On the failure of the factorization condition for non-degenerate Fourier integral operators (Q2781294)

In the context of the article under review, a Fourier integral operator \(T\) has the form NEWLINE\[NEWLINETu(x) =\int_Y\int_{\mathbb{R}^n} e^{i\Phi(x,y,\theta)}a(x,y,\theta) u(y)dyd\theta,\quad x\in X,NEWLINE\]NEWLINE where \(X\) and \(Y\) are open subsets of \(\mathbb{R}^n\) and the amplitude function \(a(x,y,\theta)\) belongs to the Hörmander class \(S^m_{1,0}\) locally uniformly in \(x,y\). The assumption \(\partial_y \partial_\theta \Phi(x,y,\theta)\) is a non-degenerate real function positively homogeneous in \(\theta\) of degree one. Under these conditions the integral above has a meaning as an oscillatory integral. The continuity properties of the operator are related to the geometric properties of a manifold defined locally as NEWLINE\[NEWLINE\Lambda_\Phi= \{(x,y,d_x\Phi,d_y\Phi):d_\theta\Phi(x,y,\theta)=0\}.NEWLINE\]NEWLINE Indeed, if there is a number \(k\) so that given any fixed point \(\lambda_0\) in the manifold the canonical projection \(\pi_{X\times Y}\) from \(T^*(X\times Y)\) to \(X\times Y\) can be represented as the composition of the local restriction of \(\pi_{X\times Y}\) with some smooth map \(\pi_{\lambda_0}\) with rank \((d\pi_{\lambda_0})= n+k\), then it was proved by \textit{A. Seeger, C. D. Sogge} and \textit{E. M. Stein} [Ann. Math. 134, 231-251 (1991; Zbl 0754.58037)] that the Fourier integral operator is continuous from \(L^p_{\text{com}}(Y)\) into \(L^p_{\text{loc}}(X)\) for \(1 <p< \infty\) and \(m\leq -k|\frac 1p-\frac 12|\). The geometric condition just stated is called the smooth factorization condition. Subsequently, \textit{M. Ruzhansky} [Hokkaido Math. J. 28, 357-362 (1999; Zbl 0933.35203)] showed that the order \(-k|\frac 1p-\frac 12|\) is optimal.NEWLINENEWLINENEWLINEIn the paper under review M. Ruzhansky shows that the smooth factorization condition is not necessary to obtain the sharp continuity result. The author works in \(\mathbb{R}^3\) and considers the phase function NEWLINE\[NEWLINE\Phi(x,y,0)=\langle x - y,\theta\rangle -\frac{1}{\theta_3}(y_1\theta_1 + y_2\theta_2)^2NEWLINE\]NEWLINE defined in the cone \(|\theta_1,\theta_2|\leq C|\theta_3|\) for some \(C > 0\). The factorization condition fails in this case, in which \(k = 1\). Still the author shows that the operators satisfy the continuity property in the optimal range \(m\leq -|\frac 1p-\frac 12|\). The example can be extended to higher dimensions. NEWLINENEWLINENEWLINEThe paper ends with a discussion of translation invariant operators for which the factorization condition fails.