A Fourier restriction theorem based on convolution powers

We prove a Fourier restriction estimate under the assumption that certain convolution power of the measure admits an $r$-integrable density.


Introduction
Let F be the Fourier transform defined on the Schwartz space bŷ where ξ, x is the Euclidean inner product. We are interested in Borel measures µ defined on R d for which F maps L p (R d ) boundedly to L 2 (µ); i.e.
Here " " means the left-hand side is bounded by the right-hand side multiplied by a positive constant that is independent of f . If µ is a singular measure, then such result can be interpreted as a restriction property of the Fourier transform. Such restriction estimates for singular measures were first obtained by Stein in the 1960's. If µ is the surface measure on the sphere, the Stein-Tomas theorem [12], [13] states that (1) holds for 1 ≤ p ≤ 2(d+1) d+3 . Mockenhaupt [10] and Mitsis [9] have shown that Tomas's argument in [12] can be used to obtain an L 2 -Fourier restriction theorem for a general class of finite Borel measures satisfying where 0 < α, β < d; they showed that (1) holds for 1 ≤ p < p 0 = 4(d−α)+2β 4(d−α)+β . Bak and Seeger [2] proved the same result for the endpoint p 0 and further strengthened it by replacing the L p 0 -norm with the L p 0 ,2 -Lorentz norm.
It is well known that if µ is the surface measure on a compact C ∞ manifold then the sharpness can be tested by some version of Knapp's homogeneity argument. See e.g. the work by Iosevich and Lu [6] who proved that if µ is the surface measure on a compact hypersurface and if F : d+3 , then the Fourier decay assumption (2) is satisfied with α = d − 1. For general measures satisfying (2) and (3), there is no Knapp's argument available to prove the sharpness of p 0 . Here we show that indeed for certain measures the restriction estimate (1) holds in a range of p beyond the range given above. This will follow from a restriction estimate based on an assumption on the n-fold convolution µ * n = µ * · · · * µ.
Remarks. (i) It is not easy to construct measures supported on lower dimensional sets for which Corollary 1 applies. Remarkably, Körner showed by a combination of Baire category and probabilistic argument that there exist "many" Borel probability measures µ supported on compact sets of Hausdorff dimension 1/2 so that µ * µ ∈ C c (R 1 ).

Apply Hölder's inequality again, this is bounded by
where we have used Young's inequality in the last line. Since 1 s ′ − 1 qr = 1 q ′ , we obtain (5) after taking the nth root. Proposition 1. Let µ be a Borel probability measure on R d . If µ * n satisfies (3) with 0 ≤ α ≤ d, then µ satisfies (3) with exponent α/n.
Proof. Assume to the contrary that given k, µ(B r k ) ≥ kr α/n k for some ball B r k with radius r k > 0. Let B * nr k = B r k + · · · + B r k be the n-fold Minkowski sum, then µ * n (B * nr k ) ≥ µ(B r k ) n ≥ k n r α k . On the other hand, since µ * n satisfies (3), Proposition 2. Let µ be a Borel probability measure on R d supported on a compact set of Hausdorff dimension 0 ≤ γ < d, then Proof. Assume to the contrary that μ s < ∞ for some 2 < s < 2d/γ. Then This decay in R → ∞ implies γ ≥ 2d/s (cf. [14], Corollary 8.7). Since 2d/s > γ, we obtain a contradiction.