Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator
Abstract
For P=X12+⋯ + Xn2, let TtPf(x) denote the solution to the linear Schrödinger equation at time t. In 1980, Carleson asked for the minimal regularity of an initial data function f∈ Hs(ℝn) that guarantees pointwise convergence of TtPf(x) to f(x) as t→ 0. This was resolved by Bourgain, who constructed counterexamples for the Schrödinger maximal operator to show that s≥ n/(2(n+1)) is necessary, and Du and Zhang, who proved that s> n/(2(n+1)) is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schrödinger maximal operator, where P = X12−X22 ± X32± ⋯ ± Xn2, and proved that, for all n≥ 2, s≥ 1/2 is necessary and s>1/2 is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of s≥ 1/2 for an infinite class of polynomial symbols P.
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BibTeX
@misc{chu2026counterexamples,
title = {{Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator}},
author = {Rena Chu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30048},
eprint = {2608.30048},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30048},
}
RIS
TY - UNPB
TI - Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator
AU - Rena Chu
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.30048
UR - https://arxiv.org/abs/2608.30048
AB - For P=X_1^2+⋯ + X_n^2, let T_t^Pf(x) denote the solution to the linear Schrödinger equation at time t. In 1980, Carleson asked for the minimal regularity of an initial data function f∈ H^s(ℝ^n) that guarantees pointwise convergence of T_t^Pf(x) to f(x) as t→ 0. This was resolved by Bourgain, who constructed counterexamples for the Schrödinger maximal operator to show that s≥ n/(2(n+1)) is necessary, and Du and Zhang, who proved that s> n/(2(n+1)) is sufficient. Rogers, Vargas, and Vega studied the analogous question for the non-elliptic Schrödinger maximal operator, where P = X_1^2−X_2^2 ± X_3^2± ⋯ ± X_n^2, and proved that, for all n≥ 2, s≥ 1/2 is necessary and s>1/2 is sufficient. In this paper, we construct counterexamples for generalizations of the non-elliptic case and prove a necessary condition of s≥ 1/2 for an infinite class of polynomial symbols P.
ER -