On the Π-operator in Clifford analysis

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Journal

of

Mathematical

Analysis

and

Applications

www.elsevier.com/locate/jmaa

On

the

Π-operator

in

Clifford

analysis

Ricardo Abreu Blayaa, Juan BoryReyesb, Alí GuzmánAdánc, Uwe Kählerd,∗ a

FacultaddeInformáticayMatemática,UniversidaddeHolguín,Holguín80100,Cuba

b

ESIME-Zacatenco,InstitutoPolitécnicoNacional,MéxicoDF07738,Mexico

c_{CliffordResearchGroup,DepartmentofMathematicalAnalysis,FacultyofEngineeringand}

Architecture,GhentUniversity,Galglaan 2,9000Ghent,Belgium

d _{DepartamentodeMatemática,UniversidadedeAveiro,Portugal}

a r t i c l e i n f o a b s t r a c t

Article history:

Received2July2014

Availableonline30September2015 SubmittedbyRichardM.Aron

Keywords:

Cliffordanalysis Teodorescutransform Pi-operator

Beltramiequation

In this paper we prove that a generalization of complex Π-operator in Clifford analysis,obtainedbytheuseoftwoorthogonalbases ofaEuclideanspace,possesses severalmappingandinvertibilityproperties,asstudiedbeforeforquaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalizemostof those previous results inthisdirection andadditionally other consequentresultsarepresented. Inparticular, theexpression ofthejumpofthe generalizedΠ-operatoracrosstheboundaryofthedomainisobtainedaswellasan estimateforthenormoftheΠ-operatorisgiven.Attheendanapplicationofthe generalizedΠ-operator tothesolutionof Beltramiequationsisstudiedwherewe giveconditionsforasolutiontorealizealocalandglobalhomeomorphism.

© 2015ElsevierInc.All rights reserved.

1. Introduction

In[30],Vekuaestablishedthemainpropertiesofthemostimportantintegraloperatorsincomplex anal-ysis.Inparticular,thecomplexΠ-operator,which playsamayor roleinthetheoryofgeneralizedanalytic functionsaswellasinthebranchofthecomplexanalysisdeeplyconnectedwithpartialdifferentialequations byusingfunctionalanalyticmethods,wasintroducedandstudiedindetail.Generalizationsofthecomplex Π-operator to higher-dimensional versions are already considered in recent times [7,9,10,22,23,25,27–29]. Although athorough treatmentis listed, it is neverthelessrestricted mainlyto the setting of quaternion-valued functions and there is still a strong need for developing further details.The advances achieved in higher dimensions use only the standard orthogonal basic, without achieving a significant progress. The latter is whyinthis paper we confineattention to thegeneralizedClifford analysis setting bythe helpof two arbitraryorthogonalbases.

* Correspondingauthor.

E-mailaddresses:[email protected](R. AbreuBlaya),[email protected](J. BoryReyes), [email protected](A. GuzmánAdán),[email protected](U. Kähler).

http://dx.doi.org/10.1016/j.jmaa.2015.09.038 0022-247X/© 2015ElsevierInc.All rights reserved.

Startingfrom hisdefinitionofageneralizedTeodorescu transform,see [27,28],Sprössigproposeda gen-eralization of the complex Π-operator in the Clifford analysis setting, which turns out to have most of the usefulproperties ofits complexorigin. Thedefinition is statedin termsof Teodorescu transform and Cauchy–Riemann operatorarising instandardCliffordanalysis ofaEuclidean space. In[23,25]the gener-alizationofthecomplexΠ-operatorisbasednowintwodifferentorthonormalbases associatedrespectively with theTeodorescu transformand Cauchy–Riemann operator.

Inthe study of thepropertiesof the generalizationof Π-operator, oneof thequalitativedisadvantages of the changeof standard orthonormal basicand its conjugate for two different orthonormal bases is the absenceofageneralizedBorel–Pompeiu representationformula,whichallowstoextendtothecorresponding situationsaseries offundamentalapplications of itsstandardantecedent. Recently,in[1]there was given suchaBorel–Pompeiu representationformula,butinthecontextofquaternionicanalysis.Inthispaperwe offerdirectextension oftheformulatotheCliffordanalysissetting.

An important question, which is in the focus of the present paper, is the study of the jump of the generalizedΠ-operatoracross theboundaryofthedomain.

One of the important applications of the complex Π-operator is to the solution of Beltrami systems. Higher-dimensional Beltrami systems in the framework of quaternions were first studied by Shevchenko in1962 [26], and later onby P. Cerejeiras, K. Gürlebeck, U. Kähler,and others [6,5,9,10,14–16]. Hereby, also thequestionof local and global homeomorphicsolutions wasstudied. Lateron A. Koski[17] studied thesolvability ofBeltrami equationsforVMO-coefficients.Forthestudy ofBeltrami equationsinthelast sectionwefirstgiveestimatesforthenormofthegeneralizedΠ-operatorinSection4.Thiswillallowusin thelastsectionto applyourΠ-operatorto thesolutionof Beltramiequations,andto give conditionsfora solutiontorealizealocal andglobalhomeomorphism.

2. Preliminaries

Let {e1,e2,. . . ,en} be an orthonormal basis of Rn. Consider the 2n-dimensional real Clifford algebra

R0,n generated bye1,e2,. . . ,en accordingtothe multiplicationruleseiej+ ejei =−2δi,j where δi,j is the

Kronecker’ssymbol.TheelementseA: A⊆ Nn:={1,2,. . . ,n} defineabasisofR0,n,whereeA= eh1· · · ehk

if A = {h1,. . . ,hk} (1 ≤ h1 < · · · < hk ≤ n) and e∅ = e0 = 1. Any a ∈ R0,n may thus be written as

a=_A⊆NnaAeA whereaA∈ R orstill asa=n_k=0[a]k,where[a]k=_|A|=kaAeA isaso-calledk-vector

(k∈ N0

n :=Nn∪ {0}).Ifwedenotethespaceofk-vectors byR(k)0,n,thenR0,n=

n k=0⊕R

(k) 0,n.

In suchaway, thespaces R and Rn _{will be identified with R}(0) 0,n and R

(1)

0,n respectively. Moreover, each

elementx= (x0,x1,. . . ,xn)∈ Rn+1 canbewrittenas

x = x0+ n  i=1 xiei∈ R(0)0,n⊕ R (1) 0,n

andtheyare oftencalledparavectors.Foreachx∈ R(0)_0,n⊕ R(1)_0,nitisremarkablethat

x¯x = ¯xx = x2₀+ x2₁+· · · + x2_n=|x|2. (1) Theextensionof(1)to anorm ofa∈ R0,nisstraightforwardandleadsto

|a|2_{= [a¯a]}

0= [¯aa]0=



A

a2_A.

Wewillusetheconjugation,definedby¯a =_A⊆N

naA¯eA,where ¯ eA:= (−1)kehk· · · eh1 = (−1) k(k+1) 2 e A, if eA= eh1· · · ehk.

ThefollowingpropertiesofthenormandconjugationinCliffordalgebrasarewell-knownandcanbefound inmanysources,seeforinstance[8].

Proposition 1.Leta,b∈ R0,n,then

(i) ab = ¯b ¯a,

(ii) |¯a|=|− a|=|a|,

(iii) [a¯b]0= [¯ab]0=a,bR2n,where ·,·Rm denotes thestandardscalarproductinRm, (iv) |ab|≤ 2n/2_|a||b|,

(v) if b issuchthat b¯b =|b|2,b = 0, thenb isinvertible and|ab|=|ba|=|a||b|.

Suppose Ω is abounded domain,with sufficiently smooth boundaryΓ, inRn+1_{. We will be interested}

in functions f : Ω → R0,n, which might be written as f (x) =



AfA(x)eA with fA R-valued. Property,

suchcontinuity,differentiability,integrability,andsoon,canbeascribedcoordinatewisely ordirectly.A left (unitary) module over R0,n (left R0,n-module for short) is a vector space V together with an algebra

morphism L:R0,n→ End(V ),orto sayitmoreexplicitly,thereexists alineartransformation(alsocalled

left multiplication)L(a) ofV suchthat

L(ab + c) = L(a)L(b) + L(c)

for all a ∈ R0,n, and L(1) is the identity operator.In the same way we have aright (unitary)module if

there isaso-calledright multiplicationR(a)∈ End(V ) suchthat

R(ab + c) = R(b)R(a) + R(c).

Giveneitheraleftorarightmultiplicationwecanalwaysconstructarightoraleftmultiplicationbyusing any anti-automorphismofthealgebra,forinstance

R(a) = L(a).

Abi-moduleisamodulewhichisbothaleft- andaright-module,orwithotherwords,amodulewhereleft and rightmultiplications commute,i.e.

L(a)R(b) = R(b)L(a), for all a, b∈ R0,n.

IfV isavectorspaceofR0,n-valuedfunctionweconsidertheleft(right)multiplicationdefinedbypoint-wise

multiplication

(L(a)f )(x) = a(f (x)) and (R(a)f )(x) = a(f (x)).

AlsoamappingK betweentwo rightmodulesV andW iscalled anR0,n-linearmappingif

K(f a + g) = K(f )a + K(g).

We should also mention whatis understanding inthis paper by a (left or right) Clifford Banachmodule (see[18]forexample).WesaythatX isaleftBanachR0,n-moduleifX isaleftR0,n-moduleandX isalso

arealBanachspacesuchthatforany a∈ R0,nandx∈ X:

for some k > 0. In particular, equality occurs in (2) if a ∈ R. Similarly one can define a right Banach R0,n-module.

These considerationsgive rise to the following right modules of R0,n-valued function defined over any

suitablesubsetE ofRn+1: • Ck_(E,R

0,n),k∈ N ∪ {0} – theright moduleof allR0,n-valued functions,k-times continuously

differ-entiable inE.It becomesarightBanachmodulewiththenorm

fCk= sup x∈E  |α|≤k |Dα wf (x)|

where α denotesamulti-indexandDα_{thecorrespondingpartialderivative.Inparticular,wealsohave}

C∞(E,R0,n):=

∞

k=0Ck(E,R0,n).Letusremarkthattheabovenormisequivalenttothenormcoming

from theinductivelimit.

• By using the corresponding Hölder-semi-norm we can introduceC0,μ_(E,R

0,n), μ ∈ (0,1] asthe right

Banachmoduleofallμ-HöldercontinuousandR0,n-valued functionsinE.

• Lp(E,R0,n) (1≤ p <∞) denotes the right module of all equivalence classesof Lebesgue measurable

functionsf : E −→ R0,n forwhich|f|p isintegrableover E.Withthenorm

fLp(E,R0,n):= ⎛ ⎝ E |f(ξ)|p_dξ ⎞ ⎠ 1/p <∞,

Lp(E,R0,n) becomesarightBanachmodule.

• C₀∞(E,R0,n) –thespaceofallinfinitelydifferentiablefunctionswithcompactsupportinE.An

impor-tant propertyofthespaceC₀∞(E,R0,n) isitsdensityinthespacesC0(E,R0,n) andLp(E,R0,n).

Furthermore, the above proposition allows us to introduce the following sesquilinearform (also called symmetricinnerproduct intheliterature)fortwofunctionsf,g : Ω→ R0,n

(f, g) := f g.

Inparticular, thissesquilinearform isareal-bilinear mappingandaClifford-linear mappinginthesecond argument.Furthermore, thissesquilinearform givesriseto anR0,n-valuedsesquilinearform

f, g2=



E

f (ξ)g(ξ) dξ.

Thisform satisfiesthefollowingproperties: 1. ·,·₂issesquilinear

2. f, gb₂=f, g₂b withb∈ R0,n

3. f, g₂=g, f₂

Furthermore,therightR0,n-moduleL2(E,R0,n) equippedwiththissesquilinearformiscompleteunderthe

norm f =     E [f (ξ)g(ξ)]0dξ, (3)

which makesitaClifford–Hilbertmodule. Manyfactsfrom classicHilbertspacescarryover tothe notion of aHilbert module. Inparticular, ageneralization of Riesz’ representation theorem is valid in the sense thatalinearfunctionalφ iscontinuousifandonlyifitcanberepresentedbyanelementfφ∈ V suchthat

φ(g) =fφ, g2.

Additionally, there are some importantinequalities involving the sesquilinear form and the norm coming from thereal-valuedinner product:

• Cauchy–Schwarzinequality:|f,g2|≤ 2n/2fLp(E,R0,n)gLq(E,R0,n)with

1 p + 1 q = 1 • afL2(E,R0,n)≤ 2 n/2_|a|f

L2(E,R0,n)foralla∈ R0,n,butafL2(E,R0,n)=|a|fL2(E,R0,n)whenevera

isaparavectoror belongstotheparavectorgroup • fL2(E,R0,n)≤ |f,f2|≤ 2

n/2_f

L2(E,R0,n)

• fL2(E,R0,n)≤ supg_L2(E,R0,n)≤1|f,g2|≤ 2

n/2_f

L2(E,R0,n)

The same statements are also true for any other Clifford Hilbert module coming from a tensor product betweentheelements oftheCliffordbasisandelements ofarealor complexHilbertspace[12].

Inthesameway wecanintroduceS asthecorresponding Schwartzspaceofrapidly decayingfunctions. Its dual spaceS given by thecontinuous linearfunctionals isthespace oftempered distributions.Again, this canbe either definedcomponentwisely or viathe sesquilinear form f,g2 butthe space S is again

considered as a right module. Let us remark that strictly speaking if we consider S as a Fréchet right module its algebraicdual S is thespace of allleft-R0,n-linearfunctionals over S, butit canbe identified

withelementsofaright-linearmodulebymeansofthestandardanti-automorphismintheabovementioned way.

ThisallowsustointroducetheSobolev spaceWk

p(E,R0,n),k∈ N∪ {0},1≤ p<∞,as therightmodule

of allfunctionals f ∈ S whose derivatives1 _Dα

wf for|α|≤ k belongto Lp(E,R0,n),withnorm

fWp(E,R0,n):= ⎛ ⎝  α≤k Dα wf p Lp(E,R0,n) ⎞ ⎠ 1/p .

LetusremarkthatweonlyconsiderSobolevspaceswithnon-negativeexponent,whichmeansthatduality discussions between Sobolev spaces are notrequired for this paper, butcan be dealtwith similarly as to theaboveargument.Formoredetailswerefertotheclassicbook[4].

Letψ :={ψ0,ψ1,. . . ,ψn}⊂ R(0)_0,n⊕ R(1)_0,n. Byabuseofnotation,we willwriteψ :={ψ0_,ψ1_{,. . . ,ψ}n_}.On

theset C1_(Ω,R

0,n) wedefinerespectivelytheleftandtheright Cauchy–Riemannoperators by:

ψ_{D[f ] :=} n  i=0 ψi∂f ∂xi , Dψ[f ] := n  i=0 ∂f ∂xi ψi. (4)

Forsimplicity ofnotationwecontinuetowrite ∂i insteadof

∂ ∂xi

.

LetΔn+1be the(n+ 1)-dimensionalLaplaceoperator.It iseasytoprovethattheequalities ψ_D·ψ_{D =}ψ_D·ψ_{D = D}ψ_{· D}ψ_{= D}ψ_{· D}ψ _{= Δ} n+1, (5) 1 _Dα wf = ∂|α|f ∂xα0 0 ···∂xαnn

hold,ifandonlyif

ψi· ψj_{+ ψ}j_{· ψ}i _{= 2δ}

i,j, i, j∈ N0n.

Notethatlastequalityyields

2δi,j= ψi· ψj+ ψj· ψi= ψi· ψj+ ψi· ψj = 2 ψi· ψj 0= 2  ψi, ψj_Rn+1, (6)

thus,factorization(5)holds ifand onlyifψ representsanorthonormalbasisofRn+1∼=R(0)0,n⊕ R (1) 0,n.

Any set ψ with the property (6) is called structural set. It is evidentthat ψ and ψ are structural sets simultaneously.Onecanfindbasicpropertiesofthestructuralsets in[23,24]andthereferencesgiventherein.

Remark 2.1. In the case n = 1 (R0,1 ∼= C), the only structural sets that we can find have the form

ψ ={eiθ_,±ieiθ_{} where0≤ θ ≤ 2π.Inotherwords,incomplexanalysis,allstructuralsetsareequivalent to}

oneofthesets {1,i} or{1,−i}.

Forfixedψ and Ω weintroducethesets

ψ_M(Ω,R

0,n) := kerψD ={f ∈ C1(Ω,R0,n) :ψD[f ] = 0Ω},

Mψ(Ω,R0,n) := ker Dψ={f ∈ C1(Ω,R0,n) : Dψ[f ] = 0Ω}.

Sometimes,elementsofthesesetsarecalledψ-hyperholomorphicfunctions(leftorrightrespectively).Denote byΘn+1thefundamentalsolutionoftheLaplaceoperatorΔn+1

Θn+1(x) =

1

(1− n)|Sn_||x|

1−n_.

Of centralimportanceis thefundamental solution(= theCauchy kernel forthe corresponding theory) oftheoperatorsψ_{D andD}ψ_{,whichisgiven, thankstotherelation(5),by:}

Kψ(x) :=ψD[Θn+1](x) = Dψ[Θn+1](x) =

x_ψ |Sn_{| · |x|}n+1.

Here x_ψ :=n_i=0xiψi ifx=

n

i=0xiei and |Sn| is theareaoftheunitsphereSn inRn+1.Thekernel Kψ

hasthefollowingimportantproperties: 1. Kψ∈ C∞(Rn+1\{0},R0,n).

2. Kψ∈ψM(Rn+1\{0},R0,n)∩ Mψ(Rn+1\{0},R0,n).

2.1. Stokes formula

OneofthemostcrucialfactsinstandardCliffordanalysis istheexistenceofaStokesformula.Here we presentthis formulaforanarbitrarystructural setψ, seeforinstance[23]and elsewhere.

Let Ω⊂ Rn+1 _{be abounded domain with a sufficientlysmooth boundary Γ.Thefollowing theorem is}

Theorem 1(Gauss’formula). Forf ∈ C1_{(Ω,R) andany i∈ N}0 n wehave  Γ ni(ξ)· f(ξ) dΓξ =  Ω ∂i[f ](ξ) dξ

where ni(ξ) isthei-th component oftheoutwardunitnormal vectoron Γ atthepointξ∈ Γ.

As aconsequenceaStokesformulaforR0,n-valuedfunctionsyieldsthefollowing theorem.

Theorem 2(Stokesformula). Letf,g∈ C1_(Ω,R

0,n) andψ beastructuralset.Then

 Γ g(ξ)· nψ(ξ)· f(ξ) dΓξ=  Ω  Dψ[g](ξ)· f(ξ) + g(ξ) ·ψD[f ](ξ)dξ, (7) where nψ(ξ)=ni=0ni(ξ)ψi.

Proof. UsingtheGauss’sformulacoordinatewisely directlywehave:  Ω  Dψ[g]f + gψD[f ]dξ =  Ω n  i=0  ∂i[g]ψif + g ψi∂i[f ]  dξ =  Ω n  i=0 ∂i  g ψifdξ = n  i=0  Ω ∂i  g ψifdξ = n  i=0  Γ ni  g ψifdΓξ =  Γ g  _n  i=0 niψi  f dΓξ =  Γ g nψf dΓξ. 2

Remark 2.2. In the case n = 1, for the structural sets {1,i} and {1,−i}, and taking into account the commutativityinC,onecanobtainthecomplexversionsof(7),see forexample[30, p. 24].

2.2. Integraloperators

TheCauchykernelgeneratesthefollowingtwo importantintegrals:

ψ_T Ω[f ](x) :=−  Ω Kψ(ξ− x) · f(ξ) dξ, x ∈ Rn+1, and ϕ,ψ_K Γ[f ](x) :=  Γ Kϕ(ξ− x) · nψ(ξ)· f(ξ) dΓξ, x /∈ Γ.

While the first is a generalization of the usual Teodorescu transform, the second represents a boundary

exotic operator whichconnects twoarbitrarystructural sets ϕ andψ. When ψ = ϕ,ϕ,ψ_K

usualCauchytypeintegral ψ_K Γ[f ](x) :=  Γ Kψ(ξ− x) · nψ(ξ)· f(ξ) dΓξ.

Thesingularversionofϕ,ψ_K

Γ[f ] onΓ,denotedbyϕ,ψSΓ[f ],isgiven,as usual,by

ϕ,ψ_S

Γ[f ] := 2ϕ,ψKΓ[f ].

Theintegralwhichdefines theoperatorϕ,ψ_S

Γ[f ] isto betakeninthesense oftheCauchyprincipalvalue.

Inthecaseofϕ= ψ,itisknown(see[23])theexactrelationbetweentheboundaryvalueofψKΓ[f ] and

ψ_S

Γ[f ]:=ψ,ψSΓ[f ].

LetusintroducethenotationsΩ+:= Ω and Ω− :=Rn+1\ Ω,thenthefollowingresultholds. Theorem3(Plemelj–Sokhotski formulas).Letf ∈ C0,μ(Γ,R0,n), μ∈ (0,1]. Thenwehave:

ψ_K± Γ[f ](t) := lim Ω±_x→t∈Γ ψ_K Γ[f ](x) = 1 2 _ψ SΓ[f ](t)± f(t)  .

TheStokesformulaleadsimmediatelytothreeimportantconsequences,whicharewidelyknownandcan befound inmanysources.

Theorem 4 (Cauchy integral theorem). Let f ∈ ψ_M(Ω,R

0,n)∩ C0(Ω,R0,n) and g ∈ Mψ(Ω,R0,n)∩

C0(Ω,R0,n). Then



Γ

g(ξ)· nψ(ξ)· f(ξ) dΓξ = 0. (8)

Theorem5(Borel–Pompeiu(Cauchy–Green) formula).Letf ∈ C1_(Ω,R

0,n).Then  Γ Kψ(ξ− x) · nψ(ξ)· f(ξ) dΓξ−  Ω Kψ(ξ− x) ·ψD[f ](ξ)dξ =  f (x) if x∈ Ω, 0 if x∈ Rn+1\ Ω.

Theorem6(Cauchy integralformula). Letf ∈ψM(Ω,R0,n)∩ C0(Ω,R0,n).Then

 Γ Kψ(ξ− x) · nψ(ξ)· f(ξ) dΓξ=  f (x) if x∈ Ω, 0 if x∈ Rn+1_{\ Ω.} (9)

Remark2.3. Particularly,for ϕ= ψ = ψst :={1,e1,. . . ,en},Theorems 4, 5 and 6havebeen knownfora

longtime,seeforinstance [7,9,11].

3. ϕ,ψ_Π

Ω-operatoranditsintegralrepresentation

From[11] itisknownthatinthecaseofthestandardstructural setψst of R0,n theoperator

ψst_T

Ω: Wpk(Ω,R0,n)→ Wpk+1(Ω,R0,n)

isacontinuousmappingfor1< p<∞,k∈ N.Usinganorthogonaltransformationofcoordinatesweobtain thesamepropertyforgeneralstructuralsets.Therefore,theoperatorϕ_Dψ_T

ΩiswelldefinedonLp(Ω,R0,n)

andonC0,μ_(Ω,R

ψst_Dψst_T

Ω= I,

where I standsfortheidentity operator.

Definition 1.Forapairofstructuralsetsϕ,ψ wedefine theoperatorϕ,ψ_Π

Ωby

ϕ,ψ_Π

Ω:=ϕDψTΩ.

Applying thedifferentialtheoryof singularintegraloperatorswe getanintegralrepresentationformula for the ϕ,ψ_Π

Ω-operator,whichgeneralizes those obtainedin[7, Theorem 4] and [9,Theorem 2]as well as

[10, Theorem 3.2]forthequaternionic case.

Theorem 7.LetΩ∈ Rn+1_{andf ∈ L}

p(Ω,R0,n), p∈ (1,∞), then forallx∈ Ω we have

ϕ,ψ_Π Ω[f ](x) =  Ω  n i=0ϕiψi |Sn||ξ − x|n+1 − (n + 1)(ξ− x)ϕ(ξ− x)ψ |Sn||ξ − x|n+3  f (ξ) dξ + n i=0ϕiψi n + 1 f (x) =  Ω ϕ,ψΛ(ξ− x)f(ξ) dξ + n i=0ϕiψi n + 1 f (x), (10) where ϕ,ψΛ(ξ− x):=−ϕDxψDξ[Θn+1(ξ− x)].

Proof. From [19, Theorem 11.1, Chapter XI] we get for f ∈ Lp(Ω,R0,n), i ∈ N0n, that ∂i

_ψ TΩ[f ]  ∈ Lp(Ω,R0,n).Moreover, wehave ∂i _ψ TΩ[f ]  (x) = −1 |Sn_| ⎧ ⎪ ⎨ ⎪ ⎩  Ω ∂i  _{(ξ− x)} ψ |ξ − x|n+1  f (ξ) dξ−  ∂B[0,1] u(x, θ) cos(r, xi)dSθf (x) ⎫ ⎪ ⎬ ⎪ ⎭, where r =|ξ − x|,r = ξ− x,θ = ξ− x |ξ − x| and u(x,θ)= (ξ− x)_ψ |ξ − x| = θψ.

It iseasilyseenthattheoutward unitnormalvector on∂B[0,1] atthepointθ ispreciselyθ.Moreover, taking intoaccountthatcos(r,xi)=

ξi− xi

|ξ − x| isthei-thcomponentoftheunitvectorparalleltor,wehave

thatcos(r,xi)= ni(θ).Then, usingtheGauss’stheorem wegettheequality

 ∂B[0,1] u(x, θ) cos(r, xi)dSθ=  ∂B[0,1] ni(θ)θψ dSθ=  B[0,1] ∂θi θ_ψdθ = ψi π (n+1)/2 Γn+1 2 + 1 . Ontheother hand,

∂i  _{(ξ− x)} ψ |ξ − x|n+1  = −ψ i|ξ − x|2_{+ (n + 1)(ξ} i− xi)(ξ− x)ψ |ξ − x|n+3 .

Fromthemappingpropertiesofψ_T

Ω itiseasytoseethat

ϕ,ψ_Π

Ω: Wpk(Ω,R0,n)→ Wpk(Ω,R0,n),

for 1< p <∞ and k ∈ N∪ {0}. Applying thetheorem of Calderon and Zygmund(see [19, Chapter XI, 3.1])fork = 0 weobtainthefollowingresult:

Theorem8.Suppose1< p<∞.ThenforΩ⊆ Rn+1 wehave that

ϕ,ψ_Π

Ω: Lp(Ω,R0,n)→ Lp(Ω,R0,n)

isacontinuous mapping.

Particularly,forϕ= ψ,animportantconsequenceofTheorem 7follows.

Corollary 1. Let ψ be a structural set and Ω ⊂ Rn+1 _{a bounded domain. Then, for all f ∈ L}

p(Ω,R0,n), p∈ (1,∞), wehave ψ,ψ_Π Ω[f ](x) =ψDψTΩ[f ](x) =  f (x) if x∈ Ω, 0 if x∈ Rn+1_{\ Ω.}

Thisproperty canbe extendedto thecaseΩ=Rn+1_.

Corollary2.Letψ beastructuralset.Then, forallf ∈ Lp(Rn+1,R0,n), p∈ (1,∞), wehave

ψ,ψ_Π

Rn+1[f ](x) =ψDψT_Rn+1[f ](x) = f (x).

Proof. Supposethatf ∈ C₀∞(Rn+1_,R

0,n),thenthereexists asuitablecompactsubsetK⊂ Rn+1suchthat

f (x)= 0 for allx∈ K./ Using Corollary 1

ψ,ψ_Π

Rn+1[f ](x) =ψDψT_Rn+1[f ](x) =ψDψT_K[f ](x) =



f (x) if x∈ K, 0 if x∈ Rn+1\ K.

Then we get that ψDψT_Rn+1[f ](x) = f (x) for all x ∈ Rn+1. Finally, taking into account the density of C₀∞(Rn+1,R0,n) inLp(Rn+1,R0,n) andthecontinuityoftheoperatorψ,ψΠRn+1 inthespaceL_p(Rn+1,R_0,n),

weobtainthedesiredresult. 2

3.1. Generalized Borel–Pompeiuformula

We are now proceeded to derive an essentialintegral formula, to be called generalized Borel–Pompeiu formula,whichexpresses a veryprofoundrelationamong theoperatorsϕ_D,ψ_T

Ω andϕ,ψKΓ.

Theorem9(GeneralizedBorel–Pompeiuformula). Letf ∈ C1_(Ω,R

0,n). Then: ϕ,ψ_Π Ω[f ](x) =  Γ Kϕ(ξ− x) · nψ(ξ)· f(ξ) dΓξ−  Ω Kϕ(ξ− x) ·ψD[f ](ξ) dξ, x /∈ Γ. (11)

Remark3.1.Theaboveformulamaybewritteninashorterwayas:

ϕ,ψ_Π

Proof. Letx∈ Ω.UsingtheStokesformula(7)forthestructuralsetψ andthefunctionsg(ξ)= Θn+1(ξ−x)

andf (ξ),inthemultiplyconnecteddomainΩx, := Ω\ B[x,] (> 0 small) wehavetheequality

 Γ Θn+1(ξ− x) · nψ(ξ)· f(ξ) dΓξ−  ∂B(x, ) Θn+1(ξ− x) · nψ(ξ)· f(ξ) dΓξ =  Ωx, Kψ(ξ− x) · f(ξ) dξ +  Ωx, Θn+1(ξ− x) ·ψD[f ](ξ) dξ. (12)

Further,foreveryξ∈ ∂B(x,) wehave !!

!Θn+1(ξ− x) · nψ(ξ)· f(ξ)!!! =

|f(ξ)|

(n− 1)|Sn|n−1.

By hypothesis f is a continuous function in the compact set Ω and then, there exists M > 0 such that

|f(ξ)|< M ,∀ξ∈ Ω∪ Γ.Consequently, !! !! !! !  ∂B(x, ) Θn+1(ξ− x) · nψ(ξ)· f(ξ) dΓξ !! !! !! ! < M (n− 1)|Sn_|n−1  ∂B(x, ) dΓξ= M (n− 1), whichimpliesthat

lim

→0



∂B(x, )

Θn+1(ξ− x) · nψ(ξ)· f(ξ) dΓξ = 0. (13)

Ontheother hand,

−ψ_{T [f ](x) =}  Ω Kψ(ξ− x) · f(ξ) dξ := lim →0  Ωx, Kψ(ξ− x) · f(ξ) dξ, (14) and  Ω Θn+1(ξ− x) ·ψD[f ](ξ) dξ := lim →0  Ωx, Θn+1(ξ− x) ·ψD[f ](ξ) dξ. (15)

Letting→ 0 in (12)andusing (13), (14)and(15)weseethat

ψ_T Ω[f ](x) =  Ω Θn+1(ξ− x) ·ψD[f ](ξ) dξ−  Γ Θn+1(ξ− x) · nψ(ξ)· f(ξ) dΓξ.

Applying theoperatorϕ_{D in bothsides oftheaboveequalityweobtain:} ϕ_Dψ_T Ω[f ](x) =  Γ Kϕ(ξ− x) · nψ(ξ)· f(ξ) dΓξ−  Ω Kϕ(ξ− x) ·ψD[f ](ξ) dξ, and (11)isproved.

The proof forthecase x∈ Rn+1_{\ Ω issimilar. First, weuse againformula (7) butnow inΩ andthen}

thetaskis onlyto applytheoperatorϕ_{D inboth sides oftheresultingequality. 2}

4. Propertiesof theϕ,ψ_Π

Ω-operator

The ideas presented in [8,11] allow us to establish the following results about the decomposition of

L2(Ω,R0,n).

Theorem10.Letψ beastructuralset.Then,theHilbertspaceL2(Ω,R0,n) permitsthefollowingorthogonal

decompositionwith respecttotheinnerproduct(3)

L2(Ω,R0,n) =  L2(Ω,R0,n)∩ψM(Ω,R0,n)  ⊕ψ_D(W◦1 2 (Ω,R0,n)). (16) There ◦

W21(Ω,R0,n) denotesthesubspace ofallfunctionsof W21(Ω,R0,n) whosetracesin Γ vanish.

Theaboveorthogonaldecompositiongeneratesapairofmutuallycomplementaryorthoprojections

ψ_{P : L} 2(Ω,R0,n)−→ L2(Ω,R0,n)∩ψM(Ω,R0,n), ψ_{Q : L} 2(Ω,R0,n)−→ψD( ◦ W₂1(Ω,R0,n)).

Inthefollowingwestateandproveacollectionofpropertiesoftheϕ,ψ_Π

Ω-operatorextendingtheresults

givenin[7,9,14,31]fortheparticular caseofϕ= ψst andψ = ϑ= ψst.

Theorem 11. Let ϕ, ψ, ϑ be three arbitrary structuralsets and f ∈ W_p1(Ω,R0,n), 1 < p <∞. Then, the

followingequalities2 _{holdsin Ω}

ϕ_Dϕ,ψ_{Π[f ] =}ψ_{D[f ], (17)} ϕ,ψ_Πϑ_{D[f ] =}ϕ_Dψ,ϑ_{Π[f ]−}ϕ_Dψ,ϑ_K Γ[f ], (18) ϕ_K Γϕ,ψΠ[f ] = (ϕ,ψΠ−ϕTΩψD)[f ], (19) (ϕDϕ,ψΠ−ψ,ϑΠϑD)[f ] =ψDϑKΓ[f ], (20) ϕ_K Γϕ,ψΠψKΓ[f ] =ϕ,ψΠψKΓ[f ]. (21)

Proof. The proofs of these equalities are based in the use of the classical or generalized version of the Borel–Pompeiuformula. (17):ϕ_Dϕ,ψ_{Π[f ]=}ϕ_D(ϕ,ψ_K Γ+ϕTΩψD)[f ]=ψD[f ], (18):ϕ,ψ_Πϑ_{D[f ]=}ϕ_Dψ_Tϑ_{D[f ]=}ϕ_D(ψ,ϑ_Π−ψ,ϑ_K Γ)[f ], (19):ϕ_K Γϕ,ψΠ[f ]= (ϕ,ψΠ−ϕTΩϕDϕDψTΩ)[f ]= (ϕ,ψΠ−ϕTψDψDψT )[f ]= (ϕ,ψΠ−ϕTψD)[f ], (20):(ϕ_Dϕ,ψ_Π−ψ,ϑ_Πϑ_{D)[f ]= (}ψ_D−ψ_{D +}ψ_Dϑ_K Γ)[f ]=ψDϑKΓ[f ] (see (17)and(18)), (21):ϕ_K Γϕ,ψΠψKΓ[f ]= (ϕ,ψΠ−ϕTΩψD)(I−ψTΩψD)[f ]=ϕ,ψΠ(I−ψTΩψD)[f ]=ϕ,ψΠψKΓ[f ]. 2

Remark4.1. Someparticularchoicesofϑ allow todifferentversionsoftheaboveformulas.Indeed,making

ϑ= ψ andϑ= ϕ in theequalities (18)and (20)respectivelywehave

ϕ,ψ_Πψ_{D[f ] =}ϕ_{D[f ]−}ϕ_Dψ_K

Γ[f ],

(ϕDϕ,ψΠ−ψ,ϕΠϕD)[f ] =ψDϕKΓ[f ]. (22)

Moreover, ifwecompare (19)and(21)withthegeneralizedBorel–Pompeiuformulaweget

Corollary 3.Letf ∈ W1

p(Ω,R0,n). Then,wehave inΩ that

ϕ_K

Γϕ,ψΠ[f ] =ϕ, ¯¯ψKΓ[f ](x). (23)

Thefollowing resultextendsthatprovedin[7,9,14,31].Alsogeneralizetheparticularcasegivenin[10].

Proposition 2.Forany pairϕ,ψ ofstructuralsets wehave

ϕ,ψ_{Π : im}ψ_{Q→ im}ϕ_{Q, (24)} ϕ,ψ_{Π : im}ψ_{P→ im}ϕ_{P. (25)}

Proof. Itissufficienttouse(22)and(17). 2

In addition,note thatproperty (25)canbe writteninamorecomplete formusing theequality (17)as follows.

Proposition 3.ϕ,ψ_{Π[f ]∈ im}ϕ_{P ⇔ f ∈ im}ψ_P.

Ournextresults againgeneralizethoseprovedin[7,9,10].

Theorem 12.Letf ∈ Wk

p(Ω,R0,n), 1< p<∞,k∈ N,then

ψ,ϕ_Πϕ,ψ_{Π[f ] =}ψ_Dϕ_T

ΩϕDψTΩ[f ] =ψD(I−ϕKΓ)ψTΩ[f ] = f−ψDϕKΓψTΩ[f ].

Observethatinterchangingϕ andψ intheaboveequalityweget

ϕ,ψ_Πψ,ϕ_{Π[f ] = f−}ϕ_Dψ_K

ΓϕTΩ[f ].

Therefore, as aconsequenceof theprevious theorem,weobtainimmediately the one-sidedinvertibility of theϕ,ψ_{Π operatorinthosespacesoffunctionswhere tr}ψ_T

Ωor trϕTΩvanishes.

Proposition 4.Foranarbitrary structuralsetψ, wehave

f ∈ imψQ⇔ trΓψTΩ[f ] = 0Γ,

where 0Γ standsforthefunctionx∈ Γ→ 0∈ R.

Proof. (⇒)Letf ∈ imψQ, thenthere exists u∈

◦

W₂1(Ω,R0,n) suchasf =ψD[u].Then,

ψ_T

Ω[f ] =ψTΩψD[u] = u−ψKΓ[u] = u⇒ trΓψTΩ[f ] = 0Γ.

(⇐)Let trΓψTΩ[f ]= 0Γ.Wehave

0Γ = trΓψTΩ[f ] = trΓψTΩψP[f ] + trΓψTΩψQ[f ]⇒ trΓψTΩψP[f ] = 0Γ.

Then, using that the function ψ_T

ΩψP[f ] is harmonic in Ω, we can conclude thatit is identicallyzero

in Ω. Therefore,ψ_Dψ_T

ΩψP[f ] = 0Ω⇒ψP[f ] = 0Ω⇒ f ∈ imψQ. 2

Proposition5(One-sidedinvertibility oftheϕ,ψ_{Π operator).} ϕ,ψ_Πψ,ϕ_{Π = I}

imϕ_Q in imϕQ,

ψ,ϕ_Πϕ,ψ_{Π = I}

imψ_Q in imψQ.

IntheremainderofthissectionweconcentrateinthecaseΩ=Rn+1_.

Theorem13. Letϕ,ψ be twoarbitrarystructuralsets andf,g∈ L2(Rn+1,R0,n),then

_ϕ,ψ

Πf, g₂=f,ψ,ϕΠg₂. (26) Proof. If g ∈ C₀∞(Rn+1,R0,n), then, there exists a compact subset K ⊂ Rn+1 such that g(x)= 0 for all

x∈ K./ Itisobviousthat,ifwetakeaballB suchthatK⊂ B strictly,wehaveψ_T

Rn+1ϕD[g]=ψT_BϕD[g].

Then,accordingtotheformula(11),

ψ_T

Rn+1ϕD[g] =ψT_BϕD[g] =ψDϕT_B[g]−ψ,ϕK_∂B[g] =ψDϕT_B[g]

=ψDϕT_Rn+1[g] =ψ,ϕΠ[g].

Itiseasyto checkforallpairs u,v∈ L2(Rn+1,R0,n) that

_ψ

T u, v₂=−u,ψT v2,

andwiththeadditionalsuppositionofv∈ C₀∞(Rn+1_,R

0,n) weget

ϕ_{Du, v}

2=−



u,ϕDv₂.

Then,forallpairs offunctionsf ∈ L2(Rn+1,R0,n) andg∈ C0∞(Rn+1,R0,n) weobtain

_ϕ,ψ

Πf, g₂=ϕDψT f, g₂=−ψT f,ϕDg₂=f,ψTϕDg2=



f,ψ,ϕΠg₂.

Finally,bydensityofthespaceC₀∞(Rn+1_,R

0,n) inL2(Rn+1,R0,n) andusingthecontinuityoftheoperator

ψ,ψ_Π

Rn+1 inthisspace, thedesiredresultfails. 2

Theformula(26)showsthatϕ,ψΠ andψ,ϕΠ areadjointoperatorsinthespace L2(Rn+1,R0,n).

Theorem14. Letϑ, ϕ,ψ bethree arbitrary structuralsetsandf ∈ L2(Rn+1,R0,n).Then

ϑ,ϕ_Πϕ,ψ_{Π[f ] =}ϑ,ψ_{Π[f ]. (27)}

Proof. Suppose thatf ∈ C₀∞(Rn+1_,R

0,n),then there exists a compactsubsetK ⊂ Rn+1such thatf ≡ 0

inRn+1_{\ K. Let B}

r := B(0,r), where r > 0 is afixed number such thatK ⊂ Br. Let x∈ Rn+1 be an

arbitrarypointandR > máx{|x|,r}.Then,byvirtueoftheclassicalBorel–Pompeiuformulainsidetheball

BR:= B(0,R) ψ_T Rn+1[f ](x) =ϕK_∂BRψT_Rn+1[f ](x) +ϕT_BRϕDψT_Rn+1[f ](x). But !!ϕ_K ∂BR ψ_T Rn+1[f ](x)!! ≤ 1 |Sn|  ∂BR 1 |ξ − x|n ·!! ψ_{T [f ](ξ)}!!_dS ξ, (28)

and foranyξ∈ ∂BR,|ξ − x|≥ dist(x,∂BR)= R− |x|,then

1

|ξ − x|n ≤

1

(R− |x|)n. (29)

Ontheother hand !!ψ_T

Rn+1[f ](ξ)!!=!!ψT_Br[f ](ξ)!! ≤ f1

|Sn|dist(ξ, Br)

−n₌ f1

|Sn|(R − r)n. (30)

Using (29)and(30)in(28)weobtain !!ϕ_K ∂BR ψ_T Rn+1[f ](x)!! ≤Cf 1 Rn (R− |x|)n(R− r)n ⇒ lim R→∞ ϕ_K ∂BR ψ_T Rn+1[f ](x) = 0. Then, ψ_T Rn+1[f ](x) =ϕT_Rn+1ϕDψT_Rn+1[f ](x).

Applying inbothsidesof thisrelationtheoperatorϑD weobtainforf ∈ C₀∞(Rn+1,R0,n) that

ϑ,ψ_{Π[f ] =}ϑ_Dψ_T

Rn+1[f ](x) =ϑ,ϕΠϕ,ψΠ[f ](x).

We now apply again the density argument of C₀∞(Rn+1_,R

0,n) in L2(Rn+1,R0,n) and the continuity of

ϕ,ψ_Π

Rn+1 inthisspace andtheproofiscomplete. 2

Remark4.2.Amoregeneralresultcanbeproved.Letς,ϑ,ϕ,ψ befour arbitrarystructuralsets.Thenthe following equalityholds

ς,ϑ_Πϕ,ψ_{Π[f ] =}ς_Dϑ,ϕ_Πψ_{T [f ].}

Fortheparticular caseofϑ= ψ we obtainfrom (27)theboth-sideinvertibilityoftheoperatorϕ,ψΠ.

Corollary 4.Letϕ,ψ be twoarbitrarystructuralsets andf ∈ L2(Rn+1,R0,n). Then

ψ,ϕ_Πϕ,ψ_{Π[f ] = f. (31)}

Remark 4.3.The relation(31)proves thatinthe spaceL2(Rn+1,R0,n) the operatorϕ,ψΠ hasaboth-side

inverse ϕ,ψ_Π−1 _{which coincides with its adjoint} ψ,ϕ_{Π. Indeed, from (31) we obtain that} ψ,ϕ_{Π is the left}

inverseofϕ,ψ_{Π,butinterchangingϕ andψ wegetthat}ψ,ϕ_{Π is alsotherightinverse.}

ByvirtueofTheorem 13andTheorem 14 weobtainthefollowing importantresult.

Theorem 15.Letϑ, ϕ, ψ bethreearbitrary structuralsets andf,g∈ L2(Rn+1,R0,n).Then,

_ϕ,ϑ

Πf,ϕ,ψΠg₂=f,ϑ,ψΠg₂. (32) As aconsequence,fortheparticularcaseofϑ= ψ wegetthat

ϕ,ψ_{Π : L}

2(Rn+1,R0,n)→ L2(Rn+1,R0,n),

Proposition6.Letϕ, ψ betwoarbitrary structuralsetsandf,g∈ L2(Rn+1,R0,n). Then

_ϕ,ψ

Πf,ϕ,ψΠg₂=f, g₂,

""ϕ,ψ_Πf""

L2(Rn+1,R0,n)=fL2(Rn+1,R0,n).

Byasimpleextensionargumentwecanalsoobtainthat""ϕ,ψ_Πf""

L2(Ω,R0,n)= 1 foranyboundedsmooth

domain Ω.

OneofthemostinterestingconjectureswithrespecttothecomplexΠ-operatoristhefamousconjecture byIwaniec[13]aboutthenormofthecomplexΠ-operator:

ΠLp=



1

p−1 1 < p≤ 2

p− 1 2 < p < ∞

ApplyingtheFouriertransformFf(ξ)= _2π1 #_Rne−i x,ξf (x)dx toourϕ,ψΠ-operatorwegetfortheFourier

symbolofouroperator

Fϕ,ψ_Πf_{(ξ) =} n  i=0 n  j=0 ϕiψjξiξj |ξ|2(Ff)(ξ)

FromtheobservationoftheFouriersymbolwegetthefollowingconnectionbetweentheϕ,ψ_{Π andtheRiesz}

transformsRjf (y)= # Rn xj−yj |x−y|n+1f (x)dx: ϕ,ψ_{Πf =} n  i=0 n  j=0 ϕiψjRiRjf.

ConsidernowtheCauchyproblem fortheheatequation Ut+ ΔU = 0 withU (0,x)= u(x).Its solution

isgiven byU (t,x)= (F−1e−|ξ|2t_{Fu)(x).Also, wewilldenote byV thecorresponding solutionwithinitial}

dataV (0,x)= v(x).Consideringthesesquilinearform

U, V  = ∞  0  Rn+1 U (x, t)V (x, t)dxdt weobtain ψi_∂ iU, ϕj∂jV = ∞  0  Rn+1 ψi₍F−1₍−iξ i)e−|ξ|2tFu)(x) ϕj(F−1(−iξj)e−|ξ| 2_t Fv)(x)dxdt =  Rn+1

((−iξi)Fu)(ξ) ψiϕj(−iξj)Fv)(ξ) ∞  0 e−2|ξ|2tdtdξ =  Rn+1 (ξiFu)(ξ) ψiϕj(ξjFv)(ξ) 1 2|ξ|2dξ = 1 2  Rn+1 ψi_R iu ϕjRjvdx.

Now, sincewehave ϕ,ψ_{Πu, v} 2= n  i=0 n  j=0 ϕiψjRiRju, v2 = n  j=0 ψjRju, n  i=0 ϕi_R iv2 = n  j=0 ψj∂jU, n  i=0 ϕi_∂ iV =ψDU,ϕ_DV we get |ϕ,ψ_{Πu, v} 2| = !! !! ! ∞  0  Rn+1 ψ_DUϕ_{DV dxdt} !! !! !. (33)

HerewecouldusetheideaofestimatesviaMartingales(see[3]andreferencestherein).Butforthepurpose ofthispaperitisenoughtoapplytheideaofPetermichl,SlavinandWick[21]ofusingtheBellmanfunction technique.Theproof ofthefollowing theoremis astraightforwardapplicationofthecorresponding proofs in[21]and[20].

Theorem 16.Forthenormof theoperator

ϕ,ψ

Π : Lp(Rn+1,R0,n)→ Lp(Rn+1,R0,n), 1 < p <∞,

we havetheestimate

ϕ,ψ_Π Lp≤ 2 n/2_{(n + 1)(p}∗_{− 1)} with p∗= max $ p,_p−1p % .

The principal point for the proof is thatfor each partial derivative Petermichl, Slavin and Wick [21] provide thefollowingtheorem(adapted toourcase).

Theorem17. Letu,v∈ C₀∞(Rm_{,H) withH beingaHilbertmoduleandp≥ 2.Supposethat} 1

p+ 1 q= 1 then 2 m  i,j=1   Rm+1+ ∂iU (x, t)H∂iV (x, t)Hdxdt≤ (p∗− 1)uLp(Rm,H)vLq(Rm,H).

Strictly speakingPetermichl,Slavin andWick[21] provedtheabovetheorem forthecaseof H beinga HilbertspaceoveraGrassmannalgebra,buttheproofcanbeeasilyadaptedtothecaseofaClifford–Hilbert modulewith theobvious modifications.

Starting withformula(33)wecannow write

|ϕ,ψ_{Πu, v} 2| ≤ 2n/2  _n  i=0  Rn+2 + |∂iU||∂iV|dxdt +  i=j  Rn+2 + |∂iU||∂jV|dxdt 

Sincetheϕ,ψ_{Π-operatorcommuteswiththepartialderivativesweobtainthefollowing theorem.}

Theorem18. Forthenormoftheoperator

ϕ,ψ_{Π : W}k p(Rn+1,R0,n)→ Wpk(Rn+1,R0,n) with 1< p<∞,k∈ N,weget ϕ,ψ_Π Wk p ≤ 2 n/2_{(n + 1)(p}∗_{− 1).}

Sinceitiswell-knownthatϕ,ψ_{Π≥ (p}∗_{− 1) wegetthefollowing estimate}

p∗− 1 ≤ ϕ,ψΠWk p ≤ 2

n/2_{(n + 1)(p}∗_{− 1) (34)}

with1< p<∞ andk∈ N.

In the last section during our study of the application to higher-dimensional Beltrami equations this resultwillbe quitehelpful.

5. Jumpofϕ,ψ_{Π operator}

Theaimofthissectionistofindanexpressionforthejumpofthefunctionϕ,ψ_Π

Ω[f ] acrosstheboundary

Γ of the domain Ω. To this end, we will need to assume the boundary Γ tobe a Lyapunov surfaceand

f ∈ C0,μ_(Γ,R

0,n).Withoutlossofgeneralitywemayregardf tobecontinuousoutofΓ andtheclassbeing

conserved.Thereforetherepresentation(11)holdsforx∈ Γ./

We claim that,without difficulties Plemelj–Sokhotski formulas remain truealso for the exotic Cauchy

typeintegralϕ,ψ_K

Γ.Thesuccessofourmethodwill dependonusing theHöldercontinuityoftheoutward

unitnormalvectoronaLyapunovsurface.Indeed: Theorem19. Letf ∈ C0,μ_(Γ,R 0,n), μ∈ (0,1]. Then,wehave ϕ,ψ_K± Γ[f ](t) := lim Ω±_x→t∈Γ ϕ,ψ_K Γ[f ](x) = 1 2 _ϕ,ψ SΓ[f ](t)± nϕ(t)nψ(t)f (t)  .

Proof. Notethat,as aconsequenceofnϕ(ξ)nϕ(ξ)= 1,wehave ϕ,ψ_K

Γ[f ](x) =



Γ

Kϕ(ξ− x) [nϕ(ξ)nϕ(ξ)] nψ(ξ)f (ξ) dΓξ =ϕKΓ[nϕnψf ](x).

Next,thePlemelj–Sokhotskiformulas appliedtoϕKΓ[nϕnψf ] showthat ϕ,ψ_K± Γ[f ](t) = 1 2 _ϕ,ψ SΓ[f ](t)± nϕ(t)nψ(t)f (t)  . 2

To proceed with, it is worth noting that the function ϕ_T

Ω[f ] is continuous on the whole Rn+1 if f ∈

Lp(Ω,R0,n),p> n+ 1.Thisresultisobtainedin[8,Proposition 8.1] inthecaseofstandardstructuralset

ϕ= ϕst.Theproofforanarbitraryϕ issimilar.

After the above preparations, we are now ready to give the expression for the jump of the function

ϕ,ψ_Π

Ω[f ] onthecontourΓ,using thepreviousPlemelj–Sokhotskiformulas.Precisely

_ϕ,ψ ΠΩ[f ] + (t)−ϕ,ψΠΩ[f ] − (t) = nϕ(t)nψ(t)f (t), t∈ Γ. (35)

Remark5.1.Inthecaseofϕ= ψ = ψst,formula(35)canbefoundin[14,Deduction 3.9,p. 22].

Remark5.2. Letn= 1,andΓ (being inthis caseacurve)consist ofpointst(s)= x(s)+ iy(s),0≤ s≤ l, where l is the length of Γ and s is the length of the arc counted from a fixed point on Γ to t. For the structural setsψst ={1,i} andψst ={1,−i}, wehave

nψst =−i dt

ds and nψst = i d¯t ds.

Then, applying(35)thejumpofψst,ψst_{Π is givenby}

& ψst,ψst_{Π[f ]} '+ (t)−&ψst,ψst_{Π[f ]} '− (t) =&n_ψ st(t) '2 f (t) =− ( d¯t ds )2 f (t).

This relationisexactlythatobtainedin[30, p. 62]forthejumpoftheclassicalcomplexΠ-operator.

6. Applicationoftheϕ,ψ_Π

Ω-operatortothesolutionofBeltramiequations

Let us now consider the following Beltrami equation ψ_{Dw(z) = q(z)}ϕ_{Dw(z), z ∈ R}n+1_{. This type of}

Beltrami equationswas first studiedin[16] in thecase ofquaternionic analysis, where it wasshownthat these equations have a solution of the form w(z) = Φ(z)+ T h(z), where Φ(z) is a monogenic function and h(z) a solution of the singular integral equation h = qϕ_{DΦ+ q}ϕ,ψ_{Πh, if only q}ϕ,ψ_{Π < 1 in the}

appropriate norm. Thissystem includesthecaseof D and D aswell as thecaseof Shevchenko[26].Here and inthe following we consider thecase ofq taking valuesinthe paravectorgroup P.This includesthe mostimportantcasesforapplications.Toobtainaglobalsolutionw werequireq∈ W1

p(Rn+1,P),p> n+ 1,

qW1

p(Rn+1,R0,n)≤ qc< 1/

ϕ,ψ_{Π.Weremarkthatin[16]theconditiononthestructuralsetsis}n

k=0ϕkψk

whichisnotsatisfiedbythestandardchoiceψk _{= ϕ}k_{,k = 0,. . . ,n.}

One of the principal questions is if and when the solution of a Beltrami equation is a local or global quasi-conformal homeomorphism. Thebasis for such adiscussion is the following theorem [5] adapted to ourcase:

Theorem 20.(See[6].)Letf be monogenicin z0,i.e.ϕDf (z)= 0 inaneighborhoodof z0,thenwehave

rk Jf(z0) < n + 1⇔ ∃p : |p| = 1 ∧ϕDf (pz)|z=z0= 0,

where rk Jf(z0) denotestherankof theJacobianatthepointz0.

We fix now the monogenic function Φ(z) such that ψDΦ = 0 and ϕDΦ = 1. Using the affine transformation ξ = (z − z0) we can reduce our study to the point z0 = 0. For each neighborhood

Uδ(0)={ξ ∈ Rn+1, |ξ|< δ} weconsider thefunctionqδ(ξ)= q(ξ)b(ξ) with

b(ξ) = ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 1 |ξ| < 1 2δ,

2&|ξ|_δ '3−9₂&|ξ|_δ '2+ 3&|ξ|_δ '−1₂ 1₂δ≤ |ξ| ≤ δ,

0 |ξ| > δ. It holds qδW1 p(Rn+1,R0,n)≤ qWp1(Rn+1,R0,n)bWp1(Rn+1,R0,n)≤ Cϕδ n_q W1 p(Rn+1,R0,n), Cϕ =  ₂n +15 2n+1_(n+1)+ 2 n+4+ 3 2(n+3)− 6 n+2  |Sn_|.

Let us denote by W1

p,0(Uδ(0),R0,n) the space of all Wp1(Rn+1,R0,n)-functions with support in

Uδ(0). Then we have for the operator ϕ,ψΠδ, defined by ϕ,ψΠδh = qδϕ,ψΠh, the mapping property ϕ,ψ_Π δ : Wp,0,1 (Uδ(0),R0,n) → Wp,01 (Uδ(0),R0,n), 1 < p < ∞. Moreover, from qW1 p(Rn+1,R0,n) ≤ qc < 1/ϕ,ψ_Π W1 p(Rn+1,R0,n), p > n+ 1, and qδWp1(Rn+1,R0,n) ≤ Cϕδ n_q W1

p(Rn+1,R0,n) it follows that for all

δ≤ *n1_C

ϕ theoperator

ϕ,ψ_Π

δ isacontractionoverWp1(Uδ(0),R0,n),p> n+ 1.

This implies that w = Φ+ψT hδ, where hδ is a solution of hδ = qδ +ϕ,ψΠδhδ, is a solution of the

Beltrami-typeequationψDw(ξ)= qδ(ξ)ϕDw(ξ) overRn+1and, moreover,alsoasolutionofthe

Beltrami-typeequationψ_{Dw(ξ)= q(ξ)}ϕ_{Dw(ξ) over U}

δ(0). Finally,after applyingtheinversetransformation ofour

affinetransformationwegetasolutionofouroriginalBeltrami-typeequationψ_{Dw(z)= q(z)}ϕ_{Dw(z) over}

Uδ(z0).Additionally, byBanach’sfixed-pointtheorem andδ < *n1_C

ϕ weobtaintheestimate hδW1 p(Rn+1,R0,n)< 2CϕqWp1(Rn+1,R0,n)δ n_{< 2} Cϕ ϕ,ψ_Π W1 p δn.

Therefore,wegetthatforall> 0 thereexistsaδ such that

hδW1 p(Rn+1,R0,n)<  Cpϕ,ψΠW1 p , (36)

whereCp denotestheembeddingconstantofWp1(Rn+1,R0,n),p> n+ 1,inC(Rn+1,R0,n),andwecanuse

Theorem 20.

Fromthis theoremwehavethecondition

rk Jw|ξ=0< n + 1⇔ ∃p : |p| = 1 ∧ϕDξw(pξ)|ξ=0= 0.

Weconsider nowthetermsϕ_{Dξw(pξ) and}ϕ_D

ξw(pξ) atzeroandtransformingthem into



AψA(PTQA),

where QA are realsymmetric matrices andP denotes ourrotationwrittenas avector. Since itis enough

thatoneofthematricesQAispositiveornegativedefinitewecanrestrictourselvestothescalarpart.This

leadstothefollowingmatrix: ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ 3 + C00 C01 C02 . . . C0n C10 3 + C11 C12 . . . C13 .. . ... ... . .. ... Cn0 Cn1 Cn2 . . . 3 + Cnn ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (37)

Hereby, we referto [6]for thecoefficients and details.Weonly remark thatthe coefficientsCij of system

(37)fulfil|Cij|≤ Cpψ,ψΠW1 ph

δ W1

p forallindices i,j, whereCp denotes againtheembeddingconstant

ofW1

p inC(Rn+1,R0,n)[9].

Moreover,from(36)weobtainthat|Cij|< 

ψ,ψ_Π

ϕ,ψ_Π holds.Choosingnow

ψ,ψ_Π

ϕ,ψ_Π< 1/3 theelementsqij

ofourmatrix(37)satisfyqii>



j=i|qij|.

Thisallowsustoestablishthefollowingtheorem. Theorem 21. Suppose that q ∈ W1

p(Rn+1,R0,n) for a certain p > n+ 1 and qW1

p(Rn+1,R0,n) ≤ qc < 1/

ϕ,ψ_Π W1

p(Rn+1,R0,n).Supposealsothatq+1 isnotazerodivisorforanypointz0inR

n+1_.Thenthe

Beltrami-typeequation ψDw(z)= q(z)ϕDw(z) hasineach pointz0 asolution,whichrealizesalocalquasi-conformal

mapping.

Corollary 5.Asufficient conditionfortheabove theorem isqW1 p(Rn+1,R0,n)≤ qc< 1 2n/2_(n−1)(p∗₋₁₎. Now, weassumeq∈ W1 p(Rn+1,R0,n) with qW1 p(Rn+1,R0,n)≤ qc< 1 (n− 1) min&ϕ,ψ_Π W1 p, ψ,ψ_Π W1 p ' for somep> n+ 1.ThismeansthatwehavehW1

p(Rn+1,R0,n)<

qc

1−qcϕ,ψΠW 1_p <

1

3ΠW 1_p andthesolution w = Φ+ T h belongs to thespaceC1_(Rn+1_,R

0,n). Moreover,ourlinearsystem (37)againturns outto be

diagonal dominant.

The well-known theorem of Hadamard [2], which states,that afunction w(z),where theJacobian de-terminant isnot equalto zeroinany pointand w(z) → ∞ if|z|→ ∞, realizesaglobal homeomorphism, together withournormestimateoftheϕ,ψ_{Π-operator(34)allowsus toestablishthefollowingtheorem.}

Theorem 22.If q∈ W1 p(Rn+1,R0,n),qW1 p(Rn+1,R0,n)≤ qc< 1 2n/2_(n−1)(p∗₋₁₎,p∗= max $ p,_p−1p %,q + 1 not being azero divisoratany pointinRn+1_{,forsomep> n+ 1 thenthefunction}

w = Φ + T h,

with h being a solution of the corresponding singular integral equation, is a solution of the Beltrami-type equation ψDw(z)= q(z)ϕDw(z) which realizesa globalhomeomorphism.

Acknowledgments

This workwassupportedbyPortuguesefundsthroughtheCIDMA–CenterforResearchand Develop-mentinMathematicsandApplications,andthePortugueseFoundationforScienceandTechnology(“FCT– FundaçãoparaaCiênciaeaTecnologia”),withinprojectUID/MAT/0416/2013.J.BoryReyeswaspartially supportedbyInstitutoPolitécnico NacionalintheframeworkofSIPprograms.

The authors would like to express their gratitude to a reviewer for drawing the attention to the fine detailsofBanachandHilbertCliffordmoduleconcepts.

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