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Journal of Mathematical Analysis and Applications www.elsevier.com/locate/jmaa

Surjective isometries on the vector-valued diﬀerentiable function spaces Lei Li ∗ , Risheng Wang School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

a r t i c l e

i n f o

Article history: Received 3 November 2014 Available online 24 February 2015 Submitted by K. Jarosz Keywords: Isometries Extreme points Diﬀerentiable functions

a b s t r a c t This paper investigates the surjective linear isometries between the diﬀerentiable function spaces C0n (Ω, E) and C0m (Σ, F ) (where Ω, Σ are open subsets of Euclidean spaces and E, F are reﬂexive, strictly convex Banach spaces), and show that such isometries can be written as weighted composition operators. © 2015 Elsevier Inc. All rights reserved.

1. Introduction The classical Banach–Stone Theorem gives a characterization of surjective linear isometries between the spaces of scalar-valued continuous functions on compact Hausdorﬀ spaces. There are several extensions of the Banach–Stone Theorem to a variety of diﬀerent settings (for a survey of this topic we refer the readers to the monographs [7,8]). M. Cambern and V.D. Pathak [3,4] considered surjective linear isometries on spaces of scalar-valued diﬀerentiable functions on the locally compact subset of R containing no isolated points and gave a representation of such mappings; and then Jarosz and Pathak [10] studied those operators on diﬀerentiable function spaces deﬁned on the compact subset X of a real line (they did not assume X contains no isolated point). Recently, F. Botelho and J. Jamison [2] extended these results to some vector-valued diﬀerentiable function space C 1 ([0, 1], H), where H is a ﬁnite-dimensional Hilbert space. In their proof, the characterization of extreme points of the unit dual ball plays a crucial role. The second author [11] gave the representation of surjective linear isometries on the scalar-valued diﬀerentiable function spaces C0n (X), where X is an open subset of Euclidean spaces Rp satisfying some other property (which is called NIP). In this paper, we will give a characterization of the surjective linear isometries between the spaces of vector-valued diﬀerentiable functions C0n (Ω, E), where Ω is an open subset of Rp and E is a reﬂexive and * Corresponding author. E-mail addresses: [email protected] (L. Li), [email protected] (R. Wang). http://dx.doi.org/10.1016/j.jmaa.2015.02.068 0022-247X/© 2015 Elsevier Inc. All rights reserved.

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strictly convex Banach space. This will extend the main results of [2–4,11] and give the Banach–Stone Theorem on the diﬀerentiable function spaces. Let Z+ be the set of non-negative integers. When p ∈ N and λ = (λ1 , λ2 , · · · , λp ) is a p-tuple in Z+ , we set |λ| = λ1 + λ2 + · · · + λp and λ! = λ1 !λ2 ! · · · λp !. Suppose that Ω is an open subset of Rp , then C0n (Ω, E) denotes the space consisting of all the E-valued functions on Ω that are vanishing at inﬁnity and of class C n , that is, those functions whose partial derivatives ∂λf =

∂ λ1 +···+λp f λ

∂xλ1 1 ∂xλ2 2 · · · ∂xp p

exist and are continuous for every λ = (λ1 , · · · , λp ) ∈ Λ, where Λ = {λ ∈ Zp+ : |λ| ≤ n}. The norm in the Banach space C0n (Ω, E) is deﬁned by ∂ λ f (x) , λ! x∈Ω

f = sup

∀ f ∈ C0n (Ω, E).

λ∈Λ

For any f ∈ C0n (Ω, E) and g ∈ C0n (Ω), it is easy to verify that C0n (Ω, E) is a C0n (Ω)-module, that is, f g ∈ C0n (Ω, E) and f g ≤ f g. For any scalar-valued function f ∈ C0n (Ω) and any element e ∈ E, the vector-valued function x ∈ Ω → f (x)e is denoted by f ⊗ e. For any (scalar-valued or vector-valued) function f , supp(f ) is denoted by the closure of cozero set of f , that is, supp(f ) = {x : f (x) = 0}. 2. Extreme points of the unit dual ball of C0n (Ω, E) In this section, we will give the complete characterization of the extreme points in the dual ball of the spaces of diﬀerentiable functions. Recall the following construction of diﬀerentiable functions due to the second author [11]. Lemma 2.1. (See [11, Proposition 1.1].) For any x0 ∈ Rp and ε, δ > 0, there exists a function f ∈ C0n (Rp ) such that f = 1, supp(f ) ⊂ B(x0 , δ) and 1 ∂ (n,0,···,0) f (x0 ) | | > 1 − ε. n! ∂xn1 We denote by BE the closed unit ball of a Banach space E. Equipped BE ∗ with the weak∗ -topology, one can isometrically embed C0n (Ω, E) onto a subspace M of C0 (W ) (in this case, we can identify C0n (Ω, E) with M), the space of continuous functions in W = Ω × (BE ∗ )λ∈Λ that vanishes at inﬁnity, by f ∈ C0n (Ω, E) → f˜ ∈ C0 (W ), where f˜ is deﬁned by f˜(x, (ϕλ )λ∈Λ ) =

ϕλ (∂ λ f (x)) , λ!

∀ (x, (ϕλ )λ∈Λ ) ∈ W.

λ∈Λ

Analogous to [2, Proposition 1.3] and [11, Lemma 1.2], one can derive the following characterization of extreme points of the unit dual ball of C0n (Ω, E). Theorem 2.2. Suppose that E is a reﬂexive and strictly convex Banach space. Then Φ is an extreme point of the unit dual ball BM∗ if and only if there exists (x, ϕ) ˜ ∈ W , with ϕ˜ = (ϕλ )λ∈Λ and all ϕλ are extreme points of BE ∗ , such that Φ(f ) =

ϕλ (∂ λ f (x)) , λ!

λ∈Λ

∀ f ∈ C0n (Ω, E).

(2.1)

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Proof. Note that any extreme point Φ of BM∗ is the restriction on M of some extreme point of the unit dual ball BC0 (W )∗ , which will also be denoted by Φ. Hence Φ = αδw for some complex number α ∈ T and some w = (x, (ϕλ )λ∈Λ ) ∈ W (see, e.g. [3, p. 397]). Here, δw is the Dirac functional deﬁned by δw (f ) = f (w) for every f ∈ C0 (W ). That is, we have Φ(f ) =

αϕλ (∂ λ f (x)) , λ!

∀ f ∈ C0n (Ω, E),

λ∈Λ

here (x, (αϕλ )λ∈Λ ) is also in W . Next we will show that every ϕλ is an extreme point of BE ∗ . Suppose on the contrary that there exists λ0 such that ϕλ0 is not an extreme point of BE ∗ , that is, ϕλ0 = 12 (φ1 + φ2 ) for some φ1 , φ2 ∈ BE ∗ , then we can derive that Φ(f ) =

1 αϕλ (∂ λ f (x)) αφ1 (∂ λ0 f (x)) [( + ) 2 λ! λ0 ! λ=λ0

+(

αϕλ (∂ λ f (x)) αφ2 (∂ λ0 f (x)) + )], λ! λ0 !

λ=λ0

and hence Φ = 12 (Φ1 + Φ2 ), where Φi = (x, (αϕiλ )λ∈Λ ), with ϕiλ = φi when λ = λ0 and ϕiλ = ϕλ when λ = λ0 , for any i = 1, 2. Since |Φi (f )| ≤

∂ λ f (x) ≤ f = f C0 (W ) λ!

(∀ i = 1, 2),

λ∈Λ

then we have Φ1 , Φ2 ∈ BC0 (W )∗ , which implies that Φ is not an extreme point. This is a contradiction. On the other hand, suppose that Φ = δ(˜x,ϕ) ˜ ∈ Ω and ϕ˜ = (ϕλ )λ∈Λ , is of the form (2.1) ˜ ∈ W , with x and Φ = 12 (Ψ1 + Ψ2 ), where Ψ1 , Ψ2 are norm one functionals in C0 (W )∗ . By Riesz representation theorem, there exist regular Borel measures μ1 , μ2 on W such that for any i = 1, 2 |μi |(W ) = Ψi = 1 and Ψi (f ) =

∀ f ∈ M.

f dμi , W

From Lemma 2.1 for any ε, δ > 0 there exists a function f¯ ∈ C0n (Rp ) with f¯ = 1 such that supp(f¯) ⊂ B(˜ x, δ) and

1 ∂ (n,0,···,0) f¯(˜ x) | | > 1 − ε. n n! ∂x1

We can choose e ∈ E with e = 1 such that ϕ(n,0,···,0) (e) > 1 − ε. Put f = f¯ ⊗ e and it is easy to verify by induction that ∂ λ f = ∂ λ f¯ ⊗ e for each λ ∈ Λ. Then we can derive that f ∈ C0n (Ω, E) with f = 1, and 1=

|ϕλ (∂ λ f (˜ |ϕλ (e)∂ λ f¯(˜ ∂ λ f (˜ x) x))| x)| ≥ = λ! λ! λ!

λ∈Λ

λ∈Λ

λ∈Λ

(n,0,···,0)

=

ϕ(n,0,···,0) (e)|∂ n!

≥ (1 − ε)2 +

λ=(n,0,···,0)

which implies that

f¯(˜ x)|

+

λ=(n,0,···,0)

|ϕλ (e)∂ f¯(˜ x)| , λ! λ

|ϕλ (e)∂ λ f¯(˜ x)| λ!

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|ϕλ (e)∂ λ f¯(˜ x)| ≤ ε. λ!

λ=(n,0,···,0)

Then we have that |Φ(f )| ≥ |

ϕλ (∂ λ f (˜ x)) | λ!

λ∈Λ

≥

1 |ϕ(n,0,···,0) (e)∂ (n,0,···,0) f¯(˜ x)| − n!

λ=(n,0,···,0)

|ϕλ (e)∂ λ f¯(˜ x)| λ!

> (1 − ε) − ε > 1 − 3ε 2

and hence |Ψ1 (f ) + Ψ2 (f )| = 2|Φ(f )| > 2(1 − 3ε).

(2.2)

Since Ψi ≤ 1, then (2.2) enables us to get that |Ψi (f )| > 1 − 6ε for any i = 1, 2, and thus we have that 1 − 6ε ≤ |Ψi (f )| = |

f dμi | W

f dμi | ≤ |μi |(B(˜ x, δ) × (BE ∗ )λ∈Λ ).

=|

(2.3)

B(˜ x,δ)×(BE ∗ )λ∈Λ

Let ε → 0, we can derive that 1 ≤ |μi |(B(˜ x, δ) × (BE ∗ )λ∈Λ ) for all δ > 0 and i = 1, 2. Since |μi | is regular and the set K := {˜ x} × (BE ∗ )λ∈Λ is compact, then 1 ≤ |μi |(K) ≤ |μi |(W ) = Ψi ≤ 1 by setting δ → 0, which implies that |μi |(W ) = |μi |(K) = 1 and |μi |(W \ K) = 0.

(2.4)

So we have Ψi (f ) =

f dμi =

W

f dμi ,

∀ f ∈ M, i = 1, 2.

(2.5)

K

For some δ0 > 0 satisfying B(˜ x, δ0 ) ∩ Ω is compact and some open neighborhood U ⊂ B(˜ x, δ0 ) of x, by n p Urysohn’s Lemma (see [9, p. 56]) there exists a function ξ ∈ C0 (R ) such that supp(ξ) ⊂ B(˜ x, δ0 ) and ξ = 1 on U . Note that if E is strictly convex, then each point of norm one is an exposed point of BE (see, e.g., [5, p. 23]). Then, for every λ ∈ Λ, the strictly convexity and reﬂexivity of E enables us to ﬁnd an element eλ ∈ E with eλ = 1 such that ϕλ (eλ ) = eλ = 1 and ϕ(eλ ) < 1 for all ϕ ∈ E ∗ with ϕ = ϕλ and ϕ = 1. When λ = (λ1 , · · · , λp ) and x = (x1 , · · · , xp ), x ˜ = (˜ x1 , · · · , x ˜p ), let (x − x ˜)λ = (x1 − x ˜1 )λ1 · · · (xp − x ˜p )λp , and then deﬁne the function θ ∈ C0n (Ω, E) by θ(x) =

(x − x ˜)λ eλ ,

∀ x ∈ Rp .

λ∈Λ

It is easy to verify that the function g = ξθ belongs to C0n (Ω, E), and thus we can derive that

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∂ λ g(˜ ϕλ (∂ λ g(˜ x)) x) = λ! λ! λ∈Λ λ∈Λ 1 1 = (Ψ1 (g) + Ψ2 (g)) = ( gdμ1 + gdμ2 ). 2 2

Φ(g) =

K

Since |μi |(K) = 1 and |g(u)| ≤

K

∂ λ g(˜ x) for all u ∈ K, then λ!

λ∈Λ

∂ λ g(˜ x) = gdμi = Φ(g), λ!

λ∈Λ

for all i = 1, 2.

(2.6)

K

Set K1 = {(˜ x, (αϕλ )λ∈Λ ) : α ∈ T}. For any w = (˜ x, (ρλ )λ∈Λ ) ∈ K \ K1 , since ϕλ (eλ ) = 1 we have that |g(w)| ≤

ϕλ (∂ λ g(˜ ∂ λ g(˜ |ρλ (∂ λ g(˜ x))| x)) x) < = = Φ(g). λ! λ! λ!

λ∈Λ

λ∈Λ

λ∈Λ

Therefore, |μi |(K \ K1 ) = 0 for i = 1, 2, and then

gdμi = K1

gdμi = Φ(g).

(2.7)

K

Finally, for any h ∈ M and w = (˜ x, (αϕλ )λ∈Λ ) ∈ K1 with α ∈ T, an easy calculation shows that h(w) =

αϕλ (∂ λ h(˜ δw (g) x)) = αΦ(h) = Φ(h). λ! Φ(g)

λ∈Λ

So together with (2.5) and (2.7) we can derive that Ψi (h) =

hdμi =

W

=

Φ(h) Φ(g)

hdμi =

K

hdμi K1

gdμi = Φ(h),

∀ i = 1, 2,

K1

which implies that Ψ1 = Ψ2 = Φ. Therefore, Φ is an extreme point in BM∗ .

2

3. The characterization of surjective isometries Suppose that p, q, n, m ∈ N, Ω ⊂ Rp and Σ ⊂ Rq are open subsets, and E, F are reﬂexive, strictly convex Banach spaces. Assume that T : C0n (Ω, E) → C0m (Σ, F ) is a surjective linear isometry. We denoted by Λ and Γ the set {λ ∈ Zp+ : |λ| ≤ n} and {γ ∈ Zq+ : |γ| ≤ m}, respectively. Let W1 = Ω × (extBE ∗ )λ∈Λ and W2 = Σ × (extBF ∗ )γ∈Γ , where extBE ∗ and extBE ∗ are equipped with weak∗ -topology in BE ∗ and BF ∗ , respectively. The conjugate operator T ∗ : C0m (Σ, F )∗ → C0n (Ω, E)∗ is also a surjective linear isometry, and maps extreme points to extreme points. That is, for any w1 = (x, (ϕλ )λ∈Λ ) ∈ W1 , we have that (T −1 )∗ (δw1 ) = δw2 for some w2 = (y, (ψγ )γ∈Γ ) ∈ W2 . Deﬁne the map Θ : W1 → W2 by Θ(w1 ) = w2 , then for any f ∈ C0n (Ω, E) one can derive that δΘ(w1 ) (T f ) = δw1 (f ).

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Lemma 3.1. Θ is a homeomorphism. Proof. By Theorem 2.2, we can identify W1 , W2 with (extBC0n (Ω,E)∗ , w∗ ), (extBC0n (Σ,F )∗ , w∗ ), respectively. Since (T −1 )∗ is also a weak∗ isomorphism, by the following commutative diagram W1 −−−−→ (extBC0n (Ω,E)∗ , w∗ ) ⏐ ⏐ ⏐ −1 ∗ ⏐ (T ) Θ W2 −−−−→ (extBC0n (Ω,F )∗ , w∗ ) we can derive that Θ is a homeomorphism. 2 Lemma 3.2. For any x0 ∈ Ω, the set ˜ for some ϕ˜ ∈ (extBE ∗ )λ∈Λ , ψ˜ ∈ (extBF ∗ )γ∈Γ } τ (x0 ) = {y ∈ Σ : Θ(x0 , ϕ) ˜ = (y, ψ) is a singleton. Proof. Suppose on the contrary that there exit y1 , y2 ∈ τ (x0 ) and y1 = y2 , then one can choose two disjoint ¯ ⊂ V1 ¯ ∈ C m (Σ) be such that supp(h) open subset V1 , V2 of Σ such that y1 ∈ V1 and y2 ∈ V2 . Let h 0 ¯ = 1 on V3 , where V3 ⊂ V1 is an open neighborhood of y1 . Assume that Θ(x0 , ϕ˜1 ) = (y1 , ψ˜1 ) and and h Θ(x0 , ϕ˜2 ) = (y2 , ψ˜2 ), where ϕ˜1 , ϕ˜2 ∈ (extBE ∗ )λ∈Λ and ψ˜1 , ψ˜2 ∈ (extBF ∗ )γ∈Γ . Let ψ˜1 = (ψγ1 )γ∈Γ , and take ¯ ⊗ e0 . an element 0 = e0 ∈ F such that ψ 1 (e0 ) = 0. Put h = h (0,···,0)

By the deﬁnition of Θ, we have that U = PΩ Θ(V2 × (extBF ∗ )γ∈Γ ) is an open subset of Ω containing x0 , and PΛ Θ(V2 × (extBF ∗ )γ∈Γ ) = (extBE ∗ )λ∈Λ . Here, PΩ : (x, ϕ) ˜ ∈ W → x ∈ Ω and PΛ : (x, ϕ) ˜ ∈ W → ϕ˜ ∈ (extBE ∗ )λ∈Λ are the canonical projections on W . Let x belong to U . For any ρ ∈ extBE ∗ , let ρ˜1 = (ρ1λ )λ∈Λ and ρ˜2 = (ρ2λ )λ∈Λ , where ρ1(0,···,0) = ρ2(0,···,0) = ρ and ρ1λ = −ρ2λ for all 1 ≤ |λ| ≤ n. Assume that Θ(x, ρ˜1 ) = (˜ y1 , κ ˜ 1 ) and Θ(x, ρ˜2 ) = (˜ y2 , κ ˜ 2 ), where y˜1 , y˜2 ∈ V2 . Then we can derive that 1 [δ(x,ρ˜1 ) + δ(x,ρ˜2 ) ](T −1 h) 2 1 = [δ(˜y1 ,˜κ1 ) + δ(˜y2 ,˜κ2 ) ](h) 2 = 0.

ρ((T −1 h)(x)) =

This implies that ∂ λ (T −1 h)(x0 ) = 0 for all |λ| ≤ n, and hence 0 = δ(y1 ,ψ˜1 ) (h) = δ(x0 ,ϕ˜1 ) (h) = 0, which is impossible. Therefore, τ (x0 ) is a singleton. 2 Since every τ (x) is a singleton for all x ∈ X by Lemma 3.2, so we can deﬁne a well-deﬁned mapping, which is also denoted by τ , such that ˜ Θ(x, ϕ) ˜ = (τ (x), ψ) for any x ∈ Ω and ϕ˜ ∈ (extBE ∗ )λ∈Λ , and the element ψ˜ ∈ (extBF ∗ )γ∈Γ depends on the choice of (x, ϕ). ˜ By the homeomorphism of Θ, we can derive the following result.

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Lemma 3.3. τ : Ω → Σ is a homeomorphism. Furthermore, we have the following theorem. Theorem 3.4. For any f ∈ C0n (Ω, E) and x0 ∈ Ω we have that ∂ λ f (x0 ) ∂ γ (T f )(τ (x0 )) = . λ! γ! γ∈Γ

λ∈Λ

Proof. Since extBE ∗ is the James boundary of E [6, Fact 3.45], then there exists ϕ˜ = (ϕ˜λ )λ∈Λ ∈ (extBE ∗ )λ∈Λ such that δ(x0 ,ϕ) ˜ (f ) =

∂ λ f (x0 ) ϕ˜λ (∂ λ f (x0 )) = . λ! λ!

λ∈Λ

λ∈Λ

˜ then one can derive that Let ψ˜ = (ψ˜γ )γ∈Γ ∈ (extBF ∗ )γ∈Γ be such that Θ(x0 , ϕ) ˜ = (τ (x0 ), ψ), ∂ γ T f (τ (x0 )) ∂ λ f (x0 ) = δ(τ (x0 ),ψ) . ˜ (T f ) ≤ λ! γ!

(3.1)

γ∈Γ

λ∈Λ

By the symmetry, the homeomorphism of τ enables us to get the other direction ∂ γ T f (τ (x0 )) ∂ λ f (x0 ) ≥ . γ! λ!

γ∈Γ

2

λ∈Λ

Theorem 3.5. For any f ∈ C0n (Ω, E) and x0 ∈ Ω, we have that f (x0 ) = 0 if and only if (T f )(τ (x0 )) = 0. Proof. We will divide the proof into the following two steps. Claim 1: For a function g ∈ C0n (Ω, E), if there exists an open neighborhood U of x0 such that g = e on U and an element e ∈ E with e = 1, then (T g)(y) = 1 and ∂ γ (T g)(y) = 0 for all y ∈ τ (U ) and 1 ≤ |γ| ≤ m. Indeed, by Theorem 3.4 for any x ∈ U we have that ∂ λ g(x) ∂ γ T g(τ (x)) = = 1, λ! γ!

λ∈Λ

γ∈Γ

which implies that {∂ γ T g(τ (x)) : γ ∈ Γ} must have at most one non-zero term and T g(τ (x)) = 0 or 1 for all x ∈ U . So for any y0 ∈ τ (U ) by the continuity of T g there exists an open neighborhood V ⊂ τ (U ) of y0 such that T g(y) ≡ 0 on V or T g(y) ≡ 1 on V . In the former case, we can derive that ∂ γ T g(y) = 0 for all y ∈ V and γ ∈ Γ, which is impossible. In the latter case, we have ∂ γ T g(y) = 0 for all y ∈ τ (U ) and 1 ≤ |γ| ≤ m. Claim 2: For any i = 1, 2, let ϕ˜i = (ϕiλ )λ∈Λ be such that ϕ1(0,···,0) = −ϕ2(0,···,0) = ϕˆ and ϕ1λ = ϕ2λ for each 1 ≤ |λ| ≤ n, where ϕˆ ∈ extBE ∗ and ϕ˜1 satisﬁes that δ(x,ϕ˜1 ) (f ) =

∂ λ f (x0 ) . λ!

λ∈Λ

For any i = 1, 2, set Θ(x0 , ϕ˜i ) = (τ (x0 ), ψ˜i ) and ψ˜i = (ψγi )γ∈Γ ∈ (extBF ∗ )γ∈Γ . Then we can derive that 1 2 1 2 ψ(0,···,0) + ψ(0,···,0) = 0. Indeed, suppose on the contrary that ψ(0,···,0) + ψ(0,···,0) = 0, and then there exists

L. Li, R. Wang / J. Math. Anal. Appl. 427 (2015) 547–556

554

1 2 1 2 e˜ ∈ F with ˜ e = 1 such that (ψ(0,···,0) + ψ(0,···,0) )(˜ e) = ψ(0,···,0) + ψ(0,···,0) = 0. Let h = T −1 (k ⊗ e˜), here, m k is a function in C0 (Σ) such that k = 1 on some neighborhood of τ (x0 ). Then by Claim 1 we have that h(x) = 1 and ∂ λ h(x) = 0 for all x in some neighborhood of x0 and 1 ≤ |λ| ≤ n. This implies that

0 = (δ(x0 ,ϕ˜1 ) + δ(x0 ,ϕ˜2 ) )(h) 1 2 + ψ(0,···,0) )(T h)(τ (x0 )) = (ψ(0,···,0) 1 2 + ψ(0,···,0) )(˜ e) = 0, = (ψ(0,···,0)

which is impossible. By Theorem 3.4 we have the following equation ∂ λ f (x0 ) ∂ γ T f (τ (x0 )) = δ(x0 ,ϕ˜i ) (f ) = δ(τ (x0 ),ψ˜i ) (T f ) = , λ! γ!

∀ i = 1, 2,

γ∈Γ

λ∈Λ

and then ∂ γ T f (τ (x0 )) 1 = [δ(τ (x0 ),ψ˜1 ) + δ(τ (x0 ),ψ˜2 ) ](T f ) γ! 2

γ∈Γ

=

ψγ1 + ψγ2 (∂ γ T f (τ (x0 ))) 2γ!

γ∈Γ

=

1≤|γ|≤m

≤

1≤|γ|≤m

ψγ1 + ψγ2 γ (∂ T f (τ (x0 ))) 2γ! ∂ γ T f (τ (x0 )) , γ!

which implies that T f (τ (x0 )) = 0. 2 Remark 3.6. Given f, g ∈ C0n (Ω, E), if f = g on some open subset U ⊂ Ω, then we have that T f = T g on the open subset τ (U ) ⊂ Σ. As a consequence, we can characterize the surjective linear isometries on the spaces of diﬀerentiable functions, which can be regarded as a smooth version Banach–Stone Theorem. This following theorem is also a generalization of [2, Theorem 3.5]. Theorem 3.7. Suppose that n, m, p, q are positive integers and Ω ⊂ Rp , Σ ⊂ Rq are open subsets, E, F are reﬂexive, strictly convex Banach spaces. Assume that T : C0n (Ω, E) → C0m (Σ, F ) is a surjective linear isometry. Then there exist a homeomorphism τ : Ω → Σ and, for each y ∈ Σ, a the mapping Jy : E → F such that (i) (ii) (iii) (iv)

p = q and n = m. For y ∈ Y , every Jy : E → F is a bijective linear isometry. τ : X → Y is a C m -diﬀeomorphism. For any f ∈ C0n (Ω, E) we have T f (y) = Jy f (τ −1 (y)),

∀ y ∈ Σ.

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Proof. By Lemma 3.3 there exists a homeomorphism τ : Ω → Σ, which implies that p = q. For any y ∈ Σ, by Theorem 3.5, we can deﬁne Jy : E → F by Jy (e) = T (he )(y),

∀ e ∈ E,

here, he ∈ C0n (Ω, E) with value e on a neighborhood of τ −1 (y). By Remark 3.6, Jy is well-deﬁned and then one can derive that T f (y) = Jy f (τ −1 (y)),

∀ y ∈ Σ, f ∈ C0n (Ω, E).

(3.2)

Claim 1. For any y ∈ Σ, there exists e ∈ E such that Jy (e) = T (he )(y) = 0. Indeed, if e ∈ F and e = 0, then there exists a function g ∈ C0n (Ω, E) such that T g = he , here he is a function in C0m (Σ, F ) such that he = e on a neighborhood of y. Eq. (3.2) enables us to get that g(τ −1 (y)) = 0, and when put e = (g ◦ τ −1 )(y) in E, we can derive that Jy (e) = e = 0. Claim 2. τ −1 : Σ → Ω is a C m -diﬀerentiable function. Fix any y0 ∈ Σ, we can choose e ∈ E and a function he ∈ C0n (Ω, E) deﬁned as above such that T (he )(y0 ) = 0, and then there exists an open neighborhood V of y0 such that T (he )(y) = 0 for all y ∈ V . By (3.2) for any f ∈ C0n (Ω) we can derive that T (f ⊗ e)(y) = T (he )(y)f (τ −1 (y)),

∀ y ∈ Σ.

For any i = 1, 2, · · · , p, let ρi : Rp → R is the projection deﬁned by ρi (x1 , · · · , xp ) = xi ,

∀ (x1 , · · · , xp ) ∈ Rp .

Then for any 1 ≤ i ≤ p we have that T (ρi ⊗ e)(y) = T (he )(y)(ρi ◦ τ −1 )(y) for all y ∈ Σ, and this implies that ρi ◦ τ −1 is a C m -diﬀerentiable function by [1, Lemma 3.2]. Therefore, τ −1 is C m -diﬀerentiable. Claim 3. For any y ∈ Σ, by the argument of Claim 1, we have that Jy is bijective. By the argument of Theorem 3.5, we can derive that T (he ) is also a constant function, which implies that Jy is an isometry. Claim 4. m = n, and then τ −1 is a C n -diﬀeomorphism. Indeed, since both T and T −1 are isometries, then we can assume that n ≤ m. Take any y0 ∈ Σ, then there exist an element e ∈ E and an open neighborhood V of y0 such that T (he )(y) = 0 for all y ∈ V . By [1, p. 502] one can choose an open neighborhood V1 ⊂ V of y0 such that τ −1 |V1 is a C m -diﬀeomorphism. Suppose on ¯ ¯ ∈ C n (Σ) \ C m (Σ) such that {y ∈ Σ : h(y) the contrary that n < m and then choose a function h = 0} ⊂ V1 . 0 0 Deﬁne h : Ω → E by ¯ (x)) ⊗ e, h(x) = h(τ

∀ x ∈ Ω.

Then h ∈ C0n (Ω, E) and so ¯ (τ −1 (y))) = T (he )(y)h(y) ¯ (T h)(y) = T (he )(y)h(τ

∀ y ∈ Σ.

¯ is C m -diﬀerentiable in V1 , which Since T (he )(y) = 0 for all y ∈ V1 , by [1, Lemma 3.2] we can derive that h m ¯ implies that h is C -diﬀerentiable in Σ. This is a contradiction. 2 Acknowledgment We would like to express our thanks to the referee for several helpful comments which improved the presentation of this paper. The ﬁrst author was supported in part by NSF of China (11301285, 11371201).

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L. Li, R. Wang / J. Math. Anal. Appl. 427 (2015) 547–556

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