Contents lists available atScienceDirect
Journal of Mathematical Analysis and Applications
www.elsevier.com/locate/jmaa
Kelvin-Möbius-invariant harmonic function spaces on the real unit ball
H.Turgay Kaptanoğlua,∗, A.Ersin Üreyenb
a BilkentÜniversitesi,MatematikBölümü,06800Ankara,Turkey
bEskişehirTeknikÜniversitesi,FenFakültesi,MatematikBölümü,YunusemreKampüsü,26470 Eskişehir,Turkey
a r t i cl e i n f o a b s t r a c t
Articlehistory:
Received11December2020 Availableonline7May2021 Submittedby M.M.Peloso
Keywords:
Kelvin-Möbiustransform Kelvin-Möbius-invariantspace HarmonicBergman-Besovspace WeightedharmonicBlochspace Atomicdecomposition
Complexinterpolation
WedefinetheKelvin-Möbiustransformofafunctionharmonicontheunitballof Rnanddetermineharmonicfunctionspacesthatareinvariantunderthistransform.
Whenn≥ 3,inthecategoryofBanachspaces,theminimalKelvin-Möbius-invariant spaceistheBergman-Besovspaceb1−(1+n/2)andthemaximalinvariantspaceisthe Blochspaceb∞(n−2)/2.ThereexistsauniquestrictlyKelvin-Möbius-invariantHilbert space, andit istheBergman-Besovspaceb2−2.Thereisaunique Kelvin-Möbius- invariantHardyspace.
©2021ElsevierInc.Allrightsreserved.
1. Introduction
LetD⊂ C betheopenunitdiscandAut(D) beitsautomorphism group,thatis,thegroupofholomor- phic,bijective maps of D. TheBloch space B consists of all holomorphicfunctions f on D suchthat the seminorm
ρB(f ) := sup{(1 − |z|2)|f(z)| : z ∈ D}
isfinite.TheBloch spaceis Möbiusinvariantinthesense thatiff ∈ B,then foreveryϕ∈ Aut(D), f ◦ ϕ isinB andρB(f◦ ϕ)= ρB(f ).Thequantity(1− |z|2)f(z) issometimescalled theinvariantderivativeof f atz since itsmodulusisMöbiusinvariantinthisparticularsense.
Moregenerally,letE be alinear spaceof holomorphicfunctionsonD that iscomplete withrespect to aseminorm ρE. Roughlyspeaking,E is calledMöbius invariantifforeveryf ∈ E andϕ∈ Aut(D), f ◦ ϕ
* Correspondingauthor.
E-mailaddresses:kaptan@fen.bilkent.edu.tr(H.T. Kaptanoğlu),aeureyen@eskisehir.edu.tr(A.E. Üreyen).
URL:http://www.fen.bilkent.edu.tr/~kaptan/(H.T. Kaptanoğlu).
https://doi.org/10.1016/j.jmaa.2021.125298 0022-247X/©2021ElsevierInc. Allrightsreserved.
belongs toE and ρE(f◦ ϕ)= ρE(f ) (or theweakerconditionρE(f◦ ϕ)∼ ρE(f ) holds). Wesay“roughly speaking”becausethereshouldbeafewtechnicalrestrictionsonE;fordetailssee[3],[5] and[25].Among allMöbius-invariantspaces,theBlochspaceB isthelargest[25] andtheBesovspaceB1isthesmallest[5].
IfoneconsidersHilbertspaces,theDirichletspaceistheuniqueMöbius-invariantHilbertspace[3].Similar results withdomains otherthanD are obtainedinvarioussources, in[24] and[32] thedomain istheunit ball ofCn,in[21] thedomainisthepolydiscand [4] considersboundedsymmetricdomains.
The purpose ofthis paper is to studythe harmonic analogue ofthis problem onthe unitball B ofRn withrespecttothestandardnorm|x|2= x· x= x21+· · · + x2n.Leth(B) betheFréchetspaceofallcomplex- valued harmonic functionson B endowed with the topologyof uniform convergence on compact subsets, and M(B) bethegroupofMöbius transformationsofB;seesubsection2.2.Intheharmoniccasethefirst problem one encountersis thatiff ∈ h(B) andϕ∈ M(B), then f◦ ϕ neednot be harmonic.To remedy thisproblemnotethatanyϕ∈ M(B) canbewrittenasacompositionofanorthogonaltransformationand aninversion.Composingaharmonicfunctionwithanorthogonaltransformationpreservesharmonicity.So, harmonicity islostinf◦ ϕ becauseofcompositionwithaninversion.Ontheotherhand,onecancompose with an inversionand still preserveharmonicity providedonemultipliesbya correctionfactor andthis is called theKelvintransform. Thereforeweneed tocombinecompositionwithϕ with theKelvintransform and thisleadsustothefollowingdefinition.
Definition 1.1.Letf ∈ h(B) andϕ∈ M(B).TheKelvin-MöbiustransformKϕ(f ) off isdefinedas
Kϕ(f )(x) :=
1− |ϕ(x)|2 1− |x|2
(n−2)/2
f (ϕ(x)).
Thefactormultiplyingf◦ ϕ stemsfromtheKelvintransform;seesubsection2.3.Iff isharmoniconB, thensoisKϕ(f ).Thisisverifiedlater,butintheliteraturetherearealsodifferentproofs.See,forexample, [2,Corollary 2.3],[9,§2.3],[19,Proposition3.1] or[23,Theorem2].
If the dimensionis n = 2,the first factordisappears and we justhave Kϕ(f )= f ◦ ϕ. Because of this there aredifferencesbetweenthecasesn= 2 andn≥ 3.Whenn= 2,theharmonic caseisverysimilar to theholomorphic casestudiedin[3,5,25];neverthelesstherearedetailsthatrequireattention.Wedealwith thatinadifferentwork[22] andthroughout thispaperweconsider thecasen≥ 3.
Remark1.2.Weshowinsubsection2.3thatKϕ: h(B)→ h(B) isinvertible withKϕ−1=Kϕ−1.
WenowdefineaKelvin-Möbius-invariantharmonicfunctionspacefollowing[25] and[31].Let(E,· E) be aBanachspaceofharmonicfunctionsonB.Anon-zerocontinuouslinearfunctionalL onh(B) iscalled decent on E ifL is alsocontinuousonE withrespecttothenorm· E.
Wedenote by1 the constantfunctionwhosevalueis1.
Definition 1.3. Letn≥ 3. A Banachspace (E,· E) ofharmonic functionson B iscalled Kelvin-Möbius invariant if thefollowingpropertieshold:
(i) E contains1.
(ii) Thereexistsadecent linearfunctional onE.
(iii) Foreveryf ∈ E andϕ∈ M(B),theKelvin-MöbiustransformKϕ(f ) belongs toE and
Kϕ(f )E≤ CfE, (1)
forsomeconstantC independentoff andϕ.
SinceE islinearwecanwrite thefirstconditionas (i)E containsconstantfunctions.
Iftheconstantin(1) isC = 1, thenbyRemark1.2,Kϕ(f )E =fE foreveryf ∈ E andϕ∈ M(B) andinthis casewecall E strictly Kelvin-Möbiusinvariantas donein[5, Definition1].Inthegeneralcase we only have Kϕ(f )E ∼ fE, where we use the notation A ∼ B tomean that A/B is bounded from aboveandbelowbysomeconstantsthatareindependentoftheparametersinvolved.IfA/B isjustbounded above,wewriteA B.
Remark1.4.IfE isKelvin-Möbiusinvariant,thenwecandefineanew,equivalentnormonE withwhichE becomesstrictlyKelvin-Möbiusinvariant(see[5,p. 111]).Thusweakeningstrictly Kelvin-Möbiusinvariance andusingKelvin-Möbiusinvariancewith“∼”isnotveryimportant.
Remark1.5. Wenote thatintheholomorphiccasetheinvariantspaceistakenasacompletesemi-normed space. In the harmonic case the same is true when n = 2 but when n ≥ 3 we do not need semi-norms becauseoftheextramultiplyingfactorinKϕ,andE in Definition1.3is aBanachspace.
Theharmonic function spaceswe are mainlyinterested inthis workare Bergman-Besov spacesbpq and weightedBloch spacesb∞s with q,s∈ R inboth.Letν be thenormalizedLebesguemeasure onB and for q∈ R,let
dνq(x) := cq(1− |x|2)qdν(x).
Themeasureνqisfiniteonlywhenq >−1 andinthiscasewepicktheconstantcqsothatνq(B)= 1.When q≤ −1, wejustset cq= 1.For0< p<∞,wedenotetheLebesgueclasseswithrespecttoνqbyLpq.
For q > −1, the well-known harmonic Bergman space bpq is defined as bpq := h(B)∩ Lpq with norm
fbpq :=fLpq. Thenext definitionextends this class to all q ∈ R. To denote partial derivatives we use multi-indicesandwrite
∂αf := ∂|α|f
∂xα11· · · ∂xαnn
,
wherethemulti-indexα = (α1,. . . ,αn) isann-tupleofnonnegative integersand|α|= α1+· · · + αn. Definition1.6. Let0< p<∞ and q∈ R.PickanonnegativeintegerN suchthat
q + pN >−1. (2)
Theharmonic Bergman-Besovspace bpq consistsofallf ∈ h(B) suchthat (1− |x|2)N∂αf ∈ Lpq
foreverymulti-indexα with|α|= N .A norm(quasinormwhen0< p< 1)onbpq is
fpbpq :=
|α|<N
|∂αf (0)|p+
|α|=N
B
(1− |x|2)N∂αf (x)p dνq(x). (3)
When0< p < 1,while westill use thenotation · bqp, bpq isnot anormed space, butitis acomplete metricspacewithrespectto themetricd(f,g)=f − gpbpq.
The space bpq does not depend on the choice of N . Different choices of N satisfying (2) give rise to equivalentnormsandinthenotation·bpq wedonotindicatethedependenceonN .Thepartialderivatives intheabovedefinitioncanbereplacedwithradialderivativesorvariousothersuitabledifferentialoperators.
Fig. 1. Equationoftheray l isq = p(n− 2)/2− n,p≥ 1.ByLemma1.7,if(p,q) is onthesolidpartoftherayl, thenbpq is Kelvin-Möbiusinvariant.
The region q > −1 is the Bergman region inwhich we takeN = 0, and q ≤ −1 is the proper Besov region. Whenp= 2, b2q isaHilbertspaceendowedwiththeinnerproduct
f, g b2q :=
|α|<N
∂αf (0)∂αg(0) +
|α|=N
B
∂αf (x)∂αg(x)(1− |x|2)2Ndνq(x), (4)
withN satisfying(2).Inthespecialcaseq =−n,thespacesbp−narestudiedin[20,29].Forthewholefamily bpq withq∈ R,see[17] (whenp≥ 1)and[13] (when0< p< 1).
WefirstdeterminewhichharmonicBergman-BesovspacesareKelvin-Möbiusinvariant.IntheBergman region q >−1,thisisveryeasy.
Lemma 1.7. Let n ≥ 3. Let 1 ≤ p < ∞ and q > −1. Then bpq is Kelvin-Möbius invariant if and only if q = p(n− 2)/2− n and inthis casebpq isstrictlyKelvin-Möbiusinvariant.
It is clearthat1∈ bpq, andit is wellknown thatpoint-evaluationfunctionals are bounded onBergman spaces. Tocheckcondition(iii)ofDefinition1.3,letf ∈ bpq andϕ∈ M(B).Then
Kϕ(f )pbpq = cq
B
|f(ϕ(x))|p
1− |ϕ(x)|2 1− |x|2
p(n−2)/2
(1− |x|2)qdν(x).
Using p(n− 2)/2= q + n, changing variables as x = ϕ(x),˜ and using (12) below, we immediately obtain
Kϕ(f )bpq =fbpq.Weshowtheonly-ifpartofLemma1.7inSection4withintheproofofTheorem1.8.
Lemma1.7tellsthatif(p,q) isonthesolidpartoftherayl inFig.1,thenthespacebpqisKelvin-Möbius invariant. ThissuggeststhattheBanachspacescorresponding tothedashedpartoftherayl mayalsobe Kelvin-Möbiusinvariant. Thisistrueanditisourfirsttheorem.
Theorem 1.8. Let n≥ 3. Letp≥ 1 and q∈ R. Thenbpq isa Kelvin-Möbiusinvariant space if andonly if q = p(n− 2)/2− n.
Of coursetheinterestingpartofTheorem 1.8isthecaseq≤ −1. Weprovethis partinSection4using complexinterpolation.
ByRemark1.4,whenq = p(n− 2)/2− n,thespacebpq isstrictly Kelvin-Möbiusinvariantwhenendowed with asuitableequivalentnorm.
Remark1.9.Theorem1.8istruealsowhenn= 2.Inthiscaseq = p(n− 2)/2− n=−2 islessthan−1 and thereisnoKelvin-Möbius-invariantharmonicBergman space.Whenn= 2, asintheholomorphiccase,all Kelvin-Möbius-invariantspacesbelongtotheproperBesovregionand lieonthehorizontallineq =−2.
Observe that when p = 2, the corresponding q in Theorem 1.8 is q = −2 and the Hilbert space (b2−2, ·,· b2−2) isKelvin-Möbiusinvariant.Thespaceb2−2 isstrictly Kelvin-Möbiusinvariantwhen endowed withasuitableinnerproduct ·,· I whichwenowdescribe.
Form≥ 0, letHm(S) bethe vectorspace ofsphericalharmonicsof degreem andδmbe itsdimension (see subsection3.1 formore details).Let{Ymj : j = 1,. . . ,δm} beanorthonormal basisof Hm(S). If f is harmoniconB,thenf hastheexpansion
f (x) =
∞ m=0
δm
j=1
fmjYmj(x) (x∈ B),
wherefmj’sarecomplexnumbers.Theaboveseriesconvergesabsolutelyanduniformly oncompactsubsets ofB.By[16,Theorem 3.8] or[17,Theorem5.1],thespaceb2−2 canalsobedescribed as
b2−2=
f =
∞ m=0
δm
j=1
fmjYmj ∈ h(B) : f2I =
∞ m=0
δm
j=1
m + n/2− 1
n/2− 1 |fmj|2<∞
.
Theabovenorm isequivalenttothenorm(s)giveninDefinition1.6and isinducedbytheinner product
f, g I :=
∞ m=0
δm
j=1
m + n/2− 1
n/2− 1 fmjgjm (f, g∈ b2−2). (5) When endowed with the above inner product ·,· I, the space b2−2 is strictly Kelvin-Möbius-invariant HilbertspaceanditistheonlystrictlyKelvin-Möbius-invariantHilbertspace.Thisisoursecondtheorem.
Theorem1.10.(i) TheHilbert space(b−22 , ·,· I) is strictlyKelvin-Möbiusinvariant.
(ii)If(H, ·,· H) isastrictlyKelvin-Möbius-invariantHilbertspace,thenH = b2−2 and ·,· H = C ·,· I for someC > 0.
InProposition6.1,wegiveanintegraldescriptionoftheinnerproduct ·,· I similarto (4),butinterms ofradialderivatives.
When(p,q) isontherayq = p(n− 2)/2− n,p≥ 1,theKelvin-Möbius-invariantspacesbpq increaseasp increases.Thisisaconsequenceof[15,Theorem1.2].Ifp= 1,thecorrespondingq is −(1+ n/2),andthis suggeststhatb1−(1+n/2) mightbethesmallest Kelvin-Möbius-invariantspace.Thisistrueanditisournext theorem.
Theorem1.11.Letn≥ 3.Thespaceb1−(1+n/2)isthesmallestKelvin-Möbius-invariantspace.Moreprecisely, ifE isKelvin-Möbiusinvariant,thenb1−(1+n/2) ⊂ E andthereexistsaC > 0 suchthatfE≤ Cfb1−(1+n/2)
forevery f ∈ b1−(1+n/2).
We next determine the largest Kelvin-Möbius-invariant space. For this we need to consider weighted harmonicBlochspaces.
Definition1.12.Lets∈ R.Pickanon-negativeintegerN sothats+ N > 0.Theweightedharmonic Bloch space b∞s consists ofallf ∈ h(B) suchthatforeverymulti-indexα with|α|= N ,
sup
x∈B(1− |x|2)s+N|∂αf (x)| < ∞.
A normonb∞s is
fb∞s :=
|α|<N
|∂αf (0)| +
|α|=N
sup
x∈B(1− |x|2)s+N|∂αf (x)| .
As before,thespace b∞s doesnotdependonthe choiceof N aslongas s+ N > 0 anddifferentchoices of N give riseto equivalent norms (see [14] formoredetails). Whenn≥ 3 we are interestedinthe space b∞(n−2)/2.ChoosingN = 0 shows
b∞(n−2)/2=
f ∈ h(B) : fb∞(n−2)/2 = sup
x∈B(1− |x|2)(n−2)/2|f(x)| < ∞
. (6)
It is immediate thatthe spaceb∞(n−2)/2 is strictly Kelvin-Möbius invariant.Clearly 1∈ b∞(n−2)/2 and fora decentlinearfunctional,wecantakeanypoint-evaluationfunctional.ThatKϕ(f )b∞(n−2)/2 =fb∞(n−2)/2 is obvious.
By[15,Theorem1.3],theincreasingfamilyofKelvin-Möbius-invariantspacesbpq,p≥ 1,q = p(n−2)/2−n are all included in the weighted Bloch space b∞(n−2))/2. This suggests that b∞(n−2)/2 might be the largest Kelvin-Möbius-invariantspace. Thisistrueanditisournexttheorem.
Theorem 1.13.Let n≥ 3. The spaceb∞(n−2)/2 is thelargest Kelvin-Möbius-invariantspace. Moreprecisely, if E isKelvin-Möbiusinvariant,thenE ⊂ b∞(n−2)/2 andthereexistsaC > 0 suchthat fb∞(n−2)/2≤ CfE
forevery f ∈ E.
For completeness we alsolook at the spaces bpq, 0< p < 1.Because these are notnormed spaces, they cannot be Kelvin-Möbiusinvariant in thesense ofDefinition 1.3. Nevertheless if q = p(n− 2)/2− n,the properties(i)-(iii)inDefinition1.3doholdevenwhen0< p< 1,whereweunderstand· asaquasinorm.
Theorem 1.14.Letn≥ 3. Let0< p< 1 andq∈ R. Thentheproperties (i)-(iii) inDefinition 1.3 holdfor bpq if andonlyif q = p(n− 2)/2− n.
Wenotethatthespacesbpq,0< p< 1,q = p(n−2)/2−n aresmallerthanb1−(1+n/2)by[15,Theorem1.2].
This, however, doesnotcontradict Theorem 1.11sincethese spacesare notBanachspaces. Ananalogous resultconcerningMöbiusinvariantholomorphic function spacesontheunitballofCn isobtainedin[33].
Anotherwell-studiedspaceofharmonicfunctionsistheclassofharmonicHardyspaces.Foreachn≥ 3, oneHardy spacecanbe shownto be Kelvin-Möbiusinvariantby using[19,Theorem 1.2].Wediscuss this inRemark4.4.
Lastly,weconsidersubspacesofh(B) andshowthatthereisnonon-trivialclosedsubspaceofh(B) that is invariantundertakingKelvin-Möbiustransform.
Theorem1.15.Letn≥ 3 andA⊂ h(B) beaclosedsubspace.If Kϕ(f )∈ A foreveryf ∈ A andϕ∈ M(B), then A={0} orA= h(B).
Corollary 1.16.Letn≥ 3 and f ∈ h(B) be non-zero.Thenspan{Kϕ(f ): ϕ∈ M(B)} is denseinh(B).
The paper is organizedas follows. In Section 2we justify Definition 1.1 and present someelementary propertiesoftheKelvin-Möbiustransform.InSection3wereviewreproducing kernelsandatomicdecom- positionsofharmonicBergman-Besovspacesbpq.WeconsidertheKelvin-Möbiusinvarianceofthebpqspaces
andproveTheorems1.8and1.14inSection4.InSection5wefirstshowthatpoint-evaluationfunctionals areboundedonKelvin-Möbius-invariantspacesand asaconsequencedeterminetheminimal andmaximal invariantspaces.WeproveTheorem1.15alsointhissection.Section6isdevotedtotheHilbert-spacecase.
Werepeatthatinthispaper wetaken≥ 3.
2. Kelvin-Möbiustransform
In this section we first recall some known factsabout Möbius transformations. For more detail about thesetransformationssee[1,7,30].WethenjustifythedefinitionofKelvin-Möbiustransformandprovesome ofitselementaryproperties.
2.1. Orthogonal transformations
We denote thegroupof all orthogonaltransformations of Rn by O(n).Fixing anorthonormal basisof Rn wecanrepresenteachelementofO(n) withanorthogonalmatrixofsizen× n.Identifyingmatricesof sizen× n withelementsofRn2 induces anaturaltopologyonsuchmatriceswhichmakesO(n) acompact topologicalgroup.TheelementsofO(n) whosecorrespondingmatriceshavedeterminant1 formasubgroup of O(n) denoted by SO(n).We denote the normalized Haar measure on SO(n) by μ0, where normalized meansμ0(SO(n))= 1.
Foraproof ofthelemmabelowsee [11,Theorem3.1] or[26,Lemma1.4.7(3)].
Lemma2.1. Letf be continuouson S andη∈ S. Then
SO(n)
f (U (η)) dμ0(U ) =
S
f (ζ) dσ(ζ).
2.2. Möbiustransformations
AMöbiustransformationofRˆn:= Rn∪{∞} isafinitecompositionofinversionsinspheresandreflections inplanes.Wedenote thegroupofallMöbiustransformationsthatmaptheunitballB toitself byM(B).
Fora∈ B,theinvolutiveMöbiustransformation ϕa thatexchangesa and0 isdefinedby
ϕa(x) := (1− |a|2)(a− x) + |x − a|2a
[x, a]2 ,
where
[x, a] :=
1− 2x · a + |x|2|a|2.
Themap ϕacanbe decomposedinto simplemaps. LetJ betheinversion withrespect tothe unitsphere S,
J (x) := x∗:= x
|x|2. Notethatwhena= 0,
[x, a] =|a| |x − a∗|. (7)
More generally,for c ∈ Rn and r > 0, let S(c,r) be the sphere with center c and radius r, and let Jc,r denote theinversionwithrespectto S(c,r),
Jc,r(x) := c + r2J (x− c).
For0= a∈ B,ϕa canbe decomposedas
ϕa=−Pa◦ Ja∗,ρ, where
ρ :=
1− |a|2
|a| (8)
and
Pa(x) := x− 2x· a
|a|2a
is thereflection abouttheplane passingthroughtheoriginand perpendicularto thevector a.For details see [1, Section2.6] whichusesTa=−ϕa.ThesphereS(a∗,ρ) isorthogonalto S andthusJa∗,ρ(B)= B.
Letϕ∈ M(B) bearbitraryand a= ϕ−1(0).If a= 0,thenϕ is anorthogonaltransformation.Ifa= 0, then by[30,Theorem2.1.2] thereexistsanorthogonaltransformationU ∈ O(n) such thatϕ= U◦ ϕaand ϕ can bedecomposedas
ϕ = U◦ (−Pa)◦ Ja∗,ρ. (9)
Thus, sincePa isalso orthogonal, we canwriteϕ as compositionof anorthogonal transformation andan inversion.
Welistsomeidentitiesinvolvingϕ.Themostusefulidentity is
1− |ϕ(x)|2=(1− |a|2)(1− |x|2)
[x, a]2 (a = ϕ−1(0)). (10)
Theratio[x,y]2/((1− |x|2)(1− |y|2)) isMöbiusinvariant,thatis,foreveryx,y∈ B andϕ∈ M(B), [ϕ(x), ϕ(y)]2
(1− |ϕ(x)|2)(1− |ϕ(y)|2) = [x, y]2
(1− |x|2)(1− |y|2). (11) Thederivativeϕ(x) ofϕ= (ϕ1,. . . ,ϕn): B→ B,is then× n matrix
ϕ(x) = ∂ϕi
∂xj n
i,j=1
.
Theabsolute valueoftheJacobiandeterminantofϕ is([30,Theorem3.3.1])
| det(ϕ(x))| =
1− |ϕ(x)|2 1− |x|2
n
. (12)
2.3. Kelvin-Möbiustransform
TheKelvintransformoff withrespectto theunitsphereS isdefinedby KS(f )(x) := 1
|x|n−2f (J (x)).
Moregenerally,Kelvintransform of f withrespectto thesphereS(c,r) isdefinedby(see[18,p. 39])
KS(c,r)(f )(x) := rn−2
|x − c|n−2f (Jc,r(x)).
Iff isharmoniconadomain Ω⊂ Rn,thenKS(c,r)(f ) isharmoniconJc,r(Ω).
Letϕ∈ M(B),ϕ−1(0)= a= 0 andϕ havethedecomposition(9).By(7) and(8),
KS(a∗,ρ)(f )(x) =(1− |a|2)(n−2)/2
[x, a]n−2 f (Ja∗,ρ(x)).
BecauseJa∗,ρ(B)= B,iff isharmoniconB,thenKS(a∗,ρ)(f ) isharmoniconB.Replacingf withf◦U◦(−Pa) wededucethatiff ∈ h(B),then sois
(1− |a|2)(n−2)/2
[x, a]n−2 f (ϕ(x)) =
1− |ϕ(x)|2 1− |x|2
(n−2)/2
f (ϕ(x)),
wherewe alsousetheformula(10).ThefunctionaboveistheKelvin-Möbiustransform of f . Ifϕ= U ∈ O(n),thenwejusthave
KU(f ) = f◦ U. (13)
Forfuturereferenceletus alsorecordthatiff = 1,then
Kϕa(1)(x) = (1− |a|2)(n−2)/2
[x, a]n−2 . (14)
SomebasicpropertiesofthetransformKϕ arelistedinthefollowing lemma.
Lemma 2.2. Let ϕ,ψ ∈ M(B). The Kelvin-Möbius transform Kϕ : h(B) → h(B) satisfies the following properties:
(i) Kϕ islinear.
(ii) Kψ◦ Kϕ=Kϕ◦ψ.
(iii) Kϕ isone-to-one andontoandK−1ϕ =Kϕ−1. (iv) K−1ϕa =Kϕa.
(v) Kϕ iscontinuous.
Proof. Part (i)isclear.Part(ii)is purecomputation.Part(iii) followsfrom part(ii) andpart(iv)istrue becauseϕa isaninvolution.Toseepart(v)supposefm→ f inh(B),thatis,fm convergestof uniformly oncompactsubsetsofB.Wehave
Kϕ(fm)(x) =
1− |ϕ(x)|2 1− |x|2
(n−2)/2
fm(ϕ(x))
and ifx lies inthecompact set |x|≤ r < 1, then there existss< 1 suchthat|ϕ(x)| ≤ s.Thefirst factor ontheright isboundedandsecond factorconvergesuniformly tof (ϕ(x)) sincefm convergesuniformly to f on|x|≤ s.HenceKϕ(fm) convergesuniformlytoKϕ(f ) on |x|≤ r.
3. HarmonicBergman-Besov spaces
This section is for review purposes. Werecall some known factsabout zonal harmonics and harmonic Bergman-Besov spaces bpq thatwill be used in the sequel. For moredetail about zonal harmonics see [6, Chapter 5],andaboutthespacesbpq see[17] and [13].
3.1. Sphericalandzonal harmonics
Let L2(S) be the Hilbert space of square integrable functions on S with respect to the inner product f,g =
Sf g dσ, where σ is thenormalizedsurface areameasure onS. LetHm(Rn) denote thecomplex vector space of all homogeneousharmonic polynomials of degree m inn real variables. Restriction of an element of Hm(Rn) to S iscalled a(surface) sphericalharmonic ofdegreem. ThecollectionHm(S) ofall sphericalharmonicsofdegreem isafinite-dimensionalsubspaceofL2(S) withdimensionδm.
For m≥ 0,let{Ymj,j = 1,. . . ,δm} beanorthonormalbasisof Hm(S).If m= k,thenHm(S)⊥ Hk(S) inL2(S),so
S
Ymj(ξ)Yki(ξ) dσ(ξ) = 0 (15)
unless m= k and j = i. For η ∈ S, the point-evaluation functional f → f(η) is bounded on Hm(S) and therefore thereexistsauniqueZm(·,η)∈ Hm(S) suchthatforallf ∈ Hm(S)
f (η) =
S
f (ξ)Zm(ξ, η) dσ(ξ).
Thezonal harmonicZm(ξ,η) isreal-valued,itissymmetricinitsvariables,andintermsof Ymj itequals
Zm(ξ, η) =
δm
j=1
Ymj(ξ)Ymj(η) (ξ, η∈ S).
TheaboveformulaextendstoRn× Rn byhomogeneity
Zm(x, y) =
δm
j=1
Ymj(x)Ymj(y) (x, y∈ Rn), (16)
where Ymj(x)=|x|mYmj(ξ) forx=|x|ξ.
Whenn≥ 3,asafunctionofx,1/[x,a]n−2isharmoniconB by(7).Itshomogeneousexpansionisgiven inthefollowinglemma.
Lemma 3.1.Letn≥ 3 anda∈ B. Then 1 [x, a]n−2 =
∞ m=0
n/2− 1
m + n/2− 1Zm(x, a), where theseriesconvergesuniformly forx∈ B.
Proof. Lemma is clear when a = 0, so we assume a = 0. For d> 0, the Gegenbauer polynomial Gdm of degreem isdefinedbythegenerating function
1
(1− 2rt + t2)d =
∞ m=0
Gdm(r) tm.
Writingx=|x|ξ,a=|a|η,wesee that 1
[x, a]n−2 = 1
(1− 2ξ · η |x||a| + |x|2|a|2)n/2−1 =
∞ m=0
Gn/2m −1(ξ· η)|x|m|a|m.
Itisknownthat(see, forexample,[17,Equation(14.8)])
Gn/2m −1(ξ· η) = n/2− 1
m + n/2− 1Zm(ξ, η).
Thus
1 [x, a]n−2 =
∞ m=0
n/2− 1
m + n/2− 1Zm(ξ, η)|x|m|a|m=
∞ m=0
n/2− 1
m + n/2− 1Zm(x, a),
wherethelastequalityfollowsfromthehomogeneityofZm.Forfixeda∈ B,theseriesconvergesuniformly forx∈ B since|Zm(ξ,η)| mn−2 (see[6,Proposition5.27(e)andExercise10,p. 107]).
3.2. Reproducingkernels of harmonic Bergman-Besovspaces
Forall q∈ R,thespace b2q is areproducingkernel Hilbertspace. Wedenote the reproducingkernel by Rq(x,·). When q > −1, the natural inner product onb2q is f,g b2q :=
Bf g dνq and with respect to this innerproduct (see[12,p. 164])
Rq(x, y) =
∞ m=0
(n/2 + q + 1)m (n/2)m
Zm(x, y) (q >−1),
wherethePochhammersymbol(a)b isdefinedby
(a)b:= Γ(a + b) Γ(a)
whena anda+ b areoffthepoleset −N ofthegammafunctionΓ.
When q ≤ −1, it is necessary to consider derivatives of the functions in the inner product and there are various choices.Different choicesof the inner product wouldgive riseto different reproducing kernels and vice versa.We follow theapproach of[16] and extend thereproducing kernels to wholeq∈ R in the followingway. Define
γm(q) :=
⎧⎪
⎪⎪
⎨
⎪⎪
⎪⎩
(n/2 + q + 1)m
(n/2)m , if q >−(1 + n/2);
(m!)2
(1− (n/2 + q))m(n/2)m
, if q ≤ −(1 + n/2);
(17)
and
Rq(x, y) :=
∞ m=0
γm(q)Zm(x, y).
For an (integral) inner product that makes this Rq(x,y) the reproducing kernel of b2q (q ∈ R) see [17, Theorem 5.2].Every Rq(x,y) issymmetricinitsvariablessinceeveryZm(x,y) is.
Using the reproducing kernels abovewe candefine radialdifferential operators Dst of order t for every t,s∈ R.Iff ∈ h(B) withhomogeneousexpansionf =∞
m=0fm,thenwedefine Dstf :=
∞ m=0
γm(s + t) γm(s) fm.
Theoperators Dst arecompatiblewithreproducing kernelsinthesensethatforeveryt,s∈ R
DtsRs(x, y) = Rs+t(x, y), (18)
where the operator acts on the first variable. Because of this inthe study of the properties of harmonic Bergman-Besov spaces it is more convenient to use the operators Dst rather than the partial derivatives.
Similar toDefinition1.6 thespacesbpq canbe describedintermsof theoperators Dst. For0< p<∞ and q∈ R,pickt∈ R suchthatq + pt>−1 and anys∈ R.Thenf ∈ bpq ifandonlyif
(t,s)fpbpq :=
B
(1− |x|2)tDstf (x)p dνq(x) <∞ (19)
andthenorm(quasinormwhen0< p< 1)(t,s)·pbpqisequivalenttothenorm·bpq in(3);see[17,Theorems 1.1 and 1.2] and [13, Theorem 1.1].We need theDst intheproof of the only-ifpartof Theorems 1.8 and 1.14.
As suggestedbyTheorem1.10,forourpurposesthemostimportantkernelisR−2.Inthis case,by(17) and Lemma3.1, thefollowing closedformulaholds.
Lemma 3.2.Letn≥ 3.Then
R−2(x, y) = 1
[x, y]n−2 (x, y∈ B).
Combining theabovelemmawith (14) weseethat
Kϕa(1)(x) = (1− |a|2)(n−2)/2
[x, a]n−2 = (1− |a|2)(n−2)/2R−2(x, a). (20) For aproof ofthe following estimate of the weightedintegralsof powers of Rq(x,y),see [17,Theorem 1.5].
Lemma 3.3.Letq∈ R,a> 0, andb>−1.Set c= a(n+ q)− (n+ b).Then
B
|Rq(x, y)|a(1− |y|2)bdν(y)∼
⎧⎪
⎪⎪
⎪⎪
⎨
⎪⎪
⎪⎪
⎪⎩ 1
(1− |x|2)c, if c > 0;
1 + log 1
1− |x|2, if c = 0;
1, if c < 0.
3.3. Atomicdecomposition ofharmonic Bergman-Besovspaces
Every function in the space bpq can be writtenas an infinite sum of reproducing kernels (atoms). For harmonicBergmanspaces(q >−1)thisisprovedin[10] andisextendedtoallq∈ R in[17,Theorem10.1]
whenp≥ 1 andin[13,Theorem1.4] when0< p< 1.Todescribethisatomicdecompositionweneedsome definitions.
Fora,b∈ B,thepseudo-hyperbolicmetric dH(a,b) isdefinedbydH(a,b):=|ϕa(b)|.Itiswellknownthat dH isMöbiusinvariant,
dH(ϕ(a), ϕ(b)) = dH(a, b) (ϕ∈ M(B), a, b ∈ B). (21) Fora∈ B and 0< r < 1,letD(a,r):={x∈ B : dH(x,a)< r} bethe pseudo-hyperbolicballwith center a andradius r. Asequence (am) inB is calledr-separated if theballs D(am,r) arepairwise disjoint.The sequence(am) iscalled anr-lattice ifB =∪∞m=1D(am,r) and(am) isr/2-separated.
Theorem3.4. Let0< p<∞ andq∈ R.Choose s suchthat q + 1 < p(s + 1), if p≥ 1;
q + n < p(s + n), if 0 < p < 1.
Thereexistsan r0> 0 such that if(am) isanr-latticewith r < r0,thenthefollowinghold:
(i) Forevery (cm)∈ p,thefunction
f (x) =
∞ m=1
cm(1− |am|2)s+n−(q+n)/pRs(x, am) (22)
isin bpq andfbpq (cm)p.The series in(22) convergestof absolutely anduniformlyon compact subsets ofB and also in· bpq.
(ii) For every f ∈ bpq,there exists(cm)∈ p with(cm)p fbpq suchthattherepresentation (22) holds.
4. Kelvin-Möbius-invariantharmonicBergman-Besov spaces
InthissectionweconsiderharmonicBergman-Besovspacesbqpforthewholerange0< p<∞,q∈ R and proveTheorems1.8and1.14.Thatis,weshowthattheproperties(i)-(iii)inDefinition1.3holdforbpqifand onlyifq = p(n− 2)/2− n. Notethatclearly1∈ bpq foreveryp andq andpoint-evaluation functionalsare boundedonallbpqby[17,Theorem13.1] and[13,Theorem5.1].Thereforeallweneedtocheckiscondition (iii)ofDefinition1.3.
WefirstprovetheifpartsofTheorems1.8and1.14anddefertheonly-ifparts totheendofthesection.
Webeginwiththecase0< p≤ 1 which wehandlebyusingatomicdecomposition.
Proposition4.1. Letn≥ 3,0< p≤ 1 andq = p(n− 2)/2− n.ThenthereexistsaconstantC > 0 suchthat
Kϕ(f )bpq ≤ Cfbqp forevery f ∈ bpq andϕ∈ M(B).
Proof. We apply Theorem 3.4 with s = −2 which is possible since n ≥ 3. Let r0 be as asserted in that theorem and pickanr-lattice (am) withr < r0. Letf ∈ bpq and ϕ∈ M(B) bearbitrary.ByTheorem 3.4 (ii),there exists(cm)∈ p with(cm)p fbpq suchthat
f (x) =
∞ m=1
cm(1− |am|2)(n−2)/2R−2(x, am) =
∞ m=1
cmKϕam(1)(x),
where the second equalityfollows from (20).Apply Kϕ tof and passitthrough thesumto each termof theseries.ThisispossiblebyLemma2.2(v)andtheuniformconvergenceoftheseriesoncompactsubsets of B.ApplyingalsoLemma2.2(ii)weobtain
Kϕ(f ) =
∞ m=1
cmKϕ◦ Kϕam(1) =
∞ m=1
cmKϕam◦ϕ(1).
Let bm= ϕ−1(am). By(21), (bm) isan r-latticetoo. Because (ϕam◦ ϕ)(bm)= 0, there exist Um ∈ O(n) suchthatϕam◦ ϕ= Um◦ ϕbm and
Kϕ(f ) =
∞ m=1
cmKϕam◦ϕ(1) =
∞ m=1
cmKUm◦ϕbm(1) =
∞ m=1
cmKϕbm(1),
where the last equalityholds becauseKUm◦ϕbm(1)=Kϕbm ◦ KUm(1) and KUm(1)= 1 by(13). Using(20) again weobtain
Kϕ(f )(x) =
∞ m=1
cm(1− |bm|2)(n−2)/2R−2(x, bm).
Finally,Theorem 3.4(i)impliesKϕ(f )bpq (cm)p and weconcludethatKϕ(f )bpq fbpq. Proposition4.1showsthatproperty(iii)inDefinition1.3holdsforbpq when0< p≤ 1,q = p(n−2)/2−n and thisprovestheifpartof Theorem1.14.WeseparatetheBanachspacecasep= 1.
Corollary 4.2.The Bergman-Besovspaceb1−(1+n/2) isKelvin-Möbiusinvariant.
We nextprovetheifpartofTheorem 1.8. Inviewof Lemma1.7weonlyneed todealwith theq≤ −1 case, thatis, we needto establish thatthespaces corresponding to thedashed partof the rayl in Fig.1 are Kelvin-Möbiusinvariant.
Note thatontherayl inFig.1,whenq = 0, thecorresponding p is2n/(n− 2) andbyLemma1.7,the unweightedBergmanspaceb2n/(n0 −2) isKelvin-Möbiusinvariant.Next,observethatthecoordinatesofthe leftendpointoftherayl arep= 1,q =−(1+n/2),andthecorrespondingspaceb1−(1+n/2)isKelvin-Möbius invariant byCorollary4.2. Tofinishtheproof itsufficesto showthatthespacescorresponding to theline segmentjoiningthepoints(1,−(1+ n/2)) and(2n/(n− 2),0) inFig.1areKelvin-Möbiusinvariant.Wedo this byusingcomplexinterpolation.
ThecomplexinterpolationspacebetweentwoBergman-Besovspacesisdeterminedin[17,Theorem13.5]
whichwe repeatbelow.
Theorem 4.3.Let1≤ p0< p1<∞ and q0,q1∈ R. If 1
p= 1− θ p0
+ θ p1
(23) forsome 0< θ < 1 and
q
p= (1− θ)q0
p0
+θq1 p1
, (24)
then[bpq00,bpq11]θ,thecomplexinterpolation spacebetweenbpq00 andbpq11,isbpq.
Proof of the if part of Theorem1.8. Withp0= 1,q0=−(1+n/2),p1= 2n/(n−2),q1= 0,andp0< p< p1, ifθ satisfies(23) andq satisfies(24),thenthecomplexinterpolation spacebpq givenbyTheorem 4.3lieson therayl inFig.1.
Now consider the linear transformation Kϕ(f ) acting on f . Corollary 4.2 says thatKϕ is bounded on b1−(1+n/2), and Lemma1.7 says thatKϕ is bounded onb2n/(n−2)0 , both uniformly for ϕ∈ M(B). On the other hand, [8, Theorem 4.1.2] says that the complex interpolation method is an interpolation functor, whichmeansthatKϕisalsoboundedonbpqwith1< p< 2n/(n− 2) and q = p(n− 2)/2− n uniformlyfor ϕ∈ M(B).Consequentlythebpq onthedashedpartoftherayareallKelvin-Möbiusinvariant.
We now deal with the only-if parts of Theorems 1.8 and 1.14 and show that if q = p(n− 2)/2− n, then property (iii) inDefinition 1.3 does nothold. This includes theBergman (q > −1) caseasserted in Lemma1.7too.
Proof of the only-if parts of Theorems1.8and1.14. Consider thefunction f = 1.Wehave1bpq = 1 and by(20)
Kϕa(1)(x) = (1− |a|2)(n−2)/2R−2(x, a),
fora∈ B.Pickt∈ R suchthatq + pt>−1. ApplyingtheoperatorDt−2 andusing(18) weobtain D−2t Kϕa(1) = (1− |a|2)(n−2)/2D−2t R−2(x, a) = (1− |a|2)(n−2)/2Rt−2(x, a).
Thusby(19)
Kϕa(1)pbpq ∼ (1 − |a|2)p(n−2)/2
B
|Rt−2(x, a)|p(1− |x|2)q+ptdν(x).
We estimatethe aboveintegralusing Lemma3.3. We check three distinct casesdepending onthe signof c= p(n− 2)− q − n.
Ifp(n− 2)− q − n> 0,then
Kϕa(1)pbpq ∼ 1
(1− |a|2)p(n−2)/2−q−n, andright-handsidetendsto ∞ or0 as|a|→ 1− unlessq = p(n− 2)/2− n.
Ifp(n−2)−q−n= 0,thenKϕa(1)pbpq → 0 as|a|→ 1−since(1−|a|2)p(n−2)/2dominateslog(1/(1−|a|2)).
Ifp(n− 2)− q − n< 0,thenKϕa(f )pbpq → 0 as |a|→ 1−.
ThusKϕa(1)bpq ∼ 1bpq = 1 as|a|→ 1− onlyifq = p(n− 2)/2− n.
Remark4.4.Wefinishthissectionbylookingatanotherclassofharmonicfunctionspaces,harmonicHardy spaces.For1≤ p<∞,thespacehp consistsofallf ∈ h(B) suchthat
fphp := sup
0≤r<1
S
|f(rξ)|pdσ(ξ) <∞.
Forϕ∈ M(B), thenorm ofKϕ: hp → hp is computed in[19,Theorem 1.2].It follows fromthis theorem thatwhen n ≥ 3,hp is Kelvin-Möbius invariantif and onlyif p= 2(n− 1)/(n− 2). Wenote thatin the
