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 then define for any density $v$ on $X$ with compact support $\langle \delta_Y, v \rangle = \int_Y v$ is it possible to say that $\delta_Y \in I^m(X,Y)$ for some $m$?

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 $m$?