The Fourier extension operator on large spheres and related oscillatory integrals.pdf

a r X i v : m a t h / 0 6 0 8 1 2 9 v 2 [ m a t h .C A ] 2 1 S e p 2 0 0 6 THE FOURIER EXTENSION OPERATOR ON LARGE SPHERES AND RELATED OSCILLATORY INTEGRALS JONATHAN BENNETT AND ANDREAS SEEGER Abstract. We obtain new estimates for a class of oscillatory integral operators with folding canonical relations satisfying a curvature condition. The main lower bounds show- ing sharpness are proved using Kakeya set constructions. As a special case of the upper bounds we deduce optimal Lp(S2) → Lq(RS2) estimates for the Fourier extension op- erator on large spheres in R3, which are uniform in the radius R. Two appendices are included, one concerning an application to Lorentz space bounds for averaging operators along curves in R3, and one on bilinear estimates. 1. Introduction For functions g ∈ L1(Sd) on the d-dimensional unit sphere we define the Fourier extension operator to be the mapping E : g 7→ g?dσ where g?dσ(ξ) = ∫ Sd e?i〈x,ξ〉g(x)dσ(x), dσ denotes the rotation invariant measure on Sd induced by Lebesgue measure in Rd+1, and ξ ∈ Rd+1. We note that the adjoint of this operator is the Fourier restriction operator f 7→ f? ∣∣ Sd , where ? denotes the Euclidean Fourier transform in d + 1 dimensions. A substantial amount of recent work is concerned with weighted inequalities of the general form (1.1) ( ∫ |g?dσ|qdμ )1/q . ‖g‖Lp(Sd) for certain measures μ on Rd+1. 1 Perhaps the most notable instance of this is the case of Lebesgue measure, which corresponds to the classical Fourier restriction problem; see for example [20], [36], [39], [8] and [38]. In addition to this, the inequalities (1.1) for certain broader classes of measures μ are known to have applications to a variety of well- known and largely unsolved problems in partial differential equations, harmonic analysis and geometric measure theory; see [4], [33], [12], [13], [41], [10], [35], [26], [19], [18], and many further references contained in those papers. The content of the current paper is partially motiv