Sommen, Franciscus

Van Acker, Nadine

English

Ghent University Academic Bibliography

Advances in Applied Clifford Algebras 4, No. 1 (1994) 65FUNCTIONS OF TWO VECTOR VARIABLESF. Sommen* and N. Van AckerDepartment of Mathematical Analysis,University of Gent,Galglaan 2 B-9000 Gent, Belgium1. IntroductionIn our paper [6] we considered functions taking values in some space ofspinor- or Clifford algebra-valued polynomials. Let Pk denote the space of ho-mogeneous polynomials of degree k, and let Ω be an open subset of Rm, thenE(Ω,Pk) denotes the space of Pk-valued smooth functions in Ω. Of course,when f ∈ E(Ω,Pk), f can be split up as follows using the monogenic decom-position of a homogeneous polynomial:f(x, u) =k∑j=0ujfk−j(x, u), fk−j ∈Mk−jwhere Ml denotes the module of inner spherical monogenics of degree l (see[1], [3]). Looking at operatorsP : E(Ω,Pk)→ E(Ω,Pk′)we proved that all GL(m)−invariant operators are generated by〈u, ∂x〉, 〈u, x〉, 〈x, ∂u〉, 〈∂x, ∂u〉.Among them we recognize the gradient operator on polynomial-valued func-tions which is part of the complete gradient (see [9])〈u, ∂x〉 =m∑j=1uj∂xj .We established the so-called monogenic decomposition of these operators (see[7]) when acting on f ∈ E(Ω,Pk), i.e. the decomposition into operators trans-forming ujfk(x, u), fk(x, ·) ∈ Mk into uj′fk′(x, u), fk′(x, ·) ∈ Mk′ . The* Senior Research Associate,NFWO, Belgium66 Functions of Two Vector Variables F. Sommen, N.Van Ackergradient operator splits up into three pieces D+,k(∂x), D0,k(∂x), D−,k(∂x)and similarly the operator 〈u, x〉 leads to D+,k(x), D0,k(x), D−,k(x), i.e. forf ∈ E(Ω,Mk) :〈u, ∂x〉f =[D+,k(∂x) + uD0,k(∂x) + u2D−,k(∂x)]f,〈u, x〉f = [D+,k(x) + uD0,k(x) + u2D−,k(x)] fwhereD+,k : E(Ω,Mk)→ E(Ω,Mk+1)D0,k : E(Ω,Mk)→ E(Ω,Mk)D−,k : E(Ω,Mk)→ E(Ω,Mk−1).The explicit expressions for these operators can be found in [6]. These six oper-ators are the basic ones, since the remaining two operators 〈x, ∂u〉 and 〈∂x, ∂u〉also admit decompositions in terms of these basic operators. By definition afunction f ∈ E(Ω,Mk) is monogenic if D0,k(∂x)f = 0, while f is hyper- resp.hypo-monogenic if D+,k(∂x)f = 0 resp. D−,k(∂x) = 0. The monogenicityconcept is of course an extension of the standard definition of monogenicityfor functions f(x) of one vector variable. Moreover, we paid attention to theproblem of defining monogenicity for f ∈ E(Ω,Pk).When dealing with polynomial-valued functions, it is obvious that thespace of Clifford algebra-valued polynomials of two vector variables is providedwith the inner product:(P (x, u), Q(x, u)) = [P (∂x, ∂u)Q(x, u)]0where for a ∈ Rm, [a]0 denotes the scalar part of a. Next take a polynomialRl(x, u) ∈ Pl(Mk); the monogenic decompositions for the operators underconsideration then led to orthogonal Fischer decompositions into “monogenicpieces”, “hyper-” and “hypo-monogenic pieces”.It is clear that so far we considered the u-variable merely as a tool to rep-resent polynomial-valued functions. However if we consider invariant operatorswith variable coefficients, then we end up with operators of the formD = D(u, x, ∂u, ∂x)in which u and x play more or less the same role. Especially if we considere.g. polynomial solutions of the homogeneous equation Df = 0 , it is not clearAdvances in Applied Clifford Algebras 4, No. 1 (1994) 67whether the decomposition of f(x, u) in terms of spherical monogenics withrespect to u is the most suitable since it destroys the symmetry between x andu.To obtain a more symmetric decomposition we now study functions of twovector variables and introduce the notion of simultaneous spherical monogenics.We will establish the corresponding Fischer decomposition into simultaneousspherical monogenics for polynomials which are both homogeneous in x andu. Yet this is neither a refinement nor a symmetrization of the monogenic,hyper- or hypo-monogenic Fischer decomposition. If one would for exampledecompose monogenicly w.r.t. x and then w.r.t. u, the original monogenicityw.r.t. x is lost.2 Fischer decompositionLet Pk,l be the space of homogeneous polynomials of degree k in x and lin u. On the analogy of the one-vector-variable case we consider homogeneouspolynomial nullsolutions of the vector-derivatives ∂u =∑mi=1 ei∂uj and ∂x =∑mi=1 ei∂xj .Definition 2.1 A polynomial Pk,l ∈ Pk,l is called a simultaneous sphericalmonogenic of degree (k, l) if ∂x Pk,l = ∂u Pk,l = 0.The module of simultaneous spherical monogenics of degree (k, l) is denotedby Mk,l.Then we have the followingLemma 2.1 Every Rk,l ∈ Pk,l can be written in a canonical way asRk,l = Pk,l + x Rk−1,l + u Rk,l−1,where Pk,l ∈Mk,l and Rk−1,l ∈ Pk−1,l , Rk,l−1 ∈ Pk,l−1.Proof. The proof is similar to the one of the standard Fischer decomposition.Consider the submodule of Pk,l given byS = { q(x, u) = x Rk−1,l + u Rk,l−1 , Rk−1,l ∈ Pk−1,l , Rk,l−1 ∈ Pk,l−1}.If r(x, u) is an element of the orthogonal complement of S in Pk,l, then0 = (q(x, u), r(x, u))= (x Rk−1,l + u Rk,l−1 , r(x, u))= (Rk−1,l , ∂xr) + (Rk,l−1 , ∂ur)68 Functions of Two Vector Variables F. Sommen, N.Van Ackerfor arbitrary Rk−1,l ∈ Pk−1,l and Rk,l−1 ∈ Pk,l−1. Hence ∂xr = ∂ur = 0 , i.e.r(x, u) ∈Mk,l.The problem with this decomposition is the non-uniqueness of the last twoterms (see [2]). Nevertheless we go through with the iteration, and, as we willsee, after re-arranging the terms we end up with a good Fischer decompositionanyway ! It is clear that after iteration we find that each Rα, β ∈ Pαβ has adecomposition of the form :Rα,β(x, u) =α∑i=0β∑j=0∑cIcij(x, u)Pcα−i,β−j(x, u)where Icij(x, u) is a Spin(m)-invariant polynomial in Pij while c ranges all pos-sible products of i times x and j times u, and where each P cα−i,β−j ∈Mα−i,β−jstill depends on c. The invariance of Icij(x, u) means invariance under theH(s)-representation of the spingroup, i.e. Icij(x, u) satisfies the condition (see [8])H(s)P (x, u) = sP (s¯xs, s¯us)s¯ = P (x, u), ∀s ∈ Spin(m).Now let us have a better look at this decomposition. As the polynomialsIci,j are both homogeneous and invariant, they can be rewritten as a linearcombination of the polynomials 〈u, x〉kxlun , k + l = i , k + n = j. Thus wecan re-arrange the terms in the decomposition according to the following list ofbasic invariant polynomials (ordered according to the total degree h = 2k+l+nof homogeneity within each row):1x ux2 〈u, x〉 x ∧ u u2x3 x2u 〈u, x〉x 〈u, x〉u xu2 u3· · · ·· · · ·· · · ·This yields the following decomposition for Rα,β(x, u) ∈ Pα,β :Rα,β(x, u) =α+β∑h=0∑2k+l+n=hk+l≤α, k+n≤β〈u, x〉k xl un P k,l,nα−k−l,β−k−n(x, u)Advances in Applied Clifford Algebras 4, No. 1 (1994) 69where P k,l,nα−k−l,β−k−n ∈Mα−k−l,β−k−n is still depending on k, l, n.Theorem 2.1 Let Rα,β ∈ Pα,β. Then the simultaneous spherical monogenicsin the decompositionRα,β(x, u) =α+β∑h=0∑2k+l+n=hk+l≤α, k+n≤β〈u, x〉k xl un P k,l,nα−k−l,β−k−nare unique.Proof. Using the fact thatdimPα,β = (dimPα) · (dimPβ) =(m+α−1m−1) (m+β−1m−1),and in view of the formula giving the dimension of Mk,l (see [2]), it can beverified that(m+α−1m−1)(m+β−1m−1)=α+β∑h=0∑2k+l+n=hk+l≤α, k+n≤β[(m+α−k−l−2m−2)(m+β−k−n−2m−2)−(m+α−k−l−3m−2)(m+β−k−n−3m−2)].It is clear that in the above formula the last term on the right hand sidevanishes if k + l = α (or k + n = β).A more direct proof will be included in [5]. As an example of a decompo-sition into simultaneous spherical monogenics, we consider the zonal functionfor Pk,l which is given by 1k! l! 〈x, a〉k〈u, b〉l, i.e.(1k! l!〈x, a〉k〈u, b〉l, Rk,l(x, u))=1k! l!〈a, ∂x〉k〈b, ∂u〉lRk,l(x, u)= Rk,l(a, b)for all Rk,l ∈ Pk,l. It is then clear that the “highest monogenic piece” in thisdecomposition will be the reproducing kernel for the module Mk,l. Considerthe case where k = l = 1. We have that〈x, a〉〈u, b〉 = P1,1(x, u) + xP0,1(u) + uP1,0(x) + 〈u, x〉P0,0 + (x ∧ u)P ∗0,070 Functions of Two Vector Variables F. Sommen, N.Van Ackerwhere each Pij ∈Mij . We want to determine each of these monogenics explic-itly. It is obvious that, when letting the operator ∂x act on both sides of theabove equation, we finda〈u, b〉 = −mP0,1 + ∂x(uP1,0) + uP0,0 + (1−m)uP ∗0,0 (1)where∂x uP1,0 = −u∂xP1,0 − 2〈u, ∂x〉P1,0 = −2P1,0 |x=u .Next letting the operator ∂u act on both sides, it is easy to see thatb a = −mP0,0 −m(1−m)P ∗0,0since ∂u(P1,0 |x=u) = 0. In a similar way we let the operator ∂x∂u act on bothsides of the decomposition, leading to:a b = ∂x∂u(xP0,1)−mP0,0 −m(m− 1)P ∗0,0where again ∂x∂u xP0,1 = 0. From the equations{mP0,0 −m(m− 1)P ∗0,0 = −b a−mP0,0 −m(m− 1)P ∗0,0 = a bwe can now determine both P0,0 and P ∗0,0 :P0,0 =1m〈a, b〉,P ∗0,0 =−1m(m− 1) a ∧ b.Substituting these expressions in the equation (1), we have thata〈u, b〉 = −mP0,1 − 2P1,0 |x=u + 1mu 〈a, b〉+ 1mu (a ∧ b),or still:a〈u, b〉 = −mP0,1 − 2P1,0 |x=u − 1mub aand alsob〈x, a〉 = −mP1,0 − 2P0,1 |u=x − 1mxa b.Advances in Applied Clifford Algebras 4, No. 1 (1994) 71It is only a small exercise to calculate from{−mP0,1(u)− 2P1,0(u) = 1m u b a+ a〈u, b〉−2P0,1(x)−mP1,0(x) = 1m x a b+ b〈x, a〉the (multiple) spherical monogenics P0,1(u) and P1,0(x):P0,1(u) =1(m+ 2)(m− 2)[2〈a, u〉b+ 2mua b− u b a−m〈u, b〉a],P1,0(x) =1(m+ 2)(m− 2)[2〈x, b〉a+ 2mxb a− x a b−m〈x, a〉b].The reproducing kernel for M1,1 is now determined:P1,1(x, u) = 〈x, a〉〈u, b〉+ 1(m+ 2)(m− 2)[m (x〈u, b〉a+ u〈x, a〉b)−2 (x〈a, u〉b+ u〈b, x〉a)− 1m+ 2〈u, x〉〈a, b〉 − 1(m− 1)(m− 2)(x ∧ u)(a ∧ b)].Now consider again an invariant differential operatorD = D(u, x, ∂u, ∂x) actingon Pα,β . Then it is clear that it suffices to consider the action of this operatoron the spaces Mα,β only ; we indeed have thatDRα,β =α∑i=0β∑j=0∑cDIci,j Pcα−i,β−jwhere the operators (D ◦ Ici,j) are again invariant differential operators.Note that the Fischer decomposition into simultaneous spherical monogen-ics is unfortunately not an orthogonal one. However, in view if the well-knownorthogonality between spherical monogenics of one vector variable and in viewof the definition of simultaneous spherical monogenics, it is clear that termscoming from a different row in the previous diagram, will be orthogonal, i.e. ifh = 2k + l + n 6= 2k′ + l′ + n′ = h′then (〈u, x〉kxlunP k,l,nα−k−l,β−k−n, 〈u, x〉k′xl′un′P k′,l′,n′α−k′−l′,β−k′−n′)= 0.72 Functions of Two Vector Variables F. Sommen, N.Van AckerWe hence come toTheorem 2.2 Rα,β ∈ Pα,β admits a unique orthogonal decomposition of theformRα,β =α+β∑s=0Ms(Rα,β)where Ms(Rα,β) has the form∑ 〈u, x〉k xl un P k,l,nα−k−l,β−k−n(x, u) with 2k +l + n = s. Moreover, M0(Rα,β) is the monogenic piece in the decompositionwhile Ms(Rα,β) is “ (s+1)-monogenic of two variables” meaning that for everyoperator P (∂x, ∂u) of degree (s + 1) belonging to the algebra generated by ∂xand ∂u , P (∂x, ∂u)Ms(Rα,β) = 0.References[1] F. Brackx, R. Delanghe and F. Sommen. Clifford Analysis, ResearchNotes in Math. 76, Pitman, London (1982).[2] D. Constales. The relative position of L2 domains in complex and inClifford analysis, PhD-thesis, Gent 1989.[3] R. Delanghe, F. Sommen and V. Soucˇek. Clifford Algebra and Spinor-valued functions: a function theory for the Dirac-operator, Mathematicsand Its Applications 53, Kluwer Academic Publishers, Dordrecht (1992).[4] J. Gilbert, M. Murray. Clifford Algebras and Dirac operators in harmonicanalysis, Cambridge University Press, 1990.[5] F. Sommen. Clifford tensor calculus, to appear in the Proceedings of theXXIIth DGM Conference, Ixtapa (1993).[6] F. Sommen, N. Van Acker. Invariant differential operators on polynomial-valued functions, to appear in the Proceedings of the Third InternationalConference on Clifford Algebras and Their Applications in MathematicalPhysics, Deinze 1993.[7] F. Sommen, N. Van Acker. Monogenic differential operators, Results inMathematics vol. 22 (1992), pp. 781-798.[8] F. Sommen, N. Van Acker. SO(m)−invariant differential operators onClifford algebra-valued functions, to appear in Foundations of Physics,1993.[9] E.M. Stein, G. Weiss. Generalization of the Cauchy-Riemann equationsand representations of the rotation group, Amer. J. Math. vol. 90 (1968),pp. 163-196.[10] A. Strasburger. Invariant differential operators associated with sphericalharmonics, preprint 1/89, University Warschau.

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