  This paper presents Strichartz estimates for the linearized 1D periodic
Dysthe equation on the torus, namely estimate of the $L^6_{x,t}(\mathbb{T}^2)$
norm of the solution in terms of the initial data, and estimate of the
$L^4_{x,t}(\mathbb{T}^2)$ norm in terms of the Bourgain space norm. The paper
also presents other results such as bilinear and trilinear estimates pertaining
to local well-posedness of the 1-dimensional periodic Dysthe equation in a
suitable Bourgain space, and ill-posedness results in Sobolev spaces.
