  We prove local well-posedness of partially periodic and periodic modified
KP-I equations, namely for $\partial_t u+(-1)^{\frac{l+1}{2}}\partial^l_x
u-\partial_x^{-1}\partial_y^2 u+u^2\partial_x u=0$ in the anisotropic Sobolev
space $H^{s,s}(\mathbb{R}\times \mathbb{T})$ if $l=3$ and $s>2$, in
$H^{s,s}(\mathbb{T}\times \mathbb{T})$ if $l=3$ and $s>\frac{19}{8}$, and in
$H^{s,s}(\mathbb{R}\times \mathbb{T})$ if $l=5$ and $s>\frac{5}{2}$. All three
results require the initial data to be small.
