A probabilistic analogue of the Fourier extension conjecture

TL;DR

This work develops a probabilistic Fourier extension framework by averaging extensions over smooth Alpert multipliers, establishing a probabilistic analogue of the Fourier extension conjecture in all dimensions $n\ge 2$ for exponents with $p>\frac{2n}{n-1}$. The approach builds robust $L^{p}$ control via compactly supported smooth Alpert frames, enabling precise $L^{2}$ and averaged $L^{4}$ estimates for the associated pseudo-projections and a Fourier square function bound equivalent to the extension bound. A careful decomposition into bilinear subforms, coupled with decay principles from radial integration by parts, vanishing moments, stationary phase, and tangential integration by parts, allows the authors to bound the majority of subforms deterministically and to resolve the resonant cases through averaging over involutive Alpert multipliers. Interpolation between $L^{2}$ and averaged $L^{4}$ estimates completes the argument, yielding a probabilistic Fourier extension theorem that mirrors known deterministic limitations while extending validity through probabilistic averaging, with potential implications for related restriction problems and two-weight inequalities. Overall, the results advance the harmonic analysis of Fourier restriction by introducing probabilistic averaging over structured wavelet frames to manage resonance phenomena.

Abstract

We prove a probabilistic Fourier extension theorem that says Fourier extension holds when averaged over certain smooth Alpert multipliers. The proofs use smooth Alpert wavelets with the classical techniques of stationary phase and interpolation of L^2 and L^4 estimates. The correct L^4 bounds for resonant forms require an expectation over Alpert multipliers.

Theorems & Definitions (64)

• Conjecture 1
• Lemma 2
• Lemma 3
• Conjecture 4
• Theorem 5: Probabilistic extension conjecture
• Theorem 7
• Remark 8
• Definition 11
• Definition 12
• proof : Proof of Lemma \ref{['prob trunc']}