2022/08/22 by Bassam Shayya, Shayya, Bassam
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.2208.10305
openalex publication_date 2022/08/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Suppose S is a smooth compact hypersurface in \Bbb Rn and σ is an appropriate measure on S. If Ef= fdσ is the extension operator associated with (S,σ), then the Mizohata-Takeuchi conjecture asserts that ∫ |Ef(x)|2 w(x) dx ≤ C (supT w(T)) ‖ f ‖L2(σ)2 for all functions f ∈ L2(σ) and weights w : \Bbb Rn → [0,∞), where the sup is taken over all tubes T in \Bbb Rn of cross-section 1, and w(T)= ∫T w(x) dx. This paper investigates how far we can go in proving the Mizohata-Takeuchi conjecture in \Bbb R2 if we only take the decay properties of σ into consideration. As a consequence of our results, we obtain new estimates for a class of convex curves that include exponentially flat ones such as (t,e-1/tm), 0 ≤ t ≤ cm, m ∈ \Bbb N.