Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions

Abstract

We prove that the range of exponents in the general L 2 Fourier restriction theorem due to Mockenhaupt, Mitsis, Bak, and Seeger is sharp for a large class of measures on R d . This extends to higher dimensions the sharpness result of Hambrook and Laba.

Endpoint Fourier restriction and unrectifiability

“…We conclude referring the reader to the survey [10] for an overview on the restriction problem for fractal measures. We just mention here the results on the restriction for general measures due to Mockenhaupt [13], Mitsis [12] and Bak-Seeger [3], and the proof of sharpness of the previous results by Hambrook-Laba [9].…”

Finite field analogue of restriction theorem for general measures

“…By modifying the construction of Hambrook and Laba [6], Chen [4] proved that the range is sharp for any 0 < α ≤ β < 1. For n ≥ 2, Hambrook and Laba [7] proved that the range of q is sharp when n−1 < β ≤ α < n. We summarise the known results in the following.…”

“…Thus the measure µ satisfies the conditions (6), (7) of Theorem 1.4. Let f = 1 A , then the estimate (17) implies that…”

“…Theorem 1.4. Let 0 < β < α < n. Then there exists a probability measure µ on F n such that µ satisfies (6), (7), and if q < q n,α,β then there is τ = τ n,α,β,q > 0 such that R * (2 → q) |F| τ .…”

“…Theorem 1.5. Let 0 < α < n. Then for any 0 < q < q n,α,α there exists a probability measure µ on F n such that µ satisfies (6), (7), and R * (2 → q) |F| τ for some τ = τ n,α,q > 0.…”