Well-posedness for the Cauchy Problem of the Modified Zakharov-Kuznetsov Equation

This paper is concerned with the Cauchy problem of modified Zakharov-Kuznetsov equation on $\mathbb{R}^d$. If $d=2$, we prove sharp estimate which implies local in time well-posedness Sobolev space $H^s(\mathbb{R}^2)$ for $s \geq 1/4$. $d 3$, by employing $U^p$ and $V^p$ spaces, establish small data global scaling critical $H^{s_c}(\mathbb{R}^d)$ where $s_c = d/2-1$.