We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Counterexamples for High-Degree Generalizations of the Schrödinger Maximal Operator.
- Authors
An, Chen; Chu, Rena; Pierce, Lillian B
- 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 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. Analogues of Carleson's question remain open for many other dispersive partial differential equations. 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 Schrödinger 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.
- Subjects
PARTIAL differential equations; RIEMANN hypothesis; SCHRODINGER equation; SOBOLEV spaces; FUNCTION spaces; FINITE fields; SCHRODINGER operator
- Publication
IMRN: International Mathematics Research Notices, 2023, Vol 2023, Issue 10, p8371
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnac088