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Shaoming Guo

Publications and source records attributed to Shaoming Guo.

At least 19 recordsLinked to original sources
arXiv2026

Curved Kakeya problems and the projective geometry of paths

We introduce a general framework for curved Kakeya problems in $\mathbb{R}^n$, encompassing those arising from H\"ormander-type oscillatory integrals. Every family of curves determines a spray geometry, which allows us to use the projective geometry of paths in the study of curved Kakeya problems. We focus on the two extremes of the "best" and "worst" possible behaviors of curved Kakeya sets. We characterize when the incidence structure underlying Wolff's hairbrush argument persists. In particular, we prove that the existence of many totally geodesic surfaces, as required by Wolff's hairbrush argument, is equivalent to projective flatness of the associated spray. Within this projectively flat class, Bourgain's condition provides a clean dichotomy: when it holds, the family is direction-equivalent to a Bochner--Riesz type family of lines and satisfies the Katz--Wolff condition, and thus the Wang--Zahl result is applicable; when it fails, every totally geodesic surface supports a two-dimensional Kakeya set. We also show that under an extra semi-algebraic assumption, a family of curves in $\mathbb{R}^3$ admits a curved Kakeya set of Hausdorff dimension $2$ if and only if it admits a curved Kakeya set contained in a surface. Equivalently, if this compression is absent, every associated curved Kakeya set has dimension strictly greater than $2$.

math.CA↗
arXiv2026

Restriction estimates for surfaces with negative curvature in $\mathbb R^3$

In $\mathbb R^3$, we prove that $L^q\to L^p$ restriction estimates associated with smooth surfaces with negative Gaussian curvature hold for all $p>\frac{22}{7}$ and $q'<\frac{p}{2}$. Building on Demeter--Wu's work for the model hyperbolic paraboloid, we introduce a geometric propagation principle for good lines, which controls the degenerate directions arising from straight-line segments on the surface. This overcomes a key difficulty in the general case, where such directions may vary with the geometry rather than being fixed by the coordinate axes.

math.CA↗
arXiv2026

The (local) geometry of oscillatory integrals on manifolds: Dimension three

Sogge studied Kakeya problems on two extreme types of three dimensional Riemannian manifolds: Manifolds with the most symmetries (manifolds of constant sectional curvature) and manifolds with the least symmetries, which he called manifolds with chaotic curvature and variably curved manifolds. In the same paper, Sogge proposed studying manifolds with intermediate symmetry, such as (locally) symmetric spaces. In the current paper, we propose a classification of curvature conditions in the spirit of Sogge's program. In particular, these curvature conditions give a complete geometric characterization of the contact order conditions (for Riemannian distance functions), introduced when people were studying H\"ormander-type oscillatory integral operators. One of these conditions generalizes Sogge's chaotic curvature condition to all finite orders: The chaotic curvature condition of order $\le k$ for every $k\ge 1,$ with the case $k=1$ corresponding to Sogge's original condition for variably curved manifolds. As byproducts of our main results, we show that there are no manifolds satisfying the chaotic curvature condition of order $\le 1$. We also show that both the chaotic curvature condition of order $\le 2$ and its failure can occur robustly under small smooth perturbations, and for every $k\ge 3$, a ``generic" manifold satisfies the chaotic curvature condition of order $\le k$. It turns out that the chaotic curvature condition of order $\le k$ is precisely the same as the notion of non-$(k+2)$-exceptional, where $k$-exceptional is introduced by Lytchak and Petrunin \cite{LP22} when studying convex sets and the non-existence of totally geodesic sub-manifolds. Thus our results imply, in particular, that every manifold is $3$-exceptional.

math.CA↗
arXiv2025

Curved Kakeya sets for generic phases in odd dimensions

We show that for each odd integer $n\ge 3$, there is an open dense subset of H\"ormander phase functions in $\mathbb{R}^n$ for which the associated curved Kakeya sets have Hausdorff dimension at least $\frac{n+1}{2} + d_n$ for some positive $d_n$, thereby exceeding the classical compression threshold. In particular, in $\mathbb{R}^3$, generic H\"ormander phases induce curved Kakeya sets of dimension at least $2 + \tfrac17$. As an application, on a generic three-dimensional Riemannian manifold, a local Nikodym set has Hausdorff dimension at least $2 + \tfrac17$. We achieve these results by generalizing the finite contact order condition from Dai--Gong--Guo--Zhang, originally developed in $\mathbb{R}^3$, to arbitrary dimensions. Our bounds are stronger than those of Dai--Gong--Guo--Zhang even in $\mathbb{R}^3$, since we derive curved Kakeya estimates directly via the polynomial method. Moreover, for H\"ormander-type oscillatory integral operators with positive-definite phases of finite contact order, we obtain quantitative improvements in all odd dimensions over the bounds of Guth--Hickman--Iliopoulou, while in three dimensions our oscillatory integral estimate exactly matches the result of Dai--Gong--Guo--Zhang.

math.CA↗
arXiv2024

On a planar Pierce--Yung operator

We show that the operator \begin{equation*} \mathcal{C} f(x,y) := \sup_{v\in \mathbb{R}} \Big|\mathrm{p.v.} \int_{\mathbb{R}} f(x-t, y-t^2) e^{i v t^3} \frac{\mathrm{d} t}{t} \Big| \end{equation*} is bounded on $L^p(\mathbb{R}^2)$ for every $1 < p < \infty$. This gives an affirmative answer to a question of Pierce and Yung.

math.CA↗
arXiv2024

Oscillatory integral operators and variable Schr\"odinger propagators: beyond the universal estimates

We consider a class of H\"ormander-type oscillatory integral operators in $\mathbb{R}^n$ for $n \geq 3$ odd with real analytic phase. We derive weak conditions on the phase which ensure $L^p$ bounds beyond the universal $p \geq 2 \cdot \frac{n+1}{n-1}$ range guaranteed by Stein's oscillatory integral theorem. This expands and elucidates pioneering work of Bourgain from the early 1990s. We also consider a closely related class of variable coefficient Schr\"odinger propagator-type operators, and show that the corresponding theory differs significantly from that of the H\"ormander-type operators. The main ingredient in the proof is a curved Kakeya/Nikodym maximal function estimate. This is established by combining the polynomial method with certain uniform sublevel set estimates for real analytic functions. The sublevel set estimates are the main novelty in the argument and can be interpreted as a form of quantification of linear independence in the $C^{\omega}$ category.

math.CA↗
arXiv2024

The dichotomy of Nikodym sets and local smoothing estimates for wave equations

We show that Nikodym sets and local smoothing estimates for linear wave equations form a dichotomy: If Nikodym sets for a family of curves exist, then the related maximal operator is not bounded on $L^p(\mathbb{R}^2)$ for any $p<\infty$; if Nikodym sets do not exist, then local smoothing estimates hold, and the related maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$. Whenever the maximal operator is bounded on $L^p(\mathbb{R}^2)$ for some $p<\infty$, we also determine the sharp exponent for $L^p(\mathbb{R}^2)$ bounds.

math.CA↗
arXiv2023

$L^p$ integrability of functions with Fourier support on a smooth space curve

We prove that if $f\in L^p(\mathbb{R}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\widehat{f}$ is supported on a small perturbation of the moment curve in $\mathbb{R}^k$, then $f$ is identically zero. This improves the more general result of Agranovsky and Narayanan, and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.

math.CA↗
arXiv2023

Oscillatory integral operators on manifolds and related Kakeya and Nikodym problems

We consider Carleson-Sj\"{o}lin operators on Riemannian manifolds that arise naturally from the study of Bochner-Riesz problems on manifolds. They are special cases of H\"{o}rmander-type oscillatory integral operators. We obtain improved $L^p$ bounds of Carleson-Sj\"{o}lin operators in two cases: The case where the underlying manifold has constant sectional curvature and the case where the manifold satisfies Sogge's chaotic curvature condition. The two results rely on very different methods: To prove the former result, we show that on a Riemannian manifold, the distance function satisfies Bourgain's condition if and only if the manifold has constant sectional curvature. To obtain the second result, we introduce the notion of "contact orders" to H\"{o}rmander-type oscillatory integral operators, prove that if a H\"{o}rmander-type oscillatory integral operator is of a finite contact order, then it always has better $L^p$ bounds than "worst cases" (in spirit of Bourgain and Guth, and Guth, Hickman and Iliopoulou), and eventually verify that for Riemannian manifolds that satisfy Sogge's chaotic curvature condition, their distance functions alway have finite contact orders. As byproducts, we obtain new bounds for Nikodym maximal functions on manifolds of constant sectional curvatures.

math.DG↗
arXiv2023

A multi-parameter cinematic curvature

We state a multi-parameter cinematic curvature condition, and prove $L^p$ bounds for related maximal operators. In particular, we verify a local smoothing conjecture of Zahl.

math.CA↗
arXiv2023

A dichotomy for Hörmander-type oscillatory integral operators

In this paper, we first generalize the work of Bourgain and state a curvature condition for Hörmander-type oscillatory integral operators, which we call Bourgain's condition. This condition is notably satisfied by the phase functions for the Fourier restriction problem and the Bochner-Riesz problem. We conjecture that for Hörmander-type oscillatory integral operators satisfying Bourgain's condition, they satisfy the same $L^p$ bounds as in the Fourier Restriction Conjecture. To support our conjecture, we show that whenever Bourgain's condition fails, then the $L^{\infty} \to L^q$ boundedness always fails for some $q= q(n) > \frac{2n}{n-1}$, extending Bourgain's three-dimensional result. On the other hand, if Bourgain's condition holds, then we prove $L^p$ bounds for Hörmander-type oscillatory integral operators for a range of $p$ that extends the currently best-known range for the Fourier restriction conjecture in high dimensions, given by Hickman and Zahl. This gives new progress on the Fourier restriction problem and the Bochner-Riesz problem.

math.CA↗
arXiv2022

A restricted projection problem for fractal sets in $\mathbb{R}^n$

Let $\gamma: [-1, 1]\to \mathbb{R}^n$ be a smooth curve that is non-degenerate. Take $m\le n$ and a Borel set $E\subset [0, 1]^n$. We prove that the orthogonal projection of $E$ to the $m$-th order tangent space of $\gamma$ at $\theta\in [-1, 1]$ has Hausdorff dimension $\min\{m, \dim(E)\}$ for almost every $\theta\in [-1, 1]$.

math.CA↗
arXiv2022

On the strict majorant property in arbitrary dimensions

In this work we study $d$-dimensional majorant properties. We prove that a set of frequencies in ${\mathbb Z}^d$ satisfies the strict majorant property on $L^p([0,1]^d)$ for all $p> 0$ if and only if the set is affinely independent. We further construct three types of violations of the strict majorant property. Any set of at least $d+2$ frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an open interval of $p \not\in 2 {\mathbb N}$ of length 2. Any infinite set of frequencies in ${\mathbb Z}^d$ violates the strict majorant property on $L^p([0,1]^d)$ for an infinite sequence of open intervals of $p \not\in 2 {\mathbb N}$ of length $2$. Finally, given any $p>0$ with $p \not\in 2{\mathbb N}$, we exhibit a set of $d+2$ frequencies on the moment curve in ${\mathbb R}^d$ that violate the strict majorant property on $L^p([0,1]^d).$

math.CA↗
arXiv2022

On restricted projections to planes in $\mathbb{R}^3$

Let $\gamma:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(\gamma(\theta),\gamma'(\theta),\gamma"(\theta)\big)\neq 0$. For each $\theta\in[0,1]$, let $V_\theta=\gamma(\theta)^\perp$ and let $\pi_\theta:\mathbb{R}^3\rightarrow V_\theta$ be the orthogonal projections. We prove that if $A\subset \mathbb{R}^3$ is a Borel set, then for a.e. $\theta\in [0,1]$ we have $\text{dim}(\pi_\theta(A))=\min\{2,\text{dim} A\}$. More generally, we prove an exceptional set estimate. For $A\subset\mathbb{R}^3$ and $0\le s\le 2$, define $E_s(A):=\{\theta\in[0,1]: \text{dim}(\pi_\theta(A)) 2$, then for a.e. $\theta\in[0,1]$ we have $\mathcal{H}^2(\pi_\theta (A))>0$.

math.CA↗
arXiv2022

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.

math.CA↗
arXiv2021

Decoupling for two quadratic forms in three variables: a complete characterization

We prove sharp decoupling inequalities for all degenerate surfaces of codimension two in $\mathbb{R}^5$ given by two quadratic forms in three variables. Together with previous work by Demeter, Guo, and Shi in the non-degenerate case (arXiv:1609.04107), this provides a classification of decoupling inequalities for pairs of quadratic forms in three variables.

math.CA↗