Improved Hörmander's theorem and new methods for oscillatory integral operators (Q2655590)

The following result for the oscillatory integral operator \[ T_{\lambda} (f)(x)=\int_{\mathbb{R}^n} e^{i\lambda \Phi (x,y)}\, \psi (x,y)\, f(y)\, dy \] is due to Hörmander: Suppose that the mixed Hessian of the phase function \(\Phi\) satisfies a non-degeneracy condition on the support of \(\psi\) as follows \[ \det \left (\frac{\partial^2 \Phi (x,y)}{\partial x_i \partial y_j}\right ) \neq 0, \qquad 1\leq i,j \leq n. \tag{1} \] Then \[ \|T_{\lambda}(f)\|_{L^2 (\mathbb{R}^n)}\leq C\, \lambda^{-n/2}\|f\|_{L^2 (\mathbb{R}^n)}. \] The article is devoted to the improvement of this theorem in the case \(n=1\) without imposing the non-degeneracy condition (1).