The Teodorescu Operator in Clifford Analysis

526F. Brack, H. De Schepper, M. E. Luna- Elizarrarda, et al.0D= II aD, of apolydisk B= i D, in cn, leading toj=1f1,-.5n)_.f(,..,zn) =df1 A.-.^dEm， zj∈D,(1.1)(2ri)n(-1-1).(6n-zn)or one takes an integral over the (piecewise) smooth boundary 8D of a bounded domain D inCn, yieldingf(z)=J8Df()U(6,z)d5，z∈ D,(1.2)where, with .c denoting the complex conjugate, the kernel(n-1!5-zfU(5,z)=(2m)-z|2n;dξi ..CS-1 ^d6j+1...L ^dξ1...is the s0 called Marinel-Bochner kernel (see, e.g., [27]), which is not holomorphic but stillharmonic. For the history of formula (1.2), obtained independently by Martinelli and Bochner,we refer to [26). So we may expect that a Teodorescu operator in several complex variables willinvolve a harmonic integral kernel which will formally mimic the Martinelli-Bochner kernel. Itis explained in Section 5 that this indeed is the case.There is however an alternative for generalizing the two-dimensional Cauchy integral for-mula, offered by Cliford analysis, where functions defined in Euclidean space Rm and takingvalues in a Clifford algebra are studied. The theory focuses on so called monogenic func-tions, i.e., null solutions to the elliptic Dirac operator 8x factorizing the Laplace operator: .眼= - Om. The Dirac operator being rotation invariant, the name Euclidean Cliford analysisis used nowadays to refer to this setting (see, e.g., [12, 22- -23, 25]). Here the kernel appearingin the Clifford-Cauchy formula is monogenic, up to a pointwise singularity, while the integralis taken over the complete boundary:f(X)= L E(- X)dozf(③)，X∈DlaDwithE(3-X)=1营-区am匡-Xm'am denoting the area of the unit sphere Sm C R", : denoting the main conjugation in theCliford algebra, and dσg being a Ciliford algebra valued differential form of order (m- 1). ThisCliford-Cauchy integral formula has been a corner stone in the development to the functiontheory. The function E(X) = 1 x而is the fundamental solution to the Euclidean Diracoperator 8x, i.e., it holds in the distributional sense that 8xE(X) = 8(X), where 8(X) is theDirac distribution in R". As can be expected, the Teodorescu inversion in Cliford analysisindeed involves a convolution with the Clifford-Caucl中国煤化工in Section 3,where the underlying mechanisms leading to this resuCHCNMH GIn the even dimensional case, Hermitian Clifford auanyous arnclgou。a Iculement of theEuclidean setting; it focuses on the simultaneous null solutions to the complex Hermitian DiracThe Teodorescu Operator i Cifford Analysis627operators oz and 8g, which decompose the Laplacian in the sense that 4(0r+821)2 = 4(8282+8+8) =△2n and which are invariant under the action of the special unitary group. The studyof complex Dirac operators was initiated in [28 30]. A systematic development of the functiontheory in the Hermitian Clifford analysis context, including the invariance properties withrespect to the underlying Lie groups and Lie algebras, is still in full progress (see, e.g-, [20, 1-4,16, 19, 8]). In this framework, a Cauchy integral formula for Hermitian monogenic functionstaking values in the complex Cliford algebra C2n or in complex spinor space S was establishedin [11], and further integral representation formulae were developed in [17]. However, fromthe start, it was clear that the desired formula could not have the traditional form of (1.1)or (1.2). Indeed, it is known (see [4) that in the special case where the functions consideredtake their values in a specific part of spinor space, Hermitian monogenicity is equivalent toholomorphy in the underlying complex variables. It turned out that a matrix approach wasthe key to obtaining the desired result. Moreover and as could be expected, the obtainedHermitian Cauchy integral formula reduces to the traditional Martinelli- Bochner formula (1.2)in the particular case mentioned. This also means that the theory of Hermitian monogenicfunctions not only refines Euclidean Ciford analysis (and thus harmonic analysis as wel), butalso has strong connections with the theory of functions of several complex variables, in somesense even encompassing its results.In Section 4, we construct Hermitian Teodorescu inversion forrmulae not only for the Hermi-tian Dirac operators 8z and 82, but also for the associated differential operators g, az^, ogt●,8zt^, obtained by splitting the Ciford algebra or geometric product into its“dot”and“wedge"parts. These associated differential operators are the counterparts in the multi-vector languageof Ciford analysis of well-known differential operators for real and complex differential formsin Euclidean space.As was already announced above, the results of this section are then interpreted in Section5 for scalar valued functions of several complex variables, leading to the expected connectionswith the Martinelli-Bochner approach.2 Preliminaries of Clifford AnalysisThe real Cifford algebra IRo,m is constructed over the vector space RO,m endowed witha non-degenerate quadratic form of signature (0, m) and generated by the orthonormal basis(e,... ,em). The non-commutative Cliford or geometric multiplication in Ro,m is governed bythe ruleseaeg+epla=-20aβ，a,β= 1,... ,m.(2.1)As a basis for Ro,m, one takes for any set A = {j,.. ,jh} C {,.. ,m}, the element eA =e_..en with1≤j1f^fjpv| Qz&E(z - y)u(w)dWj≠k=(EAf)u(z)+pvJn j=1k=1f ^{jOxE;(z - )u(w)dW(二号^t;)u()+pv.8z ^E(z -四) u(w)dW,from which the desired result fllws since, by (4.1), 8ε=az^ε as long asz≠必.Corollary 4.1 One has() oz●T(){u(2)+ 8zt●T()u(z)= (-1)中(2i)"u(z),(i) O2八T()[u(z) + oz+ ^T(2\叫(z)= 0.Fnally, making the appropriate combinations ofing theorem, weobtain expressions for the action of the Hermitian Dir中国煤化工ian TeodorescuYHCNMHGoperators.Proposition 4.3 For a function u∈C1(M2), one has638F. Bnackr, H. De Schepper, M. E. Luna Elizarrardas, et al.() b[T()叫=(-1)*中” (2i)"=βu+pv I。8_E(z - )(w)dW,Js(i) a2,T(1)[u] =0,(i) OT(2)[u]= 0,(v) :y(2(1)"=(2)(1 - )u+pv I 8xeE(a- w)(ydW.Remark 4.1 Note that formula (i) of Theorem 4.1 fllowls fom formulae (i) and (iv) ofProposition 4.3 due to formula (v) of Proposition 4.1. Note also that formulae (i) and (ii) ofProposition 4.3 are stronger than formula (i) of Theorem 4.1.5 The Case of Several Complex VariablesIn this section, we will restrict ourselves to complex valued functions f :Rm≌C2n→C.Such functions which a priori are functions of the real variables (x1,... ,In,y1,.. ,Un) becomea functions of n complex variables and their complex conjugates: f(z1,... ,rn,zi,... ,z况) or inshorthand f(z, z). It is only when a function is holomorphic that it becomes a function of thevariables (z1,... , zn) only. Naturally, all the results obtained in the foregoing section apply tosuch scalar valued functions.When concentrating on Proposition 4.2, we see that quite naturally, only formulae (ii) and(iv) are to be found in the literature on several complex variables (see, e.g, (27, 1.11),since there the focus is on holomorphic functions and the related Cauchy-Riemann operators0x;,j= 1,.. ,n, while there is not one unique diferential operator defining multidimensionalholomorphy. The mirror formulae (i) and (i) involving the conjugate Cauchy-Riemann oper-ators Oxs,j = 1,.. ,n are then tacitly assumed. However, the formulae from Proposition 4.2do not really deserve the qualfcation of Teodorescu inversion. It becomes interesting whensumming them up, leading to the results already contained in Theorem 4.2,8qoT()[]=;2o, I E(z- )f(&)dW=元(-1)"(2)"f(z)j=1)r之o, h三动一四(=(171-2=21912a2n ρ2nj=landa2●T2ln1=5Eag f eg(z-w)f(w)dW= (-1)“中()"f()or2近一岭. f(w)dW=(-1)* (2i)"f(z),2C2n.ρ2nexpressing the fact that T(1) and T(2) indeed are the right inverses of the associated Diracoperators 82●and 2t●, respectively. Similar formulad中国煤化工tial forms, canbe found in [28, Chapter 7]1THCNMHGAs was expected, the involved integral kernels are nOU (ant- )uolomorpm, oul still harmonic,since they are the components of the fundamental solutions to both Hermitian Dirac operators.The Teodorescu Operator in Cifornd Analysis639Moreover, they do coincide with the ones appearing in the Martinelli- Bochner representationformula for holomorphic functions (1.2) and its anti-holomorphic counterpart.References[1] Blaya, R. A., Reyes, J. 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