Counterexamples for high-degree generalizations of the Schr\"odinger maximal operator

TL;DR

This paper introduces a new number-theoretic method to analyze high-degree generalizations of the Schrödinger maximal operator, establishing lower bounds on regularity that surpass previous barriers in multiple dimensions.

Contribution

Develops a flexible, number-theoretic approach using Weil bounds to derive new lower bounds for the regularity of high-degree Schrödinger maximal operators, exceeding longstanding barriers.

Findings

For any integer k ≥ 2, if the degree k Schrödinger maximal operator is bounded, then s ≥ 1/4 + (n-1)/(4((k-1)n+1)).

In dimensions n ≥ 2, for k ≥ 3, this is the first result surpassing the 1/4 barrier.

Uses Weil bounds and the Riemann Hypothesis over finite fields in the analysis.

Abstract

In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space $H^s(\mathbb{R}^n)$ that implies pointwise convergence for the solution of the linear Schr\"odinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schr\"odinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. Analogues of Carleson's question remain open for many other dispersive PDE's. We develop a flexible new method to approach such problems, and prove that for any integer $k\geq 2$, if a degree $k$ generalization of the Schr\"odinger maximal operator is bounded from $H^s(\mathbb{R}^n)$ to $L^1(B_n(0,1))$, then $s \geq \frac{1}{4} + \frac{n-1}{4((k-1)n+1)}.$ In dimensions $n \geq 2$, for every degree $k \geq 3$, this is the first result that exceeds a long-standing barrier at $1/4$. Our methods are number-theoretic, and in particular apply the Weil bound, a consequence of the truth of the Riemann Hypothesis over finite fields.