A priori estimates for quasilinear equations related to the Monge-Ampère equation in two dimensions

We provea priori inequalities for non-subelliptic quasilinear equations related to the Monge-Ampère equation in two dimensions, for example, equations of the type1$$L_w = \partial _x^2 w + \partial _y \left[ {k\left( {x,w\left( {x,y} \right)\partial _y w} \right)} \right] = 0$$.