Invariant functions in Denjoy–Carleman classes

Armin Rainer Armin Rainer: Dipartimento di Matematica, Universitá di Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy armin.rainer@univie.ac.at

May 8, 2008

Abstract.

Let $V$ be a real finite dimensional representation of a compact Lie group $G$ . It is well-known that the algebra $R [V]^{G}$ of $G$ -invariant polynomials on $V$ is finitely generated, say by $σ_{1}, \dots, σ_{p}$ . Schwarz [38] proved that each $G$ -invariant $C^{\infty}$ -function $f$ on $V$ has the form $f = F (σ_{1}, \dots, σ_{p})$ for a $C^{\infty}$ -function $F$ on $R^{p}$ . We investigate this representation within the framework of Denjoy–Carleman classes. One can in general not expect that $f$ and $F$ lie in the same Denjoy–Carleman class $C^{M}$ (with $M = (M_{k})$ ). For finite groups $G$ and (more generally) for polar representations $V$ we show that for each $G$ -invariant $f$ of class $C^{M}$ there is an $F$ of class $C^{N}$ such that $f = F (σ_{1}, \dots, σ_{p})$ , if $N$ is strongly regular and satisfies

sup k \in N_{> 0} (\frac{M_{k m}}{N_{k}})^{\frac{1}{k}} < \infty,

where $m$ is an (explicitly known) integer depending only on the representation. In particular, each $G$ -invariant $(1 + δ)$ -Gevrey function $f$ (with $δ > 0$ ) has the form $f = F (σ_{1}, \dots, σ_{p})$ for a $(1 + δ m)$ -Gevrey function $F$ . Applications to equivariant functions and basic differential forms are given.

Key words and phrases:

Denjoy–Carleman class, invariant functions

2000 Mathematics Subject Classification:

26E10, 58C25, 57S15

The author was supported by ‘Fonds zur Förderung der wissenschaftlichen Forschung, Projekt P19392 & Projekt J2771’

1. Introduction

Let $V$ be a real finite dimensional representation of a compact Lie group $G$ . By a classical theorem due to Hilbert the algebra $R [V]^{G}$ of $G$ -invariant polynomials on $V$ is finitely generated. Choose a system of homogeneous generators $σ_{1}, \dots, σ_{p}$ of $R [V]^{G}$ and define $σ := (σ_{1}, \dots, σ_{p}) : V \to R^{p}$ . Schwarz [38] proved a smooth analog of Hilbert’s theorem for orthogonal representations $V$ of compact Lie groups $G$ : the induced mapping $σ^{*} : C^{\infty} (R^{p}) \to C^{\infty} (V)^{G}$ is surjective. Mather [27] showed that this mapping is even split surjective.

The finitely differentiable case was studied, too: $σ^{*} : C^{n} (R^{p}) \to C^{n} (V)^{G}$ is in general not surjective, but $σ^{*} C^{n} (R^{p})$ contains $C^{n q} (V)^{G}$ for a suitable integer $q$ . See [1], [3], [2], [37].

In this paper we treat Schwarz’s theorem in the framework of Denjoy–Carleman classes. These classes of smooth functions play an important role in harmonic analysis and various branches of differential equations (especially Gevrey classes). Let $M = (M_{k})_{k \in N}$ be a non-decreasing sequence of real numbers with $M_{0} = 1$ . A smooth function $f$ in an open subset $U \subseteq R^{n}$ belongs to the Denjoy–Carleman class $C^{M} (U)$ if for any compact subset $K \subseteq U$ there exist positive constants $C$ and $ϱ$ such that

| \partial^{α} f (x) | \leq C ϱ^{| α |} | α |! M_{| α |}

for all $α \in N^{n}$ and $x \in K$ . See section 2 for more on Denjoy–Carleman classes. As examples ([8], see also 3.3) show, one cannot expect in general that a smooth $G$ -invariant function $f$ on $V$ of class $C^{M}$ has the form $f = F \circ σ$ for a function $F$ of the same class $C^{M}$ .

For finite groups $G$ and (more generally) for polar representations $V$ we prove that the representation $f = F \circ σ$ holds in the context of Denjoy–Carleman classes, where $F$ has lower regularity than $f$ . More precisely: Let $G$ be a subgroup of finite order $m$ of $GL (V)$ . Let $M$ and $N$ be sequences satisfying some mild conditions which guarantee stability under composition and derivation for $C^{M}$ and $C^{N}$ (see 2.1). Assume that $N$ is strongly regular (see 2.6) and that

sup k \in N_{> 0} (\frac{M_{k m}}{N_{k}})^{\frac{1}{k}} < \infty .

Then for any $G$ -invariant function $f \in C^{M} (V)$ there exists a function $F \in C^{N} (R^{p})$ such that $f = F \circ σ$ . In particular: Any $G$ -invariant Gevrey function $f \in G^{1 + δ} (V)$ (with $δ > 0$ ) has the form $f = F \circ σ$ for a Gevrey function $F \in G^{1 + δ m} (R^{p})$ . See theorem 3.4. The result does not depend on the choice of generators $σ_{i}$ , since any two choices differ only by a polynomial diffeomorphism and the involved Denjoy–Carleman classes are stable under composition.

Note that Thilliez [40] treats a very similar problem: For a compact subset $E \subseteq R^{n}$ , an analytic mapping $Φ : U \to R^{n}$ on an open neighborhood $U$ of $E$ , and a function $f \in C^{M} (U)$ of the form $f = g \circ Φ$ with $g \in C^{\infty} (W)$ for an open neighborhood $W$ of $Φ (E)$ , the existence of a sequence $N$ such that $g \in C^{N} (W)$ is investigated. This is done by studying the complex setting: Now $E$ is compact in $C^{n}$ , $Φ$ is a $C^{n}$ -valued holomorphic mapping defined near $E$ , and $g$ is $C^{\infty}$ on $C^{n}$ and $¯ \partial$ -flat on $Φ (E)$ . However, our results are not covered by Thilliez’, since the minimal number of generators of $R [V]^{G}$ does in general not coincide with the dimension of the representation space $V$ .

We prove the main theorem in section 3. We shall deduce it from an analog theorem (see 3.3) due to Bronshtein [7, 8] which treats the standard representation of the symmetric group $S_{n}$ in $R^{n}$ . This method is inspired by Barbançon and Raïs [3] deploying Weyl’s account [43] of Noether’s [30] proof of Hilbert’s theorem.

The rest of the paper is devoted to several applications of this main theorem. In section 4 we treat the presentation in Denjoy–Carleman classes of equivariant functions between representations of a finite group.

In section 5 the main theorem 3.4 is generalized to polar representations, i.e., orthogonal finite dimensional representations $V$ of a compact Lie group $G$ allowing a linear subspace $Σ \subseteq V$ which meets each orbit orthogonally (see theorem 5.2). The trace of the $G$ -action in $Σ$ is the action of the generalized Weyl group $W$ which is a finite group. In analogy with a result due to Palais and Terng [32], which states that restriction induces an isomorphism $I_{1} : C^{\infty} (V)^{G} ≅ C^{\infty} (Σ)^{W}$ , we show that each $W$ -invariant function on $Σ$ of class $C^{M}$ has a $G$ -invariant extension to $V$ of class $C^{N}$ , where $M$ and $N$ are sequences with the aforementioned properties. More generally, Michor [28, 29] proved that restriction induces an isomorphism $I_{2} : Ω_{hor}^{p} (V)^{G} ≅ Ω^{p} (Σ)^{W}$ , where $Ω_{hor}^{p} (V)^{G}$ is the space of basic $p$ -forms on $V$ , i.e., $G$ -invariant forms that kill each vector tangent to some orbit. Our main theorem 3.4 allows to conclude that each $W$ -invariant $p$ -form on $Σ$ of class $C^{M}$ has a basic extension to $V$ of class $C^{N}$ (with $M$ and $N$ as above).

In [32] and [28, 29] the isomorphisms $I_{1}$ and $I_{2}$ are established in the more general setting of smooth proper Riemannian $G$ -manifolds $X$ with sections, where there exist closed submanifolds $Σ \subseteq X$ meeting each orbit orthogonally. In section 6 we explain that our analog results in the framework of Denjoy–Carleman classes generalize to real analytic proper Riemannian $G$ -manifolds $X$ with sections.

2. Denjoy–Carleman classes

2.1. Denjoy–Carleman classes of differentiable functions

We mainly follow [42] (see also the references therein). We use $N = N_{> 0} \cup {0}$ . For each multi-index $α = (α_{1}, \dots, α_{n}) \in N^{n}$ , we write $α! = α_{1}! \dots α_{n}!$ , $| α | = α_{1} + \dots + α_{n}$ , and $\partial^{α} = \partial^{| α |} / \partial x_{1}^{α_{1}} \dots \partial x_{n}^{α_{n}}$ .

Let $M = (M_{k})_{k \in N}$ be an increasing sequence ( $M_{k + 1} \geq M_{k}$ ) of real numbers with $M_{0} = 1$ . Let $U \subseteq R^{n}$ be open. We denote by $C^{M} (U)$ the set of all $f \in C^{\infty} (U)$ such that, for all compact $K \subseteq U$ , there exist positive constants $C$ and $ϱ$ such that

(2.1.1)

| \partial^{α} f (x) | \leq C ϱ^{| α |} | α |! M_{| α |}

for all $α \in N^{n}$ and $x \in K$ . The set $C^{M} (U)$ is the Denjoy–Carleman class of functions on $U$ . If $M_{k} = 1$ , for all $k$ , then $C^{M} (U)$ coincides with the ring $C^{ω} (U)$ of real analytic functions on $U$ . In general, $C^{ω} (U) \subseteq C^{M} (U) \subseteq C^{\infty} (U)$ .

We assume that $M = (M_{k})$ is logarithmically convex, i.e.,

(2.1.2)

M_{k}^{2} \leq M_{k - 1} M_{k + 1} for all k,

or, equivalently, $M_{k + 1} / M_{k}$ is increasing. Considering $M_{0} = 1$ , we obtain that also $(M_{k})^{1 / k}$ is increasing and

(2.1.3)

M_{l} M_{k} \leq M_{l + k} for all l, k \in N .

Hypothesis (2.1.2) implies that $C^{M} (U)$ is a ring, for all open subsets $U \subseteq R^{n}$ , which can easily be derived from (2.1.3) by means of Leibniz’s rule. Note that definition (2.1.1) makes sense also for functions $U \to R^{p}$ . For $C^{M}$ -mappings, (2.1.2) guarantees stability under composition ([35], see also [4, 4.7]).

A further consequence of (2.1.2) is the inverse function theorem for $C^{M}$ ([22]; for a proof see also [4, 4.10]): Let $f : U \to V$ be a $C^{M}$ -mapping between open subsets $U, V \subseteq R^{n}$ . Let $x_{0} \in U$ . Suppose that the Jacobian matrix $(\partial f / \partial x) (x_{0})$ is invertible. Then there are neighborhoods $U^{'}$ of $x_{0}$ , $V^{'}$ of $y_{0} := f (x_{0})$ such that $f : U^{'} \to V^{'}$ is a $C^{M}$ -diffeomorphism.

Moreover, (2.1.2) implies that $C^{M}$ is closed under solving ODEs (due to [23]).

Suppose that $M = (M_{k})$ and $N = (N_{k})$ satisfy $M_{k} \leq C^{k} N_{k}$ , for all $k$ and a constant $C$ , or equivalently,

(2.1.4)

sup k \in N_{> 0} (\frac{M_{k}}{N_{k}})^{\frac{1}{k}} < \infty .

Then, evidently $C^{M} (U) \subseteq C^{N} (U)$ . The converse is true as well (if (2.1.2) is assumed): One can prove that there exists $f \in C^{M} (R)$ such that $| f^{(k)} (0) | \geq k! M_{k}$ for all $k$ (see [42, Theorem 1]). So the inclusion $C^{M} (U) \subseteq C^{N} (U)$ implies (2.1.4).

Setting $N_{k} = 1$ in (2.1.4) yields that $C^{ω} (U) = C^{M} (U)$ if and only if

sup k \in N_{> 0} (M_{k})^{\frac{1}{k}} < \infty .

Since $(M_{k})^{1 / k}$ is increasing (by logarithmic convexity), the strict inclusion $C^{ω} (U) ⊊ C^{M} (U)$ is equivalent to

lim k \to \infty (M_{k})^{\frac{1}{k}} = \infty .

We shall also assume that $C^{M}$ is stable under derivation, which is equivalent to the following condition

(2.1.5)

sup k \in N_{> 0} (\frac{M_{k + 1}}{M_{k}})^{\frac{1}{k}} < \infty .

Note that the first order partial derivatives of elements in $C^{M} (U)$ belong to $C^{M^{+ 1}} (U)$ , where $M^{+ 1}$ denotes the shifted sequence $M^{+ 1} = (M_{k + 1})_{k \in N}$ . So the equivalence follows from (2.1.4), by replacing $M$ with $M^{+ 1}$ and $N$ with $M$ .

Definition

By a DC-weight sequence we mean a sequence $M = (M_{k})_{k \in N}$ of positive numbers with $M_{0} = 1$ which is monotone increasing ( $M_{k + 1} \geq M_{k}$ ), logarithmically convex (2.1.2), and satisfies (2.1.5). Then $C^{M} (U, R)$ is a differential ring, and the class of $C^{M}$ -functions is stable under compositions, as above.

2.2. Quasianalytic function classes

Let $F_{n}$ denote the ring of formal power series in $n$ variables (with real or complex coefficients). We denote by $F_{n}^{M}$ the set of elements $F = \sum_{α \in N^{n}} F_{α} x^{α}$ of $F_{n}$ for which there exist positive constants $C$ and $ϱ$ such that

| F_{α} | \leq C ϱ^{| α |} M_{| α |}

for all $α \in N^{n}$ . A class $C^{M}$ is called quasianalytic if, for open connected $U \subseteq R^{n}$ and all $a \in U$ , the Taylor series homomorphism

T_{a} : C^{M} (U) \to F_{n}^{M}, f \mapsto T_{a} f (x) = \sum α \in N^{n} \frac{1}{α!} \partial^{α} f (a) x^{α}

is injective. By the Denjoy–Carleman theorem ([14], [10]), $C^{M}$ is quasianalytic if and only if

(2.2.1)

\infty \sum k = 0 \frac{M_{k}}{(k + 1) M_{k + 1}} = \infty, or, % equivalently, \infty \sum k = 1 (\frac{1}{k! M_{k}})^{\frac{1}{k}} = \infty .

For contemporary proofs see for instance [19, 1.3.8] or [36, 19.11].

Suppose that $C^{ω} (U) ⊊ C^{M} (U)$ and $C^{M} (U)$ is quasianalytic. Then $T_{a} : C^{M} (U) \to F_{n}^{M}$ is not surjective. This is due to Carleman [10]; an elementary proof can be found in [42, Theorem 3].

2.3. Non-quasianalytic function classes

If $M$ is a DC-weight sequence which is not quasianalytic, then there are $C^{M}$ partitions of unity. Namely, there exists a $C^{M}$ function $f$ on $R$ which does not vanish in any neighborhood of 0 but which has vanishing Taylor series at 0. Let $g (t) = 0$ for $t \leq 0$ and $g (t) = f (t)$ for $t > 0$ . From $g$ we can construct $C^{M}$ bump functions as usual.

2.4. Strong non-quasianalytic function classes

Let $M$ be a DC-weight sequence with $C^{ω} (U, R) ⊊ C^{M} (U, R)$ . Then the mapping $T_{a} : C^{M} (U, R) \to F_{n}^{M}$ is surjective, for all $a \in U$ , if and only if there is a constant $C$ such that

(2.4.1)

\infty \sum k = j \frac{M_{k}}{(k + 1) M_{k + 1}} \leq C \frac{M_{j}}{M_{j + 1}} for any integer j \geq 0.

See [34] and references therein. (2.4.1) is called strong non-quasianalyticity condition.

2.5. Moderate growth

A DC-weight sequence $M$ has moderate growth if

(2.5.1)

sup j, k \in N_{> 0} (\frac{M_{j + k}}{M_{j} M_{k}})^{\frac{1}{j + k}} < \infty .

2.6. Strong regularity

Moderate growth (2.5.1) together with strong non-quasianalyticity (2.4.1) is called strong regularity: Then a version of Whitney’s extension theorem holds for the corresponding function classes.

2.7. Whitney’s extension theorem

Let $K \subseteq R^{n}$ be compact. Denote by $J^{\infty} (K)$ the $C^{\infty}$ Whitney jets on $K$ . We say that $F = (F_{α})_{α \in N^{n}} \in J^{\infty} (K)$ is a $C^{M}$ -jet on $K$ , or belongs to $J^{M} (K)$ , if there exist positive constants $C$ and $ϱ$ such that

(2.7.1)

| F_{α} (x) | \leq C ϱ^{| α |} | α |! M_{| α |}

for all $α \in N^{n}$ and $x \in K$ and

(2.7.2)

| F_{β} (x) - \partial^{β} T_{a}^{p} F (x) | \leq C ϱ^{p} | β |! M_{p + 1} | x - a |^{p + 1 - | β |}

for all $p \in N$ , all $β \in N^{n}$ with $| β | \leq p$ and all $x \in K$ , where

T_{a}^{p} F (x) = \sum | β | \leq p \frac{1}{β!} F_{β} (a) (x - a)^{β} .

If $M$ is strongly regular then a version of Whitney’s extension theorem holds (see [9], [5], and [11]): the mapping $J_{K} : C^{M} (R^{n}) \to J^{M} (K), f \mapsto (\partial^{α} f |_{K})_{α \in N^{n}}$ is surjective.

Note that, if $f \in C^{\infty} (R^{n})$ such that $F = J_{K} f$ satisfies (2.7.1) and if $K$ is Whitney $1$ -regular, then (2.7.2) is automatically fulfilled (see [5, 3.12]). Recall that $K$ is Whitney $1$ -regular if any two points $x$ and $y$ in $K$ can be connected by a path in $K$ of length $\leq C | x - y |$ , where the constant $C$ depends only on $K$ .

2.8. Gevrey functions

Let $δ > 0$ and put $M_{k} = (k!)^{δ}$ , for $k \in N$ . Then $M = (M_{k})$ is strongly regular. The corresponding class $C^{M}$ of functions is the Gevrey class $G^{1 + δ}$ .

2.9. More examples

Let $δ > 0$ and put $M_{k} = (log (k + e))^{δ k}$ , for $k \in N$ . Then $M = (M_{k})$ is quasianalytic for $0 < δ \leq 1$ and non-quasianalytic (but not strongly) for $δ > 1$ .

Let $q > 1$ and put $M_{k} = q^{k^{2}}$ , for $k \in N$ . The corresponding $C^{M}$ -functions are called $q$ -Gevrey regular. Then $M = (M_{k})$ is strongly non-quasianalytic but not of moderate growth, thus not strongly regular.

2.10. Spaces of $C^{m}$ -functions

Let $U \subseteq R^{n}$ be open. For any $ϱ > 0$ and $K \subseteq U$ compact with smooth boundary, define

C_{ϱ}^{M} (K) := {f \in C^{\infty} (K) : ∥ f ∥_{ϱ, K} < \infty}

with

∥ f ∥_{ϱ, K} := sup {\frac{| \partial^{α} f (x) |}{ϱ^{| α |} | α |! M_{| α |}} : α \in N^{n}, x \in K} .

It is easy to see that $C_{ϱ}^{M} (K)$ is a Banach space. In the description of $C_{ϱ}^{M} (K)$ , instead of compact $K$ with smooth boundary, we may also use open $K \subset U$ with $¯ ¯¯¯ ¯ K$ compact in $U$ , like [42]. Or we may work with Whitney jets on compact $K$ , like [21].

The space $C^{M} (U)$ carries the projective limit topology over compact $K \subseteq U$ of the inductive limit over $ϱ \in N_{> 0}$ :

C^{M} (U) = lim \leftarrow - K \subseteq U (lim - \to ϱ \in N_{> 0} C_{ϱ}^{M} (K)) .

One can prove that, for $ϱ < ϱ^{'}$ , the canonical injection $C_{ϱ}^{M} (K) \to C_{ϱ^{'}}^{M} (K)$ is a compact mapping (see [21]). Hence ${lim - \to}_{ϱ} C_{ϱ}^{M} (K)$ is a Silva space, i.e., an inductive limit of Banach spaces such that the canonical mappings are compact.

2.11. Polynomials are dense in $C^{m} (u)$

Let $M$ be a DC-weight sequence and let $U \subseteq R^{n}$ be open. It is proved in [20, 3.2] (see also [17, 3.2]) that the space of entire functions $H (C^{n})$ is dense in $C^{M} (U)$ . Since the polynomials are dense in $H (C^{n})$ and the inclusion $H (C^{n}) \to C^{M} (U)$ is continuous, we obtain that the polynomials are dense in $C^{M} (U)$ . For convenience we give a proof.

Lemma.

Let $M$ be a DC-weight sequence and let $U \subseteq R^{n}$ be open. Then $H (C^{n})$ is dense in $C^{M} (U)$ .

Proof. Let $f \in C^{M} (U)$ and $K \subseteq U$ compact. Let $0 < c < 1$ such that $Q = K + B_{c} (0) \subseteq U$ , where $B_{c} (0) = {x \in R^{n} : | x | \leq c}$ . Let $χ \in C^{\infty} (U)$ with $0 \leq χ \leq 1$ , $χ |_{Q} = 1$ , and compact support $Q_{1} = supp (χ) \subseteq U$ . We define for $j \in N_{> 0}$

f_{j} := E_{j} * χ f \in H (C^{n}), where E_{j} : C^{n} \to C, z \mapsto (\frac{j}{π})^{\frac{n}{} 2} e^{- j ⟨ z ∣ z ⟩} .

Induction shows

\partial_{i_{N}} \dots \partial_{i_{1}} (E_{j} * χ f) = E_{j} * (χ \partial_{i_{N}} \dots \partial_{i_{1}} f) + N \sum ν = 1 (\partial_{i_{N}} \dots \partial_{i_{ν + 1}} E_{j}) * (\partial_{i_{ν}} χ) (\partial_{i_{ν - 1}} \dots \partial_{i_{1}} f),

for all $N \in N$ and $j \in N_{> 0}$ , and hence

	$\| \partial_{i_{N}} \dots \partial_{i_{1}} (f - f_{j}) \| \leq$	$\| \partial_{i_{N}} \dots \partial_{i_{1}} f - E_{j} * (χ \partial_{i_{N}} \dots \partial_{i_{1}} f) \|$
(2.11.1)			$+ N \sum ν = 1 \| (\partial_{i_{N}} \dots \partial_{i_{ν + 1}} E_{j}) * (\partial_{i_{ν}} χ) (\partial_{i_{ν - 1}} \dots \partial_{i_{1}} f) \| .$

We have for $x \in K$ and $α \in N^{n}$

	$\| E_{j} * (χ \partial^{α} f) (x)$	$- \partial^{α} f (x) \| = \| \int E_{j} (y) (χ (x - y) \partial^{α} f (x - y) - \partial^{α} f (x)) d y \|$
		$\leq \int_{B_{c} (0)} E_{j} (y) \| \partial^{α} f (x - y) - \partial^{α} f (x) \| d y$
		$+ \int_{R^{n} ∖ B_{c} (0)} E_{j} (y) (χ (x - y) \| \partial^{α} f (x - y) \| + \| \partial^{α} f (x) \|) d y .$

By the generalized mean value theorem we have for $x \in K$ , $y \in B_{c} (0)$ , and $α \in N^{n}$

| \partial^{α} f (x - y) - \partial^{α} f (x) | \leq \sqrt{n} | y | sup \begin{matrix} 1 \leq i \leq n 0 \leq t \leq 1 \end{matrix} | \partial^{α + e_{i}} f (x - t y) | .

Choose $ϱ_{1} > 0$ such that $∥ f ∥_{ϱ_{1}, Q_{1}} < \infty$ . Then for $x \in K$ , $y \in B_{c} (0)$ , and $α \in N^{n}$

	$\| \partial^{α} f (x - y) - \partial^{α} f (x) \|$	$\leq \sqrt{n} \| y \| ∥ f ∥_{ϱ_{1}, Q_{1}} ϱ_{1}^{\| α \| + 1} (\| α \| + 1)! M_{\| α \| + 1}$
		$\leq 2 \sqrt{n} \| y \| ϱ_{1} ∥ f ∥_{ϱ_{1}, Q_{1}} (2 ϱ_{1} C)^{\| α \|} \| α \|! M_{\| α \|} (by (???)),$

where $C$ is a positive constant. For all $j \in N_{> 0}$ we have

\int_{R^{n}} | y | E_{j} (y) d y \leq \frac{C_{1}}{\sqrt{j}} and \int_{R^{n} ∖ B_{c} (0)} E_{j} (y) d y \leq \frac{C_{1}}{\sqrt{j}}

for a constant $C_{1}$ independent of $j$ . Thus there exist positive constants $C_{2}$ and $ϱ_{2}$ independent of $x$ , $α$ , and $j$ such that

(2.11.2)

| E_{j} * (χ \partial^{α} f) (x)

- \partial^{α} f (x) | \leq \frac{C_{2}}{\sqrt{j}} ϱ_{2}^{| α |} | α |! M_{| α |} .

We have for $x \in K$

	$\| (\partial_{i_{N}}$	$\dots \partial_{i_{ν + 1}} E_{j}) * (\partial_{i_{ν}} χ) (\partial_{i_{ν - 1}} \dots \partial_{i_{1}} f) (x) \|$
		$\leq \| Q_{1} \| sup \begin{matrix} 1 \leq i \leq n y \in U \end{matrix} \| \partial_{i} χ (y) \| sup y \notin B_{c} (0) \| \partial_{i_{N}} \dots \partial_{i_{ν + 1}} E_{j} (y) \| sup u \in Q_{1} \| \partial_{i_{ν - 1}} \dots \partial_{i_{1}} f (u) \|,$

where $| Q_{1} |$ denotes the Lebesgue measure of $Q_{1}$ . Cauchy’s inequalities imply for each $α \in N^{n}$ and $r > 0$

| \partial^{α} E_{j} (y) | \leq \frac{α!}{r^{| α |}} sup z \in D_{r} (y) | E_{j} (z) |,

where $D_{r} (y) = {z \in C^{n} : | z_{i} - y_{i} | \leq r for all i}$ . Choosing $r = \frac{c}{4 \sqrt{n}}$ we get for $y \in R^{n} ∖ B_{c} (0)$

| \partial^{α} E_{j} (y) | \leq \frac{α!}{r^{| α |}} (\frac{j}{π})^{\frac{n}{2}} e^{- \frac{j c^{2}}{2}} .

Hence with $C_{3} = | Q_{1} | ∥ f ∥_{ϱ_{1}, Q_{1}} {sup}_{\begin{matrix} 1 \leq i \leq n y \in U \end{matrix}} | \partial_{i} χ (y) |$ we obtain for $x \in K$

	$\| (\partial_{i_{N}} \dots \partial_{i_{ν + 1}} E_{j}) *$	$(\partial_{i_{ν}} χ) (\partial_{i_{ν - 1}} \dots \partial_{i_{1}} f) (x) \|$
		$\leq C_{3} \frac{(N - ν)!}{r^{N - ν}} (\frac{j}{π})^{\frac{n}{2}} e^{- \frac{j c^{2}}{2}} ϱ_{1}^{ν - 1} (ν - 1)! M_{ν - 1}$
(2.11.3)			$\leq C_{3} (\frac{j}{π})^{\frac{n}{2}} e^{- \frac{j c^{2}}{2}} ϱ_{3}^{N} (N - 1)! M_{N},$

where $ϱ_{3} = max {\frac{1}{r}, ϱ_{1}}$ .

It follows from (2.11), (2.11.2), and (2.11) that for $ϱ_{4} = max {ϱ_{2}, ϱ_{3}}$

∥ f - f_{j} ∥_{ϱ_{4}, K} \leq \frac{C_{2}}{\sqrt{j}} + C_{3} (\frac{j}{π})^{\frac{n}{2}} e^{- \frac{j c^{2}}{2}} .

That implies the assertion. ∎

2.12. Closed ideals

Let $U \subseteq R^{n}$ be open. Let $φ \in C^{ω} (U)$ . Consider the principal ideal $φ C^{M} (U)$ generated by $φ$ .

Proposition.

Assume that $C^{M}$ is stable under derivation (2.1.5). Let $φ$ be a linear form on $R^{n}$ . Then the ideal $φ C^{M} (R^{n})$ is closed in $C^{M} (R^{n})$ . More generally, assume that $ψ = φ_{1}^{p_{1}} \dots φ_{l}^{p_{l}}$ is a finite product of linear forms $φ_{i}$ . Then $ψ C^{M} (R^{n})$ is closed in $C^{M} (R^{n})$ .

Proof. Let $f \in ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ φ C^{M} (R^{n})$ . Then $f |_{φ^{- 1} (0)} = 0$ , since evaluation at points is continuous. As $C^{M}$ is stable under derivation, the standard integral formula (after suitable linear coordinate change) implies that $f = φ g$ for a unique $g \in C^{M} (R^{n})$ . The same reasoning shows that $ψ C^{M} (R^{n})$ is closed in $C^{M} (R^{n})$ , where $ψ = φ_{1}^{p_{1}}$ .

For the general statement it suffices to show: Let $ψ_{1}$ be a polynomial and $ψ_{2}$ a power of a linear form. If $ψ_{1}$ and $ψ_{2}$ are relatively prime and both generate closed ideals in $C^{M} (R^{n})$ , then $ψ_{1} ψ_{2} C^{M} (R^{n})$ is closed in $C^{M} (R^{n})$ . For $f \in ¯ ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯ ψ_{1} ψ_{2} C^{M} (R^{n})$ we find functions $g_{1}, g_{2} \in C^{M} (R^{n})$ with $f = ψ_{1} g_{1} = ψ_{2} g_{2}$ . Since $ψ_{1}$ and $ψ_{2}$ are relatively prime, we have $g_{1} |_{ψ_{2}^{- 1} (0)} = 0$ . By the standard integral formula we obtain as above $g_{1} = ψ_{2} h$ with $h \in C^{M} (R^{n})$ . Hence the assertion. ∎

Remark.

Note that for any hyperbolic polynomial $φ$ the principal ideal $φ C^{M} (R^{n})$ is closed in $φ C^{M} (R^{n})$ (e.g. [42, 4.2]). This follows from the fact (due to [12]) that Weierstrass division holds in $C^{M}$ for hyperbolic divisors. A polynomial $φ (x^{'}, x_{n}) = x_{n}^{d} + \sum_{j = 1}^{d} a_{j} (x^{'}) x_{n}^{d - j}$ with $a_{j} \in C^{M} (R^{n - 1})$ and $a_{j} (0) = 0$ , for $1 \leq j \leq d$ , is called hyperbolic if, for each $x^{'} \in R^{n - 1}$ , all roots of $φ (x^{'}, \cdot)$ are real.

But in general the principal ideal $φ C^{M} (U)$ generated by a real analytic function $φ$ need not be closed (see [41] and [42, part 4]). Compare this with the famous results on the division of distributions due to Hörmander [18] and Lojasiewicz [25, 26].

2.13.

Let $M$ be a DC-weight sequence, and let $X$ be a real analytic manifold. We can define the space $C^{M} (X)$ of functions of Denjoy–Carleman class $C^{M}$ on $X$ by means of local coordinate systems, since $C^{M}$ contains the real analytic functions and is stable under composition. Similarly, we may consider the space $(Ω^{M})^{p} (X)$ of $p$ -forms of class $C^{M}$ on $X$ .

3. Invariant functions in Denjoy–Carleman classes

Throughout this paper we consider a compact Lie group $G$ acting smoothly on a manifold $X$ . A function $f$ on $X$ is said to be $G$ -invariant if $f (g . x) = f (x)$ for all $g \in G$ and all $x \in X$ . If $F$ is a set of functions on $X$ , then $F^{G}$ denotes the subset of $G$ -invariant elements in $F$ .

3.1. Hilbert’s theorem

(e.g. [43]) Let $G$ be a compact Lie group and let $V$ be a real finite dimensional $G$ -module. Then, by a theorem due to Hilbert, the algebra $R [V]^{G}$ of $G$ -invariant polynomials on $V$ is finitely generated. The generators can be chosen homogeneous and with positive degree.

3.2. Schwarz’s theorem

Suppose that the representation of $G$ in $V$ is orthogonal. Let $σ_{1}, \dots, σ_{p}$ be a system of generators of $R [V]^{G}$ and put $σ = (σ_{1}, \dots, σ_{p}) : V \to R^{p}$ . Schwarz [38] proved that $σ^{*} : C^{\infty} (R^{p}) \to C^{\infty} (V)^{G}$ is surjective, which is the smooth analog of 3.1. Mather [27] showed that $σ^{*} : C^{\infty} (R^{p}) \to C^{\infty} (V)^{G}$ is even split surjective, i.e., it allows a continuous linear section.

3.3. Symmetric functions in Denjoy–Carleman classes

In the case that the symmetric group $S_{n}$ acts in $R^{n}$ by permuting the coordinates, the statement of Schwarz’s theorem 3.2 is due to Glaeser [16]. In that case $σ_{i}$ is the $i$ -th elementary symmetric function, i.e., $σ_{i} (x) = \sum_{1 \leq j_{1} < \dots < j_{i} \leq n} x_{j_{1}} \dots x_{j_{i}}$ , and $σ = (σ_{1}, \dots, σ_{n}) : R^{n} \to R^{n}$ .

The representation of symmetric functions in Denjoy–Carleman (Gevrey) classes was treated by Bronshtein [7, 8]. Since we shall need it later, we present a more general version and we sketch a proof. Let $\prod_{j = 1}^{p} S_{n}$ act in $⨁_{j = 1}^{p} R^{n}$ by permuting the coordinates. Since $R [⨁_{j = 1}^{p} R^{n}]^{\prod_{j = 1}^{p} S_{n}} ≅ ⨂_{j = 1}^{p} R [R^{n}]^{S_{n}}$ , a $\prod_{j = 1}^{p} S_{n}$ -invariant function $f$ on $R^{p n}$ has the form $f = F \circ θ$ with $θ = (σ, \dots, σ)$ .

Theorem.

Assume that $M$ and $N$ are increasing logarithmically convex sequences with $M_{0} = N_{0} = 1$ . Then for any function $f \in C^{M} (R^{p n})^{\prod_{j = 1}^{p} S_{n}}$ there exists a function $F \in C^{N} (θ (R^{p n}))$ such that $f = F \circ θ$ if and only if

(3.3.1)

sup k \in N_{> 0} (\frac{M_{k n}}{N_{k}})^{\frac{1}{k}} < \infty .

Sketch of proof. We indicate and adapt the main steps in Bronshtein’s proof. The necessity of (3.3.1) is shown by considering the symmetric function $f \in C^{M} (R^{n})$ (for $n > 2$ ) given by

f (x) = \infty \sum k = 0 c_{k} (1 - n \prod j = 1 (ρ_{k} x_{j} e^{- ρ_{k}^{2} x_{j}^{2}}))^{- 1},

where $ρ_{k} = \frac{M_{k n + 1}}{M_{k n}}$ and $c_{k} = \frac{M_{k n}}{2^{k} ρ_{k}^{k n}}$ . Then $f = F \circ σ$ with

F (σ) = \infty \sum k = 0 c_{k} (1 - ρ_{k}^{n} σ_{n} e^{- ρ_{k}^{2} (σ_{1}^{2} - 2 σ_{2})})^{- 1},

and hence

| (\partial_{σ_{n}})^{m} F (0) | = \infty \sum k = 0 c_{k} ρ_{k}^{m n} m! \geq c_{m} ρ_{m}^{m n} m! = \frac{m! M_{m n}}{2^{m}} .

Since $F \in C^{N}$ this implies (3.3.1). For $n = 2$ one can find a similar example.

Without loss suppose that $f \in C^{M} (R^{2 n})^{S_{n} \times S_{n}}$ . Instead of the elementary symmetric polynomials $σ_{i}$ we use the Newton polynomials $ν_{i} (x) = \sum_{j = 1}^{n} x_{j}^{i}$ and put $ν = (ν_{1}, \dots, ν_{n})$ (see remark 3.4(3)). Then we may write $f (x, y) = F (ν (x), ν (y)) = F (u, v)$ where $u = ν (x)$ , $v = ν (y)$ , and $(x, y) \in R^{n} \times R^{n}$ . A direct computation gives

	$\partial_{u_{k}} F (u, v)$	$= \frac{(- 1)^{k + 1}}{k} n \sum i = 1 \frac{σ_{n - k} (x_{i}^{'}) \partial_{x_{i}} f (x, y)}{\prod_{j \neq i} (x_{j} - x_{i})} = n \sum i = 1 \frac{g_{k i} (x, y)}{\prod_{j \neq i} (x_{j} - x_{i})},$
	$\partial_{v_{k}} F (u, v)$	$= \frac{(- 1)^{k + 1}}{k} n \sum i = 1 σ_{n -}$

	$\| \partial_{i_{N}} \dots \partial_{i_{1}} (f - f_{j}) \| \leq$	$\| \partial_{i_{N}} \dots \partial_{i_{1}} f - E_{j} * (χ \partial_{i_{N}} \dots \partial_{i_{1}} f) \|$
(2.11.1)			$+ N \sum ν = 1 \| (\partial_{i_{N}} \dots \partial_{i_{ν + 1}} E_{j}) * (\partial_{i_{ν}} χ) (\partial_{i_{ν - 1}} \dots \partial_{i_{1}} f) \| .$

	$\| E_{j} * (χ \partial^{α} f) (x)$	$- \partial^{α} f (x) \| = \| \int E_{j} (y) (χ (x - y) \partial^{α} f (x - y) - \partial^{α} f (x)) d y \|$
		$\leq \int_{B_{c} (0)} E_{j} (y) \| \partial^{α} f (x - y) - \partial^{α} f (x) \| d y$
		$+ \int_{R^{n} ∖ B_{c} (0)} E_{j} (y) (χ (x - y) \| \partial^{α} f (x - y) \| + \| \partial^{α} f (x) \|) d y .$