I asked the same question here https://mathoverflow.net/questions/247669/harish-chandras-submersive-principle-on-closed-subsets , but did not get an answer. I want to see if I will have some luck in this site.
Harish-Chandra's submersion principle says the following. Let $X,Y$ be two manifolds of dimensions $m$ and $n$ respectively. Let $\pi: X\rightarrow Y$ be a surjective smooth map which is submersive everywhere. Let $\omega_X,\omega_Y$ be two differential forms on $X$ and $Y$ respectively of degree $m$ and $n$. Then for every $\alpha\in C_c^\infty(X)$, there exists a unique function $F_\alpha\in C_c^\infty(Y)$ such that $$\int_X \alpha(x) \phi(\pi(x))d\omega_X =\int_Y F_\alpha(y)\phi(y)d\omega_Y$$ for all $\phi\in C_c^\infty(Y)$.
[Reference: Harish-Chandra, $\textit{Invariant distributions on Lie Algebras}$, American Journal of Math, $\textbf{86}$(1964),271-309,
The p-adic version can be found in
Harish-Chandra, Notes by G.van Dijk, $\textit{Harmonic Analysis on reductive p-adic groups}$, Lecture Notes in Math, $\textbf{162}$, p.48-49.]
Since the defining property of $F_\alpha$ is given by an integral, it seems very hard to detect values of $F_\alpha$ on closed subsets of $Y$. For example, we consider the following situation. Suppose that $X=X_1\times X_2$, and we identify $X_1$ with $X_1\times 0$, where $0\in X_2$ is a fixed point. Let $\pi, Y$ be as in the first paragraph. If we know $$\int_{X_1}\alpha(x_1)\phi(\pi(x_1))d\omega_{X_1}=0$$ for all $\phi\in C_c^\infty(\pi(X_1))$ and $\pi^{-1}(Y_1)\subset X_1$, can we conclude that the restriction of $F_\alpha$ to the closed set $\pi(X_1)$ is zero?
If the above is wrong, let us consider a weaker form of the above problem. Given $\beta\in C_c^\infty(X_1)$, if $$\int_{X_1}\beta(x_1)\phi(\pi(x_1))dx_1=0,$$ for all $\phi\in C_c^\infty(\pi(X_1))$, could we find a function $\alpha\in C_c^\infty(X)$ such that $\alpha|_{X_1}=\beta$ and $F_\alpha|_{\pi(X_1)}=0$?
A slightly different way to formulate this question is as follows. Denote $Y_1=\pi(X_1)$. Thus $\pi|_{X_1}:X_1\rightarrow Y_1$ is also submersive (is this true?). Thus Harish-Chandra's submersion principle gives a map $\beta\mapsto F_\beta$ from $C_c^\infty(X_1)$ to $C_c^\infty(Y_1)$ which satisfies $$\int_{X_1}\beta(x_1)\phi(\pi(x_1))d\omega_{X_1}=\int_{Y_1}F_\beta(y)\phi(y)d\omega_{Y_1}$$ for all $\phi\in C_c^\infty(Y_1)$. Is there any chance to make the following diagram commutative(at least in the p-adic case)? $\require{AMScd}$ \begin{CD} C_c^\infty(X) @>{HC}>> C_c^\infty(Y)\\ @V res V V @VV res V\\ C_c^\infty(X_1) @>>HC> C_c^\infty(Y_1) \end{CD} where the horizontal arrows are Harish-Chandra decent, and the vertical arrows are restrictions. In the p-adic case, the restrictions are surjective.
Qing Zhang, I got the formula because I need to calculate constantly convolutions of measures supported on surfaces, see also Hormander, "The analysis of linear partial differential operators", V. I, Thm. 6.1.2.
I will kept the discussion rather sketchy. Since we are working on manifolds, we can use a partition of unity and deal with the euclidian case; furthermore, to keep this post simple, I assume that $d\omega_X=dx$ and $d\omega_y=dy$, but the general case requires to add a weight.
Given a submersion $\pi:\mathbb{R}^{m}\to\mathbb{R}^n$, for $m\ge n$, we are asked about $$ \int_{\mathbb{R}^{m}}\alpha(x)\phi(\pi(x))\,dx. $$ We want to make a change of variables and to integrate instead over $\mathbb{R}^n$, but how to do that, if in general $\pi$ is not a diffeomorphism? Notice that $\mathbb{R}^m$ is foliated by the fibers of $\pi$, namely, by $\pi^{-1}(y)$, which are manifolds of codimension $n$.
Make a partition of $\mathbb{R}^n$ into small cubes $Q_{\delta}$, as in the Riemann integral, then the inverse image $\pi^{-1}(Q_{\delta})$ looks like a neighbourhood of $\pi^{-1}(y)$, for $y$ the center of $Q_{\delta}$, with variable width, depending on $d\pi$ for each point over the fiber.
Hence, we can write \begin{align} \int_{\mathbb{R}^{m}}\alpha(x)\phi(\pi(x))\,dx &= \sum_{y}\int_{\pi^{-1}(Q_{\delta})}\alpha(x)\phi(\pi(x))\,dx,\\ &\sim \sum_{y}\phi(y)\big[\int_{\pi^{-1}(y)}\alpha(x)W_{\pi}\,dS\big]\delta^{n}, \\ &\sim \int_{\mathbb{R}^n}\big[\int_{\pi^{-1}(y)}\alpha(x)W_{\pi}\,dS]\phi(y)\,dy, \end{align} where the term $W_{\pi}$ takes into account the variability of the width of the neighbourhood along the fibre and $dS$ is the induced measure, from the euclidian metric, on the fibre. We have that $W_{\pi}=|\det(\langle\nabla\pi_i,\nabla\pi_j\rangle)|^{-1/2}$, for $\pi=(\pi_1,\dots, \pi_n)$.
The previous discussion can be made into a rigorous proof, but this is, admitely, not the best way and I only wanted to motivate. A proof can go by reducing to a neighborhood of each point and then to complete $\pi$ with the variables, say, $x_{n+1},\ldots,x_{m}$, so that we have a diffeomorphism, then we apply the classical change of variables and make up the formula to her final look.
If there is any issue, I would receive any criticism.