Maximal estimates for perturbations of the Schrödinger operator on $\mathbb{T}^d$

Technical Summary: Maximal Estimates for Perturbations of the Schrödinger Operator on $\mathbb{T}^d$

Problem Statement
This paper investigates $L^p_x L^\infty_t$ maximal estimates for exponential sums associated with $C^2$ graph hypersurfaces, motivated by the pointwise convergence problem for the Schrödinger equation on the torus $\mathbb{T}^d$. Specifically, the authors examine whether the conjectured maximal estimate for the periodic Schrödinger equation holds under $C^2$-small perturbations of the paraboloid.

The central object of study is the exponential sum:
$$u(x, t) = \sum_{q \in [Q]^d} b_q e\left(x \cdot \frac{q}{Q} + t\psi\left(\frac{q}{Q}\right)\right),$$
where $\psi \in C^2([0, 1]^d)$. When $\psi(\xi) = |\xi|^2$, this corresponds to the parabolically rescaled solution of the Schrödinger equation on the unit torus. The conjectured maximal estimate (Conjecture 1) posits that for $2 \le p \le p_{\text{conj}} = \frac{2(d+1)}{d}$, the $L^p$ norm of the maximal function is bounded by $Q^{\frac{d}{p} + \frac{1}{p_{\text{conj}}} + \epsilon} \|b_q\|_{\ell^2}$.

Previous work by Fu, Ren, and Wang [FRW23] demonstrated that this conjecture fails for $d=1$ over the class of uniformly convex sequences, showing that the maximal estimate can be significantly larger than the conjectured bound. The open question addressed here is whether this failure persists in higher dimensions ($d \ge 2$) and for general $C^2$ perturbations of the paraboloid, rather than just specific constructed sequences.

Methodology
The paper employs a combination of harmonic analysis, number theory, and homogeneous dynamics to establish both lower and upper bounds.

1. Lower Bounds via Incidence Estimates:
   To disprove the conjectured maximal estimate, the authors construct specific perturbations $\psi$ and initial data sequences $(b_q)$ that yield large maximal functions. The core of this construction relies on a result by Cairo and Zhang [CZ25] regarding the intersection of $C^2$ submanifolds with rescaled integer lattices.

   • The authors utilize a variation of the Cairo-Zhang theorem (Theorem 1.4), proven using homogeneous dynamics (specifically the action of $SL_n(\mathbb{R})$ on the space of unimodular lattices).
   • By applying Siegel's mean value theorem and second-moment methods to the space of lattices, they demonstrate the existence of a lattice transformation $g$ and a perturbed function $\psi$ such that the graph of $\psi$ contains a large number of rational points with specific denominators.
   • These "lucky" points allow the construction of a sequence $(b_q)$ where the exponential sum exhibits constructive interference (large values) on a set of significant measure, thereby violating the conjectured upper bound.

2. Upper Bounds via Decoupling:
   For the upper bound, the authors utilize the $\ell^2$-decoupling theory for compact $C^2$ hypersurfaces established by Bourgain and Demeter [BD15].

   • They apply a localized version of the global decoupling inequality to the exponential sum.
   • By analyzing the level sets of the maximal function and utilizing the local constancy property of functions with Fourier support in small caps, they derive an upper bound for the $L^p$ norm.
   • This approach confirms that while the conjectured bound fails for perturbations, the decoupling exponent $p_{\text{crit}} = \frac{2(d+2)}{d}$ remains a valid threshold for the upper bound, up to an $Q^\epsilon$ loss.

Key Contributions and Results

• Failure of the Conjecture in Higher Dimensions: The primary result (Theorem 1.1) establishes that the conjectured maximal estimate fails for $C^2$-small perturbations of the paraboloid in all dimensions $d \ge 1$. Specifically, for any $\epsilon > 0$, there exists a perturbation $\psi$ close to the paraboloid and a sequence $(b_q)$ such that the maximal estimate scales as $Q^{\frac{d(d+1)}{2(d+2)} + \frac{1}{p} + \epsilon} \|b_q\|_{\ell^2}$.

  • At the critical exponent $p_{\text{crit}}$, the lower bound is $Q^{d/2}$, whereas the conjectured bound would be $Q^{d/2 + \epsilon}$.
  • At the conjectured endpoint $p_{\text{conj}}$, the lower bound exceeds the conjectured bound by a factor of $Q^{\frac{d}{2(d+1)(d+2)}}$.

• Sharpness of Decoupling Bounds: Theorem 1.2 provides an upper bound for the maximal estimate over uniformly convex $C^2$ hypersurfaces. The authors show that the estimates are essentially sharp at the decoupling endpoint $p_{\text{crit}}$ for general dimensions, differing from the lower bound only by a $Q^\epsilon$ factor.

• New Proofs via Homogeneous Dynamics: The paper provides an alternative proof of the incidence estimates originally proven by Cairo and Zhang [CZ25]. This proof (Theorem 1.4) uses the dynamics of $SL_n(\mathbb{R})$ acting on the space of lattices, avoiding some of the specific geometric assumptions of the original work and extending the result to arbitrary codimensions.

• Refinement of Exponents: The authors define and analyze the gap between the conjectured exponent $\alpha_{\text{conj}}(p)$, the lower bound exponent $\alpha_{\text{low}}(p)$, and the upper bound exponent $\alpha_{\text{upp}}(p)$. They show that for $d=1$, the bounds are sharp (up to $Q^\epsilon$), but for $d \ge 2$, a gap exists between the conjectured estimate and the actual behavior of perturbations at $p_{\text{conj}}$.

Significance and Claims
The paper claims to resolve the question of whether the decoupling-based conjecture for Schrödinger maximal estimates is robust under $C^2$ perturbations. The authors demonstrate that the conjecture is not robust; the number-theoretic structure of the paraboloid is essential for the conjectured bounds to hold. When the phase function is perturbed within the $C^2$ class, the maximal estimate deteriorates, aligning with the behavior observed in the $d=1$ case by [FRW23].

Furthermore, the work highlights the limitations of decoupling methods in distinguishing between the paraboloid and its small perturbations. While decoupling provides the correct upper bound for the class of uniformly convex surfaces, it cannot recover the finer conjectured bounds that rely on the specific arithmetic properties of the paraboloid.

The paper concludes that for general $C^2$ hypersurfaces, the maximal estimate is governed by the geometry of the surface (via decoupling) rather than the specific arithmetic of the paraboloid, and that the conjectured exponent $p_{\text{conj}}$ is not the correct threshold for the general $C^2$ class in dimensions $d \ge 2$. The results are presented as sharp up to $Q^\epsilon$ losses at the critical exponent $p_{\text{crit}}$.