SOME SHARP BOUNDS FOR THE CONE MULTIPLIER OF NEGATIVE ORDER IN ${\mathbb R}^3$

Abstract

This paper considers the cone multiplier operator which is defined by 2). For −3/2 < µ < −3/14, sharp L p − L q estimates and endpoint estimates for S µ are obtained.

Bilinear restriction estimates for surfaces with curvatures of different signs

“…This type of estimate was used not only for the restriction problem but it also has applications for a variety of related problems (see [14,16] and references therein, and also [4,7,8,9]). The estimate (1.3) was first formulated by Bourgain [4] with φ 1 = φ 2 = |ξ|, n = 2 and separation condition (1.4), and he showed it for some p > 2 − for some > 0.…”

Remarks on the multiplier operators associated with a cylindrical distance function

“…In fact (2.8) can be deduced from Heo's results by the standard Carleson-Sjölin reduction and asymptotic expansion for kernels as before (see Lemma 2.1 and [21]). Alternatively, the estimate without even ϵ-loss can also be deduced from sharp local smoothing estimate for the wave equation which is obtained in [13] 1 (also see [22]).…”

Sharp Lp-Lq estimates for Bochner–Riesz operators of negative index in Cho1, Kim2, Lee3 et al. 2005Journal of Functional AnalysisSelf Cite

“…We first obtain some bilinear estimates for T and then using a decomposition technique introduced in [20] and technical Lemma 2.5, we get linear estimates for T . This kind of method to obtain sharp bounds for T in terms of from bilinear estimates was already used in [11,13].…”

“…But the arguments in those work can be simplified by using the following lemma to be used several times throughout this paper. We borrow from [11] the following, which is a multilinear extension of a result implicit in [4] (also see [6]). For a simple proof, we refer the readers to [11].…”

“…We borrow from [11] the following, which is a multilinear extension of a result implicit in [4] (also see [6]). For a simple proof, we refer the readers to [11]. Let L p,r denote Lorentz space equipped with norm · p,r .…”