2025/05/21 by Sanghyuk Lee, Lee, Sanghyuk, Sewook Oh +1
Mathematics · #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Primary 42B20 #Secondary 42B10
paper · pdf · doi:10.48550/arxiv.2505.15492
openalex publication_date 2025/05/21 · openalex created_date 2025/10/11 · openalex updated_date 2026/07/28
Let H⊂ \Rd+1 be a compact, convex, analytic hypersurface of finite type with a smooth measure σ on H. Let κ denote the Gaussian curvature on H. We consider the oscillatory integral (κ1/2 σ)^\wedge with the damping factor κ1/2 and prove the optimal decay estimate |(κ1/2 σ)^\wedge(ξ)|≤ C|ξ|-d/2 for d=2,3, and with an extra logarithmic factor for d=4. Our result provides an essentially complete answer, since such decay estimates generally fail for d ≥ 5, even for convex analytic hypersurfaces, as shown by Cowling--Disney--Mauceri--Müller. Furthermore, we prove the same estimates for (κ1/2+it σ)^\wedge with C growing polynomially in |t|. As consequences, we obtain the best possible estimates for the convolution, maximal, and adjoint restriction operators associated with H, incorporating the mitigating factors of optimal orders. In particular, for d=2, 3, we prove the L2--L2(d+2)/(d+4) restriction estimate with respect to the affine surface measure κ1/(d+2) σ. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich.