2,s^2)$) and fails for others (e.g. $(t,s,t^2,ts)$, which contains an affine line), leaving open which surfaces admit it. Later work classified the quadratic model completely: Guo–Oh–Zhang–Zorin-Kranich determined the sharp $\ell^q(L^p)$ decoupling exponents for arbitrary tuples of quadratic forms, and their Corollary 1.3 shows the $\ell^2(L^p)$ constants are best possible exactly for strongly non-degenerate tuples, which for two forms in two variables at $p=6$ settles the quadratic case of the classification. The surviving open problem is the general case: identify the geometric criterion — the codimension-two analogue of curvature, which the theory currently lacks — separating the $C^3$ nondegenerate surfaces in $\mathbb{R}^4$ that admit the $\ell^2(L^6)$ decoupling from those that do not, and prove both directions.

Background

For a compact manifold $S\subset\mathbb{R}^{d+n}$, the $\ell^q(L^p)$ decoupling problem quantifies how the Fourier extension operator $Eg(x)=\int g(\xi)e(x\cdot\Psi(\xi))\,d\xi$ (with $e(y)=e^{2\pi i y}$ and $S$ parametrized by $\Psi$) behaves under a partition of the parameter domain into caps: on a ball $B_N$ of radius $N$, one compares $\|Eg\|_{L^p(w_{B_N})}$ with $(\sum_\Delta\|E_\Delta g\|_{L^p(w_{B_N})}^q)^{1/q}$, where $w_{B_N}$ is a standard radial weight and the sum runs over the caps $\Delta$ of the partition. The decoupling constant is the best such constant $D(N,p,q)$, and the goal is its sharp power of $N$, up to $N^\epsilon$ losses. For hypersurfaces with nonzero Gaussian curvature, the sharp $\ell^2(L^p)$ decoupling was proved by Bourgain and Demeter (the $\ell^2$ Decoupling Conjecture), and with Guth they proved the sharp decoupling for the moment curve $(t,t^2,\dots,t^n)$, equivalent to the main conjecture in Vinogradov's mean value theorem for exponential sums.

Bourgain and Demeter's paper ‘Decouplings for surfaces in $\mathbb{R}^4$’ established the first sharp decoupling for a manifold of dimension greater than one and codimension greater than one. Theorem 1.2 proves for every $C^3$ surface $\Psi(t,s)=(\psi_1,\dots,\psi_4)$ satisfying the nondegeneracy condition $\mathrm{rank}[\Psi_t,\Psi_s,\Psi_{tt},\Psi_{ss},\Psi_{ts}]=4$ at every point — equivalently, locally non-flat: no unit vector $\gamma$ has $\langle\gamma,\Psi(t_0+\Delta t,s_0+\Delta s)-\Psi(t_0,s_0)\rangle=O(|(\Delta t,\Delta s)|^3)$ — that, for caps of side length $N^{-1/2}$, $D(N,p)\lesssim_\epsilon N^{1/2-1/p+\epsilon}$ for $2\le p\le 6$ and $D(N,p)\lesssim_\epsilon N^{1-4/p+\epsilon}$ for $p\ge 6$, both sharp. The exponent $p=6$ is the critical index, where the two sharp bounds meet at $N^{1/3}$. The motivation was Bourgain's decoupling for nondegenerate curves in $\mathbb{R}^4$, which yielded the bound $|\zeta(1/2+it)|\ll t^{53/342+\epsilon}$ for the Riemann zeta function; the curve inequality there is derived as a consequence of the surface decoupling.

At the critical index, Theorem 1.2 is an $\ell^6(L^6)$ inequality, weaker than the $\ell^2(L^6)$ inequality (9) of the paper, $\|E_{[0,1]^2}g\|_{L^6(w_{B_N})}\le C_\epsilon N^\epsilon\Big(\sum_{\Delta\subset[0,1]^2,\,\ell(\Delta)=N^{-1/2}}\|E_\Delta g\|_{L^6(w_{B_N})}^2\Big)^{1/2}$ since Hölder converts $\ell^2$ to $\ell^6$ over the $\sim N$ caps. The authors show that (9) is not a consequence of nondegeneracy alone: for $\Psi(t,s)=(t,s,t^2,ts)$, whose image contains the affine line $\{(0,s,0,0)\}$, a discretization argument would force a square-root-cancellation estimate for exponential sums supported on $N^{-1/2}$-separated points of that line, which fails for constant coefficients. Conversely, (9) holds for $\Psi(t,s)=(t,s,t^2,s^2)$, a product of two planar parabolas, by Fubini and the planar curve decoupling — an argument the authors note ‘is not applicable for more “twisted” $\Psi$’. Which nondegenerate surfaces admit (9) was left open.

The quadratic case has since been completely classified. Guo–Zorin-Kranich proved sharp decouplings for quadratic surfaces of low codimension, recovering the source theorem in the quadratic case within a unified framework; Demeter–Guo–Shi and Guo–Oh–Roos–Yung–Zorin-Kranich treated three-dimensional quadratic manifolds in $\mathbb{R}^5$, including the degenerate cases, via small-cap decouplings; and Guo–Oh–Zhang–Zorin-Kranich determined the sharp $\ell^q(L^p)$ decoupling exponent for every tuple of quadratic forms in any numbers of variables and forms. In the last paper, Corollary 1.3 identifies the tuples whose $\ell^2(L^p)$ decoupling constants are best possible as exactly the strongly non-degenerate ones — a linear-algebraic condition on the restrictions of the forms to subspaces (condition (1.21) there: $d_{d-m,n'}(Q)\ge n'd/n-m$ for all $n'$ and $m$, where $d_{d',n'}(Q)$ is the minimal number of variables on which $n'$ linear combinations of the forms can be made to depend, after restricting them to a $d'$-dimensional subspace and applying linear changes of variables in domain and target). For two forms in two variables at $p=6$ this is exactly inequality (9): it holds precisely for strongly non-degenerate pairs; and one checks directly from the definition that for homogeneous quadratic surfaces satisfying the source paper's rank-2 condition on the second-order data (its condition (7): the $2\times 3$ matrix of quadratic coefficients has rank 2 at every point) strong non-degeneracy is in turn equivalent to the surface containing no affine line, consistent with both of the paper's examples. These classifications exploit the exact affine rescaling symmetry of quadratic graphs and scale-dependent Brascamp–Lieb inequalities (a multilinear generalization of Hölder's inequality in which the scaling data is allowed to vary with the decoupling scale), machinery that does not extend to general $C^3$ phases. For general smooth surfaces of codimension greater than one there is no analogue of Gaussian curvature, so the geometric datum governing the $\ell^2(L^6)$ threshold is itself part of the problem.

Problem Statement

Determine, by an explicit geometric criterion on $\Psi$, for exactly which $C^3$ surfaces $\Psi:[0,1]^2\to\mathbb{R}^4$ satisfying the nondegeneracy condition $\mathrm{rank}[\Psi_t,\Psi_s,\Psi_{tt},\Psi_{ss},\Psi_{ts}]=4$ the critical-index square-function decoupling at the canonical scale

$\|E_{[0,1]^2}g\|_{L^6(w_{B_N})}\le C_\epsilon N^\epsilon\Big(\sum_{\Delta\subset[0,1]^2,\,\ell(\Delta)=N^{-1/2}}\|E_\Delta g\|_{L^6(w_{B_N})}^2\Big)^{1/2}$

holds for all $g:[0,1]^2\to\mathbb{C}$, all balls $B_N\subset\mathbb{R}^4$ of radius $N$, and all $N\ge 1$, with $C_\epsilon$ depending only on $\epsilon$ and on finitely many $C^3$-norm and nondegeneracy bounds for $\Psi$. Equivalently, determine an explicit geometric criterion $C$ on $\Psi$ such that the inequality above holds for every $C^3$ surface with $\mathrm{rank}[\Psi_t,\Psi_s,\Psi_{tt},\Psi_{ss},\Psi_{ts}]=4$ satisfying $C$, and fails, with the decoupling constant gaining a positive power of $N$, for every such nondegenerate surface not satisfying $C$.

References

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