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

Abstract

Abstract. We extend an affine invariant inequality for vector polynomials established in [6] to general rational functions. As a consequence we obtain sharp universal estimates for various problems in euclidean harmonic analysis defined with respect to the so-called affine arclength measure.

Endpoint Lp→Lq bounds for integration along certain polynomial curves

“…We note that Dendrinos, Folch-Gabayet and Wright have recently shown in [8] that Theorem 2 extends to d-tuples of rational functions of bounded degree. It therefore seems likely that Theorem 1 could be generalized to give bounds for convolution with affine arclength along curves parametrized by such functions, but the author has not investigated the extent to which the arguments in this article would need to be changed.…”

Restriction estimates for space curves with respect to general measures

“…Here D a h is given by (12). Then by Lemma 3.1 it follows that (32) γ h,a τ (t) = (t a 1 ϕ 1 (ht), t a 2 ϕ 2 (ht), .…”

Restriction of the Fourier transform to some complex curves

“…(For higher dimensions there is a similar representation, defined recursively, which involves integrals of φ (d) (z). See [4,15,12].) Hence, by our assumption that Lemma 4.2 holds for d ≥ 3, it follows that Next, we change variables in the integral (6.11) and use the Plancherel theorem.…”