Restriction of Fourier Transforms to Curves Ii: Some Classes With Vanishing Torsion

Abstract

We consider the Fourier restriction operators associated to certain degenerate curves in R d for which the highest torsion vanishes. We prove estimates with respect to affine arclength and with respect to the Euclidean arclength measure on the curve. The estimates have certain uniform features, and the affine arclength results cover families of flat curves.2000 Mathematics subject classification: 42B10, 42B99.

Convolution with measures on flat curves in low dimensions

“…Here are two questions which are raised by Theorem 1.1: Having no idea how to attack these interesting questions in their natural generality, we follow the usual practice of asking what can be said along such lines by imposing additional hypotheses on φ. The requirement n j =1 φ (d) (s j ) 1/n Aφ (d) s 1 + · · · + s n n , (1.2) to hold for s j ∈ (a, b), was used with n = d in [3] to obtain Fourier restriction estimates for curves (1.1). It is obvious that if β d then condition (1.2) holds with A = 1 for φ(t) = t β on the interval (0, ∞).…”

Endpoint Lp→Lq bounds for integration along certain polynomial curves

“…where the I ij are pairwise disjoint, and each I ij is a union of O d (1) open intervals. Each I ij is associated to the real number b i = Re η i , and for each i, j and s ∈ I ij ,…”

“…Theorem 1. Let d 4, let P : R → R d be a polynomial of degree N , and let T P be the operator defined in (1). Let p d := d+1 2 and q d := d+1 2 d d−1 .…”

“…A detailed history of this problem may be found in [10], for instance, along with some recent results. Other articles in this vein include the recent work of Bak, Oberlin and Seeger [2,1] and the less recent work of Drury and Marshall [12,13] and of Sjölin [26] (and of course Drury's [11]). …”

An affine-invariant inequality for rational functions and applications in harmonic analysis

“…One then expects (2) to hold for large classes of curves in R n with a corresponding uniform bound C; this has been investigated by a number of authors, see for example, [1], [2], [6], [9], [10], [11], [13] and [15]. However simple examples show that (2) can fail if L R (t) changes sign too often and so the class of rational curves is natural to consider as the number of sign changes of L R is controlled by d. Finally a nice feature is that on the critical line p = n(n + 1)/2 q, the estimate (2) is affine-invariant.…”