Finite order of an oscillatory integral distribution ($\mathrm{ord}{I}_{\Phi }(a)\le r,r>N+k$)

An oscillatory integral with a degree-one homogeneous phase function, $k$ frequency variables, and oscillatory integral amplitude in fixed symbol class order $N$ has order of a distribution at most any nonnegative integer $r>N+k$. The integer works on every compact set; the continuity constant can vary with the compact set.