Practical way to check whether a distribution is conormal

Question

Let $X$ be a $C^\infty$-manifold, $Y$ be its $C^\infty$-submanifold. Hörmander defines the set $I^m(X,Y)$ of conormal distributions as the set of all $u \in \mathscr D'(X)$ such that $$ L_1 \ldots L_N u \in {}^{\infty}H^{\mathrm{loc}}_{(-m-n/4)}(X), \quad n = \dim X, $$ for any smooth vector fields $L_1$, $\ldots$, $L_N$ on $X$ that are tangent to $Y$ and for any $N$, where ${}^\infty H^{\mathrm{loc}}_{(-m-n/4)}(X)$ is the Besov space.

Is there some practical way to check whether some distribution $u \in \mathscr D'(X)$ is in fact from $I^m(X,Y)$ or I should always check the above inclusions? What are the examples of the most simple nontrivial conormal distributions for general $X$ and $Y$? If I take, e.g. $Y = f^{-1}(0)$ for some smooth submersion $f \colon X \to \mathbb R$ and take some $C^\infty$-density $\mu$ on $Y$ and then define for any $v \in C^\infty_c(X)$ $\langle \delta_\mu, v \rangle = \int_Y v\mu$ is it possible to say that $\delta_\mu \in I^m(X,Y)$ for some $m$?

Answer 1

Let us take a look at the case where $Y\equiv x_n=0$ in $\mathbb R^n$.

Let $a(\underbrace{\overbrace{x_1,\dots,x_{n-1}}^{x'},x_n}_x; \xi_n)$ be a symbol in $S^m(\mathbb R^n\times \mathbb R)$ and let us consider the following "oscillatory integral": $$ u(x)=\int_{\mathbb R} e^{ix_n\cdot \xi_n} a(x,\xi_n) d\xi_n. $$ This integral is not absolutely converging and we could define it as a distribution. Note that we may assume that $a$ is supported where $\vert \xi_n\vert\ge 1$ since the contribution of a compactly supported $a$ is a smooth function. Then we have for $\phi$ in the Schwartz space, $l\in 2\mathbb N$, formally integrating by parts, \begin{align*} \langle u,\phi\rangle_{\mathscr S',\mathscr S}&=\int_{\mathbb R^n\times\mathbb R} \vert \xi_n\vert^{-l} \vert D_{x_n}\vert^l (e^{ix_n\cdot \xi_n}) a(x,\xi_n)\phi(x) dx d\xi_n\\ &=\sum_{\alpha+\beta=l}c_{\alpha\beta}\int_{\mathbb R^n\times\mathbb R} e^{ix_n\cdot \xi_n} \vert \xi_n\vert^{-l} (D_{x_n}^\alpha a)(x,\xi_n)(D_{x_n}^\beta\phi)(x) dx d\xi_n, \end{align*} which makes sense for $l>1+m$. We can take the last integral as the definition of the duality product. We note also that $u$ can easily be differentiated with respect to $x'$ (tangential derivatives) since $\partial_{x'}a\in S^m$. However the distribution $u$ is more difficult to differentiate wrt $x_n$ since $\xi_na\in S^{1+m}$; however calculating $$ x_n\partial_{x_n} u=\int_{\mathbb R} x_n\partial_{x_n}(e^{ix_n\cdot \xi_n}) a(x,\xi_n) d\xi_n +\int_{\mathbb R} e^{ix_n\cdot \xi_n} x_n\partial_{x_n}(a)(x,\xi_n) d\xi_n. $$ $$ =\int_{\mathbb R} x_n i\xi_n e^{ix_n\cdot \xi_n} a(x,\xi_n) d\xi_n +\int_{\mathbb R} e^{ix_n\cdot \xi_n} x_n\partial_{x_n}(a)(x,\xi_n) d\xi_n. $$ $$ =\int_{\mathbb R} \partial_{\xi_n}(e^{ix_n\cdot \xi_n} )\xi_n a(x,\xi_n) d\xi_n +\int_{\mathbb R} e^{ix_n\cdot \xi_n} x_n\partial_{x_n}(a)(x,\xi_n) d\xi_n. $$ $$ =-\int_{\mathbb R} e^{ix_n\cdot \xi_n} \underbrace{\partial_{\xi_n}(\xi_n a(x,\xi_n))}_{\in S^m} d\xi_n +\int_{\mathbb R} e^{ix_n\cdot \xi_n} \underbrace{x_n\partial_{x_n}(a)(x,\xi_n)}_{\in S^m} d\xi_n. $$ We can prove in particular that the wave-front-set of $u$ is included in the conormal bundle of $Y$, i.e. $$ WF u\subset\{(x',0;0,\xi_n)\}_{x'\in \mathbb R^{n-1}, \xi_n \in \mathbb R^*}$$ Your simple layer $u(x)= \omega(x')\otimes \delta(x_n)$ with a smooth $\omega$ is indeed the prototypical conormal distribution since $$ u(x)= \int_\mathbb R\omega(x') e^{2iπ x_n\xi_n}d\xi_n. $$ It is not difficult to generalize the previous discussion to the case involving $k$ codimensions.