Square function estimates and Local smoothing for Fourier Integral Operators

Abstract

We prove a variable coefficient version of the square function estimate of Guth-Wang-Zhang. By a classical argument of Mockenhaupt-Seeger-Sogge, it implies the full range of sharp local smoothing estimates for 2 + 1 dimensional Fourier integral operators satisfying the cinematic curvature condition. In particular, the local smoothing conjecture for wave equations on compact Riemannian surfaces is completely settled.

Remarks on dimension of unions of curves

“…) is a positive bump function such that a = 1 on U ×J ×V . Sharp local smoothing estimates for Af were shown by Gao-Liu-Miao-Xi [10] for d = 2, and Beltran-Hickman-Sogge [3] for higher dimensions. Theorem 3.1 ( [3,10]).…”

“…Sharp local smoothing estimates for Af were shown by Gao-Liu-Miao-Xi [10] for d = 2, and Beltran-Hickman-Sogge [3] for higher dimensions. Theorem 3.1 ( [3,10]). Suppose Φ t satisfies the conditions (1.8) and (1.9) on the support of a. Then…”

“…Proving a sharp estimate for a maximal average over a neighborhood of G x,t in R 2 , Zahl [34] showed that a Borel set containing curves G x,t for every 0 < t < 1 has Hausdorff dimension 2 (see also [18]). By utilizing the local smoothing estimates in [10,3], we obtain the following.…”