Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.16123

Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a $C^{1,\mathrm{Dini}}$ curve

Ji Li·Chun-Yen Shen·Eric Sawyer·Brett Wick
View original source ↗Read source document ↗Export metadata ↓

Abstract

Let $0\leqα n-3$ in a bounded tube and $n-3<β<n-2$ in $\R^n$. The endpoints fail logarithmically whenever the measure is locally finite. In dimension three, the corresponding whole-space range for the Biot--Savart field with arbitrary vector densities is $0<β<1$.

Explore related subjects

math.CA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ji Li, Chun-Yen Shen, Eric Sawyer, Brett Wick. 2026-10-12. Two-weight T1 theorem for fractional Riesz transforms: one measure supported on a $C^{1,\mathrm{Dini}}$ curve. https://arxiv.org/abs/2610.16123

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↗