Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.17752

Cone restriction via direction-restricted two-ends Furstenberg estimates

Xiangyu Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We investigate a direct two-ends approach to restriction for the truncated cone in $\R^n$, which gives the diagonal restriction range \[ p>2+\frac{4(7n+15)}{11n^2-n-24},\qquad n\ge10. \] This improves the range obtained by applying the result of Wang--Wu and then slicing.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Xiangyu Wang. 2026-10-13. Cone restriction via direction-restricted two-ends Furstenberg estimates. https://arxiv.org/abs/2610.17752

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

KEEP EXPLORING

Related papers

arXiv2026

A discussion of two arguments related to Fefferman's Fourier extension theorem in the plane

The Fourier extension conjecture of E. Stein was proved in the plane in 1970 by C. Fefferman, see also Zygmund and Carleson and Sjölin, with simplifications given by other authors later on, in particular by L. Hörmander and T. Tao. We discuss yet two more arguments for this classical theorem on the parabola. The first argument uses C. Fefferman's decoupling together with a decomposition into Haar wavelets. This sets the stage for the second argument whose point of departure is the bilinear characterization of Tao, Vargas and Vega, and relies on the bilinear interplay with the classical wave packet constructions and discrete characterizations with an induction on scales. Each of the above two arguments rely on some form of Fefferman's convolution decoupling and the special nature of the critical planar index 4 as a positive even integer, but the second might be more amenable to weighted generalizations.

math.CA↗
arXiv2026

Turán problem on the union of three intervals of equal length

This paper addresses the Turán extremal problem on the symmetric three-interval set $Ω_λ= (-1,1) \cup (λ-1,λ+1) \cup (-λ-1,-λ+1)$, $λ\geq 0$. We establish a discrete approximation principle relating the continuous Turán problem on $Ω_λ$ to the discrete Turán problem on an associated finite discrete set. This enables the continuous problem to be exactly expressed as a limit of finite nonnegative trigonometric polynomial problems, which are equivalent to finite semidefinite programs. Using this framework, we compute the Turán constant for integer and rational parameters $λ$. We further analyze the dependence of the Turán constant on $λ$ and derive upper bounds for all $λ> 5$.

math.CA↗
arXiv2026

Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals

We continue our study of multilinear rough homogeneous singular integrals by establishing weighted estimates for their maximal truncations. Let $m\ge2$ and let $Ω\in L^q(\mathbb{S}^{mn-1})$, $1<q\le\infty$, have mean zero. For $1<p_1,\dots,p_m<\infty$ with $p_j\ge q'$ and $1/p=\sum_{j=1}^m1/p_j$, we prove $$\big\Vert \mathcal{L}_Ω^*(f_1,\dots,f_m)\big\Vert_{L^p(v_{\vec{\boldsymbol{w}}})} \lesssim \VertΩ\Vert_{L^q(\mathbb{S}^{mn-1})} \prod_{j=1}^m\Vert f_j\Vert_{L^{p_j}(w_j)}$$ whenever $\vec{\boldsymbol{w}}=(w_1,\dots,w_m)\in A_{(p_1/q',\dots,p_m/q')}$, where $q'$ is the Hölder conjugate of $q$ and $v_{\vec{\boldsymbol{w}}}=\prod_{j=1}^m w_j^{p/p_j}$. The estimate holds under the same multiple weight condition as the corresponding principal value estimate.

math.CA↗