A local smoothing estimate in higher dimensions

Let N be a large parameter, C a constant, and let ΓN (C) denote the C-neighborhood of the cone segment {ξ : 2−CN ≤ |ξ| ≤ 2N}. For fixed N , we take a partition of unity subordinate to a covering of Sd−1 by caps Θ of diameter about N− 1 2 , and use this to form a (smooth) partition of unity yΘ on ΓN(C) in the natural way. We will write ΓN,Θ(C) = supp yΘ. Let ΞΘ be a function whose Fourier transform coincides with yΘ on ΓN(1). If the support of f̂ is contained in ΓN(1), we define