Competencias laborales de los puestos claves

continuously toV by (ii). Hence each of the functions in (D.27) mapsP × Qcon- tinuously toV, implying thatΨ ∈Cm

∗ (P × Q,V).

Corollary D.23. If, under the assumptions of Proposition D.22 we are also given a functionΥ ∈Cm

∗ (P,Q)withΥ(P)⊂ Q, the mapΘ(p) :=F(Φ(p), Υ(p))be-

longs toCm

∗ (P,V)and for eachn≤m, the derivativeDnΘ(p,p˙n)is a combination

of terms

Di1Dk2F (Φ(p), Υ(p)), Dj1Υ(p,p˙j1), . . . , DjiΥ(p,p˙ji), Dℓ1Φ(p,p˙ℓ1), . . . , DℓkΦ(p,p˙ℓk)

wherejs, ℓs≥1 andPis=1js+Pks=1ℓs=n.

Proof. Observe that Θ = Ψ ◦κ where Ψ comes from Proposition D.22 and

κ:P →P×Qisκ(p) = (p, Υ(p)). Sinceκ∈Cm

∗ (P,P ×Q), κ(P)⊂ P × Qand

Ψ ∈Cm

∗ (P × Q, V), the statement follows from Theorem D.21.

The following main chain rule is a ‘symmetric’ version of the above, which is capable of being iterated. It will be stated in the following situation. Let P,

X = Xm ֒→ . . . ֒→ X0, Y = Ym ֒→ . . . ֒→ Y0 and Z = Zm ֒→ . . . ֒→ Z0 be normed linear spaces, U ⊂X, V ⊂ P, andY ⊂ Y are open. We will use fXn to denote the closure ofX inXn, and similarly forYen andZen. Also, we useX(and similarlyY_andZ_{) for the sequence (Xm, . . . ,X0).}

The class of functions we will consider may be informally described as those

G: U × V →Y for whichD₁jDℓ

2Gis a continuous mapU × V ×P ℓ

×fXj_n →Yn+ℓ, i.e., ℓ derivatives in the parameter p∈ V lead to a loss of regularity of orderℓ in the scale of Banach spaces. Since this description has several interpretations, we give a rather detailed one as a formal definition.

DefinitionD.24. For any 0≤k≤m, we defineCek(U × V,X_,Y_{) as the set of} all mapsG:U × V →Y such that

(a) G∈Ck

∗(U × V,Y0).

(b) For eachj+ℓ≤k, the function

(x, p,x˙1, . . . , ,x˙j,p˙1, . . . , ,p˙ℓ)→D1jD2ℓG((x, p),p˙1, . . . , ,p˙ℓ,x˙1, . . . , ,x˙j), which is by (a) defined as a mapU × V ×Xj×Pℓ → Y0 has a (necessarily unique) extension to a continuous mapping U × V ×fXj_ℓ×Pℓ _→ Y0. This extension is also denotedDj₁Dℓ

2G.

(c) For each 0≤j≤k−ℓand each 0≤n≤m−ℓthe restriction ofDj1Dℓ2G(which has been already extended by (b)) toU × V ×fXj_n+ℓ×Pℓhas values inYn and is continuous as a mapping between these spaces.

Notice that, clearly,Cei(U × V,X_,Y_))⊂Cek₍

U × V,X_,Y_{) fork≤i. For proving} that G ∈ Cek_{(U × V,X,Y) the following simplification of this definition is rather} useful.

Lemma D.25. Assume that 0 ≤ k ≤ m. Then G: U × V → Y belongs to

e

Ck_{(U × V,X,Y)iff}

(i) as a map of U × V to Y0, G has derivatives Dj1Dℓ2G((x, p),p˙ℓ,x˙j) for all

(ii) for 0≤j ≤k−ℓand all 0≤n≤m−ℓ there is continuous map Ψj,ℓ,n:U ×

V ×Xfn+ℓ×P → Yn such that D1jDℓ2G((x, p),p˙ℓ,x˙j) = Ψj,ℓ,n(x, p,x,˙ p˙) for every (x, p)∈ U × V,p˙∈P, andx˙ ∈X.

Proof. If G ∈ Cek_{(U × V,X,Y), (i) and (ii) are obvious. For the opposite} implication, assuming (i) and (ii) we see that for each j+ℓ ≤ k, (x, p,x,˙ p˙) →

Dj1Dℓ2G((x, p),p˙ℓ,x˙j) is a continuous mapU ×V ×X×P →Y0. HenceG∈C∗k(U ×

V,Y0) by Corollary D.15, yielding D.24(a). Lemma D.13 and the polarization formula establish the function

(x, p,x˙1, . . . ,x˙j,p˙1, . . . ,p˙ℓ)→D1jD2ℓG((x, p),p˙1, . . . ,p˙ℓ,x˙1, . . . ,x˙j) as a combination of terms (x, p,x˙1, . . . ,x˙j,p˙1, . . . ,p˙ℓ)→Dj1Dℓ2G((x, p),( X k∈I σkp˙k)ℓ,( X k∈J τkx˙k)j)

where I ⊂ {1, . . . , ℓ}, J ⊂ {1, . . . , j}, and σk, τk =±1. This shows that for each 0≤n≤m−ℓ, the derivativeD₁jDℓ

2Gcan be extended to a continuous mapΨej,ℓ,n, fromU × V ×Xfj_n+ℓ×PℓtoYn. Withn= 0 this shows D.24(b). For 0≤n≤m−ℓ we see fromX=Xm֒→Xn+ℓ֒→Xℓthat bothΨej,ℓ,n and the restriction ofΨej,ℓ,0 toU × V ×fXj_n+ℓ×Pℓ_{are continuous as maps ofU := U × V ×Xf}j

n+ℓ×P ℓ

,k · kXℓ

to Y0. SinceX is dense in (Xfn+ℓ,k · kXn+ℓ), and so also in (fXn+ℓ,k · kXℓ), the mapsΨej,ℓ,n andΨej,ℓ,0 coincide on a dense subset ofU, hence on all ofU, proving

D.24(c).

Remark D.26. Clearly, the claim remains true if one replaces

Dj₁Dℓ

2G((x, p),p˙ℓ,x˙j)

with the derivatives taken in the opposite order (see Remark D.16). In the present and the following appendices, in the notationCem_{(U ×V,X,Y) we indicate, somehow} pedantically but usefully for clarity in proofs, the sequencesX_,Y_{of Banach spaces.} When using this notion in particular applications, the sequences X _and Y _will be clear from the context and we will skip them from the notation writing just

e

Cm_{(U × V). ⋄}

For working with functions fromCem_{(U × V,X,Y) it is useful to know that they} have properties stronger than those given in the definition.

Lemma D.27. Let G∈Cem_{(U × V,X,Y) and0≤j, n≤m−ℓ. Then} (1) for fixedx∈ U, the map p→G(x, p)belongs toCℓ

∗(V,Yem−ℓ); (2) for fixedp∈ V andp˙1, . . . ,p˙ℓ∈P, the (extended) map

(x,x˙1, . . . ,x˙j)→D1jDℓ2G((x, p),p˙1, . . . ,p˙ℓ,x˙1, . . . ,x˙j) belongs toCm−ℓ−j

Xj+1n+ℓ

(U ×fXjn+ℓ,Yen).

Proof. (1) By Corollary D.9 and D.24(c) withn=m−ℓ, the mapp→G(x, p) belongs toCℓ

∗(P,Ym−ℓ). Hence the derivativeDℓ2Gis an iterated limit of elements ofY taken in the norm ofYm−ℓ, and so it belongs to Yem−ℓ.

(2) By Lemma D.20 it suffices to show that the function

D.4. CHAIN RULE WITH PARAMETER AND A GRADED LOSS OF REGULARITY 127

belongs toC_Xn+ℓm−ℓ(U,Yen). But this follows by the same argument as in the proof of

(1).

RemarkD.28. Since (2) puts the values of the (extended) derivatives into the corresponding closures ofY,Gbelongs toCm_{(U ×V,X,Y) iff and only if it belongs} to this space when Xn and Yn are replaced by fXn and Yen, respectively. So, at least in proofs, we may always assume thatX is dense inXn andY inYn. ⋄ TheoremD.29. LetG∈Cem_{(U ×V,X,Y),G(U ×V)⊂ Y,F ∈Ce}m_(Y×V,Y,Z) and define F ⋄G: U × V → Z by F ⋄G(x, p) := F(G(x, p), p). Then F ⋄G ∈

e

Cm_{(U × V,X,Z).}

Proof. By Remark D.28, we may assumeXfn=Xn, and similarly forYn and

Zn. SetH :=F⋄G. For fixedx∈ U, the functionp→H(x, p) is of the form of a compositionF(Φ(p), Υ(p)) where the outer functionF:Y × V →Z and the inner functions Φ(p) = G(x, p) andΥ(p) =psatisfy the assumptions of Corollary D.23 withQ=P,Q=P andV =Z0. Hencep→H(x, p) belongs toCm

∗ (P,Z0) and

for eachℓ≤m, the derivativeDℓ

2H((x, p),p˙ℓ) is a combination of terms (D.28) Dk 1D2iF (G(x, p), p),p˙i, D m1 2 G((x, p),p˙m1), . . . , D mk 2 G((x, p),p˙mk) wherems≥1 andi+Pk_s=1ms=ℓ.

We now fixp,p˙ and differentiate the function in (D.28) with respect to x. We set K(x) := G(x, p), Dm1 2 G((x, p),p˙m1), . . . , D2mkG((x, p),p˙mk) and L(y,y˙1, . . . ,y˙k) =D1kDi2F (y, p),p˙i,y˙1, . . . ,y˙k.

Then the expression in (D.28) is given by the composition (L◦K)(x). Sincems≤

l−i≤m−iwe havem−ms≥iand it follows from Lemma D.27 (2) (applied to thes-th component ofK withn=m−ms) that

K∈C_∗m−l(U;Y ×Yki). Application of Lemma D.27 (2) toF yields that

L∈Cm−i−k

Yik+1

(Y ×Y_ik,Z0)⊂Cm−ℓ

Yik+1

(Y ×Y_ik,Z0).

where the inclusion follows from the relationl≥i+k. Hence, Theorem D.21 shows

L◦K ∈Cm−l

∗ (U,Z0) and for each j ≤ m−ℓ the derivative of Dj(L◦K) (and

hence the derivativeD₁jDℓ

2H) exists and is given by a sum of terms of the form (D.29) D1kD2iF (G(x, p), p),p˙i, Dj1 1 D ℓ1 2 G((x, p),p˙ℓ1,x˙j1), . . . , D jk 1 D ℓk 2 G((x, p),p˙ℓk,x˙jk) wherejs+ℓs≥1,i+Pks=1ℓs=ℓandPks=1js=j.

Finally, we rely on Lemma D.27 once more. For any s = 1, . . . , k, the map (x, p,x,˙ p˙)→Djs

1 Dℓ2sG((x, p),p˙ℓs,x˙js) is a continuous map fromU ×V ×Xns+ℓs×P toYns wheneverns≤m−ℓs. Choosingns=n+ℓ−ℓsfor any fixedn≤m−ℓ, we get a mapU ×V ×Xn+ℓ×P →Yn+ℓ−ℓs. Using thatℓs≤ℓ−i, the derivatives have been extended so that the function of (x, p,x,˙ p˙) defined in (D.29) is a composition of continuous maps

and

Y ×P ×Pi×Ynk+i→Zn.

Hence (x, p,x,˙ p˙)→Dj1Dℓ2H((x, p),p˙ℓ,x˙j) is continuous as a map ofU ×V ×Xn+ℓ×

P toZn and we conclude from Lemma D.25 thatH ∈Cem(U × V,X,Z). Remark D.30. Letp0∈ V and assume thatG(U ×Bδ(p0))⊂ Y,

(D.30) kDj1Dℓ2G((x, p),p˙ℓ,x˙j)kYn≤C1kx˙kj_Xn+ℓkp˙kℓ

for any (x, p,x,˙ p˙)∈ U ×Bδ(p0)×Xn+ℓ×P and any 0≤j+ℓ≤m, 0≤n≤m−l and

(D.31) kDj₁Dℓ2F((y, p),p˙ℓ,y˙j)kZn≤C2ky˙kjYn+ℓkp˙k ℓ

for any (y, p,y,˙ p˙)∈ Y × Bδ(p0)×Yn+ℓ×P and any 0≤j+ℓ≤m, 0≤n≤m−l. Then

(D.32) kD1jDℓ2H((x, p),p˙ℓ,x˙j)kZn ≤C3kx˙kj_Xn+ℓkp˙kℓ

for any (x, p,x,˙ p˙)∈ U ×Bδ(p0)×Xn+ℓ×P and any 0≤j+ℓ≤m, 0≤n≤m−l, whereC3 depends only onC1, C2and m. In fact, sinceD1jD2ℓH((x, p),p˙ℓ,x˙j) is a weighted sum of the terms in (D.29) it is easy to see that there exists a constant

C(m) such thatC3≤C(m)C1(1 +C2m). ⋄ If we the introduce the norm

kGkCem_(U×V,X_,Y₎:= inf

n

M :kDj₁Dℓ2G((x, p),p˙ℓ,x˙j)kYn≤Mkx˙kj_Xn+ℓkp˙kℓ, (D.33)

∀(x, p,x,˙ p˙)∈ U × V ×Xn+ℓ×P and any 0≤j+ℓ≤m, 0≤n≤m−l} then the remark implies thatkHk can be controlled in terms ofkFkandkGk.

D.5. A special case of a function G that is linear in its first argument

Here we discuss conditions assuring that G ∈ Cem _{in a special case of linear} dependence on the first variable:

Lemma D.31. Let G:X× V →Y and assume that: (i) For any p∈ V, the map x7→G(x, p)is linear.

(ii) For any0≤ℓ≤mand any x∈X, the map p7→G(x, p)is inCℓ

∗(V,Ym−ℓ). (iii) For anyp0∈ V there exists δ, C >0 such that

kDℓ2G((x, p),p˙ℓ)kYn≤CkxkXn+ℓkp˙kℓ

for any0≤ℓ≤m,0≤n≤m−ℓ, and(x, p,p˙)∈X_×Bδ(p0)×P. Then G∈Cem_{(X× V,X,Y). Moreover} (D.34) kGkCem_(B R×V,X,Y)≤C(m)(1 +R)M ′_, where M′:= infM :kDℓ2G((x, p),p˙ℓ)kYn≤Mkx˙kXn+ℓkp˙kℓ, for any(x, p,x,˙ p˙)∈X_{× V ×}P and any 0≤n+ℓ≤m}