Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 327 / 3 / a

A phase function is a real smooth function $\Phi$ on $X\times ({\mathbb{R}}^k\setminus \{0\})$ that is positively homogeneous of degree one in $\theta$ and has nonzero total differential $d_{x,\theta }\Phi$. The symbol classconsists of smooth amplitudes for which, for every compact $K\subset X$ and all multi-indices $\alpha ,\beta$,

Choose a smooth cutoff function $\chi$ equal to one near zero. The associated oscillatory integral is defined on a test function $f$ byOn the compact $x$-support of $f$, use an integration-by-parts operator $L$ satisfying $L{e}^{i\Phi }=e^{i\Phi }$. Repeated application of its formal adjoint lowers the effective symbol order until the integral is absolutely convergent. The resulting bounds involve only finitely many derivatives of $f$, prove that the limit is independent of $\chi$, and give the seminorm estimate required for

Applying $D_x^{\alpha }$ changes only the constants in the defining symbol estimates, while every $D_{\theta }$ lowers the power of $⟨\theta ⟩$ by one. Hence

The Leibniz rule writes every derivative of $a_1{a}_2$ as a finite sum of products of derivatives of the two factors. Multiplying the corresponding symbol estimates adds their orders, so

If $b$ is positively homogeneous of degree $M$ for large $|\theta |$, then $D_{\theta }^{\beta }b$ is positively homogeneous of degree $M-|\beta |$, while $x$-derivatives preserve the degree. These derivatives are uniformly bounded on the unit sphere when $x$ ranges over a compact subset of $X$. Rescaling $\theta$ givesat large frequency, and smoothness controls the remaining compact region. Therefore