Counterexamples to the convergence problem for periodic dispersive equations with a polynomial symbol

Abstract

In the setting of Carleson's convergence problem for the fractional Schr\"odinger equation $i\, \partial_t u + (-\Delta)^{a/2}u=0$ with $a > 1$ in $\mathbb R^d$, which has Fourier symbol $P(\xi) = |\xi|^a$, it is known that the Sobolev exponent $d/(2(d+1))$ is sufficient, but it is not known whether this condition is necessary. In this article, we show that in the periodic problem in $\mathbb T^d$ the exponent $d/(2(d+1))$ is necessary for all non-singular polynomial symbols $P$ regardless of the degree of $P$. Among the differential operators covered, we highlight the natural powers of the Laplacian $\Delta^k$ for $k \in \mathbb N$.