Almansi-type theorems for slice-regular functions on Clifford algebras
aa r X i v : . [ m a t h . C V ] A p r Almansi-type theorems for slice-regular functionson Clifford algebras
Alessandro Perotti
Department of Mathematics, University of TrentoVia Sommarive 14, I-38123 Povo Trento, [email protected]
Abstract
We present an Almansi-type decomposition for polynomials with Clifford co-efficients, and more generally for slice-regular functions on Clifford algebras. Theclassical result by Emilio Almansi, published in 1899, dealt with polyharmonicfunctions, the elements of the kernel of an iterated Laplacian. Here we considerpolynomials of the form P ( x ) = P dk =0 x k a k , with Clifford coefficients a k ∈ R n ,and get an analogous decomposition related to zonal polyharmonics. We show therelation between such decomposition and the Dirac (or Cauchy-Riemann) operatorand extend the results to slice-regular functions. Mathematics Subject Classification (2010) . Primary 30G35; Secondary 31B30,33C50, 33C55.
Keywords:
Clifford polynomials, Slice-regular functions, Zonal harmonics, Poly-harmonic functions.
A classical result of Emilio Almansi, published in 1899 (see [1, 3]), gave an expansionof polyharmonic functions in terms of harmonic functions. We recall that for everyinteger p ≥ , a real function u defined on an open set Ω in R N is called polyharmonic of order p (or p -harmonic ) if u is of class C p and ∆ p u = 0 on Ω , where ∆ is theLaplacian operator of R N . Theorem 1 (Almansi) . If ∆ p u = 0 on a star-like domain D ⊆ R N with centre 0, thenthere exist unique harmonic functions u , . . . , u p − in D such that u ( x ) = u ( x ) + | x | u ( x ) + · · · + | x | p − u p − ( x ) for x ∈ D. Let R n denote the real Clifford algebra of signature (0 , n ) , with basis vectors e , . . . , e n . Consider Clifford polynomials of the form p ( x ) = d X k =0 x k a k with Clifford coefficients a k ∈ R n on the right. If n = 2 m +1 is odd, these polynomialsare polyharmonic of order m + 1 with respect to the standard Laplacian of R n +1 ,i.e. (∆ n +1 ) m +1 p = 0 on R n +1 (see e.g. [18, Corollary 4.4.4] and the next section).1ince the Laplacian is a real operator, this property is equivalent to say that the realcomponents of a Clifford polynomial are ( m + 1) -harmonic. The classical Almansi’sTheorem 1 implies then that the real components of a Clifford polynomial have anexpansion in terms of a ( m + 1) -tuple of harmonic functions.Observe that every Clifford polynomial p is more than ( m + 1) -polyharmonic,since it is in the kernel of the n th -order differential operator ∂ (∆ n +1 ) m , where ∂ is theCauchy-Riemann operator ∂ = ∂∂x + e ∂∂x + · · · + e n ∂∂x n on R n (Theorem 2). We recall also the definitions of the Dirac operator D = e ∂∂x + · · · + e n ∂∂x n and the conjugated Cauchy-Riemann operator ∂ = ∂∂x − e ∂∂x − · · · − e n ∂∂x n on R n . Since ∂∂ = ∂∂ = ∆ n +1 = ∂ ∂x + ∂ ∂x + · · · + ∂ ∂x n , the operators ∂, ∂ factorize the Laplacian operator of the paravector subspace V = { x + x e + · · · + x n e n ∈ R n | x , . . . , x n ∈ R } ≃ R n +1 of R n . This property shows how Clifford analysis, the well-developed function theorybased on Dirac and Cauchy-Riemann operators (see [5, 9, 16] and the vast bibliographytherein), is related to the harmonic analysis of the Euclidean space.The title of the present paper resembles the one of [17], where the authors provedAlmansi-type theorems in Clifford analysis. Note that the two classes of functions, theone of slice-regular functions on R n and the one of monogenic functions in the kernelof ∂ , are really skew, in the sense that only constant functions are both monogenic andslice-regular (see [18, Corollary 3.3.3]). In particular, standard nonconstant Cliffordpolynomials of the form P dk =0 x k a k are not monogenic. However, as also the resultsobtained in this paper show, the relations between the two function theories deserveinterest.In this paper we will prove, using the symmetry properties of Clifford powers x k , arefined Almansi-type decomposition for Clifford polynomials and we will extend it tothe larger class of slice-regular functions on R n and to the case of even dimension n . Polynomial functions f ( x ) = P k x k a k with Clifford coefficients are examples of slice-regular functions. This class of functions constitutes a recent function theory inseveral hypercomplex settings, including quaternions and real Clifford algebras (see[12, 7, 13, 8, 11, 14, 15]). It was introduced by Gentili and Struppa [12] for functionsof one quaternionic variable and by Colombo, Sabadini and Struppa [7] for functionsdefined on open subsets of R n +1 , considered as the paravector space of R n .2e briefly recall the basic definitions we need for the main results and refer thereader to the cited references for more details. Let H denote the skew field of quater-nions, with basic elements i, j, k . For each quaternion J in the sphere of imaginaryunits S H = { J ∈ H | J = − } = { x i + x j + x k ∈ H | x + x + x = 1 } , let C J = h , J i ≃ C be the subalgebra generated by J . Then we have the “slice”decomposition H = [ J ∈ S H C J , with C J ∩ C K = R for every J, K ∈ S H , J = ± K .A differentiable function f : Ω ⊆ H → H is called (left) slice-regular [12] on the openset Ω if, for each J ∈ S H , the restriction f | Ω ∩ C J : Ω ∩ C J → H is holomorphic withrespect to the complex structure defined by left multiplication by J .If Ω ⊆ R n +1 is an open subset of the paravector space in R n , a function f :Ω ⊆ R n +1 → R n such that, for each J in the sphere S n − of paravector imaginaryunits, the restriction f | Ω ∩ C J : Ω ∩ C J → R n is holomorphic w.r.t. the complexstructure L J ( v ) = Jv , is called slice monogenic [7]. For example, polynomials f ( x ) = P k x k a k with Clifford coefficients on the right are slice monogenic on R n +1 . Moregenerally, convergent power series are slice monogenic on an open ball in R n +1 centredat the origin. Observe that nonconstant polynomials (or power series) do not belong tothe kernel of the Cauchy-Riemann operator ∂ of Clifford analysis.A different approach to slice regularity has been introduced in [13, 14] and canbe generalized to an ample class of real ∗ - algebras. Here we restrict to the case ofthe Clifford algebras R n , in view of their direct relations with Euclidean harmonicanalysis and Clifford analysis. Consider on R n the ∗ -algebra structure given by theClifford conjugation x x c , the unique linear antiinvolution of R n such that e ci = − e i for i = 1 , . . . , n. The real multiples of the unity of R n are identified with the real numbers. If x = x + x e + · · · + x n e n ∈ R n +1 , then x c = x − x e − · · · − x n e n and therefore the trace t ( x ) = x + x c = 2 x is real and the norm n ( x ) = xx c = | x | is real nonnegative.These relations hold on the entire quadratic cone of R n , the subset Q R n of R n definedby Q R n = { x ∈ R n | t ( x ) ∈ R , n ( x ) ∈ R } . Note that Q R = R ≃ C , Q R = R ≃ H , while Q R n is a proper subset of R n if n ≥ , properly containing the paravector space. Each element x ∈ Q R n can beuniquely written as x = Re( x ) + Im( x ) with Re( x ) = x + x c , Im( x ) = x − x c = βJ , with β = | Im( x ) | and J ∈ S R n , the“sphere” of all imaginary units J in R n (not necessarily paravectors) compatible withthe ∗ -algebra structure, that is satisfying the conditions t ( J ) = 0 , n ( J ) = 1 . In otherwords, we still have the slice decomposition Q R n = [ J ∈ S R n C J C J = h , J i is the complex “slice” of R n generated by J ∈ S R n . It holds C J ∩ C K = R for each J, K ∈ S R n , J = ± K . The quadratic cone is a real coneinvariant with respect to translations along the real axis.Let R n ⊗ R C be the complex Clifford algebra, with elements represented as w = a + ıb , a, b ∈ R n , ı = − . Every Clifford polynomial f ( x ) = P k x k a k lifts to aunique polynomial function F : C → R n ⊗ R C which makes the following diagramcommutative for all J ∈ S R n : C ≃ R ⊗ R C F −−−−→ R n ⊗ R C Φ J y y Φ J Q R n f −−−−→ R n (1)where Φ J : R n ⊗ R C → R n is defined by Φ J ( a + ıb ) := a + Jb . The lifted poly-nomial is simply F ( z ) = P k z k a k , with variable z = α + ıβ ∈ C . In this lifting,the usual product of polynomials with coefficients in R n on one fixed side (the oneobtained by imposing commutativity of the indeterminate with the coefficients whentwo polynomials are multiplied together) corresponds to the pointwise product in thealgebra R n ⊗ R C .If a Clifford algebra-valued function f (not necessarily a polynomial) has a holo-morphic lifting F as in (1) then f is called (left) slice-regular .More precisely, let D ⊆ C be a set that is invariant with respect to complex con-jugation. In R n ⊗ R C consider the complex conjugation that maps w = a + ıb to w = a − ıb (with a, b ∈ R n ). If a function F : D → R n ⊗ R C satisfies F ( z ) = F ( z ) for every z ∈ D , then F is called a stem function on D . Let Ω D be the circular subsetof the quadratic cone defined by Ω D = [ J ∈ S R n Φ J ( D ) . The stem function F = F + ıF : D → R n ⊗ R C induces the (left) slice function f = I ( F ) : Ω D → R n in the following way: if x = α + Jβ = Φ J ( z ) ∈ Ω D ∩ C J ,then f ( x ) = F ( z ) + JF ( z ) , where z = α + ıβ . The slice function f = I ( F ) is called (left) slice-regular if F isholomorphic. The function f = I ( F ) is called slice-preserving if F and F are real-valued (this is the case already considered by Fueter [10] for quaternionic functionsand by Gürlebeck and Sprössig [16] for radially holomorphic functions on Cliffordalgebras). In this case, the condition f ( x c ) = f ( x ) c holds for each x ∈ Ω D .When n = 2 , R is the algebra of real quaternions. If the domain D intersectsthe real axis, this definition of slice regularity is equivalent to the one proposed byGentili and Struppa [12]. For any n > , when the domain Ω intersects the real axis,the definition of slice regularity on R n is equivalent to the one of slice monogenicity,in the sense that the restriction to the paravector space of a R n -valued slice-regularfunction is a slice-monogenic function. Let
Ω = Ω D be a circular open subset of the quadratic cone of R n . The slice product of two slice functions f = I ( F ) , g = I ( G ) on Ω is the slice function induced by the4ointwise product of the stems F and G , i.e. f · g = I ( F G ) . If f is slice-preserving,then f · g coincides with the pointwise product of f and g . In this case, we will denoteit simply by f g . Note that if f, g are slice-regular on Ω , then also f · g is slice-regularon Ω .The function f ◦ s : Ω → R n , called spherical value of f , and the function f ′ s :Ω \ R → R n , called spherical derivative of f , are defined as f ◦ s ( x ) := ( f ( x ) + f ( x c )) and f ′ s ( x ) := Im( x ) − ( f ( x ) − f ( x c )) . They are slice functions, constant on every set of the form S x = α + S R n β for any x = α + Jβ ∈ Ω \ R . Moreover, they satisfy the relation f ( x ) = f ◦ s ( x ) + Im( x ) f ′ s ( x ) (2)for each x ∈ Ω \ R . Remark 1.
It can be shown that f ◦ s ( x ) is the spherical mean of f on S x , while f ′ s ( x ) is the spherical mean of Im( x ) − f ( x ) on S x . In [18] some formulas were proved, relating the Cauchy-Riemann operator, the spher-ical Dirac operator and the spherical derivative of a slice function. From this it wasobtained a result which implies, in particular, the Fueter-Sce Theorem and gives acharacterization of slice-regularity by means of the Cauchy-Riemann operator. We re-call that Fueter’s Theorem [10], generalized by Sce [21], Qian [19] and Sommen [22]on Clifford algebras and octonions, in our language states that applying to a slice-preserving slice-regular function the Laplacian operator of R (in the quaternioniccase) or the iterated Laplacian operator (∆ n +1 ) m of R n +1 (in the Clifford algebracase with n = 2 m + 1 odd), one obtains a function in the kernel, respectively, of theCauchy-Riemann-Fueter operator or of the Cauchy-Riemann operator. This result wasextended in [6] to the whole class of slice-monogenic functions defined by means ofstem functions.Since the paravector space R n +1 is contained in the quadratic cone Q R n , we canconsider the restriction of a slice function on domains of the form Ω = Ω D ∩ R n +1 in R n +1 . Thanks to the representation formula (see e.g. [13, Proposition 6]), this restric-tion uniquely determines the slice function. We will therefore use the same symbol todenote the restriction. Theorem 2. [18, Corollary 3.3.3 and Corollary 3.4.4]Let f : Ω ⊆ R n +1 → R n be a slice function of class C (Ω) . Then it holds:(a) f is slice-regular if and only if ∂f = (1 − n ) f ′ s .(b) Let n = 2 m + 1 be odd. If f : Ω ⊆ R n +1 → R n is (the restriction of) aslice-regular function, then it holds:(i) (∆ n +1 ) m f ′ s = 0 on Ω , i.e. the spherical derivative of a slice-regular func-tion is m -harmonic.(ii) The following generalization of Fueter-Sce Theorem for R n holds true: ∂ (∆ n +1 ) m f = 0 on Ω . iii) (∆ n +1 ) m +1 f = 0 , i.e. every slice regular function on R n is polyharmonicof order m + 1 . Remark 2.
In the case of even n , the fractional power (∆ n +1 ) m of the Laplacian,with m = n − , can be defined by means of the Fourier transform. Recently, using theresults of Qian [19], Altavilla, De Bie and Wutzig [2, Lemma 4.4] generalized point(i) of part (b) of the previous statement to the case of even dimension n , proving that itstill holds (∆ n +1 ) m f ′ s = 0 for every slice-regular function f for which (∆ n +1 ) m f isa well-defined differentiable function. We firstly consider the case of polynomials p ( x ) = P dk =0 x k a k with Clifford coeffi-cients in R n , with n odd. We start with a decomposition of Clifford powers x k andthen extend it by right-linearity. Proposition 1. If n = 2 m + 1 is odd, then for every k ≥ and for every x ∈ R n +1 ⊆ R n it holds x k = P mk ( x ) − x c · P mk − ( x ) = P mk ( x ) − x c P mk − ( x ) where for k ≥ the function P mk ( x ) := (cid:0) x k +1 (cid:1) ′ s is the spherical derivative of theClifford power x k +1 on R n and P m − := 0 .Proof. Applying the Leibniz-type formula ( f · g ) ′ s = f ′ s · g ◦ s + f ◦ s · g ′ s satisfied by thespherical derivative (see [13, §5]) to the power x k +1 , we get ( x k +1 ) ′ s = ( x · x k ) ′ s = ( x ) ′ s ( x k ) ◦ s + ( x ) ◦ s ( x k ) ′ s = ( x k ) ◦ s + x ( x k ) ′ s . (3)Here the slice product coincides with the pointwise product since the functions areslice-preserving. Using (2) and (3), we can write x k = ( x k ) ◦ s + Im( x )( x k ) ′ s = ( x k +1 ) ′ s − x ( x k ) ′ s + Im( x )( x k ) ◦ s = ( x k +1 ) ′ s − x c ( x k ) ′ s which is the formula to be proved.For every k ≥ , the polynomial P mk ( x ) is a homogeneous polynomial in thevariables x , . . . , x n , of (total) degree k , with real coefficients (see [18, §3.5]). Frompoint (b) of Theorem 2, it follows that P mk is m -harmonic. Moreover, being a sphericalderivative, it has an axial symmetry with respect to the real axis, i.e. P mk ◦ T = P mk forevery orthogonal transformation T of R n +1 that fixes the real points. Every polynomialin the variables x , . . . x n of the form P dk =0 P mk ( x ) a k has the same properties. It willbe called a zonal m -harmonic polynomial with pole 1 . Example 1.
Let n = 5 and m = 2 and consider the (3-harmonic) powers x k of theClifford variable in R ⊆ R . It holds, for every k ≥ , x k = P k ( x ) − x c P k − ( x ) ∀ x ∈ R ⊆ R , with zonal biharmonics P k ( x ) in R . For example, x = P ( x ) − x c P ( x ) with P ( x ) = 4 x ( x − x − x − x − x − x ) and P ( x ) = 5 x − x (cid:16)P i =1 x i (cid:17) + (cid:16)P i =1 x i (cid:17) . It holds ∆ P = − x + 8 (cid:16)P i =1 x i (cid:17) and then ∆ P = 0 , as expected. orollary 1. Let f ( x ) = P dk =0 x k a k be a polynomial function of degree d ≥ withClifford coefficients a k ∈ R n (with n = 2 m + 1 odd). Then there exist two zonal m -harmonic polynomials A and B with pole 1, of degrees d and d − respectively,such that f ( x ) = A ( x ) − x c B ( x ) for every x ∈ R n +1 . Proof.
In view of the preceding proposition, it suffices to set A ( x ) := P dk =0 P mk ( x ) a k and B ( x ) := P dk =0 P mk − ( x ) a k .The polynomial P mk ( x ) coincides, up to a real multiplicative constant c mk , withthe unique zonal m -harmonic homogeneous polynomial on R n +1 of degree k withpole (see [4, Chapter 5] for the harmonic case ( m = 1 ) and [20] for the genuinepolyharmonic case ( m > )). This means that there exists a real constant c mk such that P mk ( x ) = c mk Z mk ( x, , where Z mk ( x, y ) is the reproducing kernel of the m -harmonic homogeneous polyno-mials of degree k on R n +1 with respect to the Fischer inner product or (with differentconstants) with respect to the L -product on the unit sphere S n of R n +1 . The con-stants c mk can be determined from the property P mk (1) = k + 1 . The value P mk (1) canbe computed using the fact that the spherical derivative f ′ s of a slice-regular function f = I ( F ) extends to the real points with the values of the slice derivative ∂f∂x = I (cid:0) ∂F∂z (cid:1) of f [13]. Since ∂x j ∂x = jx j − for every j ≥ , it holds P mk (1) = ( k + 1)( x k ) | x =1 = k + 1 .In particular, when m = 1 , Z k ( x, y ) is the (solid) zonal harmonic of degree k andpole y in R (denoted by Z k ( x, y ) in [4, Chapter 5]). In this case the constants are c k = 1 / ( k + 1) (see [18, Proposition 3.5.1]), i.e. P k ( x ) = 1 k + 1 Z k ( x, . Now we extend Corollary 1 in two directions: we prove the Almansi-type decom-position for every slice-regular function of a Clifford variable and we remove the re-quirement on the dimension n to be odd.In the following statement we will consider slice functions that are m -harmonicand axially symmetric with respect to the real axis. In accordance with the terms usedfor polynomials, such functions will be called zonal m -harmonic functions with pole 1 .For brevity, we use the notation ∂ α and ∂ β to denote the partial derivatives with respectto α and β . Theorem 3.
Let n ≥ be any integer and let m = n − . Let f : Ω ⊆ R n +1 → R n be slice-regular. If n is even, we assume that the fractional Laplacian (∆ n +1 ) m f isa well-defined differentiable function. Then there exist two unique R n -valued zonal m -harmonic functions A , B with pole on Ω , such that f ( x ) = A ( x ) − x c B ( x ) ∀ x ∈ Ω .Conversely, if A and B are R n -valued functions of class C on Ω , axially symmetricwith respect to the real axis, then g ( x ) := A ( x ) − x c B ( x ) is a slice function on Ω . The unction g is slice-regular if and only if A and B satisfy the system of equations ( ∂ α A − α ∂ α B − β ∂ β B = 2 B∂ β A − α ∂ β B + β ∂ α B = 0 (4) where α = x and β = | Im( x ) | = p x + · · · + x n .Proof. Assume that f is slice-regular on Ω . Then also x · f = xf is slice-regular andit holds ( xf ) ′ s = ( x · f ) ′ s = ( x ) ′ s f ◦ s + ( x ) ◦ s f ′ s = f ◦ s + x f ′ s . (5)From (2) and (5) we get f ( x ) = f ◦ s ( x ) + Im( x ) f ′ s ( x ) = ( f ◦ s ( x ) + x f ′ s ( x )) − x c f ′ s ( x ) = ( xf ) ′ s − x c f ′ s ( x ) . Define A ( x ) := f ◦ s ( x )+ x f ′ s ( x ) = ( xf ) ′ s and B ( x ) := f ′ s ( x ) . Point (b) of Theorem 2,together with Remark 2, implies that A and B , being spherical derivatives of sliceregular functions, are zonal m -harmonic slice functions on Ω with pole 1.To prove uniqueness, we observe that if f ( x ) = A ( x ) − x c B ( x ) with A and B axially symmetric functions w.r.t. the real axis, then for every x ∈ Ω \ R it holds f ′ s ( x ) = (2 Im( x )) − ( A ( x ) − x c B ( x ) − A ( x c ) + xB ( x c )) . Since A ( x c ) = A ( x ) and B ( x c ) = B ( x ) , we get that f ′ s ( x ) = (2 Im( x )) − ( x − x c ) B ( x ) = B ( x ) . From this we deduce also A ( x ) = f ( x ) + x c f ′ s ( x ) and therefore A and B are uniquely determined by f .Conversely, assume that A and B are axially symmetric w.r.t. the real axis on Ω =Ω D . Given x = α + Jβ ∈ Ω and z = α + ıβ ∈ D , we set G ( z ) := A ( x ) − x B ( x ) = A ( x ) − αB ( x ) and G ( z ) := βB ( x ) . Thanks to the symmetry properties of A and B , G and G are well-defined functions from D into R n , and the function G := G + ıG is a stem function of class C on D , which induces the slice function g = A − x c B : I ( G )( x ) = A ( x ) − x B ( x )+ JβB ( x ) = A ( x ) − ( x − Im( x )) B ( x ) = A ( x ) − x c B ( x ) . To conclude the proof it is sufficient to see that the Cauchy-Riemann equations ∂ α G = ∂ β G , ∂ β G = − ∂ α G for G are equivalent to the system (4) for A and B .Note that the correspondence f ( A, B ) given by Theorem 3 is right R n -linear.Moreover, the pairs ( A, B ) formed by real-valued functions A and B correspond to thesubclass of slice-preserving functions f . Example 2.
For n = 3 , m = 1 , the functions A, B associated to f are R -valuedharmonic function of four real variables. For example, let f ( x ) = exp( x ) = e x cos q x + x + x + Im( x ) | Im( x ) | e x sin q x + x + x be the slice-preserving regular function induced on Q R by the complex exponential.Then f ( x ) = A ( x ) − x c B ( x ) with harmonic components A, B on R given by A ( x ) = e x (cid:0) cos p x + x + x + x √ x + x + x √ x + x + x (cid:1) ,B ( x ) = e x sin √ x + x + x √ x + x + x . xample 3. For n = 2 , m = 1 / , the function g ( x ) = x − = x c | x | − is slice-regularon R \ { } ⊂ R . The function ∆ / g is a multiple of the monogenic Cauchy kernel E ( x ) = x c | x | − on R . It holds g ′ s = − ∂g = −| x | − and ( xg ) ′ s = 0 . Therefore the decomposition of Theorem 3 is given by A ≡ , B ( x ) = −| x | − . Thesefunctions are in the kernel of the fractional Laplacian ∆ / on R \ { } . Remark 3.
The m -harmonic functions A and B in the statement of Theorem 3 canalso be computed from f by differentiation using the Cauchy-Riemann operator on R n . Since A ( x ) = ( xf ) ′ s and B ( x ) = f ′ s ( x ) , from point (a) of Theorem 2 we get A = − n ∂ ( xf ) and B = − n ∂f. Combining Theorem 3 with the classical Almansi’s Theorem, we obtain anotherdecomposition in terms of biharmonic functions in the kernel of the operator ∂ ∆ n +1 .Here we assume m > because the case m = 1 is already contained in Theorem 2. Corollary 2.
Let n = 2 m + 1 > be odd. Let Ω ⊆ R n +1 be a star-like domain withcentre 0. Let f : Ω ⊆ R n +1 → R n be slice-regular. Then there exist R n -valued zonalbiharmonic functions g , . . . , g m − on Ω , with pole , such that f ( x ) = g ( x ) + | x | g ( x ) + · · · + | x | m − g m − ( x ) ∀ x ∈ Ω .The functions g , . . . , g m − belong to the kernel of the operator ∂ ∆ n +1 .Proof. Let f = A − x c B be the decomposition given in Theorem 3. Applying The-orem 1 to the n real components of A and B , which are m -harmonic on Ω , we get R n -valued harmonic functions u , . . . , u m − and v , . . . , v m − such that A ( x ) = m − X k =0 | x | k u k ( x ) and B ( x ) = m − X k =0 | x | k v k ( x ) . Therefore f ( x ) = A ( x ) − x c B ( x ) = m − X k =0 | x | k ( u k ( x ) − x c v k ( x )) = m − X k =0 | x | k g k ( x ) . The functions g k = u k − x c v k have the same symmetry properties as the functions A and B . This follows from the uniqueness of the Almansi decomposition. Givenany orthogonal transformation T of R n +1 that fixes the real points, also A ◦ T is m -harmonic and u k ◦ T is harmonic. Since A ( T ( x )) = m − X k =0 | x | k u k ( T ( x )) = A ( x ) , it must be u k ( T ( x )) = u k ( x ) for every k . The same holds for B . To prove thelast statement of the thesis, we observe that for every R n -valued function u , it holds ∆ n +1 ( x c u ) = 2 ∂u + x c ∆ n +1 u . Therefore ∆ n +1 g k = ∆ n +1 ( − x c v k ) = − ∂v k andthen ∂ ∆ n +1 g k = − ∂∂v k = − n +1 v k = 0 . In particular, the functions g k arebiharmonic. 9 Acknowledgements
This work was supported by GNSAGA of INdAM, and by the grants “Progetto diRicerca INdAM, Teoria delle funzioni ipercomplesse e applicazioni”, and PRIN “Realand Complex Manifolds: Topology, Geometry and holomorphic dynamics” of the Ital-ian Ministry of Education.
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