Stationary phase method for a saddle point near the boundary of the domain of integration (Q920324)

Let the functions \(\phi\) (x,y) and g(x,y) be analytic at 0, while 0 is a stationary saddle point of \(\phi\) \((\phi_ x=\phi_ y=0\), \(\phi_{xx}\phi_{yy}<\phi^ 2_{xy}\) for \(x=y=0)\). Furthermore, let h(x,y) be a finite, infinitely differentiable function which is identically equal to 1 on a neighborhood of 0. Let S be a domain in \(R^ 2\) and let its boundary \(\partial S\) be defined by an analytic function and let \(\partial S\) have a bounded curvature. For any mutual position of \(\partial\) and level lines \(E_ 1\) and \(E_ 2\) of the function \(\phi\) (x,y) issuing from 0 the author obtaines the asymptotic formulas for the function \[ I(\lambda)=\lambda \int \int_{S}e^{i\lambda \phi (x,y)}g(x,y)h(x,y)dxdy \] as \(\lambda\to \infty\) which are uniform with respect to sufficiently small distances \(\rho\) of \(\partial S\) and 0.