This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\mathbb{R}^N$, which I will take to mean that the pair $I\subset R$ is locally isomorphic to the ideal corresponding to a real algebraic subscheme of $\mathbb{A}^n$ (I am happy with other definitions as well). Moreover, suppose that $X$ is stratified $X_0\subset X_1\subset \dots \subset X_n = X$ by closed strata which are also singular $C^\infty$ manifolds, and such that $$Y_k: = X_k\setminus X_{k-1}$$ is smooth.

Let's make the following definitions

Definition 1

1. A distribution $D$ on $\mathbb{R}^n$ is an additive system of functionals $$f\mapsto \int_U f\cdot D$$ on $C^\infty(U)$ as $U$ varies over bounded open subsets of $\mathbb{R}^n$. (I'm also happy with other notions of a distribution, e.g. a functional on Schwarz space.)

2. A distribution $D$ is supported on $I$ if $\int_U fD$ only depends on the restriction to $U$ of $f\text{ mod } I$. (For example the Dirac distribution $\delta_0$ on $\mathbb{R}$ is supported on the ideal $(x)\subset C^\infty(\mathbb{R})$ and the distribution $\delta_0'$ is supported on $(x^2)$.)

Now we add an additional condition requiring our distribution to be locally a sum of $C^\infty$ top-degree forms on the smooth strata $U_k$:

Definition 2

Let $X, X_1\subset X_2\subset \dots \subset X_n = X$ be singular $C^\infty$ submanifolds in $\mathbb{R}$ as above, with $I$ the ideal of $X$. A distribution $D$ on $\mathbb{R}^n$ supported on $I$ is smooth with respect to the stratification $X_1\subset \dots \subset X_n$ if there exist top-degree differential forms $\omega_k$ on $Y_k = X_k\setminus X_{k-1}$ such that

• $\omega_k$ is a $C^\infty$ forms on the open $Y_k\subset X_k$ with finite-order poles on the boundary $X_{k-1}\subset X_k.$

• if $V\subset Y_k$ is a bounded open subset of the open stratum which is bounded away from the boundary (i.e., such that $\bar{V}\subset Y_k$) then $$\int_{V} f\cdot D = \int_{V} f\mid_{Y_k}\omega_k.$$

Here I define $\int_V f\cdot D$ for a closed submanifold $V$ to be the limit $$\int_V f\cdot D:= \lim_{\epsilon\to 0} \int_{V_\epsilon} f\cdot D,$$ for $V_\epsilon$ the radius-$\epsilon$ tubular neighborhood in the standard metric on $\mathbb{R}^n.$

Obviously, the distribution $D$ is uniquely determined by the forms $\omega_k$ on the open strata. My question is: can one algebraically (read: algebro-geometrically) charaterize which collections of forms $\omega_0,\dots, \omega_n$ give valid distributions $D$?

Some notes:

1. My motivating example is the "osculating cross", the singular manifold $X\subset \mathbb{R}^2$ with ideal $I = \big((y+x^n)(y-x^n)\big)$ in $C^\infty (\mathbb{R}^2),$ and with stratification $\{x_0: = (0,0)\}\subset X.$ In this case, the open stratum $V_1 = X\setminus \{x_0\}$ is isomorphic to $\mathbb{R}^*\sqcup \mathbb{R}^*$ and you can check that a pair of rational forms $(f(x)/x^a dx, g(x)/x^b dx)$ on the two copies of $\mathbb{R}^*$ define a valid distribution on $X$ if and only if $f(x)/x^a, g(x)/x^b$ both have poles of order $\le n$ (the degree of osculation) and the sum $f(x)/x^a + g(x)/x^b$ has a pole of order $\le 1.$

2. Variant. There is a variant of the definition $\int_V f(x) \cdot D$ where instead of taking the limit of $\int_{V_\epsilon}$ over tubular neighborhoods one requires that any sequence of bounded embedded opens $U_1\supset U_2\supset \cdots$ with intersection $V$ must give the same value. Using this variant in Definition 2 is more restrictive, and reduces the possible degree of allowable poles by $1$ in the osculating cross example. I'd be happy with an algebraic characterization for this more restrictive condition as well.

This is a mild reformulation of a previous question. Let $R = C^\infty(\mathbb{R}^N)$ and let $I$ be an ideal in $R$ which cuts out an $n$-dimensional "singular $C^\infty$ manifold $X$" in $\mathbb{R}^N$, which I will take to mean that the pair $I\subset R$ is locally isomorphic to the ideal corresponding to a real algebraic subscheme of $\mathbb{A}^n$ (I am happy with other definitions as well). Moreover, suppose that $X$ is stratified $X_0\subset X_1\subset \dots \subset X_n = X$ by closed strata which are also singular $C^\infty$ manifolds, and such that $$Y_k: = X_k\setminus X_{k-1}$$ is smooth.

Let's make the following definitions

Definition 1

1. A distribution $D$ on $\mathbb{R}^n$ is an additive system of functionals $$f\mapsto \int_U f\cdot D$$ on $C^\infty(\bar{U})$ as $U$ varies over bounded open subsets of $\mathbb{R}^n$. (I'm also happy with other notions of a distribution, e.g. a functional on Schwarz space.)

2. A distribution $D$ is supported on $I$ if $\int_U fD$ only depends on the restriction to $U$ of $f\text{ mod } I$. (For example the Dirac distribution $\delta_0$ on $\mathbb{R}$ is supported on the ideal $(x)\subset C^\infty(\mathbb{R})$ and the distribution $\delta_0'$ is supported on $(x^2)$.)

Now we add an additional condition requiring our distribution to be locally a sum of $C^\infty$ top-degree forms on the smooth strata $U_k$:

Definition 2

Let $X, X_1\subset X_2\subset \dots \subset X_n = X$ be singular $C^\infty$ submanifolds in $\mathbb{R}$ as above, with $I$ the ideal of $X$. A distribution $D$ on $\mathbb{R}^n$ supported on $I$ is smooth with respect to the stratification $X_1\subset \dots \subset X_n$ if there exist top-degree differential forms $\omega_k$ on $Y_k = X_k\setminus X_{k-1}$ such that

• $\omega_k$ is a $C^\infty$ forms on the open $Y_k\subset X_k$ with finite-order poles on the boundary $X_{k-1}\subset X_k.$

• if $V\subset Y_k$ is a bounded open subset of the open stratum which is bounded away from the boundary (i.e., such that $\bar{V}\subset Y_k$) then $$\int_{V} f\cdot D = \int_{V} f\mid_{Y_k}\omega_k.$$

Here I define $\int_V f\cdot D$ for a closed submanifold $V$ to be the limit $$\int_V f\cdot D:= \lim_{\epsilon\to 0} \int_{V_\epsilon} f\cdot D,$$ for $V_\epsilon$ the radius-$\epsilon$ tubular neighborhood in the standard metric on $\mathbb{R}^n.$

Obviously, the distribution $D$ is uniquely determined by the forms $\omega_k$ on the open strata. My question is: can one algebraically (read: algebro-geometrically) charaterize which collections of forms $\omega_0,\dots, \omega_n$ give valid distributions $D$?

Some notes:

1. My motivating example is the "osculating cross", the singular manifold $X\subset \mathbb{R}^2$ with ideal $I = \big((y+x^n)(y-x^n)\big)$ in $C^\infty (\mathbb{R}^2),$ and with stratification $\{x_0: = (0,0)\}\subset X.$ In this case, the open stratum $V_1 = X\setminus \{x_0\}$ is isomorphic to $\mathbb{R}^*\sqcup \mathbb{R}^*$ and you can check that a pair of rational forms $(f(x)/x^a dx, g(x)/x^b dx)$ on the two copies of $\mathbb{R}^*$ define a valid distribution on $X$ if and only if $f(x)/x^a, g(x)/x^b$ both have poles of order $\le n$ (the degree of osculation) and the sum $f(x)/x^a + g(x)/x^b$ has a pole of order $\le 1.$

2. Variant. There is a variant of the definition $\int_V f(x) \cdot D$ where instead of taking the limit of $\int_{V_\epsilon}$ over tubular neighborhoods one requires that any sequence of bounded embedded opens $U_1\supset U_2\supset \cdots$ with intersection $V$ must give the same value. Using this variant in Definition 2 is more restrictive, and reduces the possible degree of allowable poles by $1$ in the osculating cross example. I'd be happy with an algebraic characterization for this more restrictive condition as well.