Let σ be the surface measure on a smooth hypersurface ℋ ⊂ ℝᵈ⁺¹. A fundamental subject in harmonic analysis is to determine the decay of σ̂. For nondegenerate ℋ, the stationary phase method yields the optimal decay, while sharp bounds in the degenerate case are known only in limited situations. In this work, we are concerned with the oscillatory estimate
|(κ¹ᐟ²σ)̂(ξ)| ≤ C|ξ|⁻ᵈᐟ²,
for convex analytic surfaces ℋ, where κ is the Gaussian curvature. The damping factor κ¹ᐟ² is expected to recover the optimal decay, as suggested by the stationary phase expansion, but the work of Cowling–Disney–Mauceri–Muller shows that such bounds fail in general for d ≥ 5 even when the surface is convex and analytic. However, it has remained open whether the estimate holds in lower dimensions 2 ≤ d ≤ 4. We establish it for d = 2, 3, and with a logarithmic loss for d=4. Our approach is inspired by the stationary set method of Basu–Guo–Zhang–Zorin- Kranich. We also discuss applications to convolution, maximal, and adjoint restric- tion operators. This is joint work with Sewook Oh.
