Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.13866

Characterizing higher-dimensional Hölder domains via spherical geometric functions

Emily Casey·Max Engelstein·Tatiana Toro
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we characterize the regularity of a domain in terms of the behavior of a certain spherical function. More precisely, given two domains we show that a certain spherical spectral function (first defined in \cite{FTV23}, see also \cite{AKN21}) has Hölder decay if and only if the domains are complementary and separated by a $C^{1,β}$-manifold. This extends the analysis in \cite{Ca24} to dimensions $\geq 3$ and in \cite{FTV23, CTV25} to regularity above the continuous threshold. Along the way, we introduce a spherical analogue of the Jones $β$-numbers and show that the decay of these numbers implies geometric regularity of the domain on which they are defined. We believe this result is of independent interest and partially addresses a question of Fleschler-Tolsa-Villa \cite[Question 1]{FTV23}.

Explore related subjects

math.CAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Emily Casey, Max Engelstein, Tatiana Toro. 2026-10-09. Characterizing higher-dimensional Hölder domains via spherical geometric functions. https://arxiv.org/abs/2610.13866

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↗