Oscillatory integrals with polynomial phases
Let \(S(x, y)\) be a homogeneous polynomial in \((x,y)\in \mathbb{R} \times \mathbb{R}\) of degree \(n\) and define the operator \(T\) by \[ (T\varphi) (x)= \int e^{i\lambda S(x, y)} \chi(y) \varphi(y) dy, \qquad \varphi\in C_0^\infty (\mathbb{R}), \] where \(\chi\) is a fixed \(C^\infty\) cut-off function on \(\mathbb{R}\) with compact support. The main theorem of the paper is: The operator \(T\) extends to a bounded operator on \(L^2 (\mathbb{R})\) with operator norm \(|T|\leq C|\lambda |^{-1/n}\) when \(|\alpha_{(n/2) -1}|+ \cdots+ |\alpha_0 |\neq 0\) for \(n\) even and \(|\alpha_{(n-1) /2-1} |+ \cdots+ |\alpha_0 |\neq 0\) for \(n\) odd. Here \(\alpha_j\) are the coefficients of \(S(x,y)\).
scientific article; zbMATH DE number 167385