nLab Boman's theorem

Idea

In considering certain types of generalized smooth spaces, one may try to describe the smooth structure on a space by specifying the smooth curves. Boman's Theorem shows that this is sufficient to describe the smooth structure on a smooth manifold.

Statement

The following theorem appears (as Theorem 1) in the paper Boman 1967, where it is first proved, by Jan Boman:

Theorem

Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^\infty(\mathbb{R},\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}, \mathbb{R}^d)$. Then $f \in C^\infty(\mathbb{R}^d, \mathbb{R})$.

Here, $\mathbb{R}^d$ is a Cartesian space, and $C^\infty(X,Y)$ is the set of smooth maps from $X$ to $Y$.

The theorem is quoted with a proof in Kriegl & Michor 1997 (Theorem 3.4).

Less than smooth functions

This theorem is for smooth functions, that is $C^\infty$ maps. A similar theorem could be stated for continuous functions, that is $C^0$ maps. The situation is slightly less than ideal, however, for continuously differentiable functions, that is $C^1$ maps, or more generally $C^p$ maps for $0 \lt p \lt \infty$.

Boman 1967 has this as part of Theorem 2:

Theorem

Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^p(\mathbb{R},\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}, \mathbb{R}^d)$. Then $f \in C^{p-1}(\mathbb{R}^d, \mathbb{R})$.

Note that $p$ has become $p - 1$ in the conclusion. (Boman's full Theorem 2 gives stronger results involving Lipschitz conditions.)

Boman's Theorem 8 gives the desired result if we use parametrized surfaces instead of curves:

Theorem

Let $f$ be a function from $\mathbb{R}^d$ to $\mathbb{R}$, and assume that the composed function $f \circ u$ belongs to $C^p(\mathbb{R}^2,\mathbb{R})$ for every $u \in C^\infty(\mathbb{R}^2, \mathbb{R}^d)$. Then $f \in C^p(\mathbb{R}^d, \mathbb{R})$.

Here we have $\mathbb{R}^2$ instead of $\mathbb{R}$ as the domain of $u$.

Boman's Theorem 3 guarantees such counterexamples as

$$f\colon x, y \mapsto \frac{y^3}{x^2 + y^2}$$

(continuously extended so that $f(0,0) = 0$). Given any smooth —or even $C^1$— curve $u\colon t \mapsto (g(t), h(t))$, it may be shown (by several tedious cases) that $(f \circ u)'$ is continuous. Nevertheless, $f$ is not $C^1$ at $(0,0)$. (The general pattern, expressed in Boman's Theorem 10, is to use a non-polynomial function that is homogeneous in degree $p$ and $C^p$ except at $\vec{0}$. So long as $d \gt 1$, such functions exist.) Additionally, $(f \circ u)'$ exists even if $u$ is merely differentiable, but $f$ is not even differentiable at $(0,0)$.

This does not contradict the well known theorem (often taken as a definition!) that a function is $C^1$ already if only its partial derivatives are continuous; while the partial derivatives of $f$ may be expressed as derivatives of $f \circ u$ for appropriate smooth $u$ (taken from a space of curves identifiable with $d \mathbb{R}^d$), the continuity of the partial derivatives requires not that $(f \circ u)'(t)$ be continuous in $t$ (although this will follow) but that it be continuous in $u$.

References

• Jan Boman, Differentiability of a function and of its compositions with functions of one variable, Math. Scand. 20 1967 249–268, MR237728 pdf

• Andreas Kriegl, Peter W. Michor, The convenient setting of global analysis, Math. Surveys and Monographs 53, Amer. Math. Soc. 1997. x+618 pp. ISBN: 0-8218-0780-3 html MR1471480