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We prove a high-dimensional version of the Strichartz estimates for theunitary group associated to the free Zakharov--Kuznetsov equation. As aby--product, we deduce maximal estimates which allow us to prove localwell-posedness for the generalized Zakharov--Kuznetsov equation in the wholesubcritical case whenever $d \ge 4, k \ge 4,$ complementing the recent resultsof Kinoshita and Herr--Kinoshita. Finally, we use some of those maximalestimates in order to prove pointwise convergence results for the flow of thegeneralized Zakharov--Kuznetsov equation in any dimension, in the same spiritof a recent manuscript by Compaan, Luc\`a and Staffilani.

Maximal function estimates and local well-posedness for the generalized Zakharov--Kuznetsov equation