2022/06/13 by Bassam Shayya, Shayya, Bassam
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2206.06325
openalex publication_date 2022/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose 0 < α≤ n, H: \Bbb Rn → [0,1] is a Lebesgue measurable function, and Aα(H) is the infimum of all numbers C for which the inequality ∫B H(x) dx ≤ C Rα holds for all balls B ⊂ \Bbb Rn of radius R ≥ 1. After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form ‖ Ef ‖Lp(B,Hdx) ≤ C RεAα(H)1/p ‖ f ‖Lq(σ) have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer's distance set conjecture in geometric measure theory. This paper improves on the known estimates when E is the extension operator associated with the unit paraboloid \mathcal P ⊂ \Bbb R3, reaching the full possible range of p,q exponents (up to the sharp line) for p ≥ 3 + (α-2)/(α+1) and 2 < α≤ 3.