SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 914-936, January 2021.
We prove a high-dimensional version of the Strichartz estimates for the unitary group associated to the free Zakharov--Kuznetsov equation. As a by-product, we deduce maximal estimates which allow us to prove local well-posedness for the generalized Zakharov--Kuznetsov equation in the whole subcritical case whenever $d \ge 4, k \ge 4,$ complementing the recent results of Kinoshita [Global Well-Posedness for the Cauchy Problem of the Zakharov-Kuznetsov Equation in 2D, 2019; Well-Posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation, 2019] and Herr and Kinoshita [Subcritical Well-Posedness Results for the Zakharvo-Kuznestov Equation in Dimension Three and Higher, 2020]. Finally, we use some of those maximal estimates in order to prove pointwise convergence results for the flow of the generalized Zakharov--Kuznetsov equation in any dimension, in the same spirit of Compaan, Lucà, and Staffilani [Pointwise convergence of the Schrödinger flow, Int. Math. Res. Not., 1 (2021), pp. 596--647]. With the use of recently proven estimates, we are able to establish pointwise results also for the original Zakharov--Kuznetsov equation in every dimension.

中文翻译：

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广义Zakharov-Kuznetsov方程的最大函数估计和局部适定性

SIAM数学分析杂志，第53卷，第1期，第914-936页，2021年1月。
我们证明了与自由Zakharov-Kuznetsov方程相关的unit群的Strichartz估计的高维版本。作为副产品，我们推导出最大估计值，这使得我们可以证明只要$ d \ ge 4，k \ ge 4，$补充最近的结果时，在整个亚临界情况下广义Zakharov-Kuznetsov方程的局部适定性。 Kinoshita [解决Zakharov-Kuznetsov方程在2D中的柯西问题的全球良好可能性，2019年；修正的Zakharov-Kuznetsov方程Cauchy问题的适定性，2019年； Herr和Kinoshita [Zakharvo-Kuznestov方程在3维及更高维上的亚临界适定性结果，2020年]。最后，为了证明广义Zakharov-Kuznetsov方程在任何维度上的流动都具有逐点收敛性，我们使用了一些最大估计值，这与Compaan，Lucà和Staffilani的精神相同[Schrödinger流的逐点收敛性，Int。数学。Res。1（2021），第596--647页]。使用最近得到证明的估计，我们还可以为每个维度的原始Zakharov-Kuznetsov方程建立逐点结果。