Abstract: In 1980 Carleson posed a question on the minimal regularity of an initial data function in a Sobolev space that implies pointwise convergence for the solution of the linear Schrödinger equation. After progress by many authors, this was recently resolved (up to the endpoint) by Bourgain, whose counterexample construction for the Schrödinger maximal operator proved a necessary condition on the regularity, and Du and Zhang, who proved a sufficient condition. In this talk we describe how Bourgain’s counterexamples can be constructed from first principles. Then we describe a new flexible number-theoretic method for constructing counterexamples, which proves a necessary condition for high-degree analogues of the Schrödinger maximal operator to be bounded from $H^s$ to local $L^1$.

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