Some Sharp Restriction Inequalities on the Sphere

Abstract

In this paper we find the sharp forms and characterize the complex-valued extremizers of the adjoint Fourier restriction inequalities on the spherein the cases (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying: (a) k = 2, q ≥ 2 and 3 ≤ d ≤ 7;(b) k = 2, q ≥ 4 and d ≥ 8; (c) k ≥ 3, q ≥ 2k and d ≥ 2. We also prove a sharp multilinear weighted restriction inequality, with weight related to the k-fold convolution of the surface measure.Building up on the work of Christ and Shao [6, 7], Foschi [12] recently obtained the sharp form of (1.2) in the Stein-Tomas endpoint case (d, p, q) = (3, 4, 2), showing that the constant functions are global extremizers.Here we extend this paradigm to other suitable triples (d, p, q). In fact, defining, our first result is the following:Theorem 1. Let (d, p, q) = (d, 2k, q) with d, k ∈ N and q ∈ R + ∪ {∞} satisfying:

A sharp trilinear inequality related to Fourier restriction on the circle

“…(ω 1 , ω 2 , ω 3 ) ∈ (S 1 ) 3 . We now invoke [7,Theorem 4] (which is originally inspired in the work of Charalambides [8]) to conclude that there exist c ∈ C \ {0} and ν ∈ C 2 such that f (ω) = c e ν·ω for σ−a.e. ω ∈ S 1 .…”

Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

“…Symmetry D: this is the shearing transformation with x 1 and x 2 being exchanged: 9) with the vector field being changed to (1, u(x 1 − λ 4 x 2 , x 2 ) + λ 4 ).…”

“…Another difference from the L 2 case in [15] is that we will write the proof by using the δ-calculus, which has been used intensively in the Fourier restriction estimates, see [19], [14] and [9] for example. One significant advantage of the δ-calculus, which we will see shortly in the proof, is that it allows us to express everything in terms of the function h from the Main Theorem, instead of going back and forth between h and its inverse as in [15].…”

Sharp Strichartz inequalities for fractional and higher-order Schrödinger equations

“…from where one checks that xP m , P n y L 2 " 2 2n`1 δ`n " m˘, see [41, Corollary 2.16, Chapter 4]. See also [6,9,17,21,30] for earlier appearances of Legendre and other families of orthogonal polynomials in sharp Fourier restriction theory.…”