An endpoint estimate for the cone multiplier

Abstract

Abstract. In this paper we consider an endpoint estimate for high-dimensional cone multipliers.

Bochner–Riesz means associated with conical surfaces

“…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].…”

On Radial and Conical Fourier Multipliers

“…Sharp bounds in H p , p < 1 and sharp bounds for the operator acting on functions of the form f (x, t) = f 0 (|x|, t) can be found in Hong's articles [10], [11]. More recently, Heo, Hong and Yang [8] proved a weak type (1, 1) inequality for a localized cone multiplier χ(τ )ρ (d−1)/2 (ξ, τ ), in dimension d ≥ 4. As a corresponding result for the global cone multiplier one can prove that for Re (λ) = (d − 1)/2 the operator f → F −1 d+1 [ρ λ f ] is bounded from the Hardy space H 1 to L 1,∞ , under the assumption that Sph(p 1 , d) holds for some p 1 > 1.…”