RARITA-SCHWINGER TYPE OPERATORS ON SPHERES AND REAL PROJECTIVE SPACE. Junxia Li, John Ryan, and Carmen J. Vanegas

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1 ARCHIVUM MATHEMATICUM BRNO Tomus , RARITA-CHWINGER TYPE OPERATOR ON PHERE AND REAL PROJECTIVE PACE Junxia Li, John Ryan, and Carmen J. Vanegas Abstract. In this paper we deal with Rarita-chwinger type operators on spheres and real projective space. First we define the spherical Rarita-chwinger type operators and construct their fundamental solutions. Then we establish that the projection operators appearing in the spherical Rarita-chwinger type operators and the spherical Rarita-chwinger type equations are conformally invariant under the Cayley transformation. Further, we obtain some basic integral formulas related to the spherical Rarita-chwinger type operators. econd, we define the Rarita-chwinger type operators on the real projective space and construct their kernels and Cauchy integral formulas. 1. Introduction Rarita-chwinger operators are generalizations of the Dirac operator and arise in representation theory for the pin and Pin groups. ee [3, 4, 6, 14, 15]. We denote a Rarita-chwinger operator by R k, where k = 0, 1,..., m,.... When k = 0 we have the Dirac operator. The Rarita-chwinger operators R k in Euclidean space have been studied in [3, 4, 6, 14, 15]. Here we construct similar Rarita-chwinger operators together with their fundamental solutions and study their representation theory on the sphere and real projective space. First J. Ryan [12, 11] in 1997 and P. Van Lancker [13] in 1998 studied the Dirac operators on the sphere. Later, H. Liu and J. Ryan [8] studied the spherical Dirac type operators on the sphere by using Cayley transformations. ee also [1]. Using similar methods to define the Rarita-chwinger operators in R n, we can define the spherical Rarita-chwinger type operator on the sphere based on the spherical Dirac operator. We also use similar arguments as in Euclidean space to establish the conformal invariance for the projection operators and the spherical Rarita-chwinger type equations under the Cayley transformations. ee [6]. Further the fundamental solutions to the spherical Rarita-chwinger type operators are achieved by applying the Cayley transformation. In turn, tokes Theorem, Cauchy s Theorem, Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform are proved for the sphere. Furthermore, we show that tokes theorem is 2010 Mathematics ubject Classification: primary 30G35; secondary 53C27. Key words and phrases: spherical Rarita-chwinger type operators, Cayley transformation, real projective space, Almansi-Fischer decomposition, Iwasawa decomposition. Received June 6, Editor J. lovák. DOI: /AM

2 272 JUNXIA LI, J. RYAN AND C. J. VANEGA conformally invariant under Cayley transformation, and with minor modification, is equivalent to the Rarita-chwinger version of tokes Theorem in Euclidean space appearing in [3, 6] and elsewhere. By factoring out n by the group Z 2 = {±1} we obtain real projective space, RP n. On this space, we define the Rarita-chwinger type operators and construct their kernels over two different bundles over RP n. Further, we obtain some basic integral formulas from Clifford analysis associated with these operators for the two different bundles. This extends results from [7]. 2. Preliminaries A Clifford algebra, Cl n+1, can be generated from R n+1 by considering the relationship x 2 = x 2 for each x R n+1. We have R n+1 Cl n+1. If e 1,..., e n+1 is an orthonormal basis for R n+1, then x 2 = x 2 tells us that e i e j +e j e i = 2δ ij. Let A = {j 1,..., j r } {1, 2,..., n + 1} and 1 j 1 < j 2 < < j r n + 1. An arbitrary element of the basis of the Clifford algebra can be written as e A = e j1... e jr. Hence for any element a Cl n+1, we have a = A a Ae A, where a A R. The reversion is given by ã = A 1 A A 1/2 a A e A, where A is the cardinality of A. In particular, e j1... e jr = e jr... e j1. Also ãb = bã for a, b Cl n+1. The Clifford conjugation is defined by ā = A 1 A A +1/2 a A e A and satisfies e j1... e jr = 1 r e jr... e j1 and ab = bā for a, b Cl n+1. For each a = a 0 + a 1 e a 1...n+1 e 1... e n+1 Cl n+1 the scalar part of āa gives the square of the norm of a, namely a a a n+1. For more on Clifford algebras and their properties, see [9]. The Pin and pin groups play an important role in Clifford analysis. The Pin group can be defined as Pinn + 1 := {a Cl n+1 : a = y 1... y p : y 1,..., y p n, p N} and it is clearly a group under multiplication in Cl n+1. Now suppose that y n R n+1. Look at yxy = yx y y + yx y y = x y + x y where x y is the projection of x onto y and x y is perpendicular to y. o yxy gives a reflection of x in the y direction. By the Cartan Dieudonné Theorem each O On + 1 is the composition of a finite number of reflections. If a = y 1... y p Pinn + 1, then ã := y p... y 1 and axã = O a x for some O a On + 1. Choosing y 1,..., y p arbitrarily in n, we see that the group homomorphism θ : Pinn + 1 On + 1: a O a,

3 RARITA-CHWINGER OPERATOR ON PHERE 273 with a = y 1... y p and O a x = axã, is surjective. Further ax ã = axã, so 1, 1 kerθ. In fact kerθ = {±1}. ee [9]. The pin group is defined as pinn + 1 := {a Pinn + 1 : a = y 1... y p and p even} and it is a subgroup of Pinn + 1. There is a group homomorphism θ : pinn + 1 On + 1 which is surjective with kernel {1, 1}. ee [9] for details. The Dirac Operator in R n is defined to be n D := e j. x j j=1 Note D 2 = n, where n is the Laplacian in R n. If p k is a homogeneous polynomial with degree k such that Dp k = 0, we call such a polynomial a left monogenic polynomial homogeneous of degree k. Let H k be the space of Cl n+1 -valued harmonic polynomials homogeneous of degree k and M k the space of Cl n+1 -valued monogenic polynomials homogeneous of degree k. Note if h k H k, then Dh k M k 1. But Dup k 1 u = n 2k + 2p k 1 u, so H k = M k umk 1, h k = p k + up k 1. This is the so-called Almansi-Fischer decomposition of H k. ee [2, 10]. Note that if pu M k then it trivially extends to P v = pu + u n+1 e n+1 with u n+1 R and P v = pu for all u n+1 R. Consequently D n+1 P v = 0 n+1 where D n+1 = e j. u j j=1 If pu M k then for any boundary of a piecewise smooth bounded domain U R n by Cauchy s Theorem 1 nupudσu = 0. U uppose now a Pinn + 1 and u = awã then although u R n in general w belongs to the hyperplane a 1 R n ã 1 in R n+1. By applying a change of variable, up to a sign the integral 1 becomes 2 anwãp awãdσw = 0. a 1 Uã 1 As U is arbitrary then on applying tokes Theorem to 2 we see that 3 D a ãp awã = 0, quadwhere D a := D n+1 a 1 R n ã 1. uppose U is a domain in R n. Consider a function of two variables f : U R n Cl n+1 such that for each x U, fx, u is a left monogenic polynomial homogeneous of degree k in u. Let P k be the left projection map P k : H k M k,

4 274 JUNXIA LI, J. RYAN AND C. J. VANEGA then R k fx, u is defined to be P k D x fx, u. The left Rarita-chwinger equation is defined to be R k fx, u = 0. We also have a right projection P k,r : H k M k, and a right Rarita-chwinger equation fx, ud x P k,r = fx, ur k = 0, where M k stands for the space of right monogenic polynomials homogeneous of degree k. ee [6]. 3. Rarita-chwinger type operators on spheres Let R n be the span of e 1,..., e n. Consider the Cayley transformation C : R n n, where n is the unit sphere in R n+1, defined by Cx = e n+1 x+1x+e n+1 1, where x = x 1 e x n e n R n, and e n+1 is a unit vector in R n+1 which is orthogonal to R n. Now CR n = n \ {e n+1 }. uppose x s n and x s = x s1 e x sn e n + x sn+1 e n+1, then we have x = C 1 x s = e n+1 x s + 1x s e n+1 1. The Dirac operator over the n-sphere n has the form D s = wλ + n 2, where n w n and Λ = e i e j w i w j, see for instance [5, 8, 13]. w j w i i<j,i=1 Let U be a domain in R n. Consider a function f : U R n Cl n+1 such that for each x U, f x, u is a left monogenic polynomial homogeneous of degree k in u. This function reduces to fx s, u on CU R n and fx s, u takes its values in Cl n+1 where x = C 1 x s and x s CU n. Further fx s, u is a left monogenic polynomial homogeneous of degree k in u. ince u D s,xs = D s,xs u, then D s,xs fx s, u is harmonic in u. Hence by the Almansi-Fischer decomposition: D s,xs fx s, u = f 1,k x s, u + uf 2,k 1 x s, u, where f 1,k x s, u is a left monogenic polynomial homogeneous of degree k in u and f 2,k 1 x s, u is a left monogenic polynomial homogeneous of degree k 1 in u. We can also consider a function g : U R n Cl n+1 such that for each x U, g x, u is a right monogenic polynomial homogeneous of degree k in u. This function also reduces to a right monogenic polynomial homogeneous gx s, u on CU R n. Let P k be the left projection map P k : H k = M k um k 1 M k, then the n-spherical left Rarita-chwinger type operator Rk is defined to be R k fx s, u = P k D s,xs fx s, u. On the other hand, the n-spherical right Rarita-chwinger type operator R k,r is defined to be gx s, ur k,r = gx s, ud s,xs P k,r, where P k,r is the right projection P k,r : H k M k. Consequently, the left and the right n-spherical Rarita-chwinger type equations are defined to be R k fx s, u = 0 and gx s, ur k = 0 respectively.

5 RARITA-CHWINGER OPERATOR ON PHERE Conformal invariance of P k under the Cayley transformation and its inverse Consider the Cayley transformation Cx = e n+1 x+1x+e n+1 1 = e n+1 x e n+1 x + e n+1 1 = e n+1 x + e n+1 2e n+1 x + e n+1 1 = e n+1 + 2x + e n+1 1. This last term is the Iwasawa decomposition for the Cayley transformation, C. Further, C 1 x s = e n+1 x s +1x s e n+1 1 = e n+1 x s +e n+1 x e n+1 1 = e n+1 x s e n+1 +2e n+1 x s e n+1 1 = e n+1 +2x s e n+1 1, and this last term is the Iwasawa decomposition for the inverse, C 1, of the Cayley transformation. Now let fx s, u : U s R n Cl n+1 be a monogenic polynomial homogeneous of degree k in u for each x s U s, where U s is a domain in n. It is shown in [6] that P k is conformally invariant under a general Möbius transformation over R n. This trivially extends to Möbius transformations on R n+1. It follows that if we restrict x s to n, then P k is also conformally invariant under the Cayley transformation C and its inverse C 1, with x R n. It follows that we have: Theorem 1. P k,w JC, xf Cx, x + e n+1wx + e n+1 x + e n+1 2 = JC, xp k,u fx s, u, where u = x + e n+1wx + e n+1 x + e n+1 2 and JC, x = x + e n+1 x + e n+1 n weight for the Cayley transformation. is the conformal Also for U a domain in R n, and gx, u defined on U R n such that for each x U, g is monogenic in u and homogeneous of degree k in u, we have: Theorem 2. P k,w JC 1, x s g C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2 = JC 1, x s P k,u gx, u, where u = x s e n+1 wx s e n+1 x s e n+1 2 and JC 1, x s = x s e n+1 is the conformal weight for the inverse Cayley x s e n+1 n transformation. Note that in the previous theorems a 1 x := x + e n+1 x + e n+1 and a 2 x s := x s e n+1 x s e n+1 belong to Pinn + 1. o w Rn+1 and hence D a1xf = 0 and D a2x sg = 0, where for a Pinn + 1 the operator D a is defined in The intertwining formulas for R k and Rk and the conformal invariance of Rk f = 0 We can use the intertwining formulas for D x and D s,xs given in [8] to establish the intertwining formulas for R k and Rk.

6 276 JUNXIA LI, J. RYAN AND C. J. VANEGA Theorem 3. J 1 C 1, x s R k,u fx, u = R k,wjc 1, x s f C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2, where R k,u is the Rarita-chwinger operator in Euclidean space with respect to u R n, Rk,w is the spherical Rarita-chwinger type operator on n with respect to w R n+1, u = x s e n+1 wx s e n+1 and J 1 C 1, x s = x s e n+1 x s e n+1 n+2. x s e n+1 2, JC 1, x s = x s e n+1 x s e n+1 n Proof. In [8] it is shown that D x = J 1 C 1, x s 1 D s,xs JC 1, x s. Consequently, R k,u fx, u = P k,u D x fx, u = P k,u J 1 C 1, x s 1 D s,xs JC 1, x s fc 1 x s, u. Now applying Theorem 2, the previous equation becomes R k,u fx, u = J 1 C 1, x s 1 P k,w D s,xs JC 1, x s f C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2 = J 1 C 1, x s 1 Rk,wJC 1, x s f C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2. We have the similar result for the Rarita-chwinger operator under the Cayley transformation. Theorem 4. J 1 C, xrk,ugx s, u = R k,w JC, xg Cx, x + e n+1wx + e n+1 x + e n+1 2, where Rk,u is the Rarita-chwinger type operator on the sphere with respect to u and R k,w is the Rarita-chwinger operator in Euclidean space with respect to w, 4 5 u = x + e n+1wx + e n+1 x + e n+1 2, JC, x = x + e n+1 x + e n+1 n x + e n+1 J 1 C, x = x + e n+1 n+2. In other words we have the following intertwining relations for R k and R k : J 1 C 1, x s R k = R k JC 1, x s J 1 C, xr k = R k JC, x As a corollary to Theorems 3 and 4 we have the conformal invariance of equation R k,w f = 0: and

7 RARITA-CHWINGER OPERATOR ON PHERE 277 Theorem 5. Rk,u gx s, u = 0 if and only if R k,w JC, xg Cx, x + e n+1wx + e n+1 x + e n+1 2 = 0 and R k,u fx, u = 0 if and only if Rk,wJC 1, x s f C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2 = The fundamental solutions of Rk and some basic integral formulas The reproducing kernel of M k with respect to integration over n 1 is given by see [2], [6] where Z k u, v := σ P σ uv σ vv, P σ u = 1 k! Σu i 1 u 1 e 1 1 e i 1... u ik u 1 e 1 1 e i k, V σ v = k Gv v j2 2..., vjn n j j n = k, i k {2,..., n}, Gv = 1 v ω n v n, and ω n is the surface area of the unit sphere in R n. Here summation is taken over all permutations of the monomials without repetition. This function is left monogenic in u and it is a right monogenic polynomial in v. It is homogeneous of degree k in both u and v. ee [2] and elsewhere. Consider the kernel of the Rarita-chwinger operator in Euclidean n-space E k x y, u, v = 1 x y x yux y 6 ω n c k x y n Z k x y 2, v = 1 JC 1, x s 1 x s y s x yux y 7 ω n c k x s y s n JC 1, y s 1 Z k x y 2, v, where c k = n k. ee for instance [6]. Note that x yux y Rn as u, x and y R n. Now applying the Cayley transformation to the above kernel, we obtain Ek x s, y s, u, v : = 1 JC 1, x s JC 1, x s 1 x s y s ω n c k x s y s n 8 JC 1, y s 1 Z k auã, v = 1 x s y s ω n c k x s y s n JC 1, y s 1 Z k auã, v, JC 1, x s 1 x s y s JC 1, y s 1 where a = ax s, y s = JC 1, x s 1 x s y s JC 1, y s 1. Ek x s, y s, u, v is the fundamental solution to Rk fx s, u = 0 on n. This function is left monogenic in u and it is also right monogenic in v.

8 278 JUNXIA LI, J. RYAN AND C. J. VANEGA 9 In the same way we obtain that 1 Z k u, ãvajc 1, y s 1 x s y s ω n c k x s y s n is a non trivial solution to gx s, vr k,r = 0. In fact, this function is E k x s, y s, u, v. Applying the same arguments in [6] to prove the representations 8 and 9 are the same up to a reflection, we have 1 Z k u, ãvajc 1, y s 1 x s y s ω n c k x s y s n = 1 ãz k auã, vajc 1, y s 1 x s y s ω n c k x s y s n = 1 JC 1, y s 1 x s y s JC 1, x s 1 ω n c k x s y s n JC 1, x s 1 Z kauã, v JC 1, x s 1 JC 1, x s 1 = JC 1, x s 1 1 x s y s JC 1, x s 1 ω n c k x s y s n JC 1, y s 1 Z k auã, v JC 1, x s 1 JC 1, x s 1. Theorem 6 tokes Theorem for the n-spherical Dirac operator D s [8]. uppose U s is a domain on n and f, g : U s R n Cl n+1 are C 1, then for a sufficiently smooth hypersurface in U s bounding a subdomain of U s, we have gx s, unx s fx s, udσx s = gxs, ud s,xs fxs, u + gx s, u D s,xs fx s, u dx s, where dx s is the n-dimensional area measure on, dσx s is the n 1-dimensional scalar Lebesgue measure on and nx s is the normal vector tangent to the sphere at x s, orthogonal to and pointing outward. Definition 1 [6]. For any Cl n+1 -valued polynomials P u, Qu, the inner product P u, Qu u with respect to u R n is given by P u, Qu u = P uqudsu, n 1 where n 1 is the unit sphere in R n. For any p k M k, one obtains p k u = Z k u, v, p k v v = Z k u, vp k vdsv. n 1 ee [2].

9 RARITA-CHWINGER OPERATOR ON PHERE 279 Theorem 7 tokes Theorem for the n-spherical Rarita-chwinger type operator Rk. Let U s,, be as in Theorem 6. Then for f, g C 1 U s R n, M k, we have gxs, urk, fx s, u u + gx s, u, Rk fx s, u dxs u = gxs, u, P k nx s fx s, u u dσx s V s = gxs, unx s P k,r, fx s, u u dσx s V s = gxs, unx s fx s, u dσx u s where dx s is the n-dimensional area measure on, nx s and dσx s are as in Theorem 6. Proof. The proof follows similar lines to the proof of Theorem 6 in [6]. First, by the traditional Clifford version of tokes Theorem gx s, unx s fx s, u u dσx s = gxs, ud s,xs, fx s, u u + gx s, u, D s,xs fx s, u u dxs. By applying the Almansi-Fischer decomposition to gx s, ud s,xs and D s,xs fx s, u and Definition 1 the right side of the previous equation becomes gxs, urk, fx s, u + gx u s, u, Rk fx s, u dxs. u The other identities follow from arguments given in the proof of Theorem 6 in [6]. Corollary 1 Cauchy s Theorem. If Rk fx s, u = 0 and gx s, urk = 0 for f, g C 1 U s R n, M k, then we have gx s, u, P k nx s fx s, u u dσx s = 0, where is a sufficiently smooth hypersurface in U s bounding a subdomain of U s. Now let us look at tokes Theorem for Rarita-chwinger operators R k in R n. uppose U is a domain on R n and f, g : U R n Cl n+1 are C 1, then for V a sufficiently smooth hypersurface in U bounding a relatively compact subdomain V of U, we have [g x, ur k, f x, u u + g x, u, R k f x, u u ]dx n V = V g x, u, P k nxf x, u u dσx,

10 280 JUNXIA LI, J. RYAN AND C. J. VANEGA where dσx is the scalar Lebesgue measure on V. Now consider the integral on the right hand side g x, u, P k nxf x, u u dσx V = g x, up k nxf x, udsudσx V n 1 = g C 1 x s, up k,u JC 1, x s nx s C V n 1 JC 1, x s f C 1 x s, udsudσx s, where x s = Cx, C V bounds a domain CV in n, dσx s is the scalar Lebesgue measure on C V and JC 1, x s = x s e n+1 x s e n+1 n. ince P k,u interchanges with JC 1, x s, the previous integral becomes g C 1 x s, x s e n+1wx s e n+1 C V x n 1 s e n+1 2 JC 1, x s P k,w nx s JC 1, x s f C 1 x s, x s e n+1 wx s e n+1 10 x s e n+1 2 dswdσx s where u = x s e n+1 wx s e n+1 x s e n+1 2. Consider the integral on the left hand side 11 V [ g x, ur k, f x, u u + g x, u, R k f x, u u] dx n = V n 1 [g x, ur k,r,u f x, u + g x, ur k,u f x, u]dsudx n Applying Theorem 3, the integral now is equal to [g C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2 JC 1, x s Rk,r,wJC 1, x s CV 12 n 1 f C 1 x s, x s e n+1 wx s e n+1 x s e n g C 1 x s, x s e n+1 wx s e n+1 x s e n+1 2 JC 1, x s Rk,wJC 1, x s f C 1 x s, x s e n+1 wx s e n+1 ] x s e n+1 2 dsw dx s where CV = is a domain in n. tokes Theorem for Rarita-chwinger operators R k in R n tells us that 10 is equal to 12. Therefore tokes Theorem for Rarita-chwinger type operators is conformally invariant under the Cayley transformation.

11 RARITA-CHWINGER OPERATOR ON PHERE 281 Now let us consider tokes Theorem for Rk in n. gxs, urk, fx s, u u + gx s, u, Rk fx s, u dxs u = gxs, u, P k nx s fx s, u u dσx s, where,, dx s and dσx s are stated as in Theorem 7. First look at gxs, u, P k nx s fx s, u dσx u s = gx s, u, P k nx s fx s, udsuσx s n 1 = g Cx, u P k,u J Cxnx JC, xf Cx, u dsu dσx, C 1 n 1 where JC, x = x + e n+1 x + e n+1 n, x = C 1 x s and C 1 bounds a domain C 1 in R n. ince we can interchange P k,u with JC, x, the previous integral is equal to 13 C 1 n 1 g where u = x + e n+1wx + e n+1 x + e n+1 2. econd we look at Cx, x + e n+1wx + e n+1 x + e n+1 2 J CxP k,w nx JC, x f Cx, x + e n+1wx + e n+1 x + e n+1 2 dswdσx, gxs, urk, fx s, u + gx u s, u, R k fx s, u u dxs = gxs, ur k,r,u fxs, u + gx s, u Rk,ufx s, u dsu dx s. n 1 Applying Theorem 4, the integral becomes g Cx, x + e n+1wx + e n+1 C 1 x + e n 1 n+1 2 JC, xrk,r,w JC, xf Cx, x + e n+1wx + e n+1 x + e n g Cx, x + e n+1wx + e n+1 x + e n+1 2 JC, x Rk,wJC, xf Cx, x + e n+1wx + e n+1 14 x + e n+1 2 dsw dx s.

12 282 JUNXIA LI, J. RYAN AND C. J. VANEGA tokes Theorem for Rk on the sphere shows that 13 is equal to 14. Thus tokes Theorem for Rarita-chwinger operators is also conformally invariant under the inverse of the Cayley transformation. Theorem 8 Borel-Pompeiu Theorem. uppose U s, and are as in Theorem 6 and y s. Then for f C 1 U s R n, M k we have fy s, u = JC 1, y s E k x s, y s, u, v, P k nx s fx s, v v dσx s V s JC 1, y s E k x s, y s, u, v, Rk fx s, v dx v s where u = JC 1, y s 1 ujc 1, y s 1 JC 1, y s 1 2, dx s is the n-dimensional area measure on n, nx s and dσx s are as in Theorem 6. Proof. In this proof we will use the representation of E k x s, y s, u, v given by 9. Let B s y s, ɛ be the ball centered at y s n with radius ɛ. We denote C 1 B s y s, ɛ by By, r, and C 1 B s y s, ɛ by By, r, where y = C 1 y s R n and r is the radius of By, r in R n. Consider B s y s, ɛ, then we have E k x s, y s, u, v, Rk fx s, v dx v s = E k x s, y s, u, v, Rk fx s, v dx v s \B sy s,ɛ + B sy s,ɛ E k x s, y s, u, v, R k fx s, v v dx s. Because of the degree of homogeneity of x s y s in Ek, the second integral on the right hand goes to zero as ɛ goes to zero. Applying Theorem 7 to the first integral on the right hand we obtain E k x s, y s, u, v, Rk fx s, v v dx s \B sy s,ɛ = E k x s, y s, u, v, P k nx s fx s, v v dσx s V s E k x s, y s, u, v, P k nx s fx s, v v dσx s. B sy s,ɛ ince fx s, v = fx s, v fy s, v + fy s, v and taking into account the degree of homogeneity of x s y s in Ek and the continuity of f, we can replace the second integral on the right hand by E k x s, y s, u, v, P k nx s fy s, v v dσx s. B sy s,ɛ

13 RARITA-CHWINGER OPERATOR ON PHERE 283 Applying Theorem 7, this integral is equal to E k x s, y s, u, v, nx s fy s, v v dσx s B sy s,ɛ = = B sy s,ɛ B sy s,ɛ Ek x s, y s, u, vnx s fy s, v dσx s dsv n 1 1 Z k u, ãva ω n 1 n c k JC 1, y s 1 x s y s x s y s n nx sfy s, vdsv dσx s. Now applying the inverse of the Cayley transformation to the last integral, we have 1 x ywx y Z k u, By,r ω n 1 n c k x y 2 JC 1, y s 1 JC, y 1 x y x y n JC, x 1 JC, ywjc, y JC, xnxjc, xf Cy, JC, y 2 dswdσx, where dσx is the n 1-dimensional scalar Lebesgue measure on By, r in R n JC, ywjc, y and v = JC, y 2, w R n+1. In fact, v is a vector in R n which is obtained by reflecting w in R n+1 and its last component is a constant. Place JC, x = JC, x JC, y + JC, y, but JC, x JC, y tends to zero as x approaches y. Thus the previous integral can be replaced by 1 x ywx y x y Z k u, By,r ω n 1 n c k x y 2 x y n nx JC, ywjc, y JC, yf Cy, JC, y 2 dsw dσx. Here nx = becomes By,r y x x y 1 Z k u, ω n 1 n c k is the unit out normal vector. Now the last integral x ywx y 1 x y 2 x y n 1 JC, ywjc, y JC, yf Cy, JC, y 2 dsw dσx, Using Lemma 5 in [6], the integral is now JC, ywjc, y Z k u, wjc, yf Cy, JC, y 2 dsw n 1 JC, yujc, y = JC, yf Cy, JC, y 2 = JC 1, y s 1 f y s, JC 1, y s 1 ujc 1, y s 1 JC 1, y s 1 2,

14 284 JUNXIA LI, J. RYAN AND C. J. VANEGA since JC, y = JC 1, y s 1. Now by setting u = JC 1, y s 1 ujc 1, y s 1 JC 1, y s 1 2 and multiplying both sides of the above equation by JC 1, y s, we obtain fy s, u = JC 1 JC, ywjc, y, y s Z k u, wjc, yf Cy, JC, y 2 dsw. n 1 Therefore when ɛ tends to zero we get the desired result. Corollary 2. Let Ψ be a function in C R n, M k and suppψ. Then Ψy s, u = JC 1, y s E k x s, y s, u, v, Rk Ψx s, v dx v s, where u = JC 1, y s 1 ujc 1, y s 1 JC 1, y s 1 2. Corollary 3 Cauchy Integral Formula for Rk. If R k fx s, u = 0, then for y s we have fy s, u = JC 1, y s E k x s, y s, u, v, P k nx s fx s, v dσx v s V s = JC 1, y s E k x s, y s, u, vnx s P k,r, fx s, v dσx v s, where u = JC 1, y s 1 ujc 1, y s 1 JC 1, y s 1 2. Definition 2 Cauchy Transform for Rk. For a domain n and a function fx s, u: R n Cl n+1, which is monogenic in u, the T k -transform of f is defined to be T k fy s, v = E k x s, y s, u, v, fx s, u u dx s, for y s. Theorem 9. For a function ψ in C n R n, M k we have P k JC 1, y s D s,ys n E k x s, y s, u, v, ψx s, u u dx s = ψy s, v. Proof. By [8], the integral P k JC 1, y s D s,ys n E k x s, y s, u, v, ψx s, u u dx s can be replaced by P k JC 1, y s B sy s,ɛ nx s E k x s, y s, u, v, ψx s, u u dx s,

15 RARITA-CHWINGER OPERATOR ON PHERE 285 which in turn is equal to P k JC 1, y s B sy s,ɛ 1 x s y s nx s ω n 1 n c k x s y s n JC 1, y s 1 Z k auã, vψx s, udsu dx s. ince ψx s, u = ψx s, u ψy s, u + ψy s, u then using the continuity of ψ, we can replace the previous integral by P k JC 1 1 x s y s, y s nx s B sy s,ɛ ω n 1 n c k x s y s n JC 1, y s 1 Z k auã, vψy s, u dsu dx s. Now applying the inverse of the Cayley transformation to the previous integral it becomes P k JC 1 1, y s JC, xnxjc, x By,r ω n 1 n c k JC, x 1 x y x y n JC, y 1 JC 1, y s 1 x yux y JC, ywjc, y Z k x y 2, v ψ Cy, JC, y 2 dsu dσx = P k JC 1 1 x y, y s JC, xnx By,r ω n 1 n c k x y n x yux y JC, ywjc, y Z k x y 2, v ψ Cy, JC, y 2 dsu dσx, JC, ywjc, y where u = JC, y 2. Using the fact JC, x = JC, x JC, y + JC, y, and JC, x JC, y tends to zero as x approaches y, the integral can be replaced by P k JC 1, y s 1 x y JC, ynx By,r ω n 1 n c k x y n x yux y JC, ywjc, y Z k x y 2, v ψ Cy, JC, y 2 dsu dσx 1 x yux y = P k By,r ω n 1 n c k r n 1 Z k x y 2, v JC, ywjc, y ψ Cy, JC, y 2 dsu dσx

16 286 JUNXIA LI, J. RYAN AND C. J. VANEGA Applying Lemma 5 in [6], the integral becomes JC, ywjc, y = P k Z k u, vψ Cy, JC, y 2 dsu n 1 = P k n 1 Z k u, vψcy, u dsu = P k ψ Cy, v = ψy s, v. 7. Rarita-chwinger type operators on real projective space We consider n and Γ = {±1}, then n /Γ is RP n, the real projective space. In all that follows n will be a universal covering space of the conformally flat manifold RP n. o there is a projection map p: n RP n. Further for each x n we shall denote px by x. Furthermore if Q is a subset of U then we denote pq by Q. Consider the trivial bundle n Cl n+1, then we set up a spinor bundle E 1 over RP n by making the identification of x, X with x, X, where x n and X Cl n+1. Now we change the spherical Cauchy kernel G x y = 1 x y ω n x y n, x, y n, for the spherical Dirac operator into a kernel which is invariant with respect to {±1} in the variable x n. Hence we consider G x y + G x y. ee [7]. uppose V is a domain lying in the open northern hemisphere. We assume fx, u: V R n Cl n+1 is a C 1 function in x and monogenic in u. We observe that the projection map p: n RP n induces a well defined function f x, u : V R n E 1 such that f x, u = fp 1 x, u, where V is a well defined domain in RP n and x = px. We define the Rarita-chwinger type operators on RP n, which we will call the real projective Rarita-chwinger type operators, in the following form R RP n k f x, u = P k D RP n,x f x, u, where D RP n,x is the Dirac operator in RP n with respect to the variable x. ee [7]. Now we introduce the spherical Rarita-chwinger kernel which is also invariant with respect to {±1} in the variable x n : E,1 k x, y, u, v := E k x, y, u, v + Ek x, y, u, v. Through the projection map p over x, y n we obtain a kernel E RP n,1 k x, y, u, v for RP n defined by E RP n,1 k x, y, u, v = E,1 k p 1 x, p 1 y, u, v. Now suppose that is a suitably smooth hypersurface lying in the open northern hemisphere of n bounding a subdomain W of V with closure of W in V.

17 RARITA-CHWINGER OPERATOR ON PHERE 287 Theorem 10. If Rk fx, u = 0 then for y W fy, w = JC 1, y E,1 k x, y, u, v, P knxfx, u dσx, u where w = JC 1, y 1 vjc 1, y 1 JC 1, y 1 2, nx is the unit outer normal vector to at x lying in the tangent space of n at x and Σ is the usual Lebesgue measure on. Due to the projection map we have also Theorem 11. f y, ˆv = JC 1, y E RP n,1 k x, y, u, v, P k dpnxf x, u u dσ x, where ˆv = JC 1, y 1 vjc 1, y 1 JC 1, y 1 2, x = px, y = py and is the projection of. Further Σ is a induced measure on from the measure Σ on and dp is the derivative of p. Now we will assume that the domain V is such that x V for each x V and the function f is two fold periodic, so that fx = f x. Now the projection map p gives rise to a well defined domain V on RP n and a well defined function f x, u: V E 1 such that f x, u = f±x, u for p±x = x. Then if f x, u = 0, we also have f y, ˆv = JC 1, y E RP n,1 k x, y, u, v, P k dpnxf x, u u dσ x, where ˆv is stated as in Theorem 11. R RP n k If now we suppose that the hypersurface satisfies = then both y and y belong to the subdomain V and in this case JC 1, y E RP n,1 k x, y, u, v, P k dpnxf x, u u dσ x = 2f y, ˆv. We can also construct a second spinor bundle E 2 over RP n by making the identification of x, X with x, X, where x n and X Cl n+1, we introduce the kernel: E,2 k x, y, u, v := E k x, y, u, v Ek x, y, u, v. This kernel induces through the projection map on the variable x, y n, the kernel on RP n E RP n,2 k x, y, u, v = E,2 k p 1 x, p 1 y, u, v. In this case a solution of Rarita-chwinger type equation on RP n f x, u: V R n E 2 will lift to a solution of spherical-rarita-chwinger type equation: fx, u: V R n Cl n+1 such that fx, u = f x, u.

18 288 JUNXIA LI, J. RYAN AND C. J. VANEGA uppose that V as before is a domain on n and is a hypersurface in V bounding a subdomain W of V. uppose further that fx, u: V R n Cl n+1 is a solution of the spherical Rarita-chwinger type equation such that fx, u = f x, u. If lies entirely in one open hemisphere then fy, w = JC 1, y E,2 k x, y, u, v, P knxfx, u u dσx, for each y W, where w = JC 1, y 1 vjc 1, y 1 JC 1, y 1 2. Via the projection p this integral formula induces the following f y, ˆv = JC 1, y E RP n,2 k x, y, u, v, P k dpnxf x, u u dσ x, where ˆv is stated as in Theorem 11. On the other hand if is such that = then E,2 k x, y, u, v, P knxfx, u u dσx = 0. References [1] Balinsky, A., Ryan, J., ome sharp L 2 inequalities for Dirac type operators, IGMA, ymmetry Integrability Geom. Methods Appl. 2007, 10, paper 114, electronic only. [2] Brackx, F., Delanghe, R., ommen, F., Clifford Analysis, Pitman, London, [3] Bureš, J., ommen, F., ouček, V., Van Lancker, P., Rarita-chwinger type operators in Clifford analysis, J. Funct. Anal , [4] Bureš, J., ommen, F., ouček, V., Van Lancker, P., ymmetric analogues of Rarita-chwinger equations, Ann. Global Anal. Geom , [5] Cnops, J., Malonek, H., An introduction to Clifford analysis, Textos Mat. ér. B 1995, vi+64. [6] Dunkl, C., Li, J., Ryan, J., Van Lancker, P., ome Rarita-chwinger operators, submitted 2011, [7] Krausshar, R.., Ryan, J., Conformally flat spin manifolds, Dirac operators and automorphic forms, J. Math. Anal. Appl , [8] Liu, H., Ryan, J., Clifford analysis techniques for spherical PDE, J. Fourier Anal. Appl , [9] Porteous, I., Clifford Algebra and the Classical Groups, Cambridge University Press, Cambridge, [10] Ryan, J., Iterated Dirac operators in C n, Z. Anal. Anwendungen , [11] Ryan, J., Clifford analysis on spheres and hyperbolae, Math. Methods Appl. ci , [12] Ryan, J., Dirac operators on spheres and hyperbolae, Bol. oc. Mat. Mexicana , [13] Van Lancker, P., Clifford Analysis on the phere, Clifford Algebra and their Application in Mathematical Physics Aachen, 1996, Fund. Theories Phys., 94, Kluwer Acad. Publ., Dordrecht, 1998, pp

19 RARITA-CHWINGER OPERATOR ON PHERE 289 [14] Van Lancker, P., Higher pin Fields on mooth Domains, Clifford Analysis and Its Applications Brackx, F., Chisholm, J.. R., ouček, V., eds., Kluwer, Dordrecht, 2001, pp [15] Van Lancker, P., Rarita-chwinger fields in the half space, Complex Var. Elliptic Equ , Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, UA Departamento de Matemáticas, Universidad imón Bolívar, Caracas, Venezuela