arXiv subjects

Minxing Shen

Publications and source records attributed to Minxing Shen.

7 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↗

arXiv2025

$L^p\to L^q$ estimates for Stein's spherical maximal operators

In this article we consider a modification of the Stein's spherical maximal operator of complex order $α$ on ${\mathbb R^n}$: $$ {\mathfrak M}^α_{[1,2]} f(x) =\sup\limits_{t\in [1,2]} \big| {1\over Γ(α) } \int_{|y|\leq 1} \left(1-|y|^2 \right)^{α-1} f(x-ty) dy\big|. $$ We show that when $n\geq 2$, suppose $\|{\mathfrak M}^α_{[1,2]} f \|_{L^q({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})}$ holds for some $α\in \mathbb{C}$, $p,q\geq1$, then we must have that $q\geq p$ and $${\rm Re}\,α\geq σ_n(p,q):=\max\left\{\frac{1}{p}-\frac{n}{q},\ \frac{n+1}{2p}-\frac{n-1}{2}\left(\frac{1}{q}+1\right),\frac{n}{p}-n+1\right\}.$$ Conversely, we show that ${\mathfrak M}^α_{[1,2]}$ is bounded from $L^p({\mathbb R^n})$ to $L^q({\mathbb R^n})$ provided that $q\geq p$ and ${\rm Re}\,α>σ_2(p,q)$ for $n=2$; and ${\rm Re}\,α>\max\left\{σ_n(p,q), 1/(2p)- (n-2)/(2q) -(n-1)/4\right\}$ for $n>2$. The range of $α,p$ and $q$ is almost optimal in the case either $n=2$, or $α=0$, or $(p,q)$ lies in some regions for $n>2$.

math.CA↗

arXiv2024

The Spherical Maximal Operators on Hyperbolic Spaces

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We prove that when $n\geq 2$, for $\alpha\in\mathbb{R}$ and $1 1-n+n/p$ for $1 \max \{{(2-n)/p}-{1/(p p_n)},{(2-n)/p}- (p-2)/[p p_n(p_n-2)]\} $ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$.

math.FA↗

arXiv2023

Lp bounds for Stein's spherical maximal operators

Let ${\frak M}^α$ be the spherical maximal operators of complex order $α$ on ${\mathbb R^n}$. In this article we show that when $n\geq 2$, suppose \begin{eqnarray*} \|{\frak M}^α f \|_{L^p({\mathbb R^n})} \leq C\|f \|_{L^p({\mathbb R^n})} \end{eqnarray*} holds for some $α$ and $p\geq 2$, then we must have ${\rm Re}\,α\geq \max \{1/p-(n-1)/2,\ -(n-1)/p \}.$ When $n=2$, we prove that $\|{\frak M}^α f \|_{L^p({\mathbb R^2})} \leq C\|f \|_{L^p({\mathbb R^2})}$ if ${\rm Re}\ \ α>\max\{1/p-1/2,\ -1/p\}$, and hence the range of $α$ is sharp in the sense the estimate fails for ${\rm Re}\ α<\max\{1/p-1/2, -1/ p\}.$

math.AP↗

arXiv2023

On smoothing estimates for Schrödinger equations on product spaces $\mathbb{T}^m\times \mathbb{R}^n$

Let $Δ_{\mathbb{T}^m\times \mathbb{R}^n}$ denote the Laplace-Beltrami operator on the product spaces $\mathbb{T}^m\times \mathbb{R}^n$. In this article we show that $$ \left\|e^{itΔ_{\mathbb{T}^m\times \mathbb{R}^n}}f\right\|_{L^p(\mathbb{T}^m\times \mathbb{R}^n\times [0,1])} \leq C \|f\|_{W^{α,p}(\mathbb{T}^m\times\mathbb{R}^n)} $$ holds if $p\geq 2(m+n+2)/(m+n)$ and $α> (m+2n)(1/2-1/p)-2/p$. Furthermore, we apply the $\ell^2$-decoupling inequalities to establish local $L^p$-smoothing estimates for the Schrödinger operator $e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}$ in modulation spaces $M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)$: $$ \|e^{itΔ_{\mathbb{T}^m\times\mathbb{R}^n}}f\|_{L^p(\mathbb{T}^m\times\mathbb{R}^n\times [0,1])}\leq C \|f\|_{M_{p,q}^α(\mathbb{T}^m\times\mathbb{R}^n)} $$ for some range of $α$ and $p, q$. The smoothing estimates in $L^p$-Sobolev and modulation spaces are sharp up to the endpoint regularity, in a certain range of $p$ and $q$.

math.CA↗

arXiv2021

Boundedness criterion for integral operators on the fractional Fock-Sobolev spaces

We provide a boundedness criterion for the integral operator $S_φ$ on the fractional Fock-Sobolev space $F^{s,2}(\mathbb C^n)$, $s\geq 0$, where $S_φ$ (introduced by Kehe Zhu) is given by \begin{eqnarray*} S_φF(z):= \int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w}) dλ(w) \end{eqnarray*} with $φ$ in the Fock space $F^2(\mathbb C^n)$ and $dλ(w): = π^{-n} e^{-|w|^2} dw$ the Gaussian measure on the complex space $\mathbb{C}^{n}$. This extends the recent result in Cao--Li--Shen--Wick--Yan. The main approach is to develop multipliers on the fractional Hermite-Sobolev space $W_H^{s,2}(\mathbb R^n)$.

math.CV↗

arXiv2020

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock Space

We show that for an entire function $φ$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_φF(z)=\int_{\mathbb{C}^n} F(w) e^{z \cdot\bar{w}} φ(z- \bar{w})\,dλ(w), \ \ \ \ \ z\in \mathbb{C}^n, \end{eqnarray*} is bounded on ${\mathscr F}^2(\mathbb{C}^n)$ if and only if there exists a function $m\in L^{\infty}(\mathbb{R}^n)$ such that $$ φ(z)=\int_{\mathbb{R}^n} m(x)e^{-2\left(x-\frac{i}{2} z \right)\cdot \left(x-\frac{i}{2} z \right)} dx, \ \ \ \ \ \ z\in \mathbb{C}^n. $$ Here $dλ(w)= π^{-n}e^{-\left\vert w\right\vert^2}dw$ is the Gaussian measure on $\mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_φ$. Moreover, we obtain the reducing subspaces of $S_φ$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${\mathscr F}^2(\mathbb{C})$ on the complex plane ${\mathbb C}$ (Integr. Equ. Oper. Theory {\bf 81} (2015), 451--454).

math.CV↗