On Radial and Conical Fourier Multipliers

Abstract

We investigate connections between radial Fourier multipliers on R d and certain conical Fourier multipliers on R d+1 . As an application we obtain a new weak type endpoint bound for the Bochner-Riesz multipliers associated with the light cone in R d+1 , where d ≥ 4, and results on characterizations of L p → L p,ν inequalities for convolutions with radial kernels.

Local smoothing estimates for some half-wave operators

“…Compared to the Bochner-Riesz problem, little is known for the local smoothing estimate for the wave operator and the cone multiplier problem. See [1,7] and [3,5,6,10] for the Bochner-Riesz and the cone multiplier problems, respectively.…”

Bochner–Riesz means associated with conical surfaces

“…In low-dimensional cases d 1 = 2, 3, 4 and d 2 = 1, the most recent results in this direction are due to Garrigós and Seeger [1], and Garrigós et al [2]. And in high-dimensional cases d 1 ≥ 5 and d 2 = 1, the results are due to Heo [3], and Heo et al [4,5]. See also [6][7][8][9][10][11][12].…”

“…Although the case d 1 ≥ 4, d 2 = 1 was already studied in [5] by Heo et al to describe the technical and geometrical differences between the cases d 2 = 1 and d 2 ≥ 2, we treat the case d 2 = 1 again in Section 3. In fact the result for the case d 1 ≥ 4, d 2 = 1 and λ > λ(p) can be deduced from the estimate (2.5) which was the main inequality of [4].…”

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]). …”

“…+ f by following the argument in [22] (the proof of Corollary 1.3). 2 Hence, in particular, (3.10) holds for 1 < p < 2d d+3 when d ≥ 3.…”