 Cutoff independence of an oscillatory integral

For an amplitude of symbol order $N$ in $k$ frequency variables, applying a phase integration operator $r>N+k$ times makes the pairing with a test function absolutely integrable. Derivatives of a large-frequency cutoff are supported where $|\theta |\asymp R$ and give an error bounded by $C{R}^{N-r+k}$ times finitely many test derivatives. That error tends to zero, making the resulting distribution independent of the cutoff and of the admissible number of integrations.