A new integral formula for the inverse Fueter mapping theorem
Fabrizio Colombo, Dixan Peña Peña, Irene Sabadini, Frank Sommen
aa r X i v : . [ m a t h . C V ] F e b A new integral formula for the inverse Fuetermapping theorem
Fabrizio Colombo ∗ e-mail: [email protected] Dixan Pe˜na Pe˜na ∗∗ e-mail: [email protected] Irene Sabadini ∗ e-mail: [email protected] Frank Sommen ∗∗ e-mail: [email protected] ∗ Dipartimento di Matematica, Politecnico di MilanoVia Bonardi, 9, 20133 Milano, Italy ∗∗ Clifford Research Group, Department of Mathematical AnalysisFaculty of Engineering and Architecture, Ghent UniversityGalglaan 2, 9000 Gent, Belgium
Abstract
In this paper we provide an alternative method to construct the Fueterprimitive of an axial monogenic function of degree k , which is comple-mentary to the one used in [4]. As a byproduct, we obtain an explicitdescription of the kernel of the Fueter mapping. We also apply ourmethod to obtain the Fueter primitives of the Cauchy kernels withsingularities on the unit sphere. Keywords : Axial monogenic functions; Fueter’s theorem.
Mathematics Subject Classification : 30G35, 32A25, 30E20.
Let us denote by R ,m ( m ∈ N ) the real Clifford algebra generated by thestandard basis { e , . . . , e m } of the Euclidean space R m (see [2]). The multi-plication in R ,m is determined by the relations e j e k + e k e j = − δ jk , j, k = 1 , . . . , m a of R ,m may be written as a = X A a A e A , a A ∈ R , in terms of the basis elements e A = e j . . . e j k , defined for every subset A = { j , . . . , j k } of { , . . . , m } with j < · · · < j k . For the empty set, one puts e ∅ = 1, the latter being the identity element. Conjugation in R ,m is givenby a = P A a A e A , where e A = e j k · · · e j with e j = − e j , j = 1 , . . . , m .Observe that R m +1 may be naturally embedded in the real Clifford algebra R ,m by associating to any element ( x , x , . . . , x m ) ∈ R m +1 the paravectorgiven by x + x = x + m X j =1 x j e j . A function f : Ω → R ,m defined and continuously differentiable in an openset Ω in R m +1 (resp. R m ), is said to be monogenic if( ∂ x + ∂ x ) f = 0 (resp. ∂ x f = 0) in Ω , where ∂ x = P mj =1 e j ∂ x j is the Dirac operator in R m (see e.g. [1, 7, 9]). Thedifferential operator ∂ x + ∂ x , called generalized Cauchy-Riemann operator,gives a factorization of the Laplacian, i.e.∆ = m X j =0 ∂ x j = ( ∂ x + ∂ x )( ∂ x − ∂ x ) . Thus the monogenic functions can be considered a subclass of the class ofharmonic functions in m + 1 variables.Throughout the paper we assume h ( z ) = u ( x, y ) + iv ( x, y ) to be a holo-morphic function in some open subset Ξ of the upper half of the complexplane C and P k ( x ) shall denote a homogeneous monogenic polynomial of de-gree k in R m . Let us recall the following generalization of Fueter’s theoremobtained in [18]: Theorem 1
Put ω = x/r , with r = | x | , x ∈ R m . If m is odd, then thefunction Ft [ h ( z ) , P k ( x )] ( x , x ) = ∆ k + m − (cid:2)(cid:0) u ( x , r ) + ω v ( x , r ) (cid:1) P k ( x ) (cid:3) is monogenic in Ω = { ( x , x ) ∈ R m +1 : ( x , r ) ∈ Ξ } .
2n other words, this result provides a way to generate monogenic functionsstarting from a holomorphic function in the upper half of the complex plane.It was originally formulated by R. Fueter in the setting of quaternionic anal-ysis in [8] and later extended to the case of Clifford algebra-valued functionsin [15, 17] (see also [10, 12, 14, 16]).
Remark 1
It easily seen that Ft [ h ( z ) , P k ( x )] defines an R -linear operatorbetween holomorphic functions and monogenic functions considered as realvector spaces, i.e. Ft [ c h ( z ) + c h ( z ) , P k ( x )] = c Ft [ h ( z ) , P k ( x )] + c Ft [ h ( z ) , P k ( x )] , for all c , c ∈ R .The functions generated by this technique are monogenic functions of theform (cid:0) A ( x , r ) + ω B ( x , r ) (cid:1) P k ( x ) , (1)where A and B are R -valued continuously differentiable functions in R whichsatisfy the following Vekua-type system ( ∂ x A − ∂ r B = 2 k + m − r B∂ x B + ∂ r A = 0 . (2)Monogenic functions of the form (1) are called axial monogenic of degree k and represent an important class of functions in Clifford analysis (see [7]).It is quite natural to ask whether given an axial monogenic function ofdegree k H ( x , x ) = (cid:0) A ( x , r ) + ω B ( x , r ) (cid:1) P k ( x ) , one can find a holomorphic function h ( z ) such that Ft [ h ( z ) , P k ( x )] ( x , x ) = H ( x , x ) . The function h is called the Fueter primitive of H . This problem has beenrecently studied in [3, 4] and the Fueter primitive h has been explicitly con-structed. To this purpose, it was necessary to determine the Fueter primitives W ± k,m of the functions F + k,m ( x , x ) = Z S m − G ( x + x − ω ) P k ( ω ) dS ( ω ) , F − k,m ( x , x ) = Z S m − G ( x + x − ω ) ω P k ( ω ) dS ( ω ) , dS ( ω ) is the scalar element of surface area of S m − , and G ( x + x ) = 1 A m +1 x + x | x + x | m +1 is the monogenic Cauchy kernel. Then, it is possible to express the Fueterprimitive of H in terms of a suitable integral involving W ± k,m , A , B and P k (see [4]). This method can be used on any axially symmetric open set of R m +1 , i.e. on every open set which is invariant under rotations that fix thereal axis x .Every monogenic function f defined on an axially symmetric open set of R m +1 can be written as f = P ∞ k =0 f k where f k are axial monogenic functionsof degree k . Hence for each term in the series we can provide a Fueterprimitive as described above. We would like to note that the problem ofinverting the Fueter mapping theorem has been recently tackled in the caseof bi-axial monogenic functions (see [5]).The aim of this paper is to present an alternative proof of the fact thatthe Fueter mapping is surjective on the set of axial monogenic functions ofdegree k and to explicitly provide their Fueter primitives. The method wepresent here is complementary to the one presented in [4] in the sense thatwe here integrate with respect to the radius r instead of the axial coordinate x . For the sake of simplicity the method is developed on a rectangle; ofcourse it remains applicable on more general axially symmetric domains.As a byproduct of this method we describe the kernel of the Fueter map-ping. We also compute with this method an exact formula for the Fueterprimitives of the Cauchy kernels with singularities on the unit sphere. Thiscorresponds to the integrals of the standard Cauchy kernel over the unitsphere. Let f : [ a, b ] → R be a continuous function. From the Cauchy formula forrepeated integration, we know that an n -th antiderivative of f is given by f ( − n ) ( x ) = 1( n − Z xa ( x − t ) n − f ( t ) dt. Inspired by this formula, we wish to find the solutions of the equations (cid:18) x − ddx (cid:19) n g ( x ) = f ( x ) and (cid:18) ddx x − (cid:19) n g ( x ) = f ( x ) , (3)where f : [ a, b ] → R is a given continuous function.4 emark 2 It is worth noting that the following identities hold (see [6, 13]) (cid:18) x − ddx (cid:19) n g ( x ) = n X j =1 ( − n + j a j,n x j − n d j gdx j ( x )with a j,n = (2 n − j − n − j ( n − j )!( j − (cid:18) ddx x − (cid:19) n g ( x ) = n X j =0 ( − n + j a j +1 ,n +1 x j − n d j gdx j ( x ) . Moreover, the integers a j +1 ,n +1 = (2 n − j )!2 n − j ( n − j )! j !are the coefficients of the Bessel polynomial of degree n (see [11]).In order to find the solutions of (3), we define the following functions: φ n ( x ) = Z xa tφ n − ( t ) dt, ψ n ( x ) = x Z xa ψ n − ( t ) dt, x ∈ [ a, b ] , n ≥ , with φ ( x ) = ψ ( x ) = f ( x ). Obviously, φ n and ψ n satisfy φ ′ n ( x ) x = φ n − ( x ) , (cid:18) ψ n ( x ) x (cid:19) ′ = ψ n − ( x ) , φ n ( a ) = ψ n ( a ) = 0 , n ≥ . (4) Lemma 1
The functions φ n and ψ n are given by φ n ( x ) = 1(2 n − Z xa t ( x − t ) n − f ( t ) dt, (5) ψ n ( x ) = x (2 n − Z xa ( x − t ) n − f ( t ) dt, (6) where n !! denotes the double factorial of n .Proof. Using integration by parts we obtain Z xa t ( x − t ) n − f ( t ) dt = Z xa ( x − t ) n − φ ′ ( t ) dt = (cid:0) ( x − t ) n − φ ( t ) (cid:1)(cid:12)(cid:12) t = xt = a + 2( n − Z xa t ( x − t ) n − φ ( t ) dt. Z xa t ( x − t ) n − f ( t ) dt = 2( n − Z xa t ( x − t ) n − φ ( t ) dt. We iterate this procedure until the ( x − t ) term vanishes. Thus after n − Z xa t ( x − t ) n − f ( t ) dt = 2 n − ( n − Z xa tφ n − ( t ) dt = 2 n − ( n − φ n ( x ) , which proves (5). Formula (6) may be proved in a similar way. (cid:3) As an immediate consequence of Lemma 1 we obtain:
Theorem 2
Let f : [ a, b ] → R be a continuous function. The general solu-tion of the equation (cid:18) x − ddx (cid:19) n g ( x ) = f ( x ) is φ n ( x ) + n − X j =0 C j x j while the general solution of (cid:18) ddx x − (cid:19) n g ( x ) = f ( x ) is ψ n ( x ) + n − X j =0 ˜ C j x j +1 , where C j and ˜ C j , j = 0 , . . . , n − , are arbitrary real constants. Here plays an essential role the explicit form of Ft [ h ( z ) , P k ( x )] determinedin [12], namely: Ft [ h ( z ) , P k ( x )] ( x , x ) = (2 k + m − × (cid:16)(cid:0) r − ∂ r (cid:1) k + m − u ( x , r ) + ω (cid:0) ∂ r r − (cid:1) k + m − v ( x , r ) (cid:17) P k ( x ) . (7)6 heorem 3 (The inverse Fueter mapping theorem) Let H ( x , x ) = (cid:0) A ( x , r ) + ω B ( x , r ) (cid:1) P k ( x ) , be a given arbitrary axial monogenic function of degree k in Ω = (cid:8) ( x , x ) ∈ R m +1 : ( x , r ) ∈ [ a, b ] × [ c, d ] ⊂ R , c > (cid:9) . The Fueter primitives of H ( x , x ) exist and are given by u ( x , r ) = K N Z rc t ( r − t ) N − A ( x , t ) dt + N − X j =0 α j ( x ) r j , (8) v ( x , r ) = K N r Z rc ( r − t ) N − B ( x , t ) dt + N − X j =0 β j ( x ) r j +1 , (9) where K N = 12 N ((2 N − , N = k + m − . Moreover, the R -valued functions α j ( x ) and β j ( x ) satisfy the following dif-ferential equations α ′ j ( x ) − (2 j + 1) β j ( x )= ( − N − j − K N (cid:18) N − j (cid:19) c N − j ) − B ( x , c ) , j = 0 , . . . , N − , (10) β ′ j ( x ) + 2( j + 1) α j +1 ( x )= ( − N − j K N (cid:18) N − j (cid:19) c N − j − A ( x , c ) , j = 0 , . . . , N − , (11) β ′ N − ( x ) = − K N A ( x , c ) . (12) Proof.
Let h ( z ) = u ( x, y ) + iv ( x, y ) be a Fueter primitive of H , i.e. Ft [ h ( z ) , P k ( x )] ( x , x ) = H ( x , x ) , then from (7) we obtain (cid:0) r − ∂ r (cid:1) k + m − u ( x , r ) = A ( x , r )(2 k + m − , (cid:0) ∂ r r − (cid:1) k + m − v ( x , r ) = B ( x , r )(2 k + m − . u and v given by these formulae satisfy the Cauchy-Riemann equations, taking intoaccount that A and B fulfill the Vekua-type system (2).Let us define I ( x , r ) = Z rc t ( r − t ) N − A ( x , t ) dt, (13) I ( x , r ) = r Z rc ( r − t ) N − B ( x , t ) dt. (14)It follows from (2) and (4) that ∂ x I ( x , r ) = Z rc t ( r − t ) N − (cid:18) ∂ t B ( x , t ) + 2 Nt B ( x , t ) (cid:19) dt = − c ( r − c ) N − B ( x , c ) + (2 N − Z rc ( r − t ) N − B ( x , t ) dt + 2( N − Z rc t ( r − t ) N − B ( x , t ) dt,∂ r I ( x , r ) = Z rc ( r − t ) N − B ( x , t ) dt + 2( N − r Z rc ( r − t ) N − B ( x , t ) dt =(2 N − Z rc ( r − t ) N − B ( x , t ) dt + 2( N − Z rc t ( r − t ) N − B ( x , t ) dt, which implies that ∂ x I ( x , r ) − ∂ r I ( x , r ) = − c ( r − c ) N − B ( x , c ) . Similarly, we may verify that ∂ r I ( x , r ) + ∂ x I ( x , r ) = r ( r − c ) N − A ( x , c ) . We thus have ∂ x u ( x , r ) − ∂ r v ( x , r ) = N − X j =0 (cid:0) α ′ j ( x ) − (2 j + 1) β j ( x ) (cid:1) r j − K N c ( r − c ) N − B ( x , c ) , r u ( x , r ) + ∂ x v ( x , r ) = N − X j =0 (cid:0) β ′ j ( x ) + 2( j + 1) α j +1 ( x ) (cid:1) r j +1 + β ′ N − ( x ) r N − + K N r ( r − c ) N − A ( x , c ) . Therefore u and v satisfy the Cauchy-Riemann equations if and only if (10),(11) and (12) are fulfilled. (cid:3) This theorem thus asserts that Ft [ h ( z ) , P k ( x )] is surjective on the spaceof axial monogenic functions of degree k . Furthermore, we note that thisoperator is not injective since Ft [ z n , P k ( x )] ( x , x ) = 0 for 0 ≤ n ≤ k + m − z n , 0 ≤ n ≤ k + m −
2, is indeed the kernel of Ft [ h ( z ) , P k ( x )]. Corollary 1
Let R k + m − [ z ] be the vector space of all polynomials with realcoefficients in z of degree at most k + m − . Then ker ( Ft [ h ( z ) , P k ( x )]) = R k + m − [ z ] . Proof.
We only have to prove that ker ( Ft [ h ( z ) , P k ( x )]) ⊂ R k + m − [ z ] . If Ft [ h ( z ) , P k ( x )] ( x , x ) = 0, then from (8) and (9) we obtain u ( x, y ) = N − X j =0 α j ( x ) y j , v ( x, y ) = N − X j =0 β j ( x ) y j +1 . The differential equations (10), (11) and (12) now tell us that α j ( x ) (resp. β j ( x )) are polynomials of degree at most 2( N − j ) − N − j − u ( x,
0) = C + C x + . . . C N − x N − , v ( x,
0) = 0 , for certain real constants C , . . . , C N − . Then clearly h ( z ) ∈ R k + m − [ z ]. (cid:3) Remark 3 As Ft [ h ( z ) , P k ( x )] is an R -linear operator, it is clear that Ft [ h ( z ) , P k ( x )] = Ft [ h ( z ) , P k ( x )] ⇔ h ( z ) − h ( z ) ∈ R k + m − [ z ] . We end the paper with two examples.9 xample 1
From [12, 13] we have that Ft [1 /z, P k ( x )] ( x , x ) = ( − k + m − ((2 k + m − (cid:18) x − x | x + x | k + m +1 (cid:19) P k ( x ) . Thus if we apply (8) and (9) to this monogenic function we should be ableto obtain the Cauchy kernel in the plane. Let us illustrate this for the case k = 0, m = 5. For this case N = 2 and A ( x , r ) = x ( x + r ) , B ( x , r ) = − r ( x + r ) . It easily follows that β ( x ) = − x + 2 c x + c ) , α ( x ) = − x β ( x ) ,β ( x ) = 164( x + c ) , α ( x ) = − x β ( x ) ,I ( x , r ) = ( r − c ) x x + r )( x + c ) , I ( x , r ) = − ( r − c ) r x + r )( x + c ) , where I and I denote the functions defined in (13) and (14), respectively.Using (8) and (9) we obtain u ( x , r ) = x x + r ) , v ( x , r ) = − r x + r ) . That is h ( z ) equals (up to a multiplicative constant) the Cauchy kernel inthe plane. Example 2
In [3] we have considered the functions N + ( q ) = Z S G ( q − ω ) dS ( ω ) , and N − ( q ) = Z S G ( q − ω ) ω dS ( ω ) , q = x + rω, and their Fueter primitives W ± in order to provide the Fueter inverse of aregular function of a quaternionic variable. This corresponds to what wehave discussed in the introduction, i.e. the integrals F ± k,m , in the particularcase k = 0, m = 3. 10n a closed form, these two functions can be written as N + ( q ) = ωπr (cid:18)
11 + q − r Im(arctan q ) (cid:19) , and N − ( q ) = ωπr (cid:18) arctan q + q q − r Im( q arctan q ) (cid:19) . We shall use formulae (8) and (9) to retrieve the fact that the Fueter primitiveof N + is the function W + ( z ) = π arctan z . Note that the function N + isregular of axial type and it can thus be written as N + ( q ) = A ( x , r ) + ωB ( x , r ) where A ( x , r ) = 1 π x (1 + x − r ) + 4 x r and B ( x , r ) = 12 πr (cid:18) x − r )(1 + x − r ) + 4 x r − r ln (cid:18) x + ( r + 1) x + ( r − (cid:19)(cid:19) . We now compute I and I defined in (13) and (14) with N = 1. We get that2 πI ( x , r ) = 2 π Z rc tA ( x , t ) dt = arctan (cid:18) x − x − t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = rt = c = 2Re(arctan z ) − arctan (cid:18) x − x − c (cid:19) = 2Re(arctan z ) − πα ( x ) , πI ( x , r ) = 2 πr Z rc B ( x , t ) dt = r t ln (cid:18) x + ( t + 1) x + ( t − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = rt = c = 2Im(arctan z ) − r c ln (cid:18) x + ( c + 1) x + ( c − (cid:19) = 2Im(arctan z ) − πrβ ( x ) . On account of (8) and (9), we conclude that a Fueter primitive of N + ( q ) isthe function W + ( z ) = π arctan z as it was computed in [3].The function N − ( q ) is also of axial type and so N − ( q ) = A ( x , r ) + ωB ( x , r ) where A ( x , r ) = 12 π (cid:18) r ln (cid:18) x + ( r − x + ( r + 1) (cid:19) + 2( x + r − x − r ) + 4 x r (cid:19) B ( x , r ) = x πr (cid:18) r ln (cid:18) x + ( r − x + ( r + 1) (cid:19) + 2(1 + x + r )(1 + x − r ) + 4 x r (cid:19) . For this case we compute I and I with N = 1 and obtain2 πI ( x , r ) = 2 π Z rc tA ( x , t ) dt = (cid:18) x arctan (cid:18) x − x − t (cid:19) − t (cid:18) x + ( t + 1) x + ( t − (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = rt = c = 2Re( z arctan z ) − πα ( x ) , πI ( x , r ) = 2 πr Z rc B ( x , t ) dt = r (cid:18) x t ln (cid:18) x + ( t + 1) x + ( t − (cid:19) + arctan (cid:18) x − x − t (cid:19)(cid:19)(cid:12)(cid:12)(cid:12)(cid:12) t = rt = c = 2Im( z arctan z ) − πrβ ( x ) . Thus, using formulae (8), (9) we obtain that the Fueter primitive of N − ( q )is W − ( z ) = π z arctan z , as shown in [3]. Acknowledgments
D. Pe˜na Pe˜na acknowledges the support of a Postdoctoral Fellowship fundedby the “Special Research Fund” (BOF) of Ghent University.
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