Condition for the boundedness of oscillatory integrals with cubic polynomials (Q1313672)

The Schrödinger type equation \(D_ t\Phi(x,t)= P(D_ x)\Phi(x,t)\), \(\Phi(x,t)|_{t=0}= f(x)\), with its generalized solution \(\Phi(x,t)= \int^ \infty_{-\infty} \widehat f(\lambda)\exp[2\pi i(\lambda x+ tP(\lambda))]d\lambda\), the polynomials \(P(\lambda)\) being of the third order, is considered \((\widehat f(\lambda)\) is the Fourier transform). It is proved that for a real finite function \(f(x)\in H_{1,2}\) (Hölder's class) the above generalized solution is bounded on an arbitrary compact set of points \((t,x)\in R^ 2\).