Uniform periodic counterexamples to Carleson's convergence problem with polynomial symbols

TL;DR

This paper demonstrates that for dispersive equations on the torus with polynomial symbols, a specific Sobolev exponent is necessary for convergence, contrasting with Euclidean cases where the same exponent is only sufficient.

Contribution

It establishes the necessity of the Sobolev exponent for convergence in the periodic setting for all non-singular polynomial symbols, including powers of the Laplacian.

Findings

Sobolev exponent d/(2(d+1)) is necessary for convergence

Contrast with Euclidean case where the exponent is only sufficient

Uniform periodic counterexamples provided

Abstract

In Carleson's convergence problem for dispersive equations $i\, \partial_t u + P(D)u=0$ in the periodic setting $\mathbb T^d$, we prove that the Sobolev exponent $d/(2(d+1))$ is necessary for any non-singular polynomial symbol $P$, including the natural powers of the Laplacian $\Delta^k$. This is in contrast with the results known in the Euclidean case, in which for symbols $P(\xi) = |\xi|^a$ with $a > 1$ the exponent $d/(2(d+1))$ is sufficient, but we do not know if it is necessary.