Sharp variation-norm estimates for oscillatory integrals related to Carleson’s theorem

Abstract

We prove variation-norm estimates for certain oscillatory integrals related to Carleson's theorem. Bounds for the corresponding maximal operators were first proven by Stein and Wainger. Our estimates are sharp in the range of exponents, up to endpoints. Such variation-norm estimates have applications to discrete analogues and ergodic theory. The proof relies on square function estimates for Schrödinger-like equations due to Lee, Rogers and Seeger. In dimension one, our proof additionally relies on a local smoothing estimate. Though the known endpoint local smoothing estimate by Rogers and Seeger is more than sufficient for our purpose, we also give a proof of certain local smoothing estimates using Bourgain-Guth iteration and the Bourgain-Demeter ℓ 2 decoupling theorem. This may be of independent interest, because it improves the previously known range of exponents for spatial dimensions n ≥ 4.

Variation bounds for spherical averages

“…see [48,Ch.1]. One can also consult the paper by Bergh and Peetre [4] (who however work with a different type of variation space when r = 1) or refer to [16,Proposition 2.2]. Thus an inequality for the variation operator V I r A follows if we can control the B…”

“…Of particular interest is Lépingle's inequality on the r-variation of martingales for r > 2 [29] (see also [35], [8], [21], [33]) and its consequences on families of operators in ergodic theory and harmonic analysis; see e.g. the papers [20], [21], [34], [30], [16], [31], [32], which contain many other references.…”

New bounds for Stein's square function in $\mathbb{R}^3$

“…for any ε > 0 and p ≥ 3.25. Using the Proposition 4.2 in [GRY20] (which is a generalization of [See88]), it suffices to consider the local square function…”

A note on local smoothing estimates for fractional Schrödinger equations

“…However, for other α, an analogous implication has not been discovered and our result is new. For the case α > 1, we adapt the ideas of [GOW21a] and improve the results in [GRY20] (Theorem 1.6), [GMZ20], and in [GOW21a] (Corollary 1.5). The case 0 < α < 1 is slightly different from the case α > 1 as the manifold associated to the operator has negative Gaussian curvature.…”