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

Abstract

The sharp Wolff-type decoupling estimates of Bourgain-Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.2010 Mathematics Subject Classification. 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". Implications of this kind will be discussed in detail in §4.

Sharp estimates for oscillatory integral operators via polynomial partitioning

“…This theorem is a variable coefficient generalisation of a decoupling inequality due to Bourgain [6]. It can be established by adapting the argument of [6] using many of the techniques employed in the current article: see also [2]. 32 Summing together the contributions from the various spatial balls B K 2 , it remains to estimate the decoupled contributions }T λ f τ } L p pBRq .…”

“…Suppose T λ is a Hörmander-type operator. For all ε ą 0 the estimate }T λ f } L p pR n q À ε,φ,a λ ε }f } L p pB n´1 q (1.4) 2 Given a (possibly empty) list of objects L, for real numbers Ap, Bp ě 0 depending on some Lebesgue exponent p the notation Ap À L Bp or Bp Á L Ap signifies that Ap ď CBp for some constant C " C L,n,p ě 0 depending on the objects in the list, n and p. In addition, Ap " L Bp is used to signify that Ap À L Bp and Ap Á L Bp. holds uniformly for λ ě 1 whenever p satisfies p ě 2¨n`1 n´1 if n is odd;…”

Local smoothness of Fourier integral operators and related research

“…当 n = 2 时, 利用 Beltran 等 [41] 建立的最佳型 L 6 -估计与平凡的 L 2 -估计插值, 只能重新获得文 献 [22] 中的 L 4 -L 4 相应结果. 然而, 通过双线性方法, 可以改进文献 [22] 中对应的局部光滑性估计.…”

$${\mathbf {L}}^{{\mathbf {p}}}$$-Sobolev Estimates for a Class of Integral Operators with Folding Canonical Relations

“…We now consider (a, b) close to (a • , b • ) and construct changes of variables so that in the new coordinates theconstant coefficient decoupling theorem in Proposition 6.1 can be applied at suitable scales. The idea of applying a constant coefficient decoupling theorem in a variable coefficient situation also appears in [3].…”