Sharp Local Smoothing Estimates for Fourier Integral Operators

Abstract

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

A new approach to the Fourier extension problem for the paraboloid

“…6 is known to imply 1.1, and it was recently solved by Guth, Wang and Zhang in 2 + 1 dimensions in [19]. We refer the reader to the survey [5] for a more detailed account of the local smoothing conjecture.…”

Square function estimates and Local smoothing for Fourier Integral Operators

“…More recent partial results can be found in [8,9]. See the excellent survey article by Beltran, Hickman and Sogge [1] for a detailed discussion of all three conjectures.…”

“…When the underlying manifold is Euclidean, i.e. M = R d with flat metric, it was conjectured by Sogge [22] that for p ≥ p d = 2d d−1 , the space-time norm u L p (R d × [1,2]) is bounded by suitable Sobolev norms of the initial data, which represents a gain in regularity compared to fixed time estimates. Then for any p ≥ p d = 2d d−1 , σ < 1 p , we have…”

Characterizations of Hardy spaces for Fourier integral operators

“…In particular, by combining (1.6) with (1.5), one sees that the local smoothing estimates in (1.2) are invariant under application of the operators in Corollary 1.2. Regarding the invariance of (1.2) under operators as in (1), it should be noted that a variable coefficient version of (1.2) holds on compact manifolds [2].…”