On sharp Fourier extension from spheres in arbitrary dimensions

TL;DR

This work advances sharp restriction-type inequalities for Fourier extension from the sphere in all dimensions $d\ge 3$ by proving the existence of a positive correction term $a_\star(d)$ that yields equality only for constant data. The approach combines admissible weight decompositions, magical and non-magical identities, and a detailed spectral analysis of associated kernels via the Funk–Hecke formula and Gegenbauer/ Rodrigues calculus. A central mechanism reduces the problem to nonpositivity of a family of eigenvalues $\Lambda_{\mathrm{M}}(2\ell)$ and $\Lambda_{\mathrm{NM}}(2\ell)$, which is achieved by an inductive construction of weights $h_n$ and, for $d\le 7$, by direct sign control. The odd- and even-dimensional cases are treated separately, with an explicit inductive scheme for $d\ge 8$ and a self-contained verification for lower dimensions, contributing to the broader conjecture that the extremizers are constants when $a_\star=0$.

Abstract

We prove a new family of sharp $L^2(\mathbb S^{d-1})\to L^4(\mathbb R^d)$ Fourier extension inequalities from the unit sphere $\mathbb S^{d-1}\subset \mathbb R^d$, valid in arbitrary dimensions $d\geq 3$.

Theorems & Definitions (11)

• Theorem 1.1
• Remark 1.1
• Proposition 2.1
• proof : Proof sketch of Proposition \ref{['prop_fund_mechanism']}
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1: DX13
• Definition 3.2: Falling factorial
• proof : Proof of Lemma \ref{['lem_FoschiEvNonPos']}
• Remark 3.1