Essential norm of some extensions of the generalized composition operators between kth weighted type spaces

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R E S E A R C H

Open Access

Essential norm of some extensions

of the generalized composition operators

between

k

th weighted-type spaces

Stevo Stevi´c

*

*_{Correspondence: [email protected]}

Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia

Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

Abstract

We calculate the essential norm of some extensions of the generalized composition operators betweenkth weighted-type spaces on the unit disk in the complex plane, considerably extending some results in the literature.

MSC: Primary 47B38; secondary 47B33; 30H99

Keywords: essential norm; generalized composition operator;kth weighted-type space; unit disk

1 Introduction

LetDbe the open unit disk in the complex planeC,H(D) the class of all holomorphic

functions onD, andS(D) the class of all holomorphic self-maps ofD.

Letμ(z) be a positive continuous function onD(weight) andk∈N. Thekth

weighted-type space denoted byW(k)

μ (D) =Wμ(k)is deﬁned as follows:

W(k)

μ =

f∈H(D) :b_W(k) μ (f) <∞

,

where

b_W(k)

μ (f) :=sup z∈D

μ(z)f(k)(z). ()

The space was introduced in [] where the composition operators from the weighted Bergman space to the space were studied. Some other concrete operators on the space were later studied in [–].

Ifk= , thenb_W()

μ (·) is a norm on spaceW

()

μ , the so-called weighted-type space ([,

]). Ifk∈N, then it is easy to see thatb_W(k)

μ (·) is a semi-norm onW

(k)

μ . It is not a norm

on the space since fromb_W(k)

μ (f) =  it follows thatf

(k)₍_z_{) = ,}_z_∈_{D, and consequently} f(z) =pk–(z), wherepk–is a polynomial of degree at mostk– . However, it is a norm on

the quotient spaceW(k)

μ /Pk–, wherePk–is the space of all polynomials of degree less than

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a quotient space, let

f+Pk–_W(k)

μ /Pk–:=g∈finf+Pk–

b_W(k)

μ (g). ()

Then, iff+Pk–_W(k)

μ /Pk–= , by using () and (), we have

 = inf

g∈f+Pk–

b_W(k)

μ (g) =pk–inf∈Pk–

b_W(k)

μ (f +pk–)Wμ(k)=bWμ(k)(f),

from which it follows thatf ∈Pk–, that is,f +Pk–=Pk–= _W(k) μ /Pk–.

On the other hand, there are some natural algebraic isomorphisms between some quo-tient spaces and some spaces of holomorphic functions. Namely, we have

H(D)/Pk–∼=a

f ∈H(D) :f(j)() = ,j= ,k– =:Hk(D), and

W(k)

μ /Pk–∼=a

f ∈Wμ(k):f(j)() = ,j= ,k– 

=:W_μ(k_,)_k(D).

Indeed, for each class g+Pk–∈H(D)/Pk–(org+Pk–∈Wμ(k)/Pk–) there is a unique fg∈g+Pk–such thatfg(j)() = ,j= ,k– . Namely, ifg(z) =

∞

j=ajzj, then we can take

fg(z) = ∞

j=kajzj, that is,fg=g+pg,k–, wherepg,k–(z) =

k–

j=(–aj)zj, and the map

L(g+Pk–) :=fg

is a linear bijection fromH(D)/Pk–ontoHk(D), as well as fromWμ(k)/Pk–ontoWμ(k,k)(D). Hence, we can identify the quotient spaces with the corresponding subspaces of holomor-phic functions satisfying the conditionsf(j)() = ,j= ,k– .

From () and () it follows that

f+Pk–_W(k)

μ /Pk–=bWμ(k)(f),

this fact along with the above mentioned algebraic isomorphism shows that the spaces (Wμ(k)/Pk–, · _W(k)

μ/Pk–) and (W

(k)

μ,k(D),b_W(k)

μ (·)) are isometrically isomorphic, that is,

W(k)

μ /Pk–∼=Wμ(k,k)(D). So, they can be identiﬁed, and we can regard it to be the same if we sayf ∈Wμ(k)/Pk–orf ∈Wμ(k,k).

Let

f_W(k) μ =

k–

j=

f(j)()+sup

z∈D

μ(z)f(k)(z), ()

whereμis a weight andk∈N(for k=  we use the standard convention

l–

j=laj= ,

l∈Z). Then it is easy to see that () deﬁnes a norm on spaceW(k)

μ , and that (Wμ(k), · _W_μ(k))

is a Banach space. The normed space is a natural generalization of the weighted-type, Bloch-type and Zygmund-type spaces (see,e.g., [–]).

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LX→Y=supxX≤L(x)Y. An operatorK:X→Yis called compact if it maps bounded

subsets ofXinto relatively compact subsets ofY.

Essential norm of a bounded operatorL:X→Yis deﬁned by

Le,X→Y:= inf

K∈K(X,Y)L–KX→Y =K∈Kinf(X,Y)xsupX≤

L(x) –K(x)_Y,

that is, as the distance of operatorLto the set of compact operatorsK(X,Y). Let

(Df)(z) =f(z)

be the standard diﬀerentiation operator onH(D). ByDkwe will denote the composition of (exactly)kdiﬀerentiation operators, that is, iff∈H(D), then

Dkf =DD· · ·(D

k-times

f)· · ·.

Let

Ik(f)(z) := z



ζk



· · ·

ζ



f(ζ)dζdζ· · ·dζk, ()

wherek∈Nandf ∈H(D).

It is clear thatDk_I

kf =f for everyf ∈H(D), that is,

DkIk=IdH(D), ()

whereIdXdenotes the identity operator on spaceX. It is also easy to see that

DjIk(f)() = , forj= ,k– , ()

where we regard thatD_{is the identity operator.}

Beside this, by using the Newton-Leibnitz-type formula for holomorphic functionsk

times, we have

IkDk(f)(z) = z



ζk



· · ·

ζ



f(k)(ζ)dζdζ· · ·dζk

=f(z) – k–

j= f(j)_()

j! z

j_, _()

wherek∈Nandf ∈H(D), from which it follows that

IkDkf=f, ()

for everyf ∈H(D)/Pk–, that is,IkDk is the identity operator onH(D)/Pk–, and

conse-quently on its subspaces, such as areW(m)

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Letϕ∈S(D). Then byCϕwe denote the composition operator onH(D), which is deﬁned

byCϕ(f)(z) =f(ϕ(z)).

Letu∈H(D). Then byMg is denoted the multiplication operator on H(D), which is deﬁned byMg(f)(z) =g(z)f(z).

The product of operatorsCϕandMg, that is,

(Mg◦Cϕ)(f)(z) =g(z)f

ϕ(z),

is called the weighted composition operator and is denoted bygCϕ.

These three operators have been considerably studied on various spaces of holomorphic functions (see, for example, [, , , , ] and the references therein).

Letϕ∈S(D),g∈H(D) andk∈N. We deﬁne an operator onH(D) as follows:

C_ϕg_,_k(f)(z) := z



ζk

 · · ·

ζ



f(k)ϕ(ζ)

g(ζ)dζdζ· · ·dζk, ()

forf ∈H(D). Fork=  is obtained the generalized composition operator in [], which was later studied or generalized, for example, in [, –]. For some related operators; see, also [–] and the references therein.

Note that from () it immediately follows that

DjC_ϕg_,_k(f)() = , forj= ,k– . ()

Motivated by [, , ] here we calculate the essential norm of operator () between twokth weighted-type spaces. For some related results see also [, ].

2 Main results

In this section we prove the main results in this paper.

Theorem  Assume thatμandνare weights,k,m∈N,and that the operator L:Wμ(k)→ W(m)

ν is bounded.Then

L

e,Wμ(k)→Wν(m)=Le,Wμ(k)/Pk–→Wν(m). ()

Proof Ifk= , then we regard thatW()

μ /P–=Wμ(), so that () obviously holds. Now

as-sume thatk∈N. For each compact operatorK:W(k)

μ →Wν(m), its restriction onWμ(k)/Pk–,

that is,K:W(k)

μ /Pk–→Wν(m), is also a compact operator, from which along with the

def-inition of the essential norm of an operator, it easily follows that

L_e_,_W(k)

μ /Pk–→Wν(m)≤ Le,Wμ(k)→Wν(m). () LetK:W(k)

μ /Pk–→Wν(m)be a compact operator andf ∈Wμ(k). Then by

K(f)(z) =K

f – k–

j= ajzj

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whereaj=f(j)()/j!,j= ,k– , is deﬁned an extension of operatorKon the whole space

W(k)

μ , that is,K:Wμ(k)→Wν(m), which is obviously a compact operator. Denote the set of

such obtained operatorsKbyK. LetL:W(k)

μ →Wμ(m)be a bounded operator, then the operator

L(f) :=L

_k_–

j= ajzj

=

k–

j= ajL

zj,

where, as above,aj=f(j)()/j!,j= ,k– , mapsWμ(k)intoWμ(m), and is compact, since its

image is a ﬁnite-dimensional space. We have

L_e_,_W(k) μ →Wν(m) = inf

K∈K(W(k) μ ,Wν(m))

L–K_W(k) μ→W(νm)

≤ inf

K∈K(Wμ(k),Wν(m))∩K

L–K–L_W(k) μ →Wν(m)

= inf

K∈K(Wμ(k),Wν(m))∩K

sup

f

Wμ(k)≤

L(f) –K(f) –L(f)_W(m) ν

= inf

K∈K(W(k)

μ /Pk–,W(νm))

sup

f

W(k) μ

≤

L(f) –K

f– k–

j= ajzj

–L

_k_–

j= ajzj

W(m) ν

= inf

K∈K(Wμ(k)/Pk–,W(νm))

sup

f

W(k) μ ≤



(L–K)

f – k–

j= ajzj

W(m) ν

≤ inf

K∈K(Wμ(k)/Pk–,Wν(m))

sup

{g∈Wμ(k)/Pk–:g

Wμ(k) ≤}

(L–K)(g)_W(m) ν

= inf

K∈K(W(k)

μ /Pk–,W(νm))

L–K_W(k)

μ /Pk–→Wν(m)=Le,Wμ(k)/Pk–→W(νm). () From () and (), equality () follows.

Theorem  Assume thatμandν are weights,k,l,m∈N,m≥k,and that the operator C_ϕg_,_k:W(m+l–)

μ →Wν(m)is bounded.Then

Cϕg,ke,Wμ(m+l–)→Wν(m)=gCϕe,Wμ(m+l–k–)→Wν(m–k). ()

Proof First we prove the following inequality:

C_ϕg_,_k_e_,_W(m+l–)

μ →Wν(m)≤ gCϕe,Wμ(m+l–k–)→Wν(m–k). () We show that

C_ϕg_,_k_e_,_W(m+l–)

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which by Theorem  is equivalent to () (recall that whenm=kandl= , we naturally regard thatW(m+l–k–)

μ /Pm+l–k–=Wμ()).

Assume thatf ∈W(m+l–k–)

μ /Pm+l–k–. Then, since

Ikf_W(m+l–)

μ =fWμ(m+l–k–)< +∞ () and

Dj(Ikf)() = , forj= ,m+l– , ()

it follows thatIkf∈Wμ(m+l–)/Pm+l–, that is, operatorIkmaps the spaceWμ(m+l–k–)/Pm+l–k–

intoW(m+l–)

μ /Pm+l–. Further, it is clear thatCϕg,k:Wμ(m+l–)/Pm+l–→Wν(m) is bounded,

and since for everyh∈W(m) ν ,

b_W(m–k) ν

Dkh=b_W(m)

ν (h) <∞, () we see thatDk_maps_W(m)

ν intoWν(m–k). Moreover, we have

Dkh_W(m–k)

ν ≤ hWν(m), ()

where the strict inequality can occur here. Hence, we see that the operator

DkCgϕ,kIk=gCϕ ()

maps the spaceW(m+l–k–)

μ /Pm+l–k– intoWν(m–k), and from (), () and the

bounded-ness of the operator C_ϕg_,_k:W(m+l–)

μ /Pm+l– →Wν(m), it follows that the operator gCϕ : W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)is also bounded.

Due to () and (), we have

IkDkCϕg,kIkDkf =Cgϕ,kf, ()

for everyf∈W(m+l–)

μ /Pm+l–. Indeed, sincem≥kandl∈Nwe havem+l– ≥k, so by ()

we haveIkDkf =f, and further from () we getCϕg,kf ∈Wμ(m)/Pk–. By another application

of () is obtained (). LetK:W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)be a compact operator and

K:=IkKDk. ()

Then the operator maps the spaceWμ(m+l–)/Pm+l– intoWν(m) and is compact, since the

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Hence, by using ()-() and some simple estimates, we have

_Cg

ϕ,kf –K f_W(m) ν =

_I_k_Dk_Cg

ϕ,kIkDkf –IkKDkf_W(m) ν

=Ik

DkC_ϕg_,_kIk–K

Dkf_W(m) ν

=(gCϕ–K)Dkf_W(m–k) ν

≤ gCϕ–K_W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)

Dkf_W(m+l–k–) μ

≤ gCϕ–K_W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)fWμ(m+l–). () By taking the supremum in () over the unit ball in W(m+l–)

μ /Pm+l–, and then

tak-ing the inﬁmum in such obtained inequality over the set of all compact operators K :

W(m+l–k–)

μ /Pm+l–k–→Wν(m–k), we get ().

Now we prove the following inequality:

gCϕ_e_,_W(m+l–k–)

μ →Wν(m–k)≤C g

ϕ,ke,W(m+l–)

μ →Wν(m). ()

To do this ﬁrst note that since () holds for every f ∈Wμ(m+l–k–), the operatorIk :

W(m+l–k–)

μ →Wμ(m+l–)is bounded. From this, sinceC

g

ϕ,k:Wμ(m+l–)→Wν(m) is bounded

by the assumption, andDk:Wμ(m)→Wν(m–k)is also bounded due to the inequality in (),

we see that the operatorDk_Cg

ϕ,kIk=gCϕ:Wμ(m+l–k–)→Wν(m–k)is bounded.

Note also that

C_ϕg_,_k=IkgCϕDk, ()

whereDk_:_W(m+l–)

μ →Wμ(m+l–k–)andIk:Wν(m–k)→Wν(m).

LetK:W(m+l–)

μ →Wν(m)be a compact operator. Then the operator DkKIk:Wμ(m+l–k–)→Wν(m–k)

is compact too.

Using this facts and (), it follows that

gCϕf –DkKIkf_W(m–k) ν =

DkIkgCϕDkIkf –DkKIkf_W(m–k) ν

=DkC_ϕg_,_k–KIkf_W(m–k) ν

≤Cg_ϕ_,_k–KIkf_W(m) ν

≤Cg_ϕ_,_k–K_W(m+l–)

μ →Wν(m)IkfW(μm+l–)

≤Cg_ϕ_,_k–K_W(m+l–)

μ →Wν(m)fWμ(m+l–k–). () By taking the supremum in () over the unit ball inW(m+l–k–)

μ , and then taking the

inﬁmum in such obtained inequality over the set of all compact operatorsK:W(m+l–)

μ →

W(m)

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Before we formulate our next results, we want to say that their proofs are related to the one of Theorem , but we will give all the diﬀerences for the completeness.

Theorem  Assume thatμandν are weights,k,l,m∈N,m≥k,and that the operator C_ϕg_,_k:W(m)

μ →Wν(m+l–)is bounded.Then

C_ϕg_,_k_e_,_W(m)

μ →Wν(m+l–)=gCϕe,Wμ(m–k)→Wν(m+l–k–). ()

Proof First we prove

C_ϕg_,_k

e,Wμ(m)→Wν(m+l–)≤ gCϕe,Wμ(m–k)→Wν(m+l–k–). () We show that

C_ϕg_,_k_e_,_W(m)

μ /Pm–→Wν(m+l–)≤ gCϕe,Wμ(m–k)/Pm–k–→Wν(m+l–k–), () which by Theorem  is equivalent to ().

Assume thatf ∈W(m–k)

μ /Pm–k–. Then, since

Ikf_W(m)

μ =fWμ(m–k)<∞ ()

and

Dj(Ikf)() = , forj= ,m– , ()

it follows that Ikf ∈ Wμ(m)/Pm–, that is, operator Ik maps space Wμ(m–k)/Pm–k– into W(m)

μ /Pm–. Further, it is clear that C

g

ϕ,k:Wμ(m)/Pm–→Wν(m+l–) is bounded, and since

for everyh∈W(m+l–)

ν ,

b_W(m+l–k–) μ

Dkh=b_W(m+l–)

μ (h) < +∞ () we see thatDk:Wν(m+l–)→Wν(m+l–k–). Moreover, we have

Dkh_W(m+l–k–)

μ ≤ hW(μm+l–). ()

Hence, we see that the operator () mapsW(m–k)

μ /Pm–k–toW(m+l–k–), and from (),

() and the boundedness ofC_ϕg_,_k:W(m)

μ /Pm–→Wν(m+l–)it follows that the operatorgCϕ: W(m–k)

μ /Pm–k–→W(m+l–k–)is also bounded. Beside this, sincem≥k, we see that ()

holds for everyf ∈W(m) μ /Pm–.

LetK:W(m–k)

μ /Pm–k–→Wν(m+l–k–)be a compact operator. Then the operator

K:=IkKDk:Wμ(m)/Pm–→Wν(m+l–)

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From this, () and (), we have C_ϕg_,_kf –K f_W(m+l–)

ν =

IkDkCϕg,kIkDkf –IkKDkf_W(m+l–) ν

=Ik

DkC_ϕg_,_kIk–K

Dkf_W(m+l–) ν

=(gCϕ–K)Dkf_W(m+l–k–) ν

≤ gCϕ–K_W(m–k)

μ /Pm–k–→W(m+l–k–)

Dkf_W(m–k) μ /Pm–k–

≤ gCϕ–K_W(m–k)

μ /Pm–k–→W(m+l–k–)fWμ(m). () By taking the supremum in () over the unit ball in W(m)

μ /Pm–, and then taking

the inﬁmum in such obtained inequality over the set of all compact operators K :

W(m–k)

μ /Pm–k–→Wν(m+l–k–), we get ().

Now we prove that

gCϕ_e_,_W(m–k)

μ →Wν(m+l–k–)≤

Cg_ϕ_,_k_e_,_W(m)

μ →Wν(m+l–). ()

Since () holds for everyf ∈Wμ(m–k), we see that the operatorIk:Wμ(m–k)→Wμ(m) is

bounded. From this, since C_ϕg_,_k:W(m)

μ →Wν(m+l–) is bounded by the assumption, and Dk_:_W(m+l–)

ν →Wν(m+l–k–) is also bounded due to the inequality in (), we see that

the operatorDk_Cg

ϕ,kIk=gCϕ:Wμ(m–k)→Wν(m+l–k–)is bounded. Note also that () holds,

whereDk_:_W(m)

μ →Wν(m–k)andIk:Wμ(m+l–k–)→Wν(m+l–).

LetK:W(m)

μ →Wν(m+l–)be a compact operator. Then the operator DkKIk:Wμ(m–k)→Wν(m+l–k–)

is compact.

Using this fact along with () and (), we have

gCϕf –DkKIkf_W(m+l–k–)

ν =

Dk_I

kgCϕDkIkf –DkKIkf_W(m+l–k–) ν

=DkCg_ϕ_,_k–KIkf_W(m+l–k–) ν

≤Cgϕ,k–K

Ikf_W(m+l–) ν

≤Cg_ϕ_,_k–K_W(m)

μ →Wν(m+l–)IkfWμ(m)

≤Cg_ϕ_,_k–K_W(m)

μ →Wν(m+l–)fW(μm–k). () By taking the supremum in () over the unit ball inW(m–k)

μ , and then taking the inﬁmum

in such obtained inequality over the set of all compact operatorsK:W(m)

μ →Wν(m+l–), we

get (). From () and () is directly obtained (), as desired.

Theorem  Assume thatμandν are weights,k,l,m∈N,m≥k,and that the operator C_ϕg_,_k:W(m+l–)

μ /Pm+l–→Wν(m)/Pm–is bounded.Then

C_ϕg_,_k_e_,_W(m+l–)

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Proof First we prove the following inequality:

Cϕg,ke,Wμ(m+l–)/Pm+l–→Wν(m)/Pm–≤ gCϕe,Wμ(m+l–k–)/Pm+l–k–→Wν(m–k)/Pm–k–. ()

Assume thatf ∈W(m+l–k–)

μ /Pm+l–k–. Then, since () and () hold, we see that the

op-eratorIkmaps the spaceWμ(m+l–k–)/Pm+l–k–intoWμ(m+l–)/Pm+l–. Further, by the

assump-tionCgϕ,k:Wμ(m+l–)/Pm+l–→Wν(m)/Pm–is bounded, and since for everyh∈Wν(m)/Pm–,

() holds and

DjDkh() = , j= ,m–k– , ()

we haveDk_:_W(m)

ν /Pm–→Wν(m–k)/Pm–k–. Moreover, we see that () holds.

Hence, we see that the operator () maps the space Wμ(m+l–k–)/Pm+l–k– into W(m–k)

ν /Pm–k–, and from (), () and the boundedness of the operator Cgϕ,k :

W(m+l–)

μ /Pm+l–→Wν(m)/Pm–, it follows thatgCϕ:Wμ(m+l–k–)/Pm+l–k–→Wν(m–k)/Pm–k–

is also bounded. Sincem+l– ≥k, we see that () holds for everyf ∈W(m+l–) μ /Pm+l–.

LetK:W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)/Pm–k–be a compact operator andKbe deﬁned

as in (), whereDk_{maps the space}_W(m+l–)

μ /Pm+l–intoWμ(m+l–k–)/Pm+l–k–, andIkmaps the spaceW(m–k)

ν /Pm–k–intoWν(m)/Pm–. Then,K maps the spaceWμ(m+l–)/Pm+l– into W(m)

ν /Pm–and is compact.

From this and (), similar to (), is obtained

Cϕg,kf –K f_W(m)

ν ≤ gCϕ–KWμ(m+l–k–)/Pm+l–k–→Wν(m–k)/Pm–k–fW(μm+l–). () By taking the supremum in () over the unit ball in W(m+l–)

μ /Pm+l–, and then

W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)/Pm–k–, we get ().

Now we prove that

gCϕ_e_,_W(m+l–k–)

μ /Pm+l–k–→Wν(m–k)/Pm–k–≤ Cg_ϕ_,_k

e,Wμ(m+l–)/Pm+l–→Wν(m)/Pm–. ()

Since () holds for every f ∈ W(m+l–k–)

μ /Pm+l–k–, the operator Ik : Wμ(m+l–k–)/

Pm+l–k– → Wμ(m+l–)/Pm+l– is bounded. From this, since Cϕg,k : Wμ(m+l–)/Pm+l– → W(m)

ν /Pm– is bounded, and since Dk : Wμ(m)/Pm– →Wν(m–k)/Pm–k– is also bounded

due to () and (), we see that the operatorDkCgϕ,kIk=gCϕ:Wμ(m+l–k–)/Pm+l–k– → W(m–k)

ν /Pm–k– is bounded. Beside this, note that Cgϕ,k =IkgCϕDk :Wμ(m+l–)/Pm+l– → W(m)

ν /Pm–, whereDk:Wμ(m+l–)/Pm+l–→Wμ(m+l–k–)/Pm+l–k–andIk:Wν(m–k)/Pm–k–→ W(m)

ν /Pm–.

LetK:W(m+l–)

μ /Pm+l–→Wν(m)/Pm–be a compact operator. Then the operator DkKIk:Wμ(m+l–k–)/Pm+l–k–→Wν(m–k)/Pm–k–

is compact too.

Hence, as in () we have

gCϕf–DkKIkf_W(m–k) ν ≤C

g

ϕ,k–K_W(m+l–)

(11)

By taking the supremum in () over the unit ball in W(m+l–k–)

μ /Pm+l–k–, and then

taking the inﬁmum in such obtained inequality over the set of all compact operators

K:W(m+l–)

μ /Pm+l–→Wν(m)/Pm–, we get (). From () and () is obtained (). Theorem  Assume thatμandνare weights,k,l,m∈N,m≥k,and that the operator C_ϕg_,_k:W(m)

μ /Pm–→Wν(m+l–)/Pm+l–is bounded.Then

_Cg

ϕ,ke,Wμ(m)/Pm–→Wν(m+l–)/Pm+l–=gCϕe,Wμ(m–k)/Pm–k–→Wν(m+l–k–)/Pm+l–k–. ()

Proof First we prove that

Cϕg,ke,Wμ(m)/Pm–→Wν(m+l–)/Pm+l–≤ gCϕe,Wμ(m–k)/Pm–k–→Wν(m+l–k–)/Pm+l–k–. ()

Assume thatf ∈Wμ(m–k)/Pm–k–. Recall that, since () and () hold, the operatorIk boundedly maps spaceW(m–k)

μ /Pm–k–intoWμ(m)/Pm–. Since equality () holds for every h∈W(m+l–)

ν /Pm+l–and

DjDkh() = , ,m+l–k– , ()

we see thatDk_:_W(m+l–)

ν /Pm+l–→Wν(m+l–k–)/Pm+l–k–. Moreover, we see that () holds.

Using these two facts and the boundedness of the operator C_ϕg_,_k : W(m)

μ /Pm– → W(m+l–)

ν /Pm+l–, we see that the operator () maps the spaceWμ(m–k)/Pm–k–toW(m+l–k–)/

Pm+l–k–, and from (), () and the boundedness of the operatorCϕg,k:Wμ(m)/Pm–→ W(m+l–)

ν /Pm+l–, it follows that the operatorgCϕ:Wμ(m–k)/Pm–k–→W(m+l–k–)/Pm+l–k–

is also bounded. We also see that () holds for everyf ∈W(m) μ /Pm–.

LetK:W(m–k)

μ /Pm–k–→Wν(m+l–k–)/Pm+l–k–be a compact operator. Then the operator

K:=IkKDk:Wμ(m)/Pm–→Wν(m+l–)/Pm+l–

is compact.

Hence, as in () we have

Cϕg,kf –K f_W(m+l–)

ν ≤ gCϕ–KWμ(m–k)/Pm–k–→W(m+l–k–)/Pm+l–k–fWμ(m). () By taking the supremum in () over the unit ball in Wμ(m)/Pm–, and then taking

the inﬁmum in such obtained inequality over the set of all compact operators K :

W(m–k)

μ /Pm–k–→Wν(m+l–k–)/Pm+l–k–, we get ().

Now we prove that

gCϕ_e_,_W(m–k)

μ /Pm–k–→Wν(m+l–k–)/Pm+l–k–≤ Cg_ϕ_,_k

e,W(m)

μ /Pm–→Wν(m+l–)/Pm+l–. ()

Since () holds for everyf ∈W(m–k)

μ /Pm–k–and by using (), we see that the

oper-atorIk:Wμ(m–k)/Pm–k–→Wμ(m)/Pm–is bounded. From this, sinceC

g

ϕ,k:Wμ(m)/Pm–→ W(m+l–)

ν /Pm+l– is bounded by the assumption, andDk:Wν(m+l–)/Pm+l–→Wν(m+l–k–)/

Pm+l–k– is also bounded due to the inequality in () and (), we see that the

op-erator Dk_Cg

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note also thatC_ϕg_,_k=IkgCϕDk:Wμ(m)/Pm–→Wν(m+l–)/Pm+l–, whereDk:Wμ(m)/Pm–→ W(m–k)

ν /Pm–k–andIk:Wμ(m+l–k–)/Pm+l–k–→Wν(m+l–)/Pm+l–.

LetK:W(m)

μ /Pm–→Wν(m+l–)/Pm+l–be a compact operator. Then the operator DkKIk:Wμ(m–k)/Pm–k–→Wν(m+l–k–)/Pm+l–k–

is also compact.

Hence, as in () we have

gCϕf–DkKIkf_W(m+l–k–)

ν ≤C

g

ϕ,k–K_W(m)

μ /Pm–→Wν(m+l–)/Pm+l–fWμ(m–k). () By taking the supremum in () over the unit ball in W(m–k)

μ /Pm–k–, and then

W(m)

μ /Pm–→Wν(m+l–)/Pm+l–, we get (). From () and () is directly obtained (),

ﬁnishing the proof of the theorem.

Competing interests

The author declares that he has no competing interests.

Authors’ contributions

The author has contributed solely to the writing of this paper. He read and approved the manuscript.

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Received: 25 June 2017 Accepted: 3 August 2017

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