Linear biseparating maps between spaces of vector-valued differentiable functions and

automatic continuity

Jesús Araujo^{1}

Departamento de Matemáticas, Estadística y Computación Universidad de Cantabria Facultad de Ciencias Avda. de los Castros, s. n. E-39071 Santander, Spain

Abstract

We give a complete description of linear biseparating maps between spaces of vectorvalued differentiable functions. This description automatically implies continuity of such maps.

Key words: biseparating map, disjointness preserving, automatic continuity, vector-valued diffrentiable functions 1991 MSC: Primary 47B33; Secondary 46H40, 47B38, 46E40, 46E25

1 Introduction

It is well known that an algebraic link between spaces of continuous functions may lead to a topological link between the spaces on which the functions are defined. For instance, it turns out that if there exists a ring isomorphism T : C(X) → C(Y), then the realcompactifications of X and Y are homeomorphic ([16, pp. 115-118]). Also if h is the resultant homeomorphism from the realcompactification of Y onto that of X, then T f = f ∘h for every f ∈ C(X), so we have a complete description of it. As a result, when X and Y are realcompact, we deduce that if both spaces of continuous functions C(X) and

Email address: araujoj@unican.es (Jesús Araujo). 1 Research partially supported by the Spanish Dirección General de Investigación Científica y Técnica (DGICYT, PB98-1102).

Preprint submitted to Elsevier Science 18 June 2003

C(Y) are endowed with the compact-open topology, then every ring isomorphism between them is continuous. In this result, the key point is that every ring isomorphism sends maximal ideals into maximal ideals. This implies that a good description of maximal ideals lead to the definition of a map from Y onto X.

Of course, the pattern above has been successfully applied to many other algebras of functions. However the situation becomes more complicated if we consider spaces of functions which take values in arbitrary Banach spaces. In this context and unlike algebra or ring homomorphisms, we can still use mappings satisfying the property ∥T f∥ ∥T g∥ ≡ 0 if and only if ∥f∥ ∥g∥ ≡ 0. These maps are called biseparating, and coincide with disjointness preserving mappings whose inverses preserve disjointness too ([1]). In general these maps turn out to be efficacious substitutes for homomorphisms. Indeed, in [2], we prove that the existence of a biseparating mapping between a large class of spaces of vector-valued continuous functions A(X, E) and A(Y, F) (E, F are Banach spaces) yields homeomorphisms between some compactifications (and even the realcompactifications) of X and Y. The automatic continuity of a linear biseparating mapping is also accomplished in some cases (see [3,4]). Related results have also been given recently, for some other families of scalarvalued functions, for instance, in [5,15,19] and [20]. In this paper, we go a step beyond and work in a context which does not seem to have made its way into the literature yet, namely, linear operators between spaces of differentiable functions taking values in arbitrary Banach spaces.

As for spaces (indeed algebras) of scalar-valued differentiable functions, Myers ([25]) showed that the structure of a compact differentiable manifold of class C^{n} is determined by the algebra of all real-valued functions on M of class C^{n}. In this line, Pursell ([28]) checked that the ring structure of infinitely differentiable functions defined on an open convex set of ℝ^{n} determines such set up to a diffeomorphism. Homomorphisms between algebras of differentiable functions defined on real Banach spaces have been studied by Aron, Gómez and Llavona ([6]); in their paper a description of homomorphisms is given and the automatic continuity is obtained as a corollary in quite a general setting, which includes in particular the case when the Banach spaces are finite-dimensional (see also [17]).

On the other hand, automatic continuity results for algebras of differentiable functions have been also given for instance in [7,21,24,26] and [27]. In particular, automatic continuity of separating maps defined on spaces of differentiable functions has been studied by Kantrowitz and Neumann in [22]. For classical results and techniques in the study of automatic continuity, see [10] and [29], and the recent book [11] by H. G. Dales.

2 Preliminaries and notation

Let E be a real Banach space. If λ = (λ_{1}, λ_{2}, …, λ_{p}) is a p-tuple of nonnegative integers, we set |λ| := λ_{1} + λ_{2} + … + λ_{p}. If Ω is a nonempty open subset of ℝ^{p}, then C^{n}(Ω, E) consists of the E-valued functions f in Ω that are of class C^{n}, that is, those functions whose partial derivatives

∂^{λ}f := \frac{∂^{λ_{1}+λ_{2}+...+λ_{p}}f}{∂x_{1}^{λ_{1}}∂x_{2}^{λ_{2}} … ∂x_{p}^{λ_{p}}}

exist and are continuous for each λ = (λ_{1}, λ_{2}, …, λ_{p}) ∈ Λ, where Λ := {λ ∈ (ℕ∪{0})^{p} : |λ| ≤ n}. It is well known that, if for L^{k}(ℝ^{p}, E) we denote the space of continuous k-ℝ-linear maps of ℝ^{p} into E, then C^{n}(Ω, E) coincides with the space of maps f : Ω → E such that the differential D^{k}f : Ω → L^{k}(ℝ^{p}, E) exists and is continuous for each k = 0, …, n. It is well known that, when it exists at a point a ∈ Ω, the differential D^{k}f(a) ∈ L^{k}(ℝ^{p}, E) is a symmetric form of degree k.

Now, given a map f ∈ C^{n}(Ω, E), we define its Taylor polynomial function of degree n at a ∈ Ω as

T_{a}(x) := f(a) + Df(a)(x − a) + \frac{1}{2}D^{2}f(a)(x − a, x − a) + …
+ \frac{1}{n!}D^{n}f(a)(\underbrace{x − a, x − a, …, x − a}_{n}).

In the case when E = ℝ, C_{c}^{n}(Ω, ℝ) will denote the subring of C^{n}(Ω, ℝ) of functions with compact support.

On the other hand, in the case when Ω is also bounded C^{n}(Ω̄, E) denotes the subspace of C^{n}(Ω, E) of those functions whose partial derivatives up to order n admit continuous extension to the boundary of Ω.

For a set C ⊂ ℝ^{p}, cl_{ℝ^{p}} C and int_{ℝ^{p}} C denote its closure and its interior in ℝ^{p}, respectively. Given x_{0} ∈ ℝ^{p} and δ > 0, B(x_{0}, δ) and B̄(x_{0}, δ) stand for the open and closed balls of center x_{0} and radius δ, respectively. As for the norm in ℝ^{p}, if x = (x_{1}, x_{2}, …, x_{p}) belongs to ℝ^{p}, we set |x| := max_{j} |x_{j}|. For i ∈ {1, 2, …, p}, x_{i} : Ω → ℝ will be the projection on the i-th coordinate, that is, x_{i}(t_{1}, t_{2}, …, t_{p}) = t_{i} for every (t_{1}, t_{2}, …, t_{p}) ∈ Ω.

Notice also that if E is a Banach space over ℂ, it can also be viewed as a real space, and in this sense we consider defined the space C^{n}(Ω, E). It is immediate that all results valid in the real setting hold also in the complex case. But being also complex ensures that C^{n}(Ω, E) is both real and complex as a linear space, and consequently we can consider both real and complex linear maps from C^{n}(Ω, E) into some other vector spaces. This is the reason why in this paper we will assume that E and F are 𝕂-Banach spaces, where 𝕂 = ℝ or ℂ.

As for the spaces of linear functions, we will denote by L′(E, F) and by I′(E, F) the sets of (not necessarily continuous) linear maps and bijective linear maps from E into F, respectively. L(E, F) and I(E, F) will denote the spaces of continuous linear maps and bijective continuous linear maps from E into F.

The context. From now on we will assume that we are in one of the following two situations. All definitions, results and comments given in this paper apply to these two situations unless otherwise stated.

• Situation 1. Ω and Ω′ are (not necessarily bounded) open subsets of ℝ^{p} and ℝ^{q}, respectively (p, q ∈ ℕ). A^{n}(Ω, E) = C^{n}(Ω, E) and A^{m}(Ω′, F) = C^{m}(Ω′, F) (n, m ≥ 1). A^{n}(Ω, 𝕂) = C^{n}(Ω, 𝕂) and A^{m}(Ω′, 𝕂) = C^{m}(Ω′, 𝕂).

• Situation 2. Ω and Ω′ are bounded open subsets of ℝ^{p} and ℝ^{q}, respectively (p, q ∈ ℕ), with the property that int_{ℝ^{p}} cl_{ℝ^{p}} Ω = Ω and int_{ℝ^{q}} cl_{ℝ^{q}} Ω′ = Ω′. A^{n}(Ω, E) = C^{n}(Ω̄, E) and A^{m}(Ω′, F) = C^{m}(Ω̄′, F) (n, m ≥ 1). A^{n}(Ω, 𝕂) = C^{n}(Ω̄, 𝕂) and A^{m}(Ω′, 𝕂) = C^{m}(Ω̄′, 𝕂).

This means that when we refer to spaces Ω, Ω′, A^{n}(Ω, E), A^{m}(Ω′, F), A^{n}(Ω, 𝕂), A^{m}(Ω′, 𝕂), we assume that all of them are included at the same time in one of the above two situations.

The topologies. One of the goals of this paper is to provide some results of automatic continuity. This will be done when the spaces of functions are endowed with some natural topologies.

Definition 2.1 We say that a locally convex topology in A^{n}(Ω, E) is compatible with the pointwise convergence if the following two conditions are satisfied:

• 1. when endowed with it, A^{n}(Ω, E) is a Fréchet (or Banach) space, and • 2. if (f_{n}) is a sequence in A^{n}(Ω, E) converging to zero, then (f_{n}(x)) converges

to zero for every x ∈ Ω.

Biseparating maps. For a function f ∈ A^{n}(Ω, E), we denote by c(f) the cozero set of f, that is, the set {x ∈ Ω : f(x) ≠ 0}.

Definition 2.2 A map T : A^{n}(Ω, E) → A^{m}(Ω′, F) is said to be separating if it is additive and c(T f) ∩ c(T g) = ∅ whenever f, g ∈ A^{n}(Ω, E) satisfy c(f) ∩ c(g) = ∅. Besides T is said to be biseparating if it is bijective and both T and T^{−1} are separating.

Equivalently, we see that an additive map T : A^{n}(Ω, E) → A^{m}(Ω′, F) is separating if ∥(T f)(y)∥ ∥(T g)(y)∥ = 0 for all y ∈ Ω′ whenever f, g ∈ A^{n}(Ω, E) satisfy ∥f(x)∥ ∥g(x)∥ = 0 for all x ∈ Ω.

Let Ω_{1} := Ω, Ω′_{1} := Ω′ when we are in Situation 1, and Ω_{1} := Ω̄, Ω′_{1} := Ω̄′ if we are in Situation 2. A point x ∈ Ω_{1} is said to be a support point of y ∈ Ω′_{1} if, for every neighborhood U of x in Ω_{1}, there exists f ∈ A^{n}(Ω, E) satisfying c(f) ⊂ U such that (T f)(y) ≠ 0.

In [2], biseparating maps are studied in a more general setting. In particular, applied to our situation, we have that A^{n}(Ω, E) and A^{m}(Ω′, F) are modules over the strongly regular rings C^{n}(Ω_{1}, ℝ) and C^{n}(Ω′_{1}, ℝ), respectively (for the definition of strongly regular ring, see [2]; see also [30, Corollary 1.2] or [18, Proposition 2.12.5]). According to [2, Corollary 3.2], we conclude that there exists a homeomorphism h from Ω′_{1} onto Ω_{1} (which implies that in both Situation 1 and Situation 2 h is a homeomorphism from Ω′ onto Ω). This map h sends each point in Ω′_{1} into its support point in Ω_{1}, and is called support map for T. It turns out that the support map for T^{−1} is h^{−1} (see the proof of [2, Theorem 3.1]). Rephrasing Lemma 4.4 in [2], we have the following property.

Lemma 2.1 If y ∈ Ω′_{1} and f ∈ A^{n}(Ω, E) vanishes on a neighborhood of h(y), then (T f)(y) = 0.

Functions of class s − C^{n}. Suppose that K : Ω → L(E, F) is a continuous map, where L(E, F) is endowed with the topology of the norm. For each e ∈ E, we define K_{e} : Ω → F as K_{e}(y) := (Ky)(e) for every y ∈ Ω. We say that K is of class s − C^{1} if, for every e ∈ E, the map K_{e} admits all partial derivatives of order 1 in Ω, and for each i = 1, …, p, the map

\frac{∂_{s}}{∂x_{i}} K : Ω → L(E, F),

sending each y ∈ Ω and each e ∈ E into \frac{∂}{∂x_{i}} K_{e}(y), is continuous when considering in L(E, F) the strong operator topology, that is, the coarsest topology such that the mapping A ∈ L(E, F) ↪ Ae ∈ F is continuous for every e ∈ E.

Definition 2.3 Let n ≥ 2. A map J : Ω → L(E, F) is said to be of class s − C^{n} if the following two statements are satisfied:

• 1. J is of class C^{n−1} (considering L(E, F) as a Banach space).

• 2. All partial derivatives K : Ω → L(E, F) of order n − 1 of J are of class s − C^{1}.

Examples. Next we provide two examples of biseparating linear maps between spaces of vector-valued functions of class C^{1}. The first example will give the most basic form of a biseparating linear map, which is essentially the only possible when E and F are finite-dimensional spaces, that is, the case when the operator through which it is defined is itself of class C^{1}. In the second example, the biseparating map is defined through an operator which is of class s − C^{1} but it is not of class C^{1}.

Example 2.4 Let Ω = Ω′ = (−1, 1), E = F = ℝ^{2}. For each t ∈ (−1, 1), let us consider Jt : ℝ^{2} → ℝ^{2} defined as

(Jt)(x_{1}, x_{2}) := ((2 + t)x_{1}, (2 − t)x_{2})

for (x_{1}, x_{2}) ∈ ℝ^{2}. Clearly each Jt is linear, bijective and continuous, being its inverse Kt : ℝ^{2} → ℝ^{2} defined as

(Kt)(x_{1}, x_{2}) = (\frac{x_{1}}{2 + t}, \frac{x_{2}}{2 − t})

for every (x_{1}, x_{2}) ∈ ℝ^{2}.

It is easy to see that the derivatives of J : (−1, 1) → L(ℝ^{2}, ℝ^{2}) and K : (−1, 1) → L(ℝ^{2}, ℝ^{2}) exist and, for (x_{1}, x_{2}) ∈ ℝ^{2}, (J′t)(x_{1}, x_{2}) = (x_{1}, −x_{2}) and

(K′t)(x_{1}, x_{2}) = (\frac{−tx_{1}}{(2 + t)^{2}} , \frac{tx_{2}}{(2 − t)^{2}} ) .

In this way we see that J and K are of class C^{1}. Consequently the map T; C^{1}((−1, 1), ℝ^{2}), defined as (T f)(t) = (Jt)(f(t)) is biseparating (see Proposition 6.1).

Example 2.5 Let Ω = Ω′ = (−1, 1), E = F = c_{0} be the space of sequences in 𝕂 converging to zero, endowed with the sup norm. For each t ∈ (−1, 1), let us define the map Jt : c_{0} → c_{0} as

(Jt)(x_{n}) := ((2 + \frac{2n − 1}{2n}t^{\frac{2n}{2n−1}} ) x_{n})

for each (x_{n}) ∈ c_{0}.

It is easy to see that each Jt ∈ I(c_{0}, c_{0}), that is, it is linear, continuous and bijective, and that its inverse is the map Kt ∈ I(c_{0}, c_{0}) defined as

(Kt)(x_{n}) := \frac{x_{n}}{(2 + \frac{2n−1}{2n}t^{\frac{2n}{2n−1}} )}

for each (x_{n}) ∈ c_{0}.

Let us see that the map J : (−1, 1) → L(c_{0}, c_{0}) is continuous when L(c_{0}, c_{0}) is endowed with the norm topology. We are going to show that if 0 < t < t′ < 1, then ∥Jt − Jt′∥ ≤ 2 |t − t′|. It is clear that we just need to prove that if n ∈ ℕ, then |t^{′2n/2n−1} − t^{2n/2n−1}| ≤ 2 |t′ − t|. Notice that

0 ≤ t^{′ \frac{2n}{2n−1}} − t^{\frac{2n}{2n−1}}
≤ t^{′ \frac{2n}{2n−1}} + t′t^{\frac{1}{2n−1}} − tt^{′ \frac{1}{2n−1}} − t^{\frac{2n}{2n−1}}
= (t′ − t)(t^{′ \frac{1}{2n−1}} + t^{\frac{1}{2n−1}} )
≤ 2(t′ − t).

Since a similar reasoning also holds for any t, t′ ∈ (−1, 1), we conclude that J is continuous. In the same way we can prove that K : (−1, 1) → L(c_{0}, c_{0}) is continuous.

On the other hand, we have that, for each (x_{n}) ∈ c_{0}, the maps J_{(x_{n})}, K_{(x_{n})} : (−1, 1) → c_{0}, defined as J_{(x_{n})}(t) := (Jt)(x_{n}) and K_{(x_{n})}(t) := (Kt)(x_{n}) are derivable, and it can be seen that for each t ∈ (−1, 1),

J′_{(x_{n})}(t) = (t^{\frac{1}{2n−1}} x_{n})

and

K′_{(x_{n})}(t) = (−\frac{t^{\frac{1}{2n−1}} x_{n}}{(2 + \frac{2n−1}{2n}t^{\frac{2n}{2n−1}} )^{2}} ) = (−\frac{J′_{(x_{n})}(t)}{(2 + \frac{2n−1}{2n}t^{\frac{2n}{2n−1}} )^{2}} ) .

Now, let us define J′_{s}, K′_{s} : (−1, 1) → L(c_{0}, c_{0}) as J′_{s}(t)(x_{n}) := J′_{(x_{n})}(t) and K′_{s}(t)(x_{n}) := K′_{(x_{n})}(t) for each t ∈ (−1, 1) and each (x_{n}) ∈ c_{0}. It is straightforward to see that, for every (x_{n}) ∈ c_{0}, the maps J′_{(x_{n})} and K′_{(x_{n})} are continuous, which is to say that J′_{s} and K′_{s} are continuous when L(c_{0}, c_{0}) is endowed with the strong operator topology. We conclude that J and K are of class s − C^{1}.

Finally notice that, when L(c_{0}, c_{0}) is endowed with the topology of the norm, then ∥J′_{s}(t)∥ = 1 for every t ∈ (−1, 1) \ {0}, which implies that J′_{s} is not continuous at t = 0.

If we define now the map T : C^{1}((−1, 1), c_{0}) → C^{1}((−1, 1), c_{0}) as

(T f)(t) := (Jt)(f(t)),

it turns out that T is well-defined (see Proposition 6.1), and it is routine matter to check that it is biseparating.

3 Some previous results

Lemma 3.1 Suppose that a_{0}, a_{1}, …, a_{k} ∈ C^{n}(Ω, E), and that f : Ω × ℝ → E is a polynomial in t defined as

f(x, t) = ∑_{i=0}^{k} a_{i}(x)t^{i}

for every (x, t) ∈ Ω × ℝ. Then f ∈ C^{n}(Ω × ℝ, E).

Proof. It is immediate from the fact that all partial derivatives up to order n exist and are continuous. □

Lemma 3.2 Suppose that f, g ∈ C^{n}(Ω, E) and k : Ω → ℝ satisfy

f(x) = k(x)g(x)

for every x ∈ Ω. If g(x) ≠ 0 for every x ∈ Ω, then k ∈ C^{n}(Ω, ℝ).

Proof. Fix x_{0} ∈ Ω. We are going to prove that k is of class C^{n} in a neighborhood of x_{0}. First we take f′ : E → ℝ linear and continuous, and such that f′(g(x_{0})) ≠ 0. Since f′ and g are continuous, then there exists an open neighborhood U of x_{0} such that

f′(g(x)) ≠ 0

for every x ∈ U.

Now

f′(f(x)) = k(x)f′(g(x))

for every x ∈ U. This implies that, for every x ∈ U,

k(x) = \frac{f′(f(x))}{f′(g(x))},

which is the quotient of two real-valued functions of class C^{n}.

This proves that k is of class C^{n}. □

The proof of the following result is straightforward.

Lemma 3.3 Suppose that f ∈ C^{n}(Ω, E), a ∈ Ω, and 1 ≤ k ≤ n. Then the k-th derivative of the Taylor polynomial function T_{a} of degree n of f is equal to the Taylor polynomial function of degree n − k of D^{k}f at a.

The following theorem, known as Whitney’s extension theorem, can be found, for instance, in [12, Theorem 3.1.14].

Theorem 3.4 Suppose n ∈ ℕ, A is a closed subset of ℝ^{p}, and to each a ∈ A corresponds a polynomial function

P_{a} : ℝ^{p} → E

with degree P_{a} ≤ n. Whenever C ⊂ A and δ > 0 let ρ(C, δ) be the supremum of the set of all numbers

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ · |a − b|^{i−n} · (n − i)!

corresponding to i = 0, …, n and a, b ∈ C with 0 < |a − b| ≤ δ.

If ρ(C, δ) → 0 as δ → 0+ for each compact subset C of A, then there exists a map g : ℝ^{p} → E of class C^{n} such that

D^{i}g(a) = D^{i}P_{a}(a)

for i = 0, …, n and a ∈ A.

Proposition 3.5 Let p ≥ 2. For s ∈ ℝ, s > 0, consider the following compact subsets of ℝ^{p}:

A := {(x_{1}, x_{2}, …, x_{p}) ∈ ℝ^{p} : ∑_{i=2}^{p} x_{i}^{2} ≤ x_{1}^{2}/9, |x_{1}| ≤ s},

A^{+} := {(x_{1}, x_{2}, …, x_{p}) ∈ A : x_{1} ≥ 0},

and

A^{−} := {(x_{1}, x_{2}, …, x_{p}) ∈ A : x_{1} ≤ 0}.

Suppose that Ω is an open subset of ℝ^{p} containing A and that f belongs to C^{n}(Ω, E). If

∂^{λ}f(0, 0, …, 0) = 0

for every λ ∈ Λ, then there exists a function f_{+} ∈ C^{n}(Ω, E) with compact support such that, for λ ∈ Λ,

∂^{λ}f_{+}(x) = ∂^{λ}f(x)

for every x ∈ A^{+}, and

∂^{λ}f_{+}(x) = 0

for every x ∈ A^{−}.

Proof. Suppose that for x ∈ A, T_{x} stands for the polynomial function of degree n given in the Taylor formula for f at x. Now for a ∈ A^{−}, we consider as P_{a} the polynomial identically zero, and for a ∈ A^{+}, we consider as P_{a} the polynomial T_{a}. As it is seen for instance in [9, Theorem 2.71] or [23, p. 350], if r > 0 and |b − a| < r, we have that

∥T_{a}(b) − f(b)∥ ≤ \frac{|b − a|^{n}}{n!} sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥ .

Now it is easy to see that, by Lemma 3.3, for i ∈ {1, 2, …, n},

∥D^{i}T_{a}(b) − D^{i}f(b)∥ ≤ \frac{|b − a|^{n−i}}{(n − i)!} sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥ .

This proves that if |b − a| < r,

∥D^{i}T_{a}(b) − D^{i}f(b)∥ · |b − a|^{i−n} · (n − i)! ≤ sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥

for i ∈ {0, 1, …, n}. Then we define τ(r) as the supremum of the set of all numbers

sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥

for x, a ∈ A, which is a real number, because A is compact. Clearly, since D^{n}f is continuous, if r tends to zero, τ(r) tends to zero. Now suppose that a and b belong to A and 0 < |b − a| ≤ r. Then we have the following possibilities:

• a, b ∈ A^{+}. Then we have that, by Lemma 3.3, for i ∈ {0, 1, …, n},

D^{i}P_{b}(b) = D^{i}f(b) and consequently

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ = ∥D^{i}T_{a}(b) − D^{i}f(b)∥ .

• a, b ∉ A^{+}. Then

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ = 0.

• a ∉ A^{+}, b ∈ A^{+}. Note that since we are assuming by hypothesis that D^{i}f(0, 0, …, 0) = 0, then D^{i}P_{0}(b) = 0 for all i ∈ {0, 1, …, n}. Consequently

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ = ∥D^{i}P_{b}(b)∥
= ∥D^{i}P_{0}(b) − D^{i}P_{b}(b)∥
= ∥D^{i}T_{0}(b) − D^{i}f(b)∥ .

• a ∈ A^{+}, b ∉ A^{+}. Then we have that

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ = ∥D^{i}T_{a}(b)∥
≤ ∥D^{i}T_{a}(b) − D^{i}f(b)∥ + ∥D^{i}f(b)∥
= ∥D^{i}T_{a}(b) − D^{i}f(b)∥ + ∥D^{i}f(b) − D^{i}T_{0}(b)∥ .

Note that in the third and forth cases above, |a| , |b| ≤ |b − a| ≤ r. This implies that in these two cases

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ · |b − a|^{i−n} · (n − i)! ≤ sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥
+ sup_{|x|≤r} ∥D^{n}f(x) − D^{n}f(0)∥ .

On the other hand it is easy to see that in the other two cases

∥D^{i}P_{a}(b) − D^{i}P_{b}(b)∥ · |b − a|^{i−n} · (n − i)! ≤ sup_{|x−a|≤r} ∥D^{n}f(x) − D^{n}f(a)∥ .

This facts imply that, if ρ is defined as in Theorem 3.4, ρ(A, r) ≤ 2τ(r). Also, it is clear that if C is a compact subset of A, then ρ(C, r) ≤ ρ(A, r). Consequently by Theorem 3.4, we have that there exists f_{0} ∈ C^{n}(Ω, E) such that, given any λ ∈ Λ,

∂^{λ}f_{0}(x) = ∂^{λ}f(x)

for every x ∈ A^{+}, and

∂^{λ}f_{0}(x) = 0

for every x ∈ A^{−}. Also it is clear that if we take g_{0} ∈ C_{c}^{n}(Ω, ℝ) such that g_{0} ≡ 1 on an open neighborhood of A, then f_{+} := g_{0}f_{0} ∈ C^{n}(Ω, E) satisfies the requirements of the theorem. □

4 Biseparating maps: a first approach

In this section we make a first attempt to describe all biseparating maps, but we do not take into account some important details which will be discussed in Section 6. In this way we characterize all biseparating linear maps from A^{n}(Ω, E) onto A^{m}(Ω′, F) as weighted composition bijective maps. Notice that we assume no continuity properties on T. In fact, we will suppose that our spaces A^{n}(Ω, E) and A^{m}(Ω′, F) are not endowed with any topologies.

Lemma 4.1 Suppose that Ω contains the origin, and that T : A^{n}(Ω, E) → A^{m}(Ω′, F) is a biseparating map. Assume also that f ∈ A^{n}(Ω, E) satisfies that for all λ ∈ Λ,

∂^{λ}f(0, 0, …, 0) = 0.

If (0, 0, …, 0) ∈ Ω is the support point of y ∈ Ω′, then (T f)(y) = 0.

Proof. First suppose that p > 1 and that the closed ball of center 0 and radius s is contained in Ω. If we take A^{+}, A^{−} and f_{+} as in Proposition 3.5, then f_{+} and f − f_{+} belong to A^{n}(Ω, E) and satisfy f_{+}(x) = 0 and (f − f_{+})(x) = 0 for every x ∈ A^{−} and for every x ∈ A^{+} respectively. We have that (T f)(y) = (T f_{+})(y) + (T (f − f_{+}))(y). Also, since for any neighborhood U of the origin there exists an open subset V of U such that f_{+}(x) = 0 for every x ∈ V, then we have that, taking into account that the support map for T^{−1} is h^{−1}, by Lemma 2.1, (T f_{+})(h^{−1}(x)) = 0. Since h : Ω′ → Ω is a homeomorphism, we deduce that (T f_{+})(y) = 0 and, in the same way, (T (f − f_{+}))(y) = 0. We conclude that (T f)(y) = 0.

Consider now the case when p = 1. Note that if f ∈ A^{n}(Ω, E) satisfies f(0) = 0 and 0 = f′(0) = ⋯ = f^{(n)}(0), then it is clear that fξ_{(−∞,0)} and fξ_{(0,+∞)} belong to A^{n}(Ω, E) (where ξ_{A} stands for the characteristic function of A) and, as above, (T (fξ_{(0,+∞)}))(y) = 0 = (T (fξ_{(−∞,0)}))(y). We conclude that (T f)(y) = 0. □

Proposition 4.2 Suppose that T : A^{n}(Ω, E) → A^{m}(Ω′, F) is a 𝕂-linear biseparating map. Then p = q, n = m, and there exist a diffeomorphism h of class C^{n} from Ω′ onto Ω and a map J : Ω′ → I′(E, F) such that for every y ∈ Ω′ and every f ∈ A^{n}(Ω, E),

(T f)(y) = (Jy)(f(h(y))).

Proof. First, the existence of the homeomorphism h (the support map) between Ω′ and Ω implies that p = q (see for instance [13, p. 120]).

Note that if f ∈ A^{n}(Ω, E) and y ∈ Ω′ satisfy

∂^{λ}f(h(y)) = 0

for all λ ∈ Λ, then by Lemma 4.1 we have that

(T f)(y) = 0.

Now take y ∈ Ω′ and fix e ∈ E, e ≠ 0. If #Λ stands for the cardinal of Λ, then we can define a linear map S_{y} : ℝ^{#Λ} → F as follows. Given (a_{λ_{1}}, …, a_{λ_{#Λ}}) ∈ ℝ^{#Λ}, we consider any f ∈ A^{n}(Ω, ℝ) such that

∂^{λ}f(h(y)) = a_{λ}

for every λ ∈ Λ. Then we define

S_{y}(a_{λ_{1}}, …, a_{λ_{#Λ}}) := (T fe)(y).

The map S_{y} is linear and, as we have seen above, does not depend on the function f we choose. This implies that it is well defined.

Then it is easy to see that there exist functions α_{λ} from Ω′ into F, λ ∈ Λ, such that for every y ∈ Ω′ and every f ∈ A^{n}(Ω, ℝ),

(T fe)(y) = ∑_{λ∈Λ} α_{λ}(y)∂^{λ}f(h(y)). (4.1)

From now on we consider i ∈ {1, 2, …, p} fixed.

Next we define some functions

α_{i}^{0}, α_{i}^{1}, …, α_{i}^{n+1}

from Ω′ × ℝ into F. For every y ∈ Ω′ and t ∈ ℝ,

α_{i}^{0}(y, t) := (T ê)(y),
α_{i}^{1}(y, t) := (T x_{i}e)(y) − α_{i}^{0}(y, t)t,
2!α_{i}^{2}(y, t) := (T x_{i}^{2}e)(y) − α_{i}^{0}(y, t)t^{2} − 2α_{i}^{1}(y, t)t,
3!α_{i}^{3}(y, t) := (T x_{i}^{3}e)(y) − α_{i}^{0}(y, t)t^{3} − 3α_{i}^{1}(y, t)t^{2} − 6α_{i}^{2}(y, t)t,

and in general, for k ∈ {1, 2, …, n, n + 1}

k!α_{i}^{k}(y, t) := (T x_{i}^{k}e)(y) − α_{i}^{0}(y, t)t^{k} − kα_{i}^{1}(y, t)t^{k−1} − k(k − 1)α_{i}^{2}(y, t)t^{k−2}
− … − k!α_{i}^{k−1}(y, t)t.

Claim 4.1 For l ∈ {0, 1, …, n+1}, l!α_{i}^{l} is a polynomial in t whose coefficients are a linear combination of T ê, T x_{i}e, …, T x_{i}^{l}e. Moreover, for y ∈ Ω′ fixed, the degree of the polynomial l!α_{i}^{l}(y, t) is at most l. If we also assume that (T ê)(y) ≠ 0, then the degree of l!α_{i}^{l}(y, t) is l and its leading coefficient is equal to (−1)^{l}α_{i}^{0}(y, t) (notice that this term does not depend on t).

We are going to prove it by applying induction on l. It is clear that this is true for l = 0. Suppose that this relation also holds for l ∈ {0, 1, 2, …, k} for some k ≤ n. We are going to see that it holds for l = k + 1. We have that

(k + 1)!α_{i}^{k+1}(y, t) := (T x_{i}^{k+1}e)(y) − α_{i}^{0}(y, t)t^{k+1}
− (k + 1)α_{i}^{1}(y, t)t^{k} − (k + 1)kα_{i}^{2}(y, t)t^{k−1}
− (k + 1)k(k − 1)α_{i}^{3}(y, t)t^{k−2}
− …
− (k + 1)!α_{i}^{k}(y, t)t,

which implies that it is a polynomial in t and, for fixed y ∈ Ω, its coefficient for the term t^{k+1} is α_{i}^{0}(y, t)(−1+(k+1)−\binom{k+1}{2}+\binom{k+1}{3}−…−(k+1)(−1)^{k}), which

is equal to α_{i}^{0}(y, t)(−(1 − 1)^{k+1} + \binom{k+1}{k+1}(−1)^{k+1}), that is, to (−1)^{k+1}α_{i}^{0}(y, t).

Thus the claim is proved.

Now, by Lemma 3.1, we have that for every k ∈ {0, 1, …, n + 1}, α_{i}^{k} belongs to C^{m}(Ω′ × ℝ, F).

Next define α_{0} := T ê, and for k ∈ {1, 2, …, n}, α_{k} := α_{λ} : Ω′ → F, where

λ = (\underbrace{0, 0, …, 0, k,}_{i} 0, …, 0).

Also, let h_{i} stand for the i-th coordinate function of h.

Claim 4.2 For every k ∈ {0, 1, …, n}, and for every y ∈ Ω′,

α_{k}(y) = α_{i}^{k}(y, h_{i}(y)).

First we have from Equation 4.1 that for k ∈ {0, 1, …, n} and y ∈ Ω′,

(T x_{i}^{k}e)(y) = ∑_{λ∈Λ} α_{λ}(y)∂^{λ}x_{i}^{k}(h(y)),

which can be written as

(T x_{i}^{k}e)(y) = α_{0}(y)h_{i}^{k}(y)  (4.2)
+ kα_{1}(y)h_{i}^{k−1}(y)
+ …
+ k!α_{k−1}(y)h_{i}(y)
+ k!α_{k}(y),

because

∂^{λ}x_{i}^{k}(h(y)) = 0

whenever λ ∈ Λ, λ ≠ (\underbrace{0, 0, …, j,}_{i} 0, …, 0), j ∈ \{0, 1, …, k\}.

On the other hand, it is clear that α_{0}(y) = (T ê)(y) = α_{i}^{0}(y, h_{i}(y)) for every y ∈ Ω′. Also suppose that k < n and that α_{j}(y) = α_{i}^{j}(y, h_{i}(y)) for every j ∈ {0, 1, …, k} and every y ∈ Ω′. Then, by Equation 4.2, for y ∈ Ω′

(k + 1)!α_{k+1}(y) = (T x_{i}^{k+1}e)(y)
− α_{0}(y)h_{i}^{k+1}(y)
− (k + 1)α_{1}(y)h_{i}^{k}(y)
− …
− (k + 1)!α_{k}(y)h_{i}(y),

which coincides with (k + 1)!α_{i}^{k+1}(y, h_{i}(y)), and the claim is proved.

On the other hand, notice that in the same way as we obtain Equation 4.2, we have

(T x_{i}^{n+1}e)(y) = α_{0}(y)h_{i}^{n+1}(y)  (4.3)
+ (n + 1)α_{1}(y)h_{i}^{n}(y)

+ …
+ (n + 1)!α_{n}(y)h_{i}(y)

for every y ∈ Ω′.

Claim 4.3 Suppose that y_{0} ∈ Ω′ satisfies α_{0}(y_{0}) ≠ 0. Then for every open neighborhood U of y_{0}, there exists a nonempty open subset W′_{i} of U where h_{i} is of class C^{m}.

First we define F_{i}^{0} : Ω′ × ℝ → F as F_{i}^{0} := (n + 1)!α_{i}^{n+1}, that is,

F_{i}^{0}(y, t) = (T x_{i}^{n+1}e)(y)
− α_{i}^{0}(y, t)t^{n+1}
− (n + 1)α_{i}^{1}(y, t)t^{n}
− (n + 1)nα_{i}^{2}(y, t)t^{n−1}
− …
− (n + 1)!α_{i}^{n}(y, t)t,

for every y ∈ Ω′, t ∈ ℝ.

Then, if j ∈ {1, 2, …, n + 1}, we define F_{i}^{j} : Ω′ × ℝ → F as

F_{i}^{j}(y, t) = \frac{∂^{j}F_{i}^{0}}{∂t^{j}} (y, t),

for all y ∈ Ω′, t ∈ ℝ.

Notice that from the definition of F_{i}^{0}, Claim 4.2 and Equation 4.3, we deduce that

F_{i}^{0}(y, h_{i}(y)) = 0

for every y ∈ Ω′. Also, as we stated in Claim 4.1, the coefficients of F_{i}^{0} as a polynomial of degree n + 1 in t are linear combinations of

(T ê)(y), (T x_{i}e)(y), …, (T x_{i}^{n+1}e)(y),

and consequently, by Lemma 3.1, for k ∈ {1, 2, …, n}, F_{i}^{k} belongs to C^{m}(Ω′ × ℝ, F).

Taking into account that F_{i}^{n+1}(y, t) = (n + 1)!(−1)^{n+1}α_{0}(y) for every (y, t) ∈ Ω′ × ℝ, and the fact that α_{0}(y_{0}) ≠ 0, there exists k_{0} ∈ {0, 1, …, n} such that F_{i}^{k_{0}}(y, h_{i}(y)) = 0 for every y in a neighborhood of y_{0} and F_{i}^{k_{0}+1}(y, h_{i}(y)) takes a value different from 0 for some y in every neighborhood of y_{0}. Suppose then that U is an open neighborhood of y_{0} such that F_{i}^{k_{0}}(y, h_{i}(y)) = 0 for every y ∈ U and that y_{1} ∈ U satisfies F_{i}^{k_{0}+1}(y_{1}, h_{i}(y_{1})) = f ∈ F, f ≠ 0. Now take f′ in the dual space F′ (where F is wiewed as a real Banach space) such that f′(f) ≠ 0. According to the Implicit Function Theorem ([14, p.148]), there exist a neighborhood V of (y_{1}, h_{i}(y_{1})), an open neighborhood W of y_{1}, and a function ϕ : W → ℝ of class C^{m} such that ϕ(y_{1}) = h_{i}(y_{1}) and

{(y, t) ∈ V : f′ ∘ F_{i}^{k_{0}}(y, t) = 0} = {(y, ϕ(y)) : y ∈ W }.

It is easy to prove that this implies that ϕ ≡ h_{i} on a neighborhood W′_{i} of y_{1}, that is, for every open neighborhood U of y_{0}, there exists an open subset W′_{i} of U where h_{i} is of class C^{m}. The claim is proved.

Since both T and T^{−1} are biseparating, we can assume from now on, without loss of generality, that n ≤ m.

Claim 4.4 Suppose that U is a nonempty open subset of Ω′. Then there exists a nonempty open subset W′ of U such that the restriction of h to W′ is a diffeomorphism of class C^{m}.

Notice first that the the open set {y ∈ Ω′ : α_{0}(y) ≠ 0} is dense in Ω′. Otherwise we could find g ∈ A^{m}(Ω′, F), g ≠ 0, such that c(T ê) ∩ c(g) = c(α_{0}) ∩ c(g) = ∅. Since T^{−1} is separating, this would give us Ω ∩ c(T^{−1}g) = c(ê) ∩ c(T^{−1}g) = ∅, which is impossible.

So far we have considered i ∈ {1, 2, …, p} fixed. Of course a similar process can be done for every i ∈ {1, 2, …, p}. In particular, taking into account the above paragraph, by Claim 4.3, there exists an open subset W′_{1} of U such that h_{1} is of class C^{m} in W′_{1}. For the same reason we can find an open subset W′_{2} of W′_{1} where h_{2} is of class C^{m}. Following this process we construct (nonempty) open sets W′_{1}, …, W′_{p} with W′_{1} ⊃ W′_{2} ⊃ … ⊃ W′_{p} such that h is of class C^{m} in W′_{p}. It is clear that a similar reasoning shows that the map h^{−1} is of class C^{n} in an open subset V of h(W′_{p}). Then our situation is as follows: h^{−1} is of class C^{n} in V and h is of class C^{m} in h^{−1}(V). It is well known that, since m ≥ n ≥ 1, this implies that h is a diffeomorphism of class C^{m} in W′ := h^{−1}(V), and we are done.

Claim 4.5 Let W′ be as in Claim 4.4. For every k ∈ {0, 1, …, n}, the map α_{k} belongs to C^{m}(W′, F).

First we have that α_{0} = T ê belongs to A^{m}(Ω′, F). It is also clear that if k ∈ {0, 1, …, n − 1}, then as given in Equation 4.2,

(k + 1)!α_{k+1} = T x_{i}^{k+1}e − α_{0}h_{i}^{k+1} − (k + 1)α_{1}h_{i}^{k} − … − (k + 1)!α_{k}h_{i}

on Ω′.

Consequently, since h_{i} ∈ C^{m}(W′), if α_{0}, α_{1}, …, α_{k} belong to C^{m}(W′, F), α_{k+1} also belongs to C^{m}(W′, F) and then we are done.

Claim 4.6 For every k ∈ {1, 2, …, n}, α_{k} ≡ 0 in W′.

Suppose that y_{0} ∈ W′ and α_{n}(y_{0}) ≠ 0. Since by Claim 4.5 α_{n} is continuous in W′, there exists an open neighborhood U(y_{0}) of y_{0} such that U(y_{0}) ⊂ W′ and α_{n}(y) ≠ 0 for every y ∈ U(y_{0}). Then take g ∈ C^{n}(ℝ, ℝ) such that g^{(n)} is not derivable at the point h_{i}(y_{0}). We define f ∈ A^{n}(Ω, ℝ) as

f(x) := g(x_{i})

for every x = (x_{1}, x_{2}, …, x_{p}) ∈ Ω. In this way we have that

\frac{∂^{n+1}f}{∂x_{i}^{n+1}} (h(y_{0}))

does not exist. Consequently, using a reasoning similar to that giving Equation 4.2, we have that Equation 4.1 applied to f yields

(T fe)(y) = α_{0}(y)f(h(y)) + α_{1}(y)\frac{∂f}{∂x_{i}} (h(y)) + … + α_{n}(y)\frac{∂^{n}f}{∂x_{i}^{n}} (h(y))

for every y ∈ U(y_{0}). Now we analyze the terms in the above equation, taking into account that we are assuming 1 ≤ n ≤ m, and that by Claim 4.4, h is a diffeomorphism of class C^{m} in W′. First T fe is of class C^{1} in U(y_{0}). Also, by Claim 4.5, for k ∈ {0, 1, …, n}, each α_{k} is of class C^{1} in U(y_{0}). Finally, f and all of its partial derivatives up to order n − 1 are of class C^{1} in Ω.

Thus we deduce from the above equation that

α_{n} \frac{∂^{n}f}{∂x_{i}^{n}} ∘ h

is of class C^{1} in U(y_{0}), and by Lemma 3.2 the same applies to the function

\frac{∂^{n}f}{∂x_{i}^{n}} ∘ h.

But, as we said before, h is a diffeomorphism of class C^{m} in W′, and consequently

\frac{∂^{n}f}{∂x_{i}^{n}}

admits a partial derivative with respect to the i-th coordinate at the point h(y_{0}), which is a contradiction. This implies that α_{n} ≡ 0 in W′. In a similar way we can see that α_{k} ≡ 0 in W′ for k ≥ 1, that is, α_{λ} ≡ 0 in W′ whenever λ ∈ Λ is of the form

λ = (\underbrace{0, 0, …, k,}_{i} 0, …, 0).

Claim 4.7 For every λ ∈ Λ − {(0, 0, …, 0)}, α_{λ} ≡ 0 in W′.

Here our reasoning will be similar to the one given in Claim 4.6. In this way, if i, j ∈ {1, 2, …, p}, i ≠ j, again by Equation 4.1, for every y ∈ W′,

(T (x_{i}x_{j}e))(y) = α_{0}(y)h_{i}(y)h_{j}(y) + α_{λ_{0}^{1}}(y)

where λ_{0}^{1} := (λ_{01}^{1}, λ_{02}^{1}, …, λ_{0p}^{1}), λ_{0i}^{1} = 1 = λ_{0j}^{1} and λ_{0k}^{1} = 0, whenever k ≠ i, j. Taking into account that α_{0}, h_{i} and h_{j} are of class C^{m} in W′, we easily deduce that α_{λ_{0}^{1}} is of class C^{m} in W′. Likewise, we can inductively prove that α_{λ_{0}^{l}} is of class C^{m} in W′, where λ_{0}^{l} := (λ_{01}^{l}, λ_{02}^{l}, …, λ_{0p}^{l}), λ_{0i}^{l} = l, λ_{0j}^{l} = 1 and λ_{0k}^{l} = 0, whenever k ≠ i, j, l ∈ {1, 2, …, n − 1}. Suppose that α_{λ_{0}^{n−1}}(y_{0}) ≠ 0 for some y_{0} ∈ W′. Then, as in the proof of Claim 4.6, we take an open neighborhood U(y_{0}) of y_{0} such that U(y_{0}) ⊂ W′ and α_{λ_{0}^{n−1}}(y) ≠ 0 for every y ∈ U(y_{0}).

Also, we take f(x) = g(x_{i}) for every x = (x_{1}, x_{2}, …, x_{p}) ∈ Ω, where these functions meet the same requirements as in the proof of Claim 4.6, and define

d(x) := x_{j}f(x),

for every x ∈ Ω. Clearly d just depends on the i-th and j-th coordinates, which implies that its only partial derivatives which possibly are not zero at h(y) ∈ h(U(y_{0})) are maybe those

∂^{λ}d

for

λ = (\underbrace{0, 0, …, 1,}_{j} 0, …, 0),

λ = (\underbrace{0, 0, …, k,}_{i} 0, …, 0),

k = 1, 2, …, n, or

λ = λ_{0}^{l},

l ∈ {1, 2, …, n − 1}. Taking into account that α_{λ} ≡ 0 on W′ for

λ = (\underbrace{0, 0, …, 1,}_{j} 0, …, 0)

and

λ = (\underbrace{0, 0, …, k,}_{i} 0, …, 0),

Equation 4.1 gives us, for every y ∈ U(y_{0}),

(T de)(y) = α_{0}(y)d(h(y)) + α_{λ_{0}^{1}}(y)\frac{∂f}{∂x_{i}} (h(y)) + … + α_{λ_{0}^{n−1}}(y)\frac{∂^{n−1}f}{∂x_{i}^{n−1}} (h(y)).

We deduce as in the proof of Claim 4.6 that

\frac{∂^{n−1}f}{∂x_{i}^{n−1}} (h(y))

admits a second partial derivative with respect to x_{i} at the point h(y_{0}), which is a contradiction. This implies that α_{λ_{0}^{n−1}} ≡ 0 in W′. In the same way we deduce that α_{λ_{0}^{l}} ≡ 0 in W′, for l ∈ {1, 2, …, n − 2}.

A similar pattern of proof leads us to the fact that α_{λ} ≡ 0 in W′ for every λ ≠ (0, 0, …, 0), λ ∈ Λ.

Claim 4.8 For every λ ∈ Λ − {(0, 0, …, 0)}, α_{λ} ≡ 0 in Ω′.

Notice that given an open subset U of Ω′, in Claim 4.4 we obtain a subset W′ of U. Notice also that this process can be done for any open subset of Ω′ because, as we saw in the proof of Claim 4.4, c(α_{0}) is dense in Ω′. Also in Claim 4.7 we proved that, for λ ≠ (0, 0, …, 0), α_{λ} ≡ 0 on all the subsets W′ obtained in this way. This implies clearly that all these functions α_{λ} are equal to 0 on a dense subset of Ω′. Consequently, to prove Claim 4.8, it is enough to show that all these functions are continuous.

We are going to prove it using induction on |λ|. First, for |λ| = 0, we have that α_{(0,...,0)} = T ê belongs to A^{m}(Ω′, F) and, consequently, it is continuous.

Now assume that k ≤ n − 1, and whenever |λ| ≤ k, then α_{λ} is a continuous function. Then fix λ = (λ_{1}, …, λ_{p}) ∈ Λ with |λ| = k + 1.

Next define f ∈ A^{n}(Ω, ℝ) as

f := x_{1}^{λ_{1}}x_{2}^{λ_{2}} … x_{p}^{λ_{p}}.

It is clear that given μ = (μ_{1}, …, μ_{p}) ∈ Λ, if μ_{i} > λ_{i} for some i ∈ {1, …, p}, then

∂^{μ}f(h(y)) = 0

for every y ∈ Ω′. This implies that in our situation, Equation 4.1 can be written as

(T fe)(y) = ∑_{μ<<λ} α_{μ}(y)∂^{μ}f(h(y)),

where μ << λ means μ_{i} ≤ λ_{i} for every i ∈ {1, …, p}.

As a consequence, for every y ∈ Ω′,

λ_{1}! … λ_{p}!α_{λ}(y) = α_{λ}(y)∂^{λ}f(h(y)) = (T fe)(y) − ∑_{μ<<λ,μ≠λ} α_{μ}(y)∂^{μ}f(h(y)).

On the other hand, if μ << λ and μ ≠ λ, then |μ| ≤ k, and consequently, taking into account that h is continuous and the hypothesis of induction, we deduce that α_{λ} is continuous, and the claim is proved.

Recall that all the process developed so far concerns functions of the form fe ∈ A^{n}(Ω, E), where e ∈ E − {0} and f ∈ A^{n}(Ω, ℝ). For this e, we define

a_{e} := α_{0} = T ê.

Notice that by Claim 4.8, we have (T fe)(y) = a_{e}(y)f(h(y)) (4.4)

for every y ∈ Ω′ and every f ∈ A^{n}(Ω, ℝ). Next we define a map J : Ω′ → L′(E, F) as (Jy)(0) = 0, and

(Jy)(e) := a_{e}(y)

for each y ∈ Ω′ and e ∈ E − {0}. Claim 4.9 For every f ∈ A^{n}(Ω, E) and y ∈ Ω′,

(T f)(y) = (Jy)(f(h(y))).

Fix y ∈ Ω′. Suppose that ∂^{λ}f(h(y)) = e_{λ} ∈ E

for each λ ∈ Λ. Let Λ_{∗} := {λ ∈ Λ : e_{λ} ≠ 0}. Next, for each λ ∈ Λ_{∗}, take a function f_{λ} ∈ A^{n}(Ω, ℝ) such that

∂^{λ}f_{λ}(h(y)) = 1

and

∂^{μ}f_{λ}(h(y)) = 0

for every μ ≠ λ, μ ∈ Λ. It is easy to see that, for every μ ∈ Λ,

∂^{μ} ∑_{λ∈Λ_{∗}} f_{λ}e_{λ}(h(y)) = e_{μ},

if μ ∈ Λ_{∗}, and

∂^{μ} ∑_{λ∈Λ_{∗}} f_{λ}e_{λ}(h(y)) = 0

if μ ∉ Λ_{∗}.

According to Lemma 4.1, this implies that

(T f)(y) = (T ∑_{λ∈Λ_{∗}} f_{λ}e_{λ}) (y).

Consequently, by Equation 4.4,

(T f)(y) = ∑_{λ∈Λ_{∗}} a_{e_{λ}}(y)f_{λ}(h(y)).

But by the way we have constructed the functions f_{λ}, we have that f_{λ}(h(y)) = 0 if λ ≠ (0, 0, …, 0).

On the other hand, let us denote by 0 the multiindex (0, 0, …, 0). If 0 ∉ Λ_{∗}, that is, if e_{0} = 0, we conclude from the above equality that (T f)(y) = 0 = a_{e_{0}}(y) = (Jy)(0). Finally, if 0 ∈ Λ_{∗}, taking into account that f_{0}(h(y)) = 1, we deduce that

(T f)(y) = a_{e_{0}}(y)f_{0}(h(y)) = a_{e_{0}}(y) = (Jy)(e_{0}),

and we are done.

Claim 4.10 Given y ∈ Ω′, there exists e ∈ E such that a_{e}(y) ≠ 0.

Notice that T is bijective, so if f ∈ F, f ≠ 0, there exists g ∈ A^{n}(Ω, E) with T g = f̂. In particular, by Claim 4.9, we have that (Jy)(g(h(y))) = f. In other words, if we take y ∈ c(g ∘ h) and define e := g(h(y)), we have that a_{e}(y) = (Jy)(e) = f ≠ 0.

Claim 4.11 h is a function of class C^{m}.

Fix y_{0} ∈ Ω′. By Claim 4.10, we can take e ∈ E such that a_{e}(y) ≠ 0. Since a_{e} = T ê, it is a continuous function, and we deduce that for some neighborhood V of y_{0}, a_{e}(y) ≠ 0 for every y ∈ V.

Now recall that Equation 4.4,

(T fe)(y) = a_{e}(y)f(h(y)),

holds in particular for every y ∈ V and every f ∈ A^{n}(Ω, ℝ).

Consequently, for i ∈ {1, 2, …, p},

(T x_{i}e)(y) = a_{e}(y)h_{i}(y)

for every y ∈ V. Since a_{e}(y) ≠ 0 for every y ∈ V, applying Lemma 3.2, we have that h_{i} is of class C^{m} in V. Clearly this implies that h is of class C^{m}, and we are done.

Claim 4.12 n = m and h is a diffeomorphism of class C^{m}.

Recall that we are assuming that n ≤ m. Now, we have that by Claim 4.4, for every nonempty open set V ⊂ Ω′, there is a nonempty open set V′ ⊂ V such that the restriction of h to V′ is a diffeomorphism of class C^{m}. Take y_{0} ∈ Ω′. By Claim 4.10, there exists e ∈ E and an open neighborhood V of y_{0} such that a_{e}(y) ≠ 0 for every y ∈ V. Now, as we mentioned above, there exists an open set V′, V′ ⊂ V, where the restriction of h is a diffeomorphism of class C^{m}. Assume now that n < m and take g ∈ A^{n}(Ω′, ℝ) − C^{m}(Ω′, ℝ) such that c(g) ⊂ V′. Next define f : Ω → E, as

f(x) := g(h^{−1}(x))e,

for each x ∈ Ω. We are going to prove that g is of class C^{m}, obtaining a contradiction.

It is immediate that f ∈ A^{n}(Ω, E), and applying Equation 4.4, we get

(T f)(y) = a_{e}(y)g(h^{−1}(h(y))),

that is,

(T f)(y) = a_{e}(y)g(y),

for every y ∈ Ω′.

Now we have that a_{e}(y) ≠ 0 for every y ∈ V. Finally, by Lemma 3.2, g is of class C^{m} in V, and so is in Ω′, which contradicts our assumption. This implies that m ≤ n, and since we are assuming that n ≤ m, the claim is proved.

Claim 4.13 For every y ∈ Ω′, Jy ∈ L′(E, F) is bijective.

Since m = n, all claims above also hold for T^{−1}, and this means that there exists K : Ω → L′(F, E) such that for every g ∈ A^{m}(Ω′, F) and x ∈ Ω, (T^{−1}g)(x) = (Kx)(g(h^{−1}(x))).

Fix y ∈ Ω′ and f ∈ F − {0}. Let x = h(y). Now take g ∈ A^{m}(Ω′, F) with g(y) = f. Then its is clear that f = g(y) = (T (T^{−1}g))(y), that is,

f = (Jy)((T^{−1}g)(x))
= (Jy)((Kx)(g(h^{−1}(x))))
= (Jy)((Kx)(g(y)))
= (Jy)((Kx)(f)).

This implies that (Jy)(Kx) is the identity map on F. In the same way we can prove that (Kx)(Jy) is the identity map on E. Consequently, Jy is bijective.

This ends the proof of the proposition. □

5 A result on automatic continuity

In this section we see that, if we endow the spaces with some natural topologies, then we obtain the continuity as a consequence. Notice that, according to Proposition 4.2, we can assume in particular that n = m and p = q.

Theorem 5.1 Assume that A^{n}(Ω, E) and A^{n}(Ω′, F) are endowed with any topologies which are compatible with the pointwise convergence. Suppose that T : A^{n}(Ω, E) → A^{n}(Ω′, F) is a 𝕂-linear biseparating map. Then T is continuous.

Proof. In our proof we will take advantage of the description of T given in Proposition 4.2. For this reason we will use the notation given there. We start proving the following claim.

Claim 5.1 Let U be a (nonempty) bounded open subset of Ω′ with cl_{ℝ^{p}} U ⊂ Ω′. Then the set

A := {y ∈ U : Jy ∈ I′(E, F) is not continuous}

is finite.

Suppose that this is not the case, but there exist infinitely many y ∈ U such that Jy is not continuous. We are going to construct inductively a sequence of points in A, a sequence (U_{n}) of pairwise disjoint open subsets of U, a sequence of functions (f_{n}) in C_{c}^{n}(Ω, ℝ), and a sequence (e_{n}) of norm-one elements of E, satisfying the following properties:

• 1. h(y_{n}) ∈ c(f_{n}) ⊂ h(U_{n}) for every n ∈ ℕ.

• 2. ∥f_{n}∥ := max_{λ∈Λ} sup_{x∈Ω} |∂^{λ}f_{n}(x)| = 1/2^{n} for every n ∈ ℕ.

• 3. ∥(Jy_{n})(e_{n})∥ ≥ n/ |f_{n}(h(y_{n}))| for every n ∈ ℕ.

Take any point y_{1} ∈ A such that there are accumulation points of A in Ω′ − {y_{1}}. Then consider an open subset U_{1} of U in such a way that y_{1} ∈ U_{1}, and there are infinitely many points of A outside cl_{ℝ^{p}} U_{1}. Next take f_{1} ∈ C_{c}^{n}(Ω, ℝ) such that ∥f_{1}∥ = 1, and such that h(y_{1}) ∈ c(f_{1}) ⊂ h(U_{1}). Since Jy_{1} is not continuous, there exists e ∈ E, ∥e∥ = 1, with

∥(Jy_{1})(e)∥ ≥ \frac{1}{|f_{1}(h(y_{1}))|}.

Next assume that we have {y_{1}, y_{2}, …, y_{n}} ⊂ A, U_{1}, U_{2}, …, U_{n} ⊂ U open and pairwise disjoint such that there are infinitely many points of A outside cl_{ℝ^{p}} U_{1} ∪cl_{ℝ^{p}} U_{2} ∪…∪cl_{ℝ^{p}} U_{n}, {f_{1}, f_{2}, …, f_{n}} ⊂ C_{c}^{n}(Ω, ℝ) with h(y_{i}) ∈ c(f_{i}) ⊂ h(U_{i}) and ∥f_{i}∥ = 1/2^{i}, for i ∈ {1, 2, …, n}, and e_{1}, e_{2}, …, e_{n} ∈ E all of them with norm 1, and such that ∥(Jy_{i})(e_{i})∥ ≥ i/ |f_{i}(h(y_{i}))| for i = 1, 2, …, n.

Now it is easy to see how to take y_{n+1}, U_{n+1}, f_{n+1}, and e_{n+1} so that Properties 1, 2 and 3 above hold.

Since ∥f_{n}∥ = 1/2^{n} for every n ∈ ℕ, we deduce that the map

g := ∑_{n=1}^{∞} f_{n}e_{n}

belongs to A^{n}(Ω, E). Consequently, T g should belong to A^{n}(Ω′, F). But we know by Proposition 4.2 that (T g)(y_{n}) = (Jy_{n})(g(h(y_{n}))) for every n ∈ ℕ. This implies, by Property 3 above,

∥(T g)(y_{n})∥ = ∥(Jy_{n})(f_{n}(h(y_{n}))e_{n})∥
= |f_{n}(h(y_{n}))| ∥(Jy_{n})(e_{n})∥
≥ n.

As a consequence T g is unbounded in U. Since this is not possible, we conclude that the claim is correct.

Next, it is clear that to prove that T is continuous it is enough to show that it is closed, because we are dealing with Fréchet spaces. To prove it, let us consider a sequence (f_{n}) in A^{n}(Ω, E) convergent to zero, and assume that (T f_{n}) converges to g ∈ A^{n}(Ω′, F). We are going to prove that g = 0.

Take a bounded open subset U of Ω′ with cl_{ℝ^{p}} U ⊂ Ω′. By Claim 5.1 above, we have that the subset A of points y ∈ U such that Jy is not continuous is finite. So, if y ∈ U − A, Jy belongs to I(E, F). Consequently, since f_{n}(h(y)) goes to zero (because the topology in A^{n}(Ω, E) is compatible with the pointwise convergence), then we have that (T f_{n})(y) = (Jy)(f_{n}(h(y))) also goes to zero, that is, g(y) must be zero. But taking into account that U − A is dense in U, we deduce that g ≡ 0 on U. The conclusion follows now easily and T is continuous. □

6 Biseparating maps and functions of class s−C^{n}

Our aim in this section is to give a final description of biseparating maps between spaces of vector-valued differentiable functions taking into account that we know that they must be continuous when the spaces are endowed with some natural topologies. Of course these topologies will be compatible with the pointwise convergence. Namely, it is well known that by means of the seminorms p_{K} defined as

p_{K}(f) := max_{λ∈Λ} max_{x∈K} ∥∂^{λ}f(x)∥

for f ∈ C^{n}(Ω, E), where K runs through the compact subsets of Ω, C^{n}(Ω, E) becomes a locally convex space. In fact it is a Fréchet space. In the same way, in C^{n}(Ω̄, E) we can consider the norm ∥·∥ defined as

∥f∥ := max_{λ∈Λ} sup_{x∈Ω} ∥∂^{λ}f(x)∥

for f ∈ C^{n}(Ω̄, E). With this norm, our space C^{n}(Ω̄, E) is also complete. We assume that A^{n}(Ω, E) and A^{n}(Ω′, F) are endowed with the above topologies. Remark also that, as it follows easily from the Closed Graph Theorem, the topologies compatible with the pointwise convergence in our spaces coincide with these topologies.

Next proposition states that when J : Ω′ → L(E, F) is of class s − C^{n}, we can define maps through J from A^{n}(Ω, E) → A^{n}(Ω′, F) in a natural way.

Proposition 6.1 Suppose that n = m, p = q. Let J : Ω′ → L(E, F) be a map of class s − C^{n}, and let h be a diffeomorfism of class C^{n} from Ω′ onto Ω. If, for f ∈ C^{n}(Ω, E), we define (T f)(y) := (Jy)(f(h(y))) for every y ∈ Ω′, then T f ∈ C^{n}(Ω′, F).

Proof. We consider first the map Φ : L(E, F) × E → F defined as Φ(A, e) := Ae for each (A, e) ∈ L(E, F)×E. This is clearly bilinear and continuous when we consider in L(E, F) the topology of the norm.

Suppose next that L(E, F) is endowed again with the topology of the norm, and that K : Ω′ → L(E, F) is a continuous map. Then, given g : Ω′ → E continuous, the map S_{K}^{g} : Ω′ → L(E, F) × E sending each y ∈ Ω′ into (Ky, g(y)) is continuous.

On the other hand, if we suppose that L(E, F) is endowed with the topology of the norm and that f ∈ C^{n}(Ω, E), then S_{J}^{f∘h} is of class C^{n−1} because both maps J and f ∘ h are. Consequently the composition map Φ ∘ S_{J}^{f∘h} : Ω′ → F, mapping each y ∈ Ω′ into (Jy)(f(h(y))) ∈ F is of class C^{n−1}.

Now we check the form of its first partial derivatives. We just see the partial derivative with respect to the first coordinate x_{1}. It is easy to check that

\frac{∂}{∂x_{1}} (Φ ∘ S_{J}^{f∘h})(y) = \frac{∂}{∂X} Φ(S_{J}^{f∘h}(y)) ∘ \frac{∂}{∂x_{1}} Jy + \frac{∂}{∂Y} Φ(S_{J}^{f∘h}(y)) ∘ \frac{∂}{∂x_{1}} (f ∘ h)(y)
= Φ (\frac{∂}{∂x_{1}} Jy, f(h(y))) + Φ (Jy, \frac{∂}{∂x_{1}} (f ∘ h)(y))
= (Φ ∘ S_{\frac{∂}{∂x_{1}} J}^{f∘h}) (y) + (Φ ∘ S_{J}^{\frac{∂}{∂x_{1}} f∘h}) (y).

By an inductive reasoning, we see that the partial derivatives of order n − 1 of Φ∘S_{J}^{f∘h} are just a sum of terms of the form Φ∘S_{∂^{λ}J}^{∂^{μ}(f∘h)}, where |λ| , |μ| ≤ n−1. Consequently, to prove that T f ∈ C^{n}(Ω′, F), we just have to show that each one of the terms Φ ∘ S_{∂^{λ}J}^{∂^{μ}(f∘h)} (|λ| , |μ| ≤ n − 1) is of class C^{1}. We suppose that S_{K}^{g} : Ω′ → L(E, F) × E is one of the above S_{∂^{λ}J}^{∂^{μ}(f∘h)} (that is, let us denote g = ∂^{μ}(f ∘ h) ∈ C^{1}(Ω′, E) and K = ∂^{λ}J), which is continuous when L(E, F) is endowed with the topology of the norm, as we stated above.

Now we check the form of its first partial derivatives. Take i ∈ {1, …, p}, and assume without loss of generality that i = 1. It is easy to check that

lim_{k→0} \frac{(Φ ∘ S_{K}^{g})(y + (k, 0, …, 0)) − (Φ ∘ S_{K}^{g})(y)}{k},

that is,

lim_{k→0} \frac{(K(y + (k, 0, …, 0)))(g(y + (k, 0, …, 0))) − (Ky)(g(y))}{k}

is equal to

lim_{k→0} \frac{(K(y + (k, 0, …, 0)))(g(y + (k, 0, …, 0))) − (K(y + (k, 0, …, 0)))(g(y))}{k}+

+ lim_{k→0} \frac{(K(y + (k, 0, …, 0)))(g(y)) − (Ky)(g(y))}{k},

that is, it is equal to

(Ky) (\frac{∂}{∂x_{1}} g (y)) + (\frac{∂_{s}}{∂x_{1}} Ky) (g(y)),

by the definition of \frac{∂_{s}}{∂x_{1}} Ky and the fact that K is continuous for L(E, F) endowed with the topology of the norm.

Applied to our context, we have that

\frac{∂}{∂x_{1}} (Φ ∘ S_{∂^{λ}J}^{∂^{μ}(f∘h)})(y) = (Φ ∘ S_{∂^{λ}J}^{\frac{∂}{∂x_{1}} ∂^{μ}(f∘h)}) (y) + (Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)}) (y).

Now, as we noted above, S_{∂^{λ}J}^{\frac{∂}{∂x_{1}} ∂^{μ}(f∘h)} is continuous when considering in L(E, F) the topology of the norm. As a consequence, to obtain the continuity of all partial derivatives of order n of T f, it is enough to see that Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)} is

continuous.

In order to prove this, notice first that, since J is of class s − C^{n−1}, then for the above λ the map \frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}J is continuous when L(E, F) is endowed with the strong operator topology. This means that, given e ∈ E, the map \frac{∂_{s}}{∂_{x_{1}}} (∂^{λ}J)_{e} : Ω′ → F sending each y ∈ Ω′ into (\frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy)(e) is continuous. Now take ϵ > 0 and y_{0} ∈ Ω′. We are going to show that there exists δ > 0 such that if |y − y_{0}| < δ, y ∈ Ω′, then

∥Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)}(y) − Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)}(y_{0})∥ < ϵ.

Let e_{0} := ∂^{μ}(f ∘ h)(y_{0}), and take δ_{1} > 0 such that the closed ball B̄(y_{0}, δ_{1}) is contained in Ω′. Since \frac{∂_{s}}{∂_{x_{1}}} (∂^{λ}J)_{e_{0}} is continuous, there exists an upper bound M for this function on the compact set B̄(y_{0}, δ_{1}). Also, there exists δ_{2} > 0, δ_{2} < δ_{1}, such that if |y − y_{0}| < δ_{2}, then

∥(\frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy − \frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy_{0}) (e_{0})∥ < \frac{ϵ}{2}.

On the other hand, since ∂^{μ}(f ∘ h) is continuous, there exists δ > 0, δ < δ_{2}, such that if y ∈ B(y_{0}, δ), then ∥∂^{μ}(f ∘ h)(y) − e_{0}∥ < ϵM/2.

Consequently, if |y − y_{0}| < δ, we have

∥Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)}(y) − Φ ∘ S_{\frac{∂s}{∂x_{1}} ∂^{λ}J}^{∂^{μ}(f∘h)}(y_{0})∥

is less than or equal to

∥(\frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy) (∂^{μ}(f ∘ h)(y) − e_{0})∥ + ∥(\frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy − \frac{∂_{s}}{∂_{x_{1}}} ∂^{λ}Jy_{0}) (e_{0})∥ ,

which is easily strictly less than ϵ. This proves the continuity of our functions, as it was to see. □

The following result is a direct consequence of Proposition 4.2 and Theorem 5.1. Roughly speaking, it says that Proposition 6.1 provides the only way to construct linear biseparating maps from A^{n}(Ω, E) onto A^{n}(Ω′, F). We state the result in its complete form.

Theorem 6.2 Suppose that T : A^{n}(Ω, E) → A^{m}(Ω′, F) is a 𝕂-linear biseparating map. Then p = q, n = m, and there exist a diffeomorphism h of class C^{n} from Ω′ onto Ω and a map J : Ω′ → L(E, F) of class s − C^{n} such that for every y ∈ Ω′ and every f ∈ A^{n}(Ω, E),

(T f)(y) = (Jy)(f(h(y))).

Moreover, Jy ∈ I(E, F) for every y ∈ Ω′.

Proof. We will follow the same notation as in Proposition 4.2, which provided a first description of linear biseparating maps, in particular everything related to the definition of J and h. Also, by Proposition 4.2, p = q and n = m.

First, for each λ ∈ Λ, we define a map J_{λ} : Ω′ → L′(E, F) as

(J_{λ}y)(e) := ∂^{λ}(T ê)(y),

for y ∈ Ω′ and e ∈ E. J_{λ} is clearly well defined.

The rest of the proof will apply just for the case when we are in Situation 1, but it is easy to see that slight changes in it allow to prove the theorem when we are in Situation 2. We will prove it through several claims.

Claim 6.1 For every y ∈ Ω′ and every λ ∈ Λ, J_{λ}y belongs to L(E, F).

Take y ∈ Ω′ and a sequence (e_{n}) in E converging to zero. We will see that ((J_{λ}y)(e_{n})) goes to zero. First we have that, since T is continuous by Theorem 5.1, the sequence of functions (T ê_{n}) converges to zero, which implies in particular that ((∂^{λ}T ê_{n})(y)) goes to zero. But this last sequence is precisely ((J_{λ}y)(e_{n})), so the claim is proved.

Claim 6.2 For each λ ∈ Λ, |λ| ≤ n − 1, the map J_{λ} : Ω′ → L(E, F) is continuous when L(E, F) is endowed with the topology of the norm.

We will show that if y_{0} ∈ Ω′, then J_{λ} is continuous at y_{0}. Since T is continuous, we have that, for r > 0 such that the closed ball B̄(y_{0}, r) is contained in Ω′, there exists M > 0 such that p_{B̄(y_{0},r)}(T ê) < M holds for every e ∈ E with ∥e∥ ≤ 1. This implies that, for these e, if |y − y_{0}| < r, then ∥(D∂^{λ}T ê)(y)∥ ≤ pM. Consequently, as it can be seen for instance in [8, Theorem 3.3.2], we have that

∥(∂^{λ}T ê)(y) − ∂^{λ}(T ê)(y_{0})∥ ≤ pM |y − y_{0}| ,

for every e ∈ E with ∥e∥ ≤ 1. Now, taking into account that (J_{λ}y)(e) = (∂^{λ}T ê)(y) for every y ∈ Ω′, the result follows, and the claim is proved.

Claim 6.3 For each λ ∈ Λ, |λ| ≤ n − 1, ∂^{λ}J = J_{λ}.

Of course, the result is clear if n = 1, so we suppose that n ≥ 2. We will just prove the claim in the particular case when λ = λ_{1} := (1, 0, …, 0). The proof for all other λ ∈ Λ is similar and can be achieved inductively.

Take y_{0} ∈ Ω′ and r > 0 such that the closed ball B̄(y_{0}, r) ⊂ Ω′. It is clear that if h ∈ ℝ − {0}, |h| < r, and if e is in the closed unit ball of E, then

∥\frac{J(y_{0} + (h, 0, …, 0))(e) − (Jy_{0})(e)}{h} − (J_{λ_{1}}y_{0})(e)∥

is equal to

∥\frac{(T ê)(y_{0} + (h, 0, …, 0)) − (T ê)(y_{0})}{h} − (∂^{λ_{1}}T ê)(y_{0})∥ ,

which, by [23, Corollary XIII.4.4], is less than

sup_{|y−y_{0}|<2h} ∥(∂^{λ_{1}}T ê)(y_{0}) − (∂^{λ_{1}}T ê)(y)∥ ,

that is, less than

sup_{|y−y_{0}|<2h} ∥J_{λ_{1}}y_{0} − J_{λ_{1}}y∥ .

Clearly this implies that

lim_{h→0} ∥\frac{J(y + (h, 0, …, 0)) − Jy}{h} − J_{λ_{1}}y∥ = 0.

Consequently, the partial derivative of J with respect to the first coordinate exists at each y_{0} ∈ Ω′ and is equal to J_{λ_{1}}y_{0}.

Claim 6.4 Take λ = (n_{1}, n_{2}, …, n_{i}, …, n_{p}) ∈ Λ with |λ| = n − 1. Then J_{λ} is of class s − C^{1}. Moreover, if for each i ∈ {1, …, n}, μ_{i} = (n_{1}, n_{2}, …, n_{i} + 1, …, n_{p}), then

\frac{∂_{s}}{∂x_{i}} J_{λ} = J_{μ_{i}}.

The proof that \frac{∂_{s}}{∂x_{i}} J_{λ} = J_{μ_{i}} is similar to the proof of Claim 6.3 we have done, taking into account that J_{μ_{i}} is perhaps no longer continuous when L(E, F) is endowed with its norm but ∂^{μ_{i}}T ê is continuous for every e ∈ E. Consequently, to finish we just have to show that J_{μ}, |μ| = n, is continuous when considering in L(E, F) the strong operator topology. We have to prove that if μ ∈ Λ, and (y_{n}) is a sequence in Ω′ converging to y ∈ Ω′, then (J_{μ}y_{n})(e) converges to (J_{μ}y)(e) for each e ∈ E. But this is immediate from the definition of J_{μ}. □

Remark 6.3 Notice that, in the case when we are in Situation 2, in Theorem 6.2 the map J and its partial (s-)derivatives up to order n can also be extended continuously to the boundary of Ω′ in a natural way, when considering in L(E, F) the strong operator topology.

In the special case when E = 𝕂 = F, we immediately deduce the following result.

Corollary 6.4 Suppose that T : A^{n}(Ω, 𝕂) → A^{m}(Ω′, 𝕂) is a 𝕂-linear biseparating map. Then p = q, n = m, and there exist a diffeomorphism h of class C^{n} from Ω′ onto Ω and a map a : Ω′ → 𝕂 of class C^{n} which does not vanish at any point of Ω′, such that for every y ∈ Ω′ and every f ∈ A^{m}(Ω, 𝕂),

(T f)(y) = a(y)f(h(y)).

We finish with a corollary whose proof is easy from Theorem 6.2.

Corollary 6.5 If A^{n}(Ω, E) and A^{n}(Ω′, F) are endowed with the topology of the pointwise convergence, then every linear biseparating map T : A^{n}(Ω, E) → A^{n}(Ω′, F) is continuous.

7 Final Remark

Even if in previous sections we consider two possible situations for our spaces of vector-valued functions, Proposition 4.2 and Theorems 5.1 and 6.2 can be given in a broader context.

Let Ω ⊂ ℝ^{p} and Ω′ ⊂ ℝ^{q} be (nonempty) open sets, and consider any of the following contexts.

• Case i) Ω_{1} := Ω and Ω′_{1} := Ω′.

• Case ii) Ω and Ω′ are bounded, int_{ℝ^{p}} cl_{ℝ^{p}} Ω = Ω, int_{ℝ^{q}} cl_{ℝ^{q}} Ω′ = Ω′, and Ω_{1} := cl_{ℝ^{p}} Ω and Ω_{1} := cl_{ℝ^{q}} Ω.

Suppose now that A ⊂ C^{n}(Ω, ℝ) ∩ C(Ω_{1}, ℝ) and B ⊂ C^{m}(Ω′, ℝ) ∩ C(Ω′_{1}, ℝ) are strongly regular rings (see [2] for an appropriate description). Take now A(Ω, E) ⊂ C^{n}(Ω, E)∩C(Ω_{1}, E) and B(Ω′, F) ⊂ C^{m}(Ω′, F)∩C(Ω′_{1}, F). Namely, to construct the support map h : Ω′_{1} → Ω_{1}, we need that subspaces A(Ω, E) ⊂ C^{n}(Ω, E) ∩ C(Ω_{1}, E) and B(Ω′, F) ⊂ C^{m}(Ω′, F) ∩ C(Ω′_{1}, F) are a compatible A-module and a compatible B-module (see [2]). In both cases h will be a homeomorphism from Ω′ onto Ω.

On the other hand, apart from these necessary conditions for constructing h, a careful reading of the proofs shows that the essential requirements that linear subspaces A(Ω, E) and B(Ω′, F) must meet so as to satisfy Lemma 4.1 and Proposition 4.2 are: 1) A(Ω, E) and B(Ω′, F) contain the constant functions; 2) A(Ω, E) (respectively, B(Ω′, F)) contains all functions in C^{n}(Ω, E) (respectively, C^{m}(Ω′, F)) with compact support; and 3) C_{c}^{n}(Ω, ℝ) ⊂ A and C_{c}^{m}(Ω′, ℝ) ⊂ B. In the case when Ω and Ω′ are bounded and the coordinate projections x_{i} belong to A and B, then the proof of Proposition 4.2 can be followed step by step with no changes. Otherwise (even if Ω and Ω′ are not bounded), changes are few and natural.

Finally, if we want to obtain results similar to Theorems 5.1 and 6.2 (with the same proofs) for our spaces A(Ω, E) and B(Ω′, F), besides all the above conditions, they must be endowed with a suitable norm or family of seminorms providing a topology compatible with the pointwise convergence. This will be the case, for instance, of the spaces of functions with bounded derivatives.

So we study the case of the spaces C_{∗}^{n}(Ω, E) ⊂ C^{n}(Ω, E) and C_{∗}^{m}(Ω′, F) ⊂ C^{m}(Ω′, F) consisting of all functions such that all partial derivatives up to orders n and m, respectively, are bounded. The space C_{∗}^{n}(Ω, E) (and similarly C_{∗}^{m}(Ω′, F)) becomes a Banach space with the norm defined for each f ∈ C_{∗}^{n}(Ω, E) as

∥f∥ := max_{λ∈Λ} sup_{x∈Ω} ∥∂^{λ}f(x)∥ .

This is a suitable norm in the above sense because the proof of Theorem 6.2 can be followed easily for these spaces equipped with such norm.

First, notice that, since we are assuming that n, m ≥ 1, then in particular when Ω is convex all functions in C_{∗}^{n}(Ω, E) admit a continuous extension to the closure of Ω in ℝ^{p}: suppose that (x_{n}) is a sequence in Ω converging to x_{0} in the boundary of Ω, and that f ∈ C_{∗}^{n}(Ω, E); then since the differential Df is bounded on the whole Ω by an M > 0, we have that, by [8, Theorem 3.3.2],

∥f(x_{n}) − f(x_{m})∥ ≤ M |x_{n} − x_{m}| ,

which implies that (f(x_{n})) is a Cauchy sequence. In this way we would define the extension f(x) as the limit of this sequence. It is straightforward to see that the new extended function is continuous in the closure of Ω.

On the other hand, when Ω and Ω′ are bounded and convex, then it is easy to see that C_{∗}^{n}(Ω, E) is a C^{n}(Ω̄, ℝ)-module, and a similar statement is also valid for C_{∗}^{m}(Ω′, F). As a consequence, by the comments given above, Proposition 4.2 and Theorems 5.1 and 6.2 can also be stated in this new situation (for Ω and Ω′ bounded and convex). Furthermore, in this case, as in Remark 6.3, it is also possible to say that partial derivatives up to order n − 1 of J admit a continuous extension (when L(E, F) is equipped with the strong operator topology) to the boundary of Ω′. As for the partial s-derivatives of all partial derivatives of order n − 1 of J, an elementary application of the Uniform Boundedness Theorem shows that they are bounded on Ω′.

What happens if Ω is for instance not bounded? One might be tempted to follow a similar pattern as indicated above when trying to describe linear biseparating maps defined between C_{∗}^{n}(Ω, E) and C_{∗}^{m}(Ω′, F). But, in that case, we have that C_{∗}^{n}(Ω, E) is no longer a C^{n}(Ω, ℝ)-module (as A^{n}(Ω, E) in previous sections was). Anyway, it is a C_{∗}^{n}(Ω, ℝ)-module, and we could try to follow the proof of Proposition 4.2 to get a similar description of linear biseparating maps, but even if we could manage to adapt the proof step by step (with some changes), there is a major problem from the beginning: in general the space C_{∗}^{n}(Ω, ℝ) is not a strongly regular ring, so our results cannot be applied, and in particular the existence of the support map h is not clear. Let us see an example where C_{∗}^{n}(Ω, ℝ) is not a strongly regular ring.

Example 7.1 Suppose that Ω ⊂ ℝ is the open set defined as Ω := ⋃_{k=1}^{∞}(k − 1/k, k + 1/k). Let us define K as the closure of ℕ in βΩ (the Stone-Čech compactification of Ω). Now take x_{0} ∈ βΩ − (K ∪ ℝ). It is clear that if a derivable function f : Ω → ℝ satisfies f^{βΩ} ≡ 0 on K and f^{βΩ} ≡ 1 on a neighborhood of x_{0} (where f^{βΩ} stands for its extension to βΩ), then for some sequence (x_{n}) in Ω going to infinity, the sequence (f(x_{n})) converges to 1. As a consequence from the Mean Value Theorem, we conclude that the derivative of f cannot be bounded on Ω, that is, f ∉ C_{∗}^{1}(Ω, ℝ), as we wanted to show.

We end the paper with some related questions, concerning special cases where our techniques cannot be applied.

Problem 1. Assume that there exists a biseparating map T : A^{n}(Ω, E) → A^{m}(Ω′, F) which is not linear. Can we deduce that the support map h : Ω′ → Ω is a diffeomorphism of class n? Remark that the assumption of linearity in Proposition 4.2 is necessary for its proof.

Problem 2. Suppose that C^{∞}(Ω, E) is the space of E-valued functions which are of class C^{∞} in Ω, and that C^{∞}(Ω′, F) is defined in a similar way. Describe the linear biseparating maps from C^{∞}(Ω, E) onto C^{∞}(Ω′, F). Must such a map be continuous? Notice that by the comments given in the Final Remark above, the construction of the support map h is possible, but the proof of Proposition 4.2 is no longer valid.

Problem 3. Let Ω and Ω′ be unbounded open subsets of ℝ^{p} and ℝ^{q}, respectively. Describe the linear biseparating maps from C_{∗}^{n}(Ω, E) onto C_{∗}^{m}(Ω′, F).

Problem 4. Determine all subspaces A(Ω, E) ⊂ A^{n}(Ω, E) and B(Ω′, F) ⊂ A^{m}(Ω′, F) such that the existence of a (linear) biseparating map from A(Ω, E) onto B(Ω′, F) implies that E and F are isomorphic as Banach spaces.

Acknowledgements

The author wishes to thank three anonymous referees for their valuable suggestions and, also, Professor R. Aron for drawing his attention to some papers which appear in the references.

References

[1] Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143 (2000).

[2] J. Araujo, Realcompactness and spaces of vector valued functions, Fundamenta Math. 172 (2002), 27-40.

[3] J. Araujo, Realcompactness and Banach-Stone theorems. To appear in Bull. Belg. Math. Soc. Simon Stevin. Available at http : // www . matesco . unican . es / ˜ araujo / isom3.dvi

[4] J. Araujo, Separating maps and linear isometries between some spaces of continuous functions, J. Math. Anal. Appl. 226 (1998), 23-39.

[5] J. Araujo, E. Beckenstein, and L. Narici, Biseparating maps and homeomorphic realcompactifications, J. Math. Anal. Appl. 192 (1995), 258-265.

[6] R. Aron, J. Gómez, and J. G. Llavona, Homomorphisms between algebras of differentiable functions in infinite dimensions, Michigan Math. J. 35 (1988), 163-178.

[7] W. G. Bade, P. C. Curtis, Jr., and K. B. Laursen, Automatic continuity in algebras of differentiable functions, Math. Scand. 40 (1977), 249–270.

[8] H. Cartan, ” Calcul différentiel” , Hermann, Paris, 1967.

[9] G. Chilov, ”Analyse Mathématique. Fonctions de plusieurs variables réelles”, 2nd edition. Mir, Moscow, 1975.

[10] H. G. Dales, Automatic continuity: a survey. Bull. London Math. Soc. 10 (1978), 129-183.

[11] H. G. Dales, ”Banach algebras and automatic continuity”, Oxford University Press, New York, 2000.

[12] H. Federer, ”Geometric measure theory”, Springer Verlag, 1969.

[13] V. V. Fedorchuk, The fundamentals of dimension theory. In ”General Topology I”, edited by A. V. Arkhangelskiĭ and L. S. Pontryagin, Springer Verlag, 1990, 91-195.

[14] W. Fleming, ”Functions of several variables”, 2nd edition, Springer Verlag, 1977.

[15] J. J. Font and S. Hernández, On separating maps between locally compact spaces, Arch. Math. (Basel) 63 (1994), no. 2, 158–165.

[16] L. Gillman and M. Jerison, ”Rings of continuous functions”, Van Nostrand, Princeton, 1960.

[17] J. M. Gutiérrez and J. G. Llavona, Composition operators between algebras of differentiable functions, Trans. Amer. Math. Soc. 338 (1993), 769–782.

[18] J. Horváth, ”Topological vector spaces and distributions. Volume I”, Addison-Wesley, Reading, 1966.

[19] K. Jarosz, Automatic continuity of separating linear isomorphisms, Canad. Math. Bull. 33 (1990), 139-144.

[20] J-S. Jeang and N-C. Wong, Weighted composition operators of C_{0}(X)’s, J. Math. Anal. Appl. 201 (1996), 981–993.

[21] R. Kantrowitz and M. M. Neumann, Automatic continuity of homomorphisms and derivations on algebras of vector-valued continuous functions, Czecholovak Math. J. 45 (1995), 747-756.

[22] R. Kantrowitz and M. M. Neumann, Disjointness preserving and local operators on algebras of differentiable functions, Glasgow Math. J. 43 (2001), 295-309.

[23] S. Lang, ”Real and Functional Analysis”, 3rd. edition. Springer-Verlag, New York, 1993.

[24] K. B. Laursen, Prime ideals and automatic continuity in algebras of differentiable functions, J. Funct. Anal. 38 (1980), 16-24.

[25] S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc. 5 (1954), 917-922.

[26] L. Molnar, On rings of differentiable functions. In ”Contributions to the theory of functional equations, II”, 13–16. Grazer Math. Ber., 327, Karl-Franzens-Univ. Graz, Graz, 1996.

[27] M. M. Neumann, A. Rodríguez-Palacios, and M. V. Velasco, Continuity of homomorphisms and derivations on algebras of vector-valued functions, Quart. J. Math. Oxford (2) 50 (1999), 279-300.

[28] L. E. Pursell, Rings of continuous functions on open convex subsets of R^{n}, Proc. Amer. Math. Soc. 19 (1968), 581–585.

[29] A. M. Sinclair, ”Automatic continuity of linear operators”, London Math. Soc. Lecture Notes Series 21, Cambridge University Press, 1976.

[30] J. Wloka, ”Partial differential equations”, Cambridge University Press, Cambridge-New York, 1987.