research article isomorphisms on weighed banach spaces of...

Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 178460 6 pageshttpdxdoiorg1011552013178460

Research ArticleIsomorphisms on Weighed Banach Spaces of Harmonic andHolomorphic Functions

Enrique Jordaacute and Ana Mariacutea Zarco

Departamento de Matematica Aplicada E Politecnica Superior de Alcoy Universidad Politecnica de ValenciaPlaza Ferrandiz y Carbonell 2 03801 Alcoy Spain

Correspondence should be addressed to Enrique Jorda ejordamatupves

Received 30 May 2013 Accepted 3 August 2013

Academic Editor Miguel Martin

Copyright copy 2013 E Jorda and A M Zarco This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

For an arbitrary open subset119880sub R119889 or119880sube C119889 and a continuous function V 119880 rarr ]0infin[we show that the space ℎV0 (119880) of weighedharmonic functions is almost isometric to a (closed) subspace of 1198880 thus extending a theorem due to Bonet and Wolf for spacesof holomorphic functions 119867V0

(119880) on open sets 119880sub C119889 Inspired by recent work of Boyd and Rueda we characterize in terms ofthe extremal points of the dual of ℎV0 (119880) when ℎV0

(119880) is isometric to a subspace of 1198880 Some geometric conditions on an open set119880sube C119889 and convexity conditions on a weight V on 119880 are given to ensure that neither119867V0

(119880) nor ℎV0 (119880) are rotund

1 Introduction Notation and Preliminaries

Let 119880 sube R119889 or 119880 sube C119889 be a nonempty open set A weighton 119880 is a function V 119880 rarr R which is strictly positivecontinuous and bounded Let ℎ(119880) denote the space ofcomplex valued harmonic functions on 119880 For a weight V theweighed Banach spaces of harmonic functions with weight Vare defined by

ℎV (119880) = 119891 isin ℎ (119880) 1003817100381710038171003817119891

1003817100381710038171003817V = sup119911isin119866

V (119911) 1003816100381610038161003816119891 (119911)1003816100381610038161003816 lt infin

ℎV0(119880) = 119891 isin ℎ (119880) V119891 vanishes at infinity on 119880

(1)

A function 119892 119880 rarr C is said to vanish at infinity on119880 iffor each 120576 gt 0 there exists a compact subset 119870 sub 119880 such that|119892(119909)| le 120576 for each 119909 isin 119880 119870

If 119880 sube C119889 then the weighed Banach spaces of holomor-phic functions119867V(119880) and119867V0

(119880) are defined in the samewayThe unit disc and the unit circle in C are denoted by D andΓ respectively 119880 is said to be balanced if 120582119911 isin 119880 for each119911 isin 119880 and each 120582 isin D A weight V on a balanced open subset119880 sube C119899 is said to be radial whenever V(120582119911) = V(119911) for each120582 isin Γ

Weighed Banach spaces of holomorphic functions withweighed supremum norms and composition operatorsbetween them have been studied by Bierstedt et al Bonetet al Contreras and Hernandez-Dıaz Garcıa et al MontesRodrıguez and others see [1ndash8] and the references therein

Spaces of harmonic functions have been investigated byShields and Williams [9] in connection with the growth ofthe harmonic conjugate of a function In [10] they provedresults of duality for weighed spaces of harmonic functionson the open unit disk Lusky also considers weighed spaces ofharmonic functions in [11 12] where the isomorphism classesin the case of radial weights on the disk are determined In[13] the authors have studied associated weights in weighedBanach spaces of harmonic functions and composition oper-ators with holomorphic symbol between weighed spaces ofpluriharmonic functions

In [14ndash16] Boyd and Rueda investigate the geometry ofℎV0

(119880) and 119867V0(119880) where 119880 is an open set of the complexplaneMoreover they examined the extreme points of ℎV0(119880)

lowast

and of119867V0(119880)lowast for 119880 sube C119889

In this work we prove in the second section that for anyweight V on an open subset 119880 of R119889 the weighed Banachspace ℎV0

(119880) of harmonic functions with V119891 vanishing atinfinity on 119880 is always isomorphic to a subspace of 1198880 thus

2 Journal of Function Spaces and Applications

extending a result due toBonet andWolf [17] for holomorphicfunctions Further given 0 lt 120576 lt 1 one can get a linearmapping 119879 ℎV0

(119880) rarr 1198880 such that (1 minus 120576)119891V le

119879(119891) le 119891V for each 119891 isin ℎV0(119880) that is ℎV0(119880) is

almost isometric to a subspace of 1198880 We show that for radialweights in balanced open subsets the isomorphism cannotbe isometric generalizing [14 Theorem 16 Corollary 17]to spaces of harmonic functions defined on not necessarilybounded setsWe also show in this section that if the unit ballof ℎV0(119880)

lowast has a big quantity of extremal points then ℎV0(119880)

is not isometric to any subspace of 1198880In the third sectionwemainly focus ourselves onweighed

spaces of holomorphic and harmonic functions defined onbalanced subsets 119880 sube C119889 and with radial weights We studythe extremal points of the unit sphere of119867V0

(119880)lowast in the way

that Boyd and Rueda initiated in [14 15] More precisely wegive conditions on 119880 and V under which one can find in theunit sphere of119867V0

(119880)lowast a big quantity of extremal points We

show that these conditions imply that119867V0(119880) is not rotund

Our notation for functional analysis and Banach spacetheory is standard We refer to [18] For a Banach space 119883

we denote by 119861119883 and 119878119883 its unit ball and its unit sphererespectively A (real or complex) Banach space 119883 is said tobe rotund whenever (119909 + 119910)2 lt 1 for 119909 = 119910 = 1 and119909 = 119910 A vector 119909 in the unit sphere 119878119883 of a Banach space 119883is called extremal if there are not 119910 119911 isin 119878119883 such that 119910 = 119911

and 119909 = (12)(119910 + 119911) and it is called exposed if there is119909lowast

isin 119883lowast such that 119909lowast = 1 119909lowast(119909) = 1 and 119909

lowast(119910) = 1

for each 119910 isin 119878119883 119909 If 119883 is complex this last condition isequivalent to 119909

lowast(119909) = 1 and Re 119909lowast(119910) lt 1 for 119910 isin 119861119883 119909

where Re 119911 denotes the real part of a complex number 119911 If119909 isexposed then 119909 is also extremal If 119883 is a dual Banach spaceand the functional exposing 119909 can be taken in the predualthen 119909 is called weaklowastexposed For 119909 119910 isin R119889 (or C119889) wedenote by ⟨119909 119910⟩ its canonical scalar product The Euclideannorm is denoted by sdot When we consider other norms on afinite dimensional space they are denoted by | sdot | making anabuse of notation because the modulus of a complex numberis denoted also by | sdot |

For a Hausdorff locally compact topological space 119876 wedenote by 119876 = 119876 cup infin its Alexandroff compactificationThe boundary of a subset 119880 of a topological space is denotedby 120597119880 If 119880 sube R119889 is an open subset then the space ℎV0(119880) isformed by the harmonic functions on 119880 such that V119891 can beextended continuously to by (V119891)(infin) = 0 For an open set119880 sube R119889 the isometry between ℎV0(119880) and a subspace of119862()by means of 119891 997891rarr V119891 permits to use the same argument as in[14 Proposition 1] to show that the extreme points of ℎV0(119880)

lowast

are contained in the set

120582V (119911) 120575119911 120582 isin Γ 119911 isin 119880 (2)

where 120575119911 denotes the evaluation at 119911 The following defini-tions are due to Boyd and Rueda [14ndash16] The harmonic V-boundary is defined as follows

119887V (119880) = 119911 isin 119880 V (119911) 120575119911

is extremal in the unit sphere of ℎV0(119880)lowast

(3)

The set of harmonic V-peak points is defined as

119901V (119880) = 119911 isin 119880 V (119911) 120575119911

is weaklowastexposed in the unit sphere of ℎV0(119880)lowast

(4)

Remark 1 The same argument as in [15 Theorem 6] showsthat to check that a functional of the form V(1199090)1205751199090 is weak

lowast

exposed it is enough to show that there exists 119891 in the unitball of ℎV0(119880) such that V(1199090)119891(1199090) = 1 V(119909) Re119891(119909) lt 1 foreach 119909 isin 119880 1199090

If 119880 sube C119889 then the holomorphic V-boundary and theset of holomorphic V-peak points are defined analogouslyand they are denoted by 119861V(119880) and 119875V(119880) respectively Since119867V0

(119880) sub ℎV0(119880) in this case we have

119875V (119880) sube 119861V (119880) sube 119880 (5)

and also

119875V (119880) sube 119901V (119880) sube 119887V (119880) sube 119880 (6)

We finish this section with this lemma also due to Boydand Rueda

Lemma 2 (see [14 15]) Let 119880 sube R119889(119880 sube C119889) be a nonemptyopen set and let V be a weight which vanishes at infinity on119880

(a) For any 120582 isin Γ and 119911 isin 119880 the functional 120582V(119911)120575119911 isextremal in the unit sphere of ℎV0(119880)

lowast (is extremal inthe unit sphere of119867V0

(119880)lowast) if and only if 119911 isin 119887V(119880)(119911 isin

119861V(119880))(b) If 119880 sube C119889 is balanced V is radial and 120582 isin Γ then 120582119911 isin

119887V(119880) for each 119911 isin 119887V(119880)(120582119911 isin 119861V(119880) for each 119911 isin

119861V(119880))

2 Isomorphisms on ℎV0(119880)

We extend below the main theorem of [17] to weighed spacesof harmonic functions The proof is analogous to the proofgiven by Bonet and Wolf One only has to take in accountthat there are Cauchy type inequalities which are valid for thederivatives of harmonic functions [19 24] We include it forthe sake of completeness The argument was inspired by oneused by Kalton and Werner [20 Corollary 49] to show thatthe little Bloch space 1198610 embeds almost isometrically in 1198880Lusky showed in [21] that whenever V is a radial weight onDthen ℎV0

(D) is isomorphic to a subspace of 1198880

Theorem 3 Let 119880 sub R119889 be a nonempty open set let V be aweight on 119880 and let 120576 gt 0 There exists an isomorphism 119879

ℎV0(119880) rarr 1198880 such that (1 minus 120576)119891V le 119879(119891) le 119891V for each

119891 isin ℎV0(119880)

Proof Let (119870119895)119895 be a fundamental sequence of compactsubsets of 119880 and let 120576 gt 0 We claim that there exists asequence of pairwise disjoint finite subsets (119865119895)119895 such that

Journal of Function Spaces and Applications 3

119865119895 sub 119860119895 = 119870119895 119870∘

119895minus1and satisfying that for each 119891 isin ℎV0

with 119891V = 1

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 + sup

119910isin119865119895

V (119910) 1003816100381610038161003816119891 (119910)1003816100381610038161003816 (7)

This implies that if we denote 119865 = cup119895119865119895 then we have

sup119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817V le (1 + 120576) sup

119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 (8)

and consequently if we write 119865 as a sequence (119908119895)119895 then thelinear mapping 119879 ℎV0

(119880) rarr 1198880 119891 997891rarr (V(119908119895)119891(119908119895))119895 isinjective it has closed range and norm no greater than oneand its inverse 119879

minus1 defined on a closed subspace of 1198880 hasnorm less than or equal to 1 + 120576 Since this is true for each120576 gt 0 this is equivalent to our statement

We proceed to show the claim For each 119895 isin N we define

119872119895 = sup119909isin119870119895

V (119909) 119886119895 = min(12119889 (119870119895R

119873 119870∘

119895+1) 1)

(9)

and we select 119909(119895) isin 119870119895 such that V(119909(119895)) = min119909isin119870119895V(119909) Foreach 119891 isin ℎV0

(119880) with 119891V = 1 and each 119895 isin N we have

sup119909isin119870119895

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

1

V (119909(119895)) (10)

If 119909 isin 119870119895 and 120577 isin 119863(119909 119886119895) (here 119863(119909 119886119895) denotes the closedball with center 119909 and radius 119886119895) then from the definitionof 119886119895 we deduce that 120577 isin 119870119895+1 and hence together with theprevious inequality

1003816100381610038161003816119891 (120577)1003816100381610038161003816 le

1

V (119909(119895+1)) (11)

By Cauchyrsquos inequalities for harmonic functions given in [1924] there exists a positive number119862 (independent of 119909) suchthat for each 120577 isin 119863(119909 119886119895) the 119894th derivative of 119891 satisfies

1003816100381610038161003816119863119894119891 (120577)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119894 = 1 2 119889 (12)

By compactness we can get a finite subset 119865119895 of 119860119895 such that

119860119895 sub ⋃

119909isin119865119895

1199091015840isin 119880

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816lt 120575119895

10038161003816100381610038161003816V (1199091015840) minus V (119909)

10038161003816100381610038161003816lt 120575119895

(13)

where 120575119895 has been chosen satisfying 120575119895 lt 119886119895 and

(1

V (119909(119895))+

119872119895+1119889119862

119886119895V (119909(119895+1))

) 120575119895 lt 120576 (14)

Consequently for each 119909 isin 119860119895 there exists 119908 isin 119865119895 with|119908 minus 119909| lt 120575119895 and |V(119908) minus V(119909)| lt 120575119895 On the other hand

119863(119909 120575119895) sub 119863(119909 119886119895) sub 119870119895+1 for each 119909 = (1199091 1199092 119909119889) isin

119860119895 We write 119908 = (1199081 1199082 119908119889) to get

1003816100381610038161003816119891 (119909)1003816100381610038161003816 =

1003816100381610038161003816119891 (1199091 119909119889)1003816100381610038161003816

le1003816100381610038161003816119891 (1199091 119909119889) minus 119891 (1199081 1199092 119909119889)

1003816100381610038161003816

+1003816100381610038161003816119891 (1199081 1199092 119909119889) minus 119891 (1199081 1199082 119909119889)

1003816100381610038161003816

+ sdot sdot sdot +1003816100381610038161003816119891 (1199081 1199082 119908119889)

1003816100381610038161003816

(15)

We apply the mean value theorem to each difference and (12)to get

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119889 |119909 minus 119908| +1003816100381610038161003816119891 (119908)

1003816100381610038161003816

le

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816

(16)

Now since 119908 isin 119870119895+1

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le |V (119909) minus V (119908)| 1003816100381610038161003816119891 (119909)

1003816100381610038161003816 + V (119908) 1003816100381610038161003816119891 (119909)1003816100381610038161003816

le 120575119895

1

V (119909(119895))+ V (119908)(

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816)

le

120575119895

V (119909(119895))+ 119872119895+1

119889120575119895119862

119886119895V (119909(119895+1))

+ V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

le 120576 + V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

(17)

Therefore

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 (18)

Again as in [17] since each infinite dimensional subspaceof 1198880 contains a complemented subspace isomorphic to 1198880 thefollowing result follows

Corollary 4 There are not infinite dimensional subspaces ofℎV0

(119880) which are reflexive

In order to get examples where the isomorphism abovecannot be an isometry we will consider a condition whichis inspired by the results of Boyd and Rueda [14 Theorem16 Corollary 17] for weighed spaces defined on boundedopen sets of C119889 First we observe that for any weight V forwhich ℎV0

(119880) contains the polynomials it happens that thefunctionals 120575119911 119911 isin 119880 sube ℎV0

(119880)lowast are linearly independent

If 119880 is bounded this is the case for any weight V which tendsto zero at the boundary If 119880 = R119889 then the condition issatisfied when V = 119892(|119909|) with | sdot | being a norm in R119889 and119892 [0infin[rarr ]0infin[ being a rapidly decreasing continuousfunction that is with lim119905rarrinfin119905

119899119892(119905) = 0 for any 119899 isin N First

we extend easily [14 Lemma 10] to unbounded domains


Lemma 5 Let 119880 sub R119889 be a nonempty open set and let V be acontinuous strictly positive weight which vanishes at infinity on119880 If ℎV0(119880) contains the polynomials then the map 120583 119880 rarr

(ℎV0(119880)lowast 119908lowast) and 119911 997891rarr V(119911)120575119911 is a homeomorphism onto its

image

Proof Since the polynomials belong to ℎV0(119880) we get that

120575119911 119911 isin 119880 sub ℎV0(119880)lowast are linearly independent functionals

We consider the map 120583 rarr (ℎV0(119880)lowast 119908lowast) 119911 rarr V(119911)120575119911

if 119911 isin 119880 and 120583(infin) = 0 The very definition of ℎV0(119880)implies that 120583 is continuous and it is injective by the linearindependence of the evaluations By the compactness of it follows that 120583 is a homeomorphism and hence also itsrestriction 120583 to 119880

Proposition 6 Let119880 sub R119889(119880 sub C119889) be a nonempty open setand let V be a weight for which ℎV0

(119880)(119867V0(119880)) contains the

polynomials Assume that 119887V(119880)(119861V(119880)) is not discrete ThenℎV0

(119880)(119867V0(119880)) cannot be isometric to any subspace of 1198880

Proof We give only the proof for the harmonic case Denoteby N = N cup infin the Alexandroff compactification of NLet 119879 ℎV0

(119880) rarr 1198880 be an isometry onto its image Let119865 = 119879(ℎV0

(119880)) sube 1198880 Since 1198880 is the subspace of 119862(N) formedby the functions which vanish at infin the extremal points ofthe unit sphere of 119865lowast are contained in 120582120575119899 120582 isin Γ 119899 isin

N (cf [22 Lemma V86]) Let 119876 119865 rarr ℎV0(119880) be the

inverse isometry of 119879 The transpose linear mapping 119876119905

ℎV0(119880)lowast

rarr 119865lowast is also an isometry and it maps extremal

points to extremal points Thus for each 119911 isin 119887V(119880) thereexists 120582119911 isin Γ and 119899(119911) isin N such that 119876119905(V(119911)120575119911) = 120582119911120575119899(119911)From the linear independence of 120575119911 119911 isin 119880 in ℎ

lowast

V0 we

conclude that 119899(119911) = 119899(119908) for 119911 119908 isin 119880 with 119911 =119908 and then119887V(119880) is countable and 120582119911120575119899(119911) 119911 isin 119887V(119880) is discretefor the weaklowast topology since it is a sequence in the sphereof 119865lowast which is 120590(119865lowast 119865) convergent to 0 But 119876119905 is a weaklowast-weaklowast homeomorphism Hence we conclude from Lemma 5that 119887V(119880) is discrete

Now we apply the set of extremal points of the unit ballof a dual Banach space which is never empty and Lemma 2 toget the following result

Corollary 7 If119880 sube C119889 is a balanced open set and V is a radialweight which vanishes at infinity on119880 then neither119867V0

(119880) norℎV0

(119880) is isometric to a subspace of 1198880

Although we do not have an example of weight V on119880 such that 119887V(119880) is a discrete subset of 119880 we see belowthat we can reformulate the problem of finding an exampleof space ℎV0(119880) which can be isometrically embedded in 1198880

in terms of the possible existence of an open set 119880 sube R119889

and a weight V such that 119887V(119880) is not discrete Of course theanalogue result is also true for the holomorphic case In viewof Lemma 2 this is not possible for radial weights in balanceddomains

Proposition 8 If119880 sube R119889 is an open set and V is a weight on119880such that ℎV0(119880) contains the polynomials and 119887V(119880) is discretethen ℎV0

(119880) embeds isometrically in 1198880

Proof Since the closed unit ball 119863 of ℎV0(119880)lowast is the

weaklowastclosure of the absolutely convex hull of the extremalpoints of the sphere Lemma 2(a) implies that119863 is in fact theclosure of the absolutely convex hull of V(119911)120575119911 119911 isin 119880Thus this implies that 119887V(119880) cannot be finite and since it isdiscrete we have that 119887V(119880) = (119909119899)119899 with 119909119899 tending to infinityon119880 when 119899 goes to infinity We get now that119863 is the weaklowastclosure of the absolutely convex hull of V(119909119899)120575119909119899 119899 isin Nand hence for each 119891 isin ℎV0

(119880) it holds1003817100381710038171003817119891

1003817100381710038171003817V = sup119899isinN

V (119909119899)1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816 (19)

Since 119909119899 tends to infinity on 119880 we have that the linear map

119879 ℎV0(119880) 997888rarr 1198880 119891 997891997888rarr (V (119909119899) 119891 (119909119899))119899 (20)

is an isometry

3 Geometry of Weighed Banach Spaces ofHolomorphic and Harmonic Functions andTheir Duals

In [15 Theorem 29] Boyd and Rueda showed that if 119880 sube

C119889 is a balanced bounded open set and V is a radial weightvanishing at infinity on 119880 such that 119875V(119880) = 119861V(119880)

then 119867V0(119880) is not rotund and then neither ℎV0(119880) is This

condition is trivially satisfied when 119875V(119880) = 119880 To the bestof our knowledge there are not so far concrete examples ofopen subsets 119880 sub C119889 and weights V on 119880 such that 119861V(119880)

119875V(119880) = 0 In the concrete examples of spaces119867V0(119880) given in

[15] where 119861V(119880) is calculated the equality 119861V(119880) = 119875V(119880)

is always satisfied With a similar proof to the one of [15Theorem 29] we present below a new condition for spaces119867V0

(119880) with 119880 balanced and V radial depending only on thesize of 119875V(119880) which ensures that119867V0

(119880) is not rotund

Proposition 9 Let 119880 sube C119889 be the open unit ball for a norm| sdot | in C119889 and let V = 119892(|119909|) for 119892 [0 1] rarr [0infin[ bea nonincreasing continuous function with 119892(1) = 0 If thereexists 1199110 isin 120597119880 such that the set 1199051199110 0 lt 119905 lt 1 sube 119875V(119880) then119867V0

(119880) is not rotund and then neither ℎV0(119880) is

Proof Let 120593 be a linear map on C119889 such that 1205932V = 1 SinceV1205932 is continuous on and vanishes at infinity there exists1199111 isin 119880 1199111 = 0 such that

1 =10038171003817100381710038171003817120593210038171003817100381710038171003817V

= V (1199111) 1205932(1199111) = V (minus1199111) 120593

2(minus1199111) (21)

Since |1199110| = 1 we apply the theorem of Hahn-Banach to get119910 isin C119889 such that ⟨1199110 119910⟩ = 1 and |⟨119911 119910⟩| le |119911| for each 119911 isin

C119889 Let 1199030 = |1199111| We define

ℎ 119880 997888rarr C 119911 997888rarr 1205932(⟨119911 119910⟩

1199030

1199111) (22)


We have

V (11990301199110) ℎ (11990301199110) = 119892 (1199030) 1205932(1199111)

= 119892 (10038161003816100381610038161199111

1003816100381610038161003816) 1205932(1199111) = V (1199111) 120593

2(1199111) = 1

(23)

Since 120593 is linear we also get

V (minus11990301199110) ℎ (minus11990301199110) = 119892 (1199030) 1205932(minus1199111) = V (minus1199111) 120593

2(minus1199111) = 1

(24)

For 119911 isin 119880 arbitrary we use that |11991111199030| = 1 |⟨119911 119910⟩| le |119911| and119892 is not increasing to compute

V (119911) |ℎ (119911)| = 119892 (|119911|) |ℎ (119911)| le 119892 (1003816100381610038161003816⟨119911 119910⟩

1003816100381610038161003816) |ℎ (119911)|

= V(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

1205932(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171003817120593210038171003817100381710038171003817V

= 1

(25)

Thus ℎ is in the unit ball of119867V0(119880) and

V (11990301199110) 12057511990301199110 (ℎ) = V (minus11990301199110) 120575(minus11990301199110) (ℎ) = 1 (26)

Since 11990301199110 is assumed to be a peak point we can find 119891 in theunit ball of119867V0

(119880) different from ℎ such that V(11990301199110)119891(11990301199110) =1 and V(119911) Re(119891(119911)) lt 1 for each 119911 isin 119880 11990301199110 Hence weconclude from

10038171003817100381710038171003817100381710038171003817

119891 + ℎ

2

10038171003817100381710038171003817100381710038171003817V= V (11990301199110)

(119891 (11990301199110) + ℎ (11990301199110))

2= 1 (27)

that119867V0(119880) is not rotund

Let 119892 [0 1] rarr [0infin[ be a function such that 119892(1) =

0 119892 is decreasing 119892 is twice differentiable in ]0 1[ andlog(1119892(119905)) is convex In [15 Corollary 12] the authors showedthat for a weight V defined in the Euclidean unit ball 119880 ofC119889 by V(119909) = 119892(119909) the equality 119875V(119880) = 119880 holds andthen 119867V0

(119880) cannot be rotund by [15 Theorem 29] Also if1198801 sube C119889 and 1198802 sube C119896 are the Euclidean unit balls and V1and V2 satisfy this last condition for certain 1198921 and 1198922 thenV 1198801 times 1198802 rarr]0infin[ V(119909 119910) = V1(119909)V2(119910) also satisfies119875V(119880) = 119880 (cf [15 Proposition 26]) We present belownew examples of spaces 119867V0

(119880) which cannot be rotund Inthe required assumptions besides considering a wider classof open sets we remove from the weight the condition ofdifferentiability given by Boyd and Rueda as we did in [13Theorem 1] to give conditions under which V = V V beingthe associated weight introduced by Bierstedt et al in [23]The same technique permits us to get concrete radial weightsV(119909) = 119892(119909) defined in the Euclidean ball 119880 of R119889 with119889 odd such that the spaces ℎV0(119880) are not isometric to anysubspace of 1198880 If 119889 = 2119896 then the assertion is a consequenceof Corollary 7 since 119880 is the Euclidean unit ball of C119896 and119867V0

(119880) sub ℎV0(119880)

Proposition 10 Let 119892 [0 1] rarr [0infin[ be a continuousdecreasing function such that 119892(1) = 0 and log(1119892) is strictlyconvex in [0 1[

(a) Let119880 sube C119889 be the unit ball for a norm | sdot | and V 119880 rarr

]0infin[ 119909 997891rarr 119892(|119909|) Then 119867V0(119880) is not rotund (and

then neither ℎV0(119880) is)

(b) Let 119880 sube R119889 be the Euclidean unit ball and V(119909) =

119892(119909) for 119909 isin 119880 Then ℎV0(119880) is not isometric to any

subspace of 1198880

Proof Let ℎ(119903) = log(1119892(119903)) 0 le 119903 lt 1 Since ℎ isincreasing and strictly convex for each 0 le 1199030 lt 1 the map

[0 1 [ 1199030 997888rarr] 0infin[ 119903 997891997888rarr(ℎ (119903) minus ℎ (1199030))

(119903 minus 1199030)(28)

is increasing Hence for each 0 le 1199030 lt 1 there exists 1205720 ge 0

which depends on 1199030 such that

ℎ (119903) minus ℎ (1199030) gt 1205720 (119903 minus 1199030) (29)

for all 119903 isin [0 1[1199030 Now we compute

sup0le119903lt1119903 = 1199030

119892 (119903) exp (1205720119903) = exp( sup0lt119903lt1119903 = 1199030

(minusℎ (119903) + 1205720119903))

lt exp( sup0le119903lt1119903 = 1199030

(minus1205720 (119903 minus 1199030)

minus ℎ (1199030) + 1205720119903) )

= 119892 (1199030) exp (12057201199030) (30)

We prove (a) Straszewiczrsquos theorem [24 Theorem 789]together with the Krein-Milman theorem imply that thereexists 1199110 isin 120597119880 which is exposed Let 119910 isin C119889 such that⟨1199110 119910⟩ = 1 and Re(⟨119911 119910⟩) lt |119911| if 119911 isin 119880 1199031199110 0 le 119903 lt 1Let 0 le 1199030 lt 1 be arbitrary and let 1205720 be as above Thefunction 119891 119880 rarr C defined by 1198910(119911) = (exp(1205720 ⟨119911 119910⟩))(119892(1199030) exp(1205720 1199030)) is holomorphic and V(11990301199110)1198910(11990301199110) = 1 Itfollows immediately from (30) that if 0 le 119903 lt 1 119903 = 1199030

V (1199031199110)10038161003816100381610038161198910 (1199031199110)

1003816100381610038161003816 =119892 (119903) exp (1205720119903)119892 (1199030) exp (12057201199030)

lt 1 (31)

If 119911 isin 119880 1199031199110 0 le 119903 lt 1 then we use (30) andRe(⟨119911 119910⟩) lt |119911| to get

V (119911) 10038161003816100381610038161198910 (119911)1003816100381610038161003816 = 119892 (|119911|)

exp (1205720 Re (⟨119911 119910⟩))119892 (1199030) exp (12057201199030)

lt 119892 (|119911|)exp (1205720 |119911|)

119892 (1199030) exp (12057201199030)le 1

(32)

Hence 1198910 peaks V(11990301199110)12057511990301199110 We conclude that 1199031199110 0 le 119903 lt

1 sube 119875V(119880) since 1199030 is arbitrary We apply Proposition 9 to getthat119867V0

(119880) is not rotundTo prove (b) we consider 1199090 isin R119889 with 0 lt 1199090 = 1199030 lt 1

and get 1205720 again as above Let 119860 be an orthogonal transfor-mation such that 119860(1199090) = (1199030 0 0) For 119909 = (1199091 119909119889)


we take the harmonic function ℎ(119909) = exp(12057201199091) cos(12057201199092)and we define1198910(119909) = ℎ∘119860(119909) for 119909 isin 119880 which is harmonicsee [19 Chapter 1] A similar argument as in (a) permits us toconclude that V(1199090)1198910(1199090) = 1 and V(119909)|119891(119909)| lt 1 for each119909 isin 119880 1199090 Hence 119901V(119880) = 119880 and therefore ℎV0(119880) is notisometrically embedded in 1198880 by Proposition 6

Remark 11 (a) An inspection of the proofs of Propositions 9and 10 shows that the statements remain true when 119880 = C119889

and 119892 [0infin[rarr ]0infin[ is a decreasing continuous functionwhich satisfies that log(1119892) is strictly convex and 119892(119905) =

119900(119890minus119886119905

) when 119905 goes to infinity for each 119886 gt 0(b) For any increasing function ℎ [0 1[rarr [0infin[which

is strictly convex and satisfies lim119905rarr1ℎ(119905) = infin (or ℎ

[0infin[rarr ]0infin[ with 119890119886119905 = 119900(119890ℎ(119905)

) when 119905 goes to infinity foreach 119886 gt 0) the function 119892(119905) = 119890

minusℎ(119905) satisfies the hypothesisof Propositions 9 and 10 The examples given in [15 Example13] of weights in the Euclidean unit ball can be obtained usingthis general method

Acknowledgments

The authors are grateful to J Bonet for a lot of ideas anddiscussions during all this work They are also indebted toP Rueda who besides giving them some references providedthem some unpublished work which has inspired a big partof this paper They also thank M Maestre for his carefulreading of the paper and helpful discussions Finally theythank the referees for the careful analysis of the paper andthe important suggestions which they gave us The researchof the first author was supported by MICINN andFEDERProject MTM2010ndash15200 The research of both authors ispartially supported by Programa de Apoyo a la Investigaciony Desarrollo de la UPV PAID-06-12

References

[1] K D Bierstedt J Bonet and A Galbis ldquoWeighted spaces ofholomorphic functions on balanced domainsrdquo The MichiganMathematical Journal vol 40 no 2 pp 271ndash297 1993

[2] J Bonet P Domanski and M Lindstrom ldquoEssential normand weak compactness of composition operators on weightedBanach spaces of analytic functionsrdquo Canadian MathematicalBulletin vol 42 no 2 pp 139ndash148 1999

[3] J Bonet P Domanski and M Lindstrom ldquoWeakly com-pact composition operators on analytic vector-valued functionspacesrdquo Annales Academiaelig Scientiarum Fennicaelig Mathematicavol 26 no 1 pp 233ndash248 2001

[4] J Bonet P Domanski M Lindstrom and J Taskinen ldquoCom-position operators between weighted Banach spaces of analyticfunctionsrdquo Journal of the AustralianMathematical Society SeriesA vol 64 no 1 pp 101ndash118 1998

[5] M D Contreras and A G Hernandez-Diaz ldquoWeighted com-position operators in weighted Banach spaces of analytic func-tionsrdquo Journal of the Australian Mathematical Society Series Avol 69 no 1 pp 41ndash60 2000

[6] D Garcıa M Maestre and P Rueda ldquoWeighted spaces ofholomorphic functions on Banach spacesrdquo StudiaMathematicavol 138 no 1 pp 1ndash24 2000

[7] D Garcıa M Maestre and P Sevilla-Peris ldquoCompositionoperators between weighted spaces of holomorphic functionson Banach spacesrdquo Annales Academiaelig Scientiarum FennicaeligMathematica vol 29 no 1 pp 81ndash98 2004

[8] A Montes-Rodrıguez ldquoWeighted composition operators onweighted Banach spaces of analytic functionsrdquo Journal of theLondon Mathematical Society vol 61 no 3 pp 872ndash884 2000

[9] A L Shields and D L Williams ldquoBounded projections and thegrowth of harmonic conjugates in the unit discrdquo The MichiganMathematical Journal vol 29 no 1 pp 3ndash25 1982

[10] A L Shields and D LWilliams ldquoBounded projections dualityandmultipliers in spaces of harmonic functionsrdquo Journal fur dieReine und Angewandte Mathematik vol 299-300 pp 256ndash2791978

[11] W Lusky ldquoOn weighted spaces of harmonic and holomorphicfunctionsrdquo Journal of the London Mathematical Society vol 51no 2 pp 309ndash320 1995

[12] W Lusky ldquoOn the isomorphism classes of weighted spaces ofharmonic and holomorphic functionsrdquo Studia Mathematicavol 175 no 1 pp 19ndash45 2006

[13] E Jorda andAM Zarco ldquoWeighted Banach spaces of harmonicfunctionsrdquo Revista de la Real Academia de Ciencias ExactasFisicas y Naturales Serie A 2012

[14] C Boyd and P Rueda ldquoThe -boundary of weighted spacesof holomorphic functionsrdquo Annales Academiaelig ScientiarumFennicaelig Mathematica vol 30 no 2 pp 337ndash352 2005

[15] C Boyd and P Rueda ldquoComplete weights and -peak pointsof spaces of weighted holomorphic functionsrdquo Israel Journal ofMathematics vol 155 pp 57ndash80 2006

[16] C Boyd and P Rueda ldquoIsometries of weighted spaces ofharmonic functionsrdquo Potential Analysis vol 29 no 1 pp 37ndash482008

[17] J Bonet and E Wolf ldquoA note on weighted Banach spaces ofholomorphic functionsrdquo Archiv der Mathematik vol 81 no 6pp 650ndash654 2003

[18] M Fabian P Habala P Hajek V Montesinos and V ZizlerBanach SpaceTheory CMS Books inMathematicsOuvrages deMathematiques de la SMC Springer New York NY USA 2011

[19] S Axler P Bourdon andW RameyHarmonic FunctionTheoryvol 137 of Graduate Texts in Mathematics Springer New YorkNY USA 2nd edition 2001

[20] N J Kalton and D Werner ldquoProperty (119872) 119872-ideals andalmost isometric structure of Banach spacesrdquo Journal fur dieReine und Angewandte Mathematik vol 461 pp 137ndash178 1995

[21] W Lusky ldquoOn the structure of 119867V0(119863) and ℎV0

(119863)rdquo Mathema-tische Nachrichten vol 159 pp 279ndash289 1992

[22] N Dunford and J T Schwartz Linear Operators Part IWileyClassics Library Wiley Classics Library John Wiley amp SonsNew York NY USA 1988

[23] K D Bierstedt J Bonet and J Taskinen ldquoAssociated weightsand spaces of holomorphic functionsrdquo StudiaMathematica vol127 no 2 pp 137ndash168 1998

[24] C D Aliprantis and K C Border Infinite Dimensional AnalysisA Hitchhikerrsquos Guide Springer Berlin Germany 3rd edition2006

Submit your manuscripts athttpwwwhindawicom

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Algebra

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Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


extending a result due toBonet andWolf [17] for holomorphicfunctions Further given 0 lt 120576 lt 1 one can get a linearmapping 119879 ℎV0

(119880) rarr 1198880 such that (1 minus 120576)119891V le

119879(119891) le 119891V for each 119891 isin ℎV0(119880) that is ℎV0(119880) is

almost isometric to a subspace of 1198880 We show that for radialweights in balanced open subsets the isomorphism cannotbe isometric generalizing [14 Theorem 16 Corollary 17]to spaces of harmonic functions defined on not necessarilybounded setsWe also show in this section that if the unit ballof ℎV0(119880)

lowast has a big quantity of extremal points then ℎV0(119880)

is not isometric to any subspace of 1198880In the third sectionwemainly focus ourselves onweighed

spaces of holomorphic and harmonic functions defined onbalanced subsets 119880 sube C119889 and with radial weights We studythe extremal points of the unit sphere of119867V0

(119880)lowast in the way

that Boyd and Rueda initiated in [14 15] More precisely wegive conditions on 119880 and V under which one can find in theunit sphere of119867V0

(119880)lowast a big quantity of extremal points We

show that these conditions imply that119867V0(119880) is not rotund

Our notation for functional analysis and Banach spacetheory is standard We refer to [18] For a Banach space 119883

we denote by 119861119883 and 119878119883 its unit ball and its unit sphererespectively A (real or complex) Banach space 119883 is said tobe rotund whenever (119909 + 119910)2 lt 1 for 119909 = 119910 = 1 and119909 = 119910 A vector 119909 in the unit sphere 119878119883 of a Banach space 119883is called extremal if there are not 119910 119911 isin 119878119883 such that 119910 = 119911

and 119909 = (12)(119910 + 119911) and it is called exposed if there is119909lowast

isin 119883lowast such that 119909lowast = 1 119909lowast(119909) = 1 and 119909

lowast(119910) = 1

for each 119910 isin 119878119883 119909 If 119883 is complex this last condition isequivalent to 119909

lowast(119909) = 1 and Re 119909lowast(119910) lt 1 for 119910 isin 119861119883 119909

where Re 119911 denotes the real part of a complex number 119911 If119909 isexposed then 119909 is also extremal If 119883 is a dual Banach spaceand the functional exposing 119909 can be taken in the predualthen 119909 is called weaklowastexposed For 119909 119910 isin R119889 (or C119889) wedenote by ⟨119909 119910⟩ its canonical scalar product The Euclideannorm is denoted by sdot When we consider other norms on afinite dimensional space they are denoted by | sdot | making anabuse of notation because the modulus of a complex numberis denoted also by | sdot |

For a Hausdorff locally compact topological space 119876 wedenote by 119876 = 119876 cup infin its Alexandroff compactificationThe boundary of a subset 119880 of a topological space is denotedby 120597119880 If 119880 sube R119889 is an open subset then the space ℎV0(119880) isformed by the harmonic functions on 119880 such that V119891 can beextended continuously to by (V119891)(infin) = 0 For an open set119880 sube R119889 the isometry between ℎV0(119880) and a subspace of119862()by means of 119891 997891rarr V119891 permits to use the same argument as in[14 Proposition 1] to show that the extreme points of ℎV0(119880)

lowast

are contained in the set

120582V (119911) 120575119911 120582 isin Γ 119911 isin 119880 (2)

where 120575119911 denotes the evaluation at 119911 The following defini-tions are due to Boyd and Rueda [14ndash16] The harmonic V-boundary is defined as follows

119887V (119880) = 119911 isin 119880 V (119911) 120575119911

is extremal in the unit sphere of ℎV0(119880)lowast

(3)

The set of harmonic V-peak points is defined as

119901V (119880) = 119911 isin 119880 V (119911) 120575119911

is weaklowastexposed in the unit sphere of ℎV0(119880)lowast

(4)

Remark 1 The same argument as in [15 Theorem 6] showsthat to check that a functional of the form V(1199090)1205751199090 is weak

lowast

exposed it is enough to show that there exists 119891 in the unitball of ℎV0(119880) such that V(1199090)119891(1199090) = 1 V(119909) Re119891(119909) lt 1 foreach 119909 isin 119880 1199090

If 119880 sube C119889 then the holomorphic V-boundary and theset of holomorphic V-peak points are defined analogouslyand they are denoted by 119861V(119880) and 119875V(119880) respectively Since119867V0

(119880) sub ℎV0(119880) in this case we have

119875V (119880) sube 119861V (119880) sube 119880 (5)

and also

119875V (119880) sube 119901V (119880) sube 119887V (119880) sube 119880 (6)

We finish this section with this lemma also due to Boydand Rueda

Lemma 2 (see [14 15]) Let 119880 sube R119889(119880 sube C119889) be a nonemptyopen set and let V be a weight which vanishes at infinity on119880

(a) For any 120582 isin Γ and 119911 isin 119880 the functional 120582V(119911)120575119911 isextremal in the unit sphere of ℎV0(119880)

lowast (is extremal inthe unit sphere of119867V0

(119880)lowast) if and only if 119911 isin 119887V(119880)(119911 isin

119861V(119880))(b) If 119880 sube C119889 is balanced V is radial and 120582 isin Γ then 120582119911 isin

119887V(119880) for each 119911 isin 119887V(119880)(120582119911 isin 119861V(119880) for each 119911 isin

119861V(119880))

2 Isomorphisms on ℎV0(119880)

We extend below the main theorem of [17] to weighed spacesof harmonic functions The proof is analogous to the proofgiven by Bonet and Wolf One only has to take in accountthat there are Cauchy type inequalities which are valid for thederivatives of harmonic functions [19 24] We include it forthe sake of completeness The argument was inspired by oneused by Kalton and Werner [20 Corollary 49] to show thatthe little Bloch space 1198610 embeds almost isometrically in 1198880Lusky showed in [21] that whenever V is a radial weight onDthen ℎV0

(D) is isomorphic to a subspace of 1198880

Theorem 3 Let 119880 sub R119889 be a nonempty open set let V be aweight on 119880 and let 120576 gt 0 There exists an isomorphism 119879

ℎV0(119880) rarr 1198880 such that (1 minus 120576)119891V le 119879(119891) le 119891V for each

119891 isin ℎV0(119880)

Proof Let (119870119895)119895 be a fundamental sequence of compactsubsets of 119880 and let 120576 gt 0 We claim that there exists asequence of pairwise disjoint finite subsets (119865119895)119895 such that


119865119895 sub 119860119895 = 119870119895 119870∘


with 119891V = 1

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 + sup

119910isin119865119895

V (119910) 1003816100381610038161003816119891 (119910)1003816100381610038161003816 (7)


sup119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817V le (1 + 120576) sup

119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 (8)





119872119895 = sup119909isin119870119895

V (119909) 119886119895 = min(12119889 (119870119895R

119873 119870∘

119895+1) 1)

(9)



sup119909isin119870119895

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

1

V (119909(119895)) (10)


1003816100381610038161003816119891 (120577)1003816100381610038161003816 le

1

V (119909(119895+1)) (11)


1003816100381610038161003816119863119894119891 (120577)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119894 = 1 2 119889 (12)


119860119895 sub ⋃

119909isin119865119895

1199091015840isin 119880

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816lt 120575119895

10038161003816100381610038161003816V (1199091015840) minus V (119909)

10038161003816100381610038161003816lt 120575119895

(13)


(1

V (119909(119895))+

119872119895+1119889119862

119886119895V (119909(119895+1))

) 120575119895 lt 120576 (14)



119860119895 We write 119908 = (1199081 1199082 119908119889) to get

1003816100381610038161003816119891 (119909)1003816100381610038161003816 =

1003816100381610038161003816119891 (1199091 119909119889)1003816100381610038161003816

le1003816100381610038161003816119891 (1199091 119909119889) minus 119891 (1199081 1199092 119909119889)

1003816100381610038161003816

+1003816100381610038161003816119891 (1199081 1199092 119909119889) minus 119891 (1199081 1199082 119909119889)

1003816100381610038161003816

+ sdot sdot sdot +1003816100381610038161003816119891 (1199081 1199082 119908119889)

1003816100381610038161003816

(15)


1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119889 |119909 minus 119908| +1003816100381610038161003816119891 (119908)

1003816100381610038161003816

le

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816

(16)

Now since 119908 isin 119870119895+1

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le |V (119909) minus V (119908)| 1003816100381610038161003816119891 (119909)

1003816100381610038161003816 + V (119908) 1003816100381610038161003816119891 (119909)1003816100381610038161003816

le 120575119895

1

V (119909(119895))+ V (119908)(

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816)

le

120575119895

V (119909(119895))+ 119872119895+1

119889120575119895119862

119886119895V (119909(119895+1))

+ V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

le 120576 + V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

(17)

Therefore

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 (18)













image






(119880)(119867V0(119880)) contains the








ℎV0(119880)lowast



lowast

V0 we




(119880) norℎV0









(119880) it holds1003817100381710038171003817119891

1003817100381710038171003817V = sup119899isinN

V (119909119899)1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816 (19)


119879 ℎV0(119880) 997888rarr 1198880 119891 997891997888rarr (V (119909119899) 119891 (119909119899))119899 (20)

is an isometry














1 =10038171003817100381710038171003817120593210038171003817100381710038171003817V

= V (1199111) 1205932(1199111) = V (minus1199111) 120593

2(minus1199111) (21)


C119889 Let 1199030 = |1199111| We define

ℎ 119880 997888rarr C 119911 997888rarr 1205932(⟨119911 119910⟩

1199030

1199111) (22)


We have

V (11990301199110) ℎ (11990301199110) = 119892 (1199030) 1205932(1199111)

= 119892 (10038161003816100381610038161199111

1003816100381610038161003816) 1205932(1199111) = V (1199111) 120593

2(1199111) = 1

(23)



2(minus1199111) = 1

(24)


V (119911) |ℎ (119911)| = 119892 (|119911|) |ℎ (119911)| le 119892 (1003816100381610038161003816⟨119911 119910⟩

1003816100381610038161003816) |ℎ (119911)|

= V(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

1205932(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171003817120593210038171003817100381710038171003817V

= 1

(25)


V (11990301199110) 12057511990301199110 (ℎ) = V (minus11990301199110) 120575(minus11990301199110) (ℎ) = 1 (26)



10038171003817100381710038171003817100381710038171003817

119891 + ℎ

2

10038171003817100381710038171003817100381710038171003817V= V (11990301199110)

(119891 (11990301199110) + ℎ (11990301199110))

2= 1 (27)






(119880) sub ℎV0(119880)







subspace of 1198880



(119903 minus 1199030)(28)



ℎ (119903) minus ℎ (1199030) gt 1205720 (119903 minus 1199030) (29)


sup0le119903lt1119903 = 1199030

119892 (119903) exp (1205720119903) = exp( sup0lt119903lt1119903 = 1199030

(minusℎ (119903) + 1205720119903))

lt exp( sup0le119903lt1119903 = 1199030

(minus1205720 (119903 minus 1199030)

minus ℎ (1199030) + 1205720119903) )

= 119892 (1199030) exp (12057201199030) (30)


V (1199031199110)10038161003816100381610038161198910 (1199031199110)

1003816100381610038161003816 =119892 (119903) exp (1205720119903)119892 (1199030) exp (12057201199030)

lt 1 (31)


V (119911) 10038161003816100381610038161198910 (119911)1003816100381610038161003816 = 119892 (|119911|)

exp (1205720 Re (⟨119911 119910⟩))119892 (1199030) exp (12057201199030)

lt 119892 (|119911|)exp (1205720 |119911|)

119892 (1199030) exp (12057201199030)le 1

(32)









119900(119890minus119886119905






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119865119895 sub 119860119895 = 119870119895 119870∘


with 119891V = 1

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 + sup

119910isin119865119895

V (119910) 1003816100381610038161003816119891 (119910)1003816100381610038161003816 (7)


sup119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

10038171003817100381710038171198911003817100381710038171003817V le (1 + 120576) sup

119909isin119865

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 (8)





119872119895 = sup119909isin119870119895

V (119909) 119886119895 = min(12119889 (119870119895R

119873 119870∘

119895+1) 1)

(9)



sup119909isin119870119895

1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

1

V (119909(119895)) (10)


1003816100381610038161003816119891 (120577)1003816100381610038161003816 le

1

V (119909(119895+1)) (11)


1003816100381610038161003816119863119894119891 (120577)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119894 = 1 2 119889 (12)


119860119895 sub ⋃

119909isin119865119895

1199091015840isin 119880

100381610038161003816100381610038161199091015840minus 119909

10038161003816100381610038161003816lt 120575119895

10038161003816100381610038161003816V (1199091015840) minus V (119909)

10038161003816100381610038161003816lt 120575119895

(13)


(1

V (119909(119895))+

119872119895+1119889119862

119886119895V (119909(119895+1))

) 120575119895 lt 120576 (14)



119860119895 We write 119908 = (1199081 1199082 119908119889) to get

1003816100381610038161003816119891 (119909)1003816100381610038161003816 =

1003816100381610038161003816119891 (1199091 119909119889)1003816100381610038161003816

le1003816100381610038161003816119891 (1199091 119909119889) minus 119891 (1199081 1199092 119909119889)

1003816100381610038161003816

+1003816100381610038161003816119891 (1199081 1199092 119909119889) minus 119891 (1199081 1199082 119909119889)

1003816100381610038161003816

+ sdot sdot sdot +1003816100381610038161003816119891 (1199081 1199082 119908119889)

1003816100381610038161003816

(15)


1003816100381610038161003816119891 (119909)1003816100381610038161003816 le

119862

119886119895V (119909(119895+1))

119889 |119909 minus 119908| +1003816100381610038161003816119891 (119908)

1003816100381610038161003816

le

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816

(16)

Now since 119908 isin 119870119895+1

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le |V (119909) minus V (119908)| 1003816100381610038161003816119891 (119909)

1003816100381610038161003816 + V (119908) 1003816100381610038161003816119891 (119909)1003816100381610038161003816

le 120575119895

1

V (119909(119895))+ V (119908)(

119889120575119895119862

119886119895V (119909(119895+1))

+1003816100381610038161003816119891 (119908)

1003816100381610038161003816)

le

120575119895

V (119909(119895))+ 119872119895+1

119889120575119895119862

119886119895V (119909(119895+1))

+ V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

le 120576 + V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816

(17)

Therefore

sup119909isin119860119895

V (119909) 1003816100381610038161003816119891 (119909)1003816100381610038161003816 le 120576 +max

119908isin119865119895

V (119908) 1003816100381610038161003816119891 (119908)1003816100381610038161003816 (18)













image






(119880)(119867V0(119880)) contains the








ℎV0(119880)lowast



lowast

V0 we




(119880) norℎV0









(119880) it holds1003817100381710038171003817119891

1003817100381710038171003817V = sup119899isinN

V (119909119899)1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816 (19)


119879 ℎV0(119880) 997888rarr 1198880 119891 997891997888rarr (V (119909119899) 119891 (119909119899))119899 (20)

is an isometry














1 =10038171003817100381710038171003817120593210038171003817100381710038171003817V

= V (1199111) 1205932(1199111) = V (minus1199111) 120593

2(minus1199111) (21)


C119889 Let 1199030 = |1199111| We define

ℎ 119880 997888rarr C 119911 997888rarr 1205932(⟨119911 119910⟩

1199030

1199111) (22)


We have

V (11990301199110) ℎ (11990301199110) = 119892 (1199030) 1205932(1199111)

= 119892 (10038161003816100381610038161199111

1003816100381610038161003816) 1205932(1199111) = V (1199111) 120593

2(1199111) = 1

(23)



2(minus1199111) = 1

(24)


V (119911) |ℎ (119911)| = 119892 (|119911|) |ℎ (119911)| le 119892 (1003816100381610038161003816⟨119911 119910⟩

1003816100381610038161003816) |ℎ (119911)|

= V(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

1205932(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171003817120593210038171003817100381710038171003817V

= 1

(25)


V (11990301199110) 12057511990301199110 (ℎ) = V (minus11990301199110) 120575(minus11990301199110) (ℎ) = 1 (26)



10038171003817100381710038171003817100381710038171003817

119891 + ℎ

2

10038171003817100381710038171003817100381710038171003817V= V (11990301199110)

(119891 (11990301199110) + ℎ (11990301199110))

2= 1 (27)






(119880) sub ℎV0(119880)







subspace of 1198880



(119903 minus 1199030)(28)



ℎ (119903) minus ℎ (1199030) gt 1205720 (119903 minus 1199030) (29)


sup0le119903lt1119903 = 1199030

119892 (119903) exp (1205720119903) = exp( sup0lt119903lt1119903 = 1199030

(minusℎ (119903) + 1205720119903))

lt exp( sup0le119903lt1119903 = 1199030

(minus1205720 (119903 minus 1199030)

minus ℎ (1199030) + 1205720119903) )

= 119892 (1199030) exp (12057201199030) (30)


V (1199031199110)10038161003816100381610038161198910 (1199031199110)

1003816100381610038161003816 =119892 (119903) exp (1205720119903)119892 (1199030) exp (12057201199030)

lt 1 (31)


V (119911) 10038161003816100381610038161198910 (119911)1003816100381610038161003816 = 119892 (|119911|)

exp (1205720 Re (⟨119911 119910⟩))119892 (1199030) exp (12057201199030)

lt 119892 (|119911|)exp (1205720 |119911|)

119892 (1199030) exp (12057201199030)le 1

(32)









119900(119890minus119886119905






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












image






(119880)(119867V0(119880)) contains the








ℎV0(119880)lowast



lowast

V0 we




(119880) norℎV0









(119880) it holds1003817100381710038171003817119891

1003817100381710038171003817V = sup119899isinN

V (119909119899)1003816100381610038161003816119891 (119909119899)

1003816100381610038161003816 (19)


119879 ℎV0(119880) 997888rarr 1198880 119891 997891997888rarr (V (119909119899) 119891 (119909119899))119899 (20)

is an isometry














1 =10038171003817100381710038171003817120593210038171003817100381710038171003817V

= V (1199111) 1205932(1199111) = V (minus1199111) 120593

2(minus1199111) (21)


C119889 Let 1199030 = |1199111| We define

ℎ 119880 997888rarr C 119911 997888rarr 1205932(⟨119911 119910⟩

1199030

1199111) (22)


We have

V (11990301199110) ℎ (11990301199110) = 119892 (1199030) 1205932(1199111)

= 119892 (10038161003816100381610038161199111

1003816100381610038161003816) 1205932(1199111) = V (1199111) 120593

2(1199111) = 1

(23)



2(minus1199111) = 1

(24)


V (119911) |ℎ (119911)| = 119892 (|119911|) |ℎ (119911)| le 119892 (1003816100381610038161003816⟨119911 119910⟩

1003816100381610038161003816) |ℎ (119911)|

= V(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

1205932(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171003817120593210038171003817100381710038171003817V

= 1

(25)


V (11990301199110) 12057511990301199110 (ℎ) = V (minus11990301199110) 120575(minus11990301199110) (ℎ) = 1 (26)



10038171003817100381710038171003817100381710038171003817

119891 + ℎ

2

10038171003817100381710038171003817100381710038171003817V= V (11990301199110)

(119891 (11990301199110) + ℎ (11990301199110))

2= 1 (27)






(119880) sub ℎV0(119880)







subspace of 1198880



(119903 minus 1199030)(28)



ℎ (119903) minus ℎ (1199030) gt 1205720 (119903 minus 1199030) (29)


sup0le119903lt1119903 = 1199030

119892 (119903) exp (1205720119903) = exp( sup0lt119903lt1119903 = 1199030

(minusℎ (119903) + 1205720119903))

lt exp( sup0le119903lt1119903 = 1199030

(minus1205720 (119903 minus 1199030)

minus ℎ (1199030) + 1205720119903) )

= 119892 (1199030) exp (12057201199030) (30)


V (1199031199110)10038161003816100381610038161198910 (1199031199110)

1003816100381610038161003816 =119892 (119903) exp (1205720119903)119892 (1199030) exp (12057201199030)

lt 1 (31)


V (119911) 10038161003816100381610038161198910 (119911)1003816100381610038161003816 = 119892 (|119911|)

exp (1205720 Re (⟨119911 119910⟩))119892 (1199030) exp (12057201199030)

lt 119892 (|119911|)exp (1205720 |119911|)

119892 (1199030) exp (12057201199030)le 1

(32)









119900(119890minus119886119905






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










We have

V (11990301199110) ℎ (11990301199110) = 119892 (1199030) 1205932(1199111)

= 119892 (10038161003816100381610038161199111

1003816100381610038161003816) 1205932(1199111) = V (1199111) 120593

2(1199111) = 1

(23)



2(minus1199111) = 1

(24)


V (119911) |ℎ (119911)| = 119892 (|119911|) |ℎ (119911)| le 119892 (1003816100381610038161003816⟨119911 119910⟩

1003816100381610038161003816) |ℎ (119911)|

= V(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

1205932(⟨119911 119910⟩

1199030

1199111)

100381610038161003816100381610038161003816100381610038161003816

le10038171003817100381710038171003817120593210038171003817100381710038171003817V

= 1

(25)


V (11990301199110) 12057511990301199110 (ℎ) = V (minus11990301199110) 120575(minus11990301199110) (ℎ) = 1 (26)



10038171003817100381710038171003817100381710038171003817

119891 + ℎ

2

10038171003817100381710038171003817100381710038171003817V= V (11990301199110)

(119891 (11990301199110) + ℎ (11990301199110))

2= 1 (27)






(119880) sub ℎV0(119880)







subspace of 1198880



(119903 minus 1199030)(28)



ℎ (119903) minus ℎ (1199030) gt 1205720 (119903 minus 1199030) (29)


sup0le119903lt1119903 = 1199030

119892 (119903) exp (1205720119903) = exp( sup0lt119903lt1119903 = 1199030

(minusℎ (119903) + 1205720119903))

lt exp( sup0le119903lt1119903 = 1199030

(minus1205720 (119903 minus 1199030)

minus ℎ (1199030) + 1205720119903) )

= 119892 (1199030) exp (12057201199030) (30)


V (1199031199110)10038161003816100381610038161198910 (1199031199110)

1003816100381610038161003816 =119892 (119903) exp (1205720119903)119892 (1199030) exp (12057201199030)

lt 1 (31)


V (119911) 10038161003816100381610038161198910 (119911)1003816100381610038161003816 = 119892 (|119911|)

exp (1205720 Re (⟨119911 119910⟩))119892 (1199030) exp (12057201199030)

lt 119892 (|119911|)exp (1205720 |119911|)

119892 (1199030) exp (12057201199030)le 1

(32)









119900(119890minus119886119905






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119900(119890minus119886119905






Acknowledgments


References

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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