Sharp L2→L4 extension inequality for the Euclidean hyperbolic paraboloid

Determine the sharp Fourier extension inequality at the Stein–Tomas L2→L4 endpoint for the Euclidean hyperbolic paraboloid HR := {(ξ1, ξ2, τ) ∈ R2 × R : τ = ξ1^2 − ξ2^2} by identifying the exact best constant and fully characterizing all maximizers for the inequality ||(fσ)∨||L4(R3) ≤ C ||f||L2(HR, dσ), where dσ is the natural surface measure on HR.

Background

The paper proves sharp L2→L4 Fourier extension inequalities over finite fields for several quadratic surfaces, including the hyperbolic paraboloid, and completely characterizes maximizers in that discrete arithmetic setting. In particular, Theorem 1.4 establishes the sharp constant for the finite-field hyperbolic paraboloid and classifies all extremizers.

The authors explicitly note that the corresponding problem in the Euclidean setting is still unresolved. The Euclidean analogue concerns the surface HR := {(ξ1, ξ2, τ) ∈ R2 × R : τ = ξ1^2 − ξ2^2}, for which the sharp endpoint extension inequality and extremizer classification remain open, as indicated by related works referenced in [7, 10].