Affiliation:
1. Duke University, 120 Science Drive, Durham, NC 27708, USA
Abstract
AbstractIn 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.
Funder
NSF CAREER
Sloan Research Fellowship
AMS Joan and Joseph Birman Fellowship
Publisher
Oxford University Press (OUP)
Reference68 articles.
1. On the dimension of divergence sets of dispersive equations;Barceló;Math. Ann.,2011
2. Local smoothing and convergence properties of Schrödinger type equations;Ben-Artzi;J. Funct. Anal.,1991
3. On the Schrödinger maximal function in higher dimensions;Bourgain;Tr. Mat. Inst. Steklova,2013
4. A note on the Schrödinger maximal function;Bourgain;J. Anal. Math.,2016
5. On the Vinogradov mean value;Bourgain;Proc. Steklov Inst. Math.,2017
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