We present the local and the global existence theorems for the elliptic-hyperbolic Davey-Stewartson equation which does not allows the classical energy estimates. To overcome this difficulty, we make use of the smoothing property of linear Schrödinger type equations which was obtained by S. Doi. Then under the smallness condition to $L^{2}$-norm of the initial data, we get the local solution. Moreover we show the global existence of small amplitude solutions by the a priori estimates for which the null gauge condition of Y. Tsutsumi plays an important role.