This result is neither stronger nor weaker than the Sobolev space result [@Taka], in the sense than neither result implies the other. However, Bourgain’s Fourier-Lebesgue space result also enables him to obtain a probabilistic global well-posedness result for . This will be discussed further below.

The Hamiltonian conservation together with the Galgiardo-Nirenburg-Sobolev inequality yield global existence in the energy space $H^1({\mathbb{T}}) \times L^2({\mathbb{T}})$.

In the two-dimensional case, the sharp Sobolev space local theory remains at the same level – it holds for initial data $(u_0, n^\pm_0) \in H^\frac12({\mathbb{T}}^2) \times L^2({\mathbb{T}}^2)$. However, to prove this, Kishimoto employed modified Besov-type Fourier restriction norms [@Kish2]. Similar arguments had previously been used by Bejenaru-Herr-Holmer-Tataru [@BHHT] to prove the parallel result for the Zakharov system on ${\mathbb{R}}^2$. We will use these norms as well in our work on ${\mathbb{T}}$.

In the ${\mathbb{T}}^2$ setting, Kishimoto has also obtained global existence below the energy space, in $H^{\frac9{14}+}({\mathbb{T}}^2) \times L^2({\mathbb{T}}^2)$ for Schrödinger data with sufficiently small $L^2({\mathbb{T}}^2)$ norm (in relation to that of the ground-state solution) [@Kish2]. The result is established using the I-method of Colliander-Keel-Staffilani-Takaoka-Tao [@CKSTT]. This result also applies to ${\mathbb{T}}$, since a solution to on ${\mathbb{T}}$ corrresponds to a solution on ${\mathbb{T}}^2$ which is constant 