Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation

Abstract

Abstract. We prove L q estimates related to Sogge's conjecture for a class of Fourier integral operators associated with wave equations.

Variable coefficient Wolff-type inequalities and sharp local smoothing estimates for wave equations on manifolds

“…Primary: 35S30, Secondary: 35L05. 1 3 Such inequalities are also conjectured to hold at the endpoint (that is, the case σ = 1/p) and endpoint estimates have been obtained for a further restricted range of p in high-dimensional cases: see [24] and [29].…”

“…Primary: 35S30, Secondary: 35L05. 1 3 Such inequalities are also conjectured to hold at the endpoint (that is, the case σ = 1/p) and endpoint estimates have been obtained for a further restricted range of p in high-dimensional cases: see [24] and [29].4 The examples in [32] concern certain oscillatory integral operators of Carleson-Sjölin type, defined with respect to the geodesic distance on M . Their results lead to counterexamples for local smoothing estimates via a variant of the well-known implication "local smoothing ⇒ Bochner-Riesz".…”

Square function estimates and Local smoothing for Fourier Integral Operators

“…Conjecture 1.2 and 1.3 were studied in parallel to the developments of Conjecture 1.1 [21,17]. As mentioned above, Beltran, Hickman and Sogge [2] established Conjecture 1.3 for odd spatial dimensions by proving a variable coefficient analog of Bourgain-Demeter decoupling inequality.…”

Regularity of fractional maximal functions through Fourier multipliers

“…This forces us to study the behavior in L p with large p more carefully than what is needed to understand L p mapping properties of the spherical maximal function. We take advantage of the sharp local smoothing estimate for the wave equation in L n−1 (R n ), which is available whenever n ≥ 5 thanks to recent advances in decoupling theory (see [6,14,15,24,39] and [3,19,26,29,35] for more on decoupling and local smoothing estimates). We remark that results in n = 4 could be obtained upon further progress on local smoothing estimates.…”