Distributions, conormal kernels, and cotangent relations

In submersion coordinates $(x,v)$, a compactly supported test density on $Y$ is $\psi (x,v)\mathrm{\mid }dxdv\mathrm{\mid }$. Its fiber integral is $\tilde{\psi }(x)=\int \psi (x,v)\mathrm{\mid }dv\mathrm{\mid }$. The integrand has compact support, so differentiation under the integral gives a smooth function; its support lies in the compact projection of $\mathrm{supp}\psi$. Thus $\tilde{\psi }(x)\mathrm{\mid }dx\mathrm{\mid }$ is a test density and transposing this operation defines $p^{\ast }u$.

If $q$ is smooth and proper on $\mathrm{supp}V$ and $K$ is compact in $M$, then $Q_K=q^{-1}(K)\cap \mathrm{supp}V$ is compact. Choose one smooth cutoff ${\beta }_K$ equal to one near $Q_K$. For every test function $\phi$ supported in $K$, smoothness of $q$ makes ${\beta }_K(\phi \circ q)$ smooth with support in the fixed compact set $\mathrm{supp}{\beta }_K$, so define $⟨q_{\ast }V,\phi ⟩=⟨V,{\beta }_K(\phi \circ q)⟩$. If another cutoff is used, the difference of the two test functions vanishes on a neighborhood of $\mathrm{supp}V$: it vanishes near $Q_K$ because both cutoffs equal one there, and at every point of $\mathrm{supp}V\setminus Q_K$, continuity of $q$ gives a neighborhood mapped outside the closed set $K$, where $\phi \circ q=0$. Hence [F4] gives the same pairing; choosing a common cutoff for the union of two compact test supports also proves linearity. The chain and product rules bound every fixed-support seminorm of ${\beta }_K(\phi \circ q)$ by finitely many fixed-support seminorms of $\phi$, so continuity of $V$ proves continuity of $q_{\ast }V$. A finite cover by coordinate balls and [F12] supplies ${\beta }_K$. Multiplication by a smooth density is defined by the stated transpose formula, so the typed pushforward is well-defined.

For tests supported in a fixed compact $Q\subseteq G\times X$, ambient local finiteness gives $\mathrm{\mid }⟨K_Z,\mathrm{\Phi }⟩\mathrm{\mid }\le {\sup }_Q\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }{\int }_{Z\cap Q}\mathrm{\mid }{\omega }_Z\mathrm{\mid }$. Thus the integral is finite and continuous in the fixed-support test topology. If $Z$ is closed, $Z\cap Q$ is compact in $Z$, and finitely many relatively compact charts bound this integral for every smooth density. In coordinates where $Z=\{t=0\}$, the functional is integration of the restricted test function against ${\omega }_Z$. Under a change of coordinates on $Z$, the coefficient transforms by the absolute tangential Jacobian [F5], so the functional is independent of the defining chart.

A covector annihilating $T_{z}Z$ transforms by the inverse transpose rule [F6], so $N^{\ast }Z$ and $C_Z$ are coordinate independent. If a conormal has the form $(\eta ,0)$, it annihilates the image of $d{\pi }_G:T_{(g,x)}Z\to T_{g}G$, which is all of $T_{g}G$ because ${\pi }_G$ is a submersion; therefore $\eta =0$. Applying the same argument to ${\pi }_X$ shows that $(0,\xi )$ can only be zero.

The analytic conditions are relative to the fixed atlases. On overlaps of induced analytic charts on $Z$, the nonzero coordinate Jacobian has locally constant sign, so its absolute value is analytic. The density transition rule therefore preserves analyticity of its coefficient. The defining submersions and analytic chart changes specify the same analytic embedded submanifold, while the underlying smooth distribution operations are unchanged.

On overlapping submersion charts, the full absolute Jacobian factors into the base Jacobian and the fiber-density quotient. The fiber integral therefore transforms as a base density. This proves that the transposed functional in step 1.1 is chart independent.

Let $\phi$ be any compactly supported test density on $M$; the zero test is immediate. Its support $K$ is compact. Cover $K$ by finitely many smaller coordinate balls $B_i$ whose closures lie in coordinate balls $U_i$. Using [F12] in each chart, choose ${\beta }_i$ supported in $U_i$ and equal to one on $B_i$, and put $\sigma ={\sum }_i{\beta }_i$. Then $\sigma >0$ on a neighborhood of $K$. A second finite cover by coordinate balls and [F12] gives a compactly supported smooth $\theta$ in $\{\sigma >0\}$ equal to one near $K$. Define ${\chi }_i=\theta {\beta }_i\mathrm{/}\sigma$ on $\{\sigma >0\}$ and extend by zero outside. Each ${\chi }_i$ is smooth and compactly supported in $U_i$, and ${\sum }_i{\chi }_i=\theta$. Hence $u(\phi )={\sum }_{i}u({\chi }_i\phi )$, since $\theta =1$ near $K$. Each term is a chartwise pairing with a compactly supported Euclidean distribution, so [F9] and [F10] give its tempered and smooth-function realizations. The finite sum therefore recovers the original functional and its packet pairings, without assuming $\mathrm{supp}u$ compact or choosing global coordinates.