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Shengwen Gan

Publications and source records attributed to Shengwen Gan.

At least 19 recordsLinked to original sources
arXiv2026

Sharp endpoint multilinear estimates for oscillatory integrals and spectral clusters

We prove sharp $k$-linear $L^p$ estimates for Carleson--Sj\"olin oscillatory integral operators with arbitrary separated frequency scales for all $k\ge 2$ and $1\le p\le \infty$. The estimates are sharp, including the endpoint logarithmic behavior for general Carleson--Sj\"olin phases. Moreover, we obtain log-free endpoint bilinear spectral cluster estimates on every closed three-dimensional Riemannian manifold, resolving a problem of Burq--G\'erard--Tzvetkov. As a consequence, we establish sharp $k$-linear $L^p$ spectral cluster estimates for all $k\ge 2$ and $1\le p\le \infty$.

math.AP↗
arXiv2025

Oscillatory integral operators and variable Schrödinger propagators: beyond the universal estimates

We consider a class of Hörmander-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 Schrödinger propagator-type operators, and show that the corresponding theory differs significantly from that of the Hörmander-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 $C^ω$ category.

math.CA↗
arXiv2025

Sharp local smoothing estimates for curve averages

We prove sharp local smoothing estimates for curve averages in all dimensions. As a corollary, we prove the sharp $L^p$ boundedness of the helical maximal operator in $\mathbb{R}^4$, which was previously known only for $\mathbb{R}^2$ and $\mathbb{R}^3$. We also improve previously known results in higher dimensions. The main new ingredient is a novel wave envelope estimate adapted to moment curves, which is a powerful tool in the proof of the local smoothing estimate.

math.CA↗
arXiv2025

On local smoothing estimates for wave equations

We prove sharp local smoothing estimates for wave equations on compact Riemannian manifolds in $n+1$ dimensions for odd $n$ and obtain improved estimates in even dimensions. This is achieved by deriving local smoothing estimates for certain Fourier integral operators. We also obtain improved local smoothing estimates for wave equations in Euclidean spaces.

math.AP↗
arXiv2024

Small cap square function estimates

We introduce small cap square function estimates for parabola and cone, and prove the sharp estimates. More precisely, we study the inequalities of form \[ \|f\|_p\le C_{α,p}(R) \Big\|(\sum_{γ\inΓ_α(R^{-1})}|f_γ|^2)^{1/2}\Big\|_p, \] where $Γ_α(R^{-1})$ is the set of small caps of width $R^{-α}$. We find the sharp constant $C_{α,p}(R)$.

math.CA↗
arXiv2024

Kakeya problem and projection problem for $k$-geodesics in Grassmannians

The Kakeya problem in $\mathbb{R}^n$ is about estimating the size of union of $k$-planes; the projection problem in $\mathbb{R}^n$ is about estimating the size of projection of a set onto every $k$-plane ($1\le k\le n-1$). The $k=1$ case has been studied on general manifolds in which $1$-planes become geodesics, while $k\ge 2$ cases were still only considered in $\mathbb{R}^n$. We formulate these problems on homogeneous spaces, where $k$-planes are replaced by $k$-dimensional totally geodesic submanifolds. After formulating the problem, we prove a sharp estimate for Grassmannians.

math.CA↗
arXiv2024

Exceptional set estimate through Brascamp-Lieb inequality

Fix integers $1\le k<n$, and numbers $a,s$ satisfying $0<s<\min\{k,a\}$. The problem of exceptional set estimate is to determine \[T(a,s):=\sup_{A\subset \mathbb{R}^n\ \text{dim}A=a}\text{dim}(\{ V\in G(k,n): \text{dim}(π_V(A))<s \}). \] In this paper, we prove a new upper bound for $T(a,s)$ by using Brascamp-Lieb inequality. As one of the corollary, we obtain the estimate \[T(a,\frac{k}{n}a)\le k(n-k)-\min\{k,n-k\}, \] which improves a previous result $T(a,\frac{k}{n}a)\le k(n-k)-1$ of He. By constructing examples, we can determine the explicit value of $T(a,s)$ for certain $(a,s)$: When $k\le \frac{n}{2}$, $β\in(0,1]$ and $γ\in(β,\frac{k}{n}(1+β)]$, we have \[T(1+β,γ)=k(n-k)-k.\] When $k\ge \frac{n}{2}$, $β\in(0,1]$ and $γ\in (β, (1-\frac{k}{n})+\frac{k}{n}β]$, we have \[T(n-1+β,k-1+γ)=k(n-k)-(n-k).\]

math.CA↗
arXiv2024

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

Let $γ: [-1, 1]\to \mathbb{R}^n$ 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 $γ$ at $θ\in [-1, 1]$ has Hausdorff dimension $\min\{m, \dim(E)\}$ for almost every $θ\in [-1, 1]$.

math.CA↗
arXiv2023

A Marstrand projection theorem for lines

Fix integers $1<k<n$. For $V\in G(k,n)$, let $P_V: \mathbb{R}^n\rightarrow V$ be the orthogonal projection. For $V\in G(k,n)$, define the map \[ π_V: A(1,n)\rightarrow A(1,V)\bigsqcup V. \] \[ \ell\mapsto P_V(\ell). \] For any $0<a<\text{dim}(A(1,n))$, we find the optimal number $s(a)$ such that the following is true. For any Borel set $\boldsymbol{A} \subset A(1,n)$ with $\text{dim}(\boldsymbol{A})=a$, we have \[ \text{dim}(π_V(\boldsymbol{A}))=s(a), \text{for a.e. } V\in G(k,n). \] When $A(1,n)$ is replaced by $A(0,n)=\mathbb{R}^n$, it is the classical Marstrand projection theorem, for which $s(a)=\min\{k,a\}$. A new ingredient of the paper is the Fourier transform on affine Grassmannian.

math.CA↗
arXiv2023

Restriction estimates for quadratic manifolds of arbitrary codimensions

The restriction conjecture is one of the famous problems in harmonic analysis. There have been many methods developed in the study of the conjecture for the paraboloid. In this paper, we generalize the multilinear method of Bourgain and Guth for the paraboloid, and obtain restriction estimates for all quadratic manifolds of arbitrary codimensions. In particular, our theorem recovers the main theorem of Bourgain and Guth for the paraboloid. A new ingredient is a covering lemma for varieties whose proof relies on Tarski's projection theorem in real algebraic geometry. We also provide algorithms to compute several algebraic quantities that naturally appear in the argument. These algorithms rely on a cylindrical decomposition in real algebraic geometry.

math.CA↗
arXiv2023

Hausdorff dimension of unions of $k$-planes

We prove a conjecture of Héra on the dimension of unions of $k$-planes. Let $0<k \le d<n$ be integers, and $β\in[0,k+1)$. If $\mathcal{V}\subset A(k,n)$, with $\text{dim}(\mathcal{V})=(k+1)(d-k)+β$, then $\text{dim}(\bigcup_{V\in\mathcal{V}}V)\ge d+\min\{1,β\}$. The proof combines a recent idea of Zahl and the Brascamp-Lieb inequality.

math.CA↗
arXiv2023

Exceptional set estimates in finite fields

We study the exceptional set estimate for projections in $\mathbb{F}_q^n$. For each $V\in G(k,\mathbb{F}^n_q)$, let $$ π_V: \mathbb{F}_q^n\rightarrow V $$ be the projection map. We prove the following result: If $A\subset \mathbb{F}_q^n$ with $\#A=q^a$ ($n-1\le a\le n$) and $0< s<\frac{a+n-2}{2}$, then $$ \# \{V\in G(n-1,\mathbb{F}^n_q): \#π_V(A)< q^s \}\lessapprox q^{n-2}.$$ This improves the previous range $0 \frac{a+n-2}{2}$, then the right hand side above should be at least $q^t$ for some $t>n-2$.

math.CA↗
arXiv2023

On Kakeya maps with regularity assumptions

In $\mathbb R^n$, we parametrize Kakeya sets using Kakeya maps. A Kakeya map is defined to be a map $$ϕ:B^{n-1}(0,1)\times [0,1]\rightarrow \mathbb{R}^{n}, (v,t)\mapsto (c(v)+tv,t),$$ where $ c:B^{n-1}(0,1)\rightarrow \mathbb{R}^{n-1}$. The associated Kakeya set is defined to be $ K:=\text{Im} (ϕ). $ We show that the Kakeya set $K$ has positive measure if either one of the following conditions is true. (1) $c$ is continuous and $c|_{S^{n-2}}\in C^α(S^{n-2})$ for some $α>\frac{(n-2)n}{(n-1)^2}$, (2) $c$ is continuous and $c|_{S^{n-2}}\in W^{1,p}(S^{n-2})$ for some $p>n-2$.

math.CA↗
arXiv2023

Exceptional set estimates for radial projections in $\mathbb{R}^n$

We prove two conjectures in this paper. The first conjecture is by Lund, Pham and Thu: Given a Borel set $A\subset \mathbb{R}^n$ such that $\dim A\in (k,k+1]$ for some $k\in\{1,\dots,n-1\}$. For $0<s<k$, we have \[ \text{dim}(\{y\in \mathbb{R}^n \setminus A\mid \text{dim} (π_y(A)) < s\})\leq \max\{k+s -\dim A,0\}. \] The second conjecture is by Liu: Given a Borel set $A\subset \mathbb{R}^n$, then \[ \text{dim} (\{x\in \mathbb{R}^n \setminus A \mid \text{dim}(π_x(A))<\text{dim} A\}) \leq \lceil \text{dim} A\rceil. \]

math.CA↗
arXiv2022

An exceptional set estimate for restricted projections to lines in $\mathbb{R}^3$

Let $γ:[0,1]\rightarrow \mathbb{S}^{2}$ be a non-degenerate curve in $\mathbb{R}^3$, that is to say, $\det\big(γ(θ),γ'(θ),γ''(θ)\big)\neq 0$. For each $θ\in[0,1]$, let $l_θ=\{tγ(θ):t\in\mathbb{R}\}$ and $ρ_θ:\mathbb{R}^3\rightarrow l_θ$ be the orthogonal projections. We prove an exceptional set estimate. For any Borel set $A\subset\mathbb{R}^3$ and $0\le s\le 1$, define $E_s(A):=\{θ\in[0,1]: \text{dim}(ρ_θ(A))<s\}$. We have $\text{dim}(E_s(A))\le 1+\frac{s-\text{dim}(A)}{2}$.

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 $\mathcal{H}^2(\pi_\theta (A))>0$.

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