Loomis-Whitney-type inequalities and low regularity well-posedness of the periodic Zakharov-Kuznetsov equation

Abstract: Local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in the fully periodic case with initial data in Sobolev spaces $H<sup>s$, $s&gt;1$, is proved. Frequency dependent time localization is utilized to control the derivative nonlinearity. The new ingredient to improve on previous results is a nonlinear Loomis-Whitney-type inequality.