Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples


Chu, R; Pierce, LB


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.


Chu, R., and L. B. Pierce. “Generalizations of the Schrödinger maximal operator: building arithmetic counterexamples.” Journal d’Analyse Mathematique 151, no. 1 (December 1, 2023): 59–114.
Cover:Journal d'Analyse Mathématique

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