A family of fractal Fourier restriction estimates with implications on the Kakeya problem

2022/06/26 by Shayya, Bassam
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2206.12971

Abstract

In a recent paper [Ann. of Math. 189 (2019), 837--861], Du and Zhang proved a fractal Fourier restriction estimate and used it to establish the sharp L² estimate on the Schrödinger maximal function in \Bbb Rⁿ, n ≥ 2. In this paper, we show that the Du-Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in \Bbb Rⁿ that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in \Bbb R².

Related

• Sharp L² estimate of Schrödinger maximal function in higher dimensions
• Restriction and Kakeya maximal estimates in ℝ⁴
• Improved bounds for the Kakeya maximal conjecture in higher dimensions
• Discrete Fourier restriction associated with Schrodinger equations
• Knapp-type obstructions in multilinear fractal Fourier extension