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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

A phase function is a real smooth function $\Phi$ on $X\times ({\mathbb{R}}^k\setminus \{0\})$, positively homogeneous of degree one in $\theta$, with $d_{x,\theta }\Phi \ne 0$. The symbol classconsists of smooth amplitudes satisfying, on each compact $K\subset X$,

To define the oscillatory integral, insert a cutoff $\chi (\epsilon \theta )$ equal to one near zero and setRepeated integration by parts with an operator $L$ satisfying $L{e}^{i\Phi }=e^{i\Phi }$ makes the integral absolutely convergent after enough iterations and shows that the limit defines a distribution.

Although $|u(x)|=e^{x^2}$ has superpolynomial growth, the rapidly varying phase makes $u$ an oscillatory tempered distribution. Split the integral against $\phi \in \mathcal{S}(\mathbb{R})$ into $|x|\le 1$ and the two tails. On a tail, with $\Phi (x)=e^{x^4}$,Integration by parts transfers the derivative toThis function and its derivative are integrable because $e^{x^2-x^4}$ dominates every polynomial, and the boundary term at infinity vanishes. The result is bounded by finitely many Schwartz space seminorms. The compact part has the same property. Thus the cutoff integrals converge and define a continuous linear functional:

For any smooth amplitude $v(x,\lambda )$,because $D_x=-i{\partial }_x$. Applying each term of the differential operator under the oscillatory integral givesThe identity is justified distributionally by regularizing the frequency integral and integrating by parts.

Iterate the correction: after constructing $b_{j-1}\in S^{-(j-1)}$, setThe same calculation improves the remainder by one order:Let $A_j$ and $B_N$ be the corresponding inverse oscillatory integrals. ThenWhen $N>n$, the frequency integral defining $B_N$ converges absolutely and depends continuously on $x$, so $B_N\in C(X)$. This completes the finite-order parametrix construction.

In one base and one phase variable, take $\Phi =x\theta$ and $a=x\eta (\theta )/|\theta |$, with an even smooth cutoff function equal to zero near zero and one for large $|\theta |$. This is a symbol of order $-1$. Its oscillatory integral is $-2x\log |x|$ plus a smooth function near zero. Thus it is singular there even though the restricted function $a(0,\cdot )$ has empty support. The joint closed conic support of an oscillatory amplitude retains the limiting base point and its stationary directions. A support-sensitive regularity theorem must use that conic neighborhood information, not merely vanishing of an amplitude at one base point.

For the constant-velocity linear transport equation $u_t+c\cdot {\nabla }_{x}u=0$, a Dirac delta distribution initially at zero moves along $x=ct$. Its spacetime distribution pairs by $⟨u,\phi ⟩=\int \phi (ct,t)dt$, and its initial trace is obtained by weak convergence of distributions. The Fourier representation of the Dirac delta function writes it as an oscillatory integral with phase function $(x-ct)\cdot \theta$. Its singular support is precisely the trajectory; it has order-zero distribution regularity as a measure, without becoming a smooth function.

The sine and cosine integrals accumulate $\sin t/t$ and $\cos t/t$, with different normalizations at zero and infinity. They arise in oscillatory integrals, Fourier transforms of singular kernels and Fourier series of logarithmic profiles.