Publications [#375268] of Lillian B. Pierce

Papers Published

1. Chu, R; Pierce, LB, Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples, Journal d'Analyse Mathematique, vol. 151 no. 1 (December, 2023), pp. 59-114 [doi]
   (last updated on 2024/07/16)

   Abstract:
   Let TtP2f(x) denote the solution to the linear Schrödinger equation at time t, with initial value function f, where P 2(ξ) = ∣ξ∣2. In 1980, Carleson asked for the minimal regularity of f that is required for the pointwise a.e. convergence of TtP2f(x) to f(x) as t → 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.