Fourier extension for extremal quadratic submanifolds

2016/02/15 by Philip T. Gressman, Gressman, Philip T.
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.CA

paper · pdf · doi:10.48550/arxiv.1602.04789

This paper has been withdrawn by the author as this problem has been previously solved by D. M. Oberlin (Canad. Math. Bull. Vol. 48 (2), 2005 pp. 260--266) using very similar methods

openalex publication_date 2016/02/15 · arxiv created 2016/02/16 · arxiv updated 2016/02/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note establishes the full range of L^p--L^q Fourier extension estimates for the model n-dimensional quadratic submanifold in \mathbb R^n(n+3)/2 parametrized by γ(x₁,…,x_n) := (x₁,…,x_n, (x_i x_j)_{1 ≤ i ≤ j ≤ n}). This class of submanifolds is extremal in the sense that an n-dimensional quadratic submanifold of \mathbb R^d can only satisfy nontrivial Fourier extension inequalities when d ≤ (n(n+3))/(2). The proof is via an inflation-type argument, with the unexpected twist that a significant amount of "overinflation" is necessary but in no way limits the sharpness of the argument.

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