Generalizations of the Schr\"odinger maximal operator: building arithmetic counterexamples

TL;DR

This paper extends the study of pointwise convergence of Schr"odinger solutions to more general dispersive equations with polynomial symbols, constructing counterexamples using number theory to show divergence in broader settings.

Contribution

It introduces a new class of counterexamples for pointwise convergence failure for general polynomial symbols, including indecomposable forms, using advanced number-theoretic techniques.

Findings

Counterexamples for pointwise a.e. convergence for general polynomial symbols

Use of Weil bound and Dwork-regular forms in construction

First counterexamples for indecomposable polynomial forms

Abstract

Let $T_t^{P_2}f(x)$ denote the solution to the linear Schr\"odinger equation at time $t$, with initial value function $f$, where $P_2 (\xi) = |\xi|^2$. In 1980, Carleson asked for the minimal regularity of $f$ that is required for the pointwise a.e. convergence of $T_t^{P_2} f(x)$ to $f(x)$ as $t \rightarrow 0.$ This was recently resolved by work of Bourgain, and Du and Zhang. This paper considers more general dispersive equations, and constructs counterexamples to pointwise a.e. convergence for a new class of real polynomial symbols $P$ of arbitrary degree, motivated by a broad question: what occurs for symbols lying in a generic class? We construct the counterexamples using number-theoretic methods, in particular the Weil bound for exponential sums, and the theory of Dwork-regular forms. This is the first case in which counterexamples are constructed for indecomposable forms, moving beyond special regimes where $P$ has some diagonal structure.