Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.20909

The $(n-1)$-linear restriction estimate in $\mathbb{R}^n$ with degenerate curvature

Ioan Bejenaru·Tiklung Chan
View original source ↗Read source document ↗Export metadata ↓

Abstract

We revisit the $(n-1)$-linear restriction estimate in $\mathbb{R}^n$ proved by the first author in \cite{Be-6} and refine the result in the case when the curvature assumptions contain small parameters. The standard $L^p$ estimate is replaced by a more complex norm/quasi-norm of type $\ell^p L^\frac{2}{n-2}$ which reflects the geometry of wave packet propagation and the degenerate curvature assumptions. The primary motivation for this work stems from a concrete application to the study of the quadratic nonlinear Schrödinger equation in $\mathbb{R}^{3+1}$, which is done in a companion paper \cite{BeBoCh}.

Explore related subjects

math.CAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ioan Bejenaru, Tiklung Chan. 2026-10-14. The $(n-1)$-linear restriction estimate in $\mathbb{R}^n$ with degenerate curvature. https://arxiv.org/abs/2610.20909

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reverse square function estimates for degenerate curves and their applications

Building on the classical work of Córdoba--Fefferman and the recent work of Schippa, we establish $L^4$ reverse square function estimates for functions whose Fourier support is contained in a $δ$-neighborhood of the curve $\{(ξ,|ξ|^a): |ξ|\leq 1\}$ in $\mathbb{R}^2$, for all exponents $a\in(0,\infty)\backslash\{1\}$. As applications, we derive sharp $L^4$ Strichartz estimates on the one-dimensional torus for fractional Schrödinger equations and establish new local smoothing estimates in modulation spaces. In the latter application, orthogonal Strichartz-type estimates also play a crucial role.

math.CA↗
arXiv2026

Gaussian rigidity and width of measures

Given $0\le m<n$ and a positive Radon measure $μ$ on $\mathbb R^n$ with at most polynomial growth, assume that it has finite $m$-dimensional Gaussian discrepancy, defined as $$\mathcal D(μ):=\limsup_{R\to\infty}\Bigl[R^2\sup_{x\in\mathrm{spt}μ}|G_R^μ(x)-1|\Bigr],$$ where $G_R^μ(x):=(4πR^2)^{-m/2}\int_{\mathbb R^n} e^{-|z-x|^2/(4R^2)}\,dμ(z)$. Provided that $μ$ admits a flat blow-down $\mathcal H^m\llcorner V$ (an assumption that cannot be dropped entirely), we prove that there exists a finite (smallest) $0\leδ<\infty$, the so-called width of $μ$, such that $$\mathrm{spt}μ\subseteq \bar B_δ(V+x_0),\quad δ^2\le C(m,n)\mathcal{D}(μ),$$ where $\bar B_δ(V+x_0)$ denotes the closed $δ$-neighborhood of a suitable translate of $V$. Note that the exponent is optimal because of scale invariance. In particular, when $\mathcal D(μ)=0$, the support of $μ$ is included in an $m$-dimensional affine plane. We use this fact to give a simpler proof of Preiss's rectifiability theorem: while still based on Gaussian averages of $μ$, this new proof avoids showing uniqueness (and hence conicality) of blow-downs of $m$-uniform measures, as well as studying Gaussian-weighted moments beyond second order. The strategy of proof consists of an anisotropic blow-down of $μ$, followed by an analysis of the resulting Young measure, using a rigidity lemma involving the heat semigroup.

math.CA↗
arXiv2026

Fourier decay and endpoint estimates for lacunary maximal operators

We study endpoint estimates for lacunary maximal operators associated with compactly supported measures. For every $d\geq1$, we construct a compactly supported, absolutely continuous probability measure on $\mathbb R^d$ with polynomial Fourier decay whose associated lacunary maximal operator fails to map $H^1(\mathbb R^d)$ boundedly into $L^{1,\infty}(\mathbb R^d)$. In particular, it fails to be of weak type $(1,1)$. This gives a negative answer to the weak type $(1,1)$ question posed by Seeger and Wright in 2011 and to Open Problems 14 and 15, contributed by Wright at the 2016 Conference in Harmonic Analysis, held at the University of Wisconsin--Madison in honor of Professor Michael Christ.

math.CA↗