= Cutoff independence of an oscillatory integral
{title2=$R^{N-r+k}\to0,\quad r>N+k$}
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 $CR^{N-r+k}$ 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.
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