Sharp decay of the spherical averages of the Fourier transform of open sets

For a bounded domain Ω⊂ Rᵈ, the spherical average E_Ω(R)=∫_ Sᵈ⁻¹|1_Ω(Rω)|²dω is never O(R⁻ᵃ) with $a>d+1$, and for smooth boundaries it is O(R⁻ᵈ⁻¹), as it is for the ball, whether or not the boundary is curved. In 2003, Brandolini, Hofmann and Iosevich extended this to every convex domain and to every domain with C1,1/2 boundary, and this regularity condition has not been improved since. How much regularity does the decay R⁻ᵈ⁻¹ require? We show that among Hölder classes the answer is C1,1/4. Every bounded domain with C1,1/4 boundary satisfies E_Ω(R)≤ C_ΩR⁻ᵈ⁻¹, while for every $0