First we define the spherical Rarita-Schwinger type operators and construct their fundamental solutions

Tomus 48 (2012), 271–289

RARITA-SCHWINGER TYPE OPERATORS ON SPHERES AND REAL PROJECTIVE SPACE

Junxia Li, John Ryan, and Carmen J. Vanegas

Abstract. In this paper we deal with Rarita-Schwinger type operators on spheres and real projective space. First we define the spherical Rarita-Schwinger type operators and construct their fundamental solutions. Then we establish that the projection operators appearing in the spherical Rarita-Schwinger type operators and the spherical Rarita-Schwinger type equations are conformally invariant under the Cayley transformation. Further, we obtain some basic integral formulas related to the spherical Rarita-Schwinger type operators.

Second, we define the Rarita-Schwinger type operators on the real projective space and construct their kernels and Cauchy integral formulas.

1. Introduction

Rarita-Schwinger operators are generalizations of the Dirac operator and arise in representation theory for the Spin and Pin groups. See [3, 4, 6, 14, 15]. We denote a Rarita-Schwinger operator byRk, where k= 0,1, . . . , m, . . .. When k= 0 we have the Dirac operator. The Rarita-Schwinger operatorsRk in Euclidean space have been studied in [3, 4, 6, 14, 15]. Here we construct similar Rarita-Schwinger 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. See also [1]. Using similar methods to define the Rarita-Schwinger operators in Rⁿ, we can define the spherical Rarita-Schwinger 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-Schwinger type equations under the Cayley transformations. See [6].

Further the fundamental solutions to the spherical Rarita-Schwinger type operators are achieved by applying the Cayley transformation. In turn, Stokes’ Theorem, Cauchy’s Theorem, Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform are proved for the sphere. Furthermore, we show that Stokes’ theorem is

2010Mathematics Subject Classification: primary 30G35; secondary 53C27.

Key words and phrases: spherical Rarita-Schwinger type operators, Cayley transformation, real projective space, Almansi-Fischer decomposition, Iwasawa decomposition.

Received June 6, 2012. Editor J. Slovák.

DOI: 10.5817/AM2012-4-271

conformally invariant under Cayley transformation, and with minor modification, is equivalent to the Rarita-Schwinger version of Stokes’ Theorem in Euclidean space appearing in [3, 6] and elsewhere.

By factoring out Sⁿ by the groupZ2 ={±1} we obtain real projective space, RPⁿ. On this space, we define the Rarita-Schwinger type operators and construct their kernels over two different bundles overRPⁿ. 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, Cln+1, can be generated from Rⁿ⁺¹ by considering the relationship

x²=−kxk²

for each x∈Rⁿ⁺¹. We haveRⁿ⁺¹⊆Cln+1. Ife1, . . . , en+1 is an orthonormal basis forRⁿ⁺¹, thenx²=−kxk²tells us thate_ie_j+e_je_i=−2δij. LetA={j1, . . . , j_r} ⊂ {1,2, . . . , n+ 1} and 1 ≤j₁ < j₂ < · · · < j_r ≤ n+ 1. An arbitrary element of the basis of the Clifford algebra can be written ase_A=e_j1. . . e_jr. Hence for any element a∈Cl_n+1, we havea=P

Aa_Ae_A,wherea_A∈R. The reversion is given by

˜ a=X

A

(−1)|A|(|A|−1)/2aAeA,

where|A|is the cardinality ofA. In particular,ej₁^. . . ej_r =ej_r. . . ej₁. Alsoabe = ˜b˜a fora, b∈Cln+1. The Clifford conjugation is defined by

¯ a=X

A

(−1)|A|(|A|+1)/2a_Ae_A

and satisfiese_j1. . . e_jr = (−1)^re_jr. . . e_j1 andab= ¯b¯afora, b∈Cl_n+1.

For eacha=a0+a1e1+· · ·+a1...n+1e1. . . en+1∈Cln+1 the scalar part of ¯aa gives the square of the norm of a, namelya²₀+a²₁+· · ·+a²_1...n+1. For more on Clifford algebras and their properties, see [9].

The Pin and Spin groups play an important role in Clifford analysis. The Pin group can be defined as

Pin(n+ 1) :={a∈Cl_n+1:a=y₁. . . y_p :y₁, . . . , y_p∈Sⁿ, p∈N} and it is clearly a group under multiplication in Cln+1.

Now suppose thaty∈Sⁿ ⊆Rⁿ⁺¹. Look atyxy=yx^kyy+yx^⊥yy=−x^ky+x^⊥y where x^ky is the projection of xonto y andx^⊥y is perpendicular toy. So yxy gives a reflection ofxin theydirection. By the Cartan–Dieudonné Theorem each O∈O(n+ 1) is the composition of a finite number of reflections. Ifa=y1. . . yp∈ Pin(n+ 1), then ˜a:=yp. . . y1andax˜a=Oa(x) for someOa ∈O(n+ 1). Choosing y1, . . . , yp arbitrarily inSⁿ, we see that the group homomorphism

θ: Pin(n+ 1)−→O(n+ 1) :a7−→O_a,

witha=y₁. . . y_p andO_a(x) =ax˜a, is surjective. Further−ax(−˜a) =ax˜a, so 1,

−1∈ker(θ). In fact ker(θ) ={±1}. See [9]. The Spin group is defined as Spin(n+ 1) :={a∈Pin(n+ 1) :a=y₁. . . yp and p even}

and it is a subgroup of Pin(n+ 1). There is a group homomorphism θ: Spin(n+ 1)−→SO(n+ 1)

which is surjective with kernel{1,−1}. See [9] for details.

The Dirac Operator inRⁿ is defined to be D:=

n

X

j=1

ej

∂

∂x_j . NoteD²=−∆n,where ∆n is the Laplacian inRⁿ.

If pk is a homogeneous polynomial with degreek such thatDpk = 0,we call such a polynomial a left monogenic polynomial homogeneous of degreek.

Let Hk be the space of Cln+1-valued harmonic polynomials homogeneous of degreek andMk the space of Cln+1-valued monogenic polynomials homogeneous of degree k. Note ifh_k ∈ Hk,thenDh_k ∈ Mk−1. ButDup_k−1(u) = (−n−2k+ 2)p_k−1(u), so

Hk =Mk

MuM_k−1, hk =pk+up_k−1.

This is the so-called Almansi-Fischer decomposition ofH_k. See [2, 10].

Note that if p(u) ∈ Mk then it trivially extends to P(v) = p(u+un+1en+1) withun+1 ∈R andP(v) = p(u) for allun+1 ∈R. Consequently Dn+1P(v) = 0 whereD_n+1=

n+1

X

j=1

e_j ∂

∂u_j.

If p(u)∈ Mk then for any boundary of a piecewise smooth bounded domain U ⊆Rⁿ by Cauchy’s Theorem

(1)

Z

∂U

n(u)p(u)dσ(u) = 0.

Suppose now a∈Pin(n+ 1) andu=aw˜athen althoughu∈Rⁿ in general w belongs to the hyperplane a⁻¹Rⁿ˜a⁻¹ inRⁿ⁺¹.

By applying a change of variable, up to a sign the integral (1) becomes (2)

Z

a⁻¹∂U˜a⁻¹

an(w)˜aP(aw˜a)dσ(w) = 0.

As∂U is arbitrary then on applying Stokes’ Theorem to (2) we see that (3) Da˜aP(aw˜a) = 0, quadwhere Da :=Dn+1

_a−1

Rⁿ˜a⁻¹. SupposeU is a domain inRⁿ. Consider a function of two variables

f:U×Rⁿ−→Cln+1

such that for eachx∈U,f(x, u) is a left monogenic polynomial homogeneous of degreekin u. LetPk be the left projection map

P_k: Hk → Mk,

then R_kf(x, u) is defined to beP_kD_xf(x, u). The left Rarita-Schwinger equation is defined to be

Rkf(x, u) = 0.

We also have a right projectionP_k,r: Hk → Mk, and a right Rarita-Schwinger equationf(x, u)D_xP_k,r=f(x, u)R_k= 0, whereMk stands for the space of right monogenic polynomials homogeneous of degreek. See [6].

3. Rarita-Schwinger type operators on spheres

Let Rⁿ be the span ofe1, . . . , en. Consider the Cayley transformationC:Rⁿ→ Sⁿ, whereSⁿis the unit sphere inRⁿ⁺¹, defined byC(x) = (en+1x+1)(x+en+1)⁻¹, where x=x1e1+· · ·+xnen ∈Rⁿ, and en+1 is a unit vector inRⁿ⁺¹ which is orthogonal toRⁿ. NowC(Rⁿ) =Sⁿ\ {en+1}. Supposexs∈Sⁿ andxs=xs₁e1+

· · ·+xsnen+xsn+1en+1, then we havex=C⁻¹(xs) = (−en+1xs+ 1)(xs−en+1)⁻¹. The Dirac operator over then-sphere Sⁿ has the formD_s=w(Λ +ⁿ₂), where w∈Sⁿ and Λ =

n

X

i<j,i=1

eiej(wi

∂

∂wj

−wj

∂

∂wi

), see for instance [5, 8, 13].

LetU be a domain inRⁿ. Consider a functionf?:U×Rⁿ→Cln+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 f(xs, u) on C(U)×Rⁿ and f(xs, u) takes its values in Cln+1 wherex=C⁻¹(xs) andxs∈C(U)⊂Sⁿ. Furtherf(xs, u) is a left monogenic polynomial homogeneous of degreek inu.

Since ∆uD_s,xs =D_s,xs∆u, thenD_s,xsf(x_s, u) is harmonic in u. Hence by the Almansi-Fischer decomposition:

Ds,x_sf(xs, u) =f1,k(xs, u) +uf_2,k−1(xs, u),

wheref_1,k(x_s, u) is a left monogenic polynomial homogeneous of degreekinuand f_2,k−1(x_s, u) is a left monogenic polynomial homogeneous of degreek−1 inu.

We can also consider a functiong_?:U×Rⁿ→Cl_n+1such that for eachx∈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 g(xs, u) on C(U)×Rⁿ.

Let Pk be the left projection mapPk:Hk =Mk⊕uM_k−1 → Mk, then the n-spherical left Rarita-Schwinger type operatorR_k^S is defined to be

R_k^Sf(x_s, u) =P_kD_s,xsf(x_s, u).

On the other hand, the n-spherical right Rarita-Schwinger type operatorR^S_k,r is defined to be

g(x_s, u)R_k,r^S =g(x_s, u)D_s,xsP_k,r,

whereP_k,r is the right projectionP_k,r:H_k→M¯_k. Consequently, the left and the rightn-spherical Rarita-Schwinger type equations are defined to be

R^S_kf(xs, u) = 0 and g(xs, u)R^S_k = 0 respectively.

4. Conformal invariance of P_k under the Cayley transformation and its inverse

Consider the Cayley transformationC(x) = (e_n+1x+ 1)(x+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+ 2(x+e_n+1)⁻¹. This last term is the Iwasawa decomposition for the Cayley transformation, C.

Further,C⁻¹(xs) = (−en+1xs+1)(xs−en+1)⁻¹=−en+1(xs+en+1)(x−en+1)⁻¹=

−en+1(xs−en+1+2en+1)(xs−en+1)⁻¹=−en+1+2(xs−en+1)⁻¹, and this last term is the Iwasawa decomposition for the inverse,C⁻¹, of the Cayley transformation.

Now letf(xs, u) :Us×Rⁿ →Cln+1 be a monogenic polynomial homogeneous of degreekin ufor eachxs∈Us, where Usis a domain inSⁿ.

It is shown in [6] that Pk is conformally invariant under a general Möbius transformation overRⁿ. This trivially extends to Möbius transformations on Rⁿ⁺¹. It follows that if we restrictxsto Sⁿ, thenPk is also conformally invariant under the Cayley transformationC and its inverseC⁻¹, withx∈Rⁿ.

It follows that we have:

Theorem 1.

Pk,wJ(C, x)f

C(x),(x+en+1)w(x+en+1) kx+e_n+1k²

=J(C, x)Pk,uf(xs, u), where u= (x+e_n+1)w(x+e_n+1)

kx+en+1k² and J(C, x) = x+e_n+1

kx+en+1kⁿ is the conformal weight for the Cayley transformation.

Also forU a domain inRⁿ,andg(x, u) defined onU×Rⁿ such that for each x∈U,g is monogenic inuand homogeneous of degreekinu, we have:

Theorem 2.

Pk,wJ(C⁻¹, xs)g

C⁻¹(xs),(xs−en+1)w(xs−en+1) kxs−en+1k²

=J(C⁻¹, xs)Pk,ug(x, u), whereu= (x_s−e_n+1)w(x_s−e_n+1)

kxs−en+1k² andJ(C⁻¹, x_s) = x_s−e_n+1

kxs−en+1kⁿ is the confor- mal weight for the inverse Cayley transformation.

Note that in the previous theoremsa₁(x) := x+e_n+1 kx+en+1k and a2(xs) := xs−en+1

kxs−en+1k belong to Pin(n+ 1). Sow∈Rⁿ⁺¹ and henceDa₁(x)f = 0 andD_a2(xs)g= 0, where for a∈Pin(n+ 1) the operatorD_a is defined in (3).

5. The intertwining formulas forRk andR^S_k and the conformal invariance of R^S_kf = 0

We can use the intertwining formulas for Dx andDs,x_s given in [8] to establish the intertwining formulas for R_k andR^S_k.

Theorem 3.

J−1(C⁻¹, xs)Rk,uf(x, u)

=R^S_k,wJ(C⁻¹, xs)f

C⁻¹(xs),(xs−en+1)w(xs−en+1) kx_s−e_n+1k²

, where Rk,u is the Rarita-Schwinger operator in Euclidean space with respect to u∈Rⁿ,R^S_k,w is the spherical Rarita-Schwinger type operator on Sⁿ with respect to w ∈ Rⁿ⁺¹, u = (xs−en+1)w(xs−en+1)

kx_s−e_n+1k² , J(C⁻¹, xs) = xs−en+1

kx_s−e_n+1kⁿ and J₋₁(C⁻¹, x_s) = x_s−e_n+1

kxs−en+1kⁿ⁺².

Proof. In [8] it is shown thatDx=J−1(C⁻¹, xs)⁻¹Ds,x_sJ(C⁻¹, xs). Consequently, R_k,uf(x, u) =P_k,uD_xf(x, u) =P_k,uJ₋₁(C⁻¹, x_s)⁻¹D_s,xsJ(C⁻¹, x_s)f(C⁻¹(x_s), u). Now applying Theorem 2, the previous equation becomes

Rk,uf(x, u) =J−1(C⁻¹, xs)⁻¹Pk,wDs,xsJ(C⁻¹, xs)

×f

C⁻¹(xs),(xs−en+1)w(xs−en+1) kx_s−e_n+1k²

=J₋₁(C⁻¹, x_s)⁻¹R^S_k,wJ(C⁻¹, x_s)

×f

C⁻¹(xs),(xs−en+1)w(xs−en+1) kxs−en+1k²

.

We have the similar result for the Rarita-Schwinger operator under the Cayley transformation.

Theorem 4.

J₋₁(C, x)R_k,u^S g(xs, u) =Rk,wJ(C, x)g

C(x),(x+en+1)w(x+en+1) kx+e_n+1k²

, where R^S_k,u is the Rarita-Schwinger type operator on the sphere with respect tou andRk,w is the Rarita-Schwinger operator in Euclidean space with respect tow,

u= (x+en+1)w(x+en+1)

kx+en+1k² , J(C, x) = x+en+1

kx+en+1kⁿ and J₋₁(C, x) = x+e_n+1

kx+en+1kⁿ⁺² .

In other words we have the following intertwining relations for R_k andR^S_k: J₋₁(C⁻¹, xs)Rk =R^S_kJ(C⁻¹, xs)

(4)

J₋₁(C, x)R^S_k =R_kJ(C, x) (5)

As a corollary to Theorems 3 and 4 we have the conformal invariance of equation R^S_k,wf = 0:

Theorem 5. R^S_k,ug(x_s, u) = 0 if and only if R_k,wJ(C, x)g

C(x),(x+e_n+1)w(x+e_n+1) kx+en+1k²

= 0 andRk,uf(x, u) = 0 if and only if

R^S_k,wJ(C⁻¹, xs)f

C⁻¹(xs),(xs−en+1)w(xs−en+1) kxs−en+1k²

= 0. 6. The fundamental solutions ofR^S_k and some basic integral

formulas

The reproducing kernel ofMk with respect to integration overSⁿ⁻¹ is given by (see [2], [6])

Zk(u, v) :=X

σ

Pσ(u)Vσ(v)v , where

Pσ(u) = 1

k!Σ(ui₁−u1e⁻¹₁ ei₁). . .(ui_k−u1e⁻¹₁ ei_k), Vσ(v) = ∂^kG(v)

∂v^j₂². . . ∂vn^jn

,

j₂+· · ·+j_n =k, i_k ∈ {2, . . . , n}, G(v) = −1 ωn

v

kvkⁿ, andω_n is the surface area of the unit sphere in Rⁿ. Here summation is taken over all permutations of the monomials without repetition. This function is left monogenic inuand it is a right monogenic polynomial inv. It is homogeneous of degreekin bothuandv. See [2]

and elsewhere.

Consider the kernel of the Rarita-Schwinger operator in Euclideann-space E_k(x−y, u, v) = 1

ωnck

x−y

kx−ykⁿZ_k(x−y)u(x−y) kx−yk² , v (6)

= 1

ωnck

J(C⁻¹, xs)⁻¹ xs−ys

kxs−yskⁿJ(C⁻¹, ys)⁻¹Zk

(x−y)u(x−y) kx−yk² , v

, (7)

wherec_k = n−2

n−2 + 2k. See for instance [6]. Note that (x−y)u(x−y)∈Rⁿ asu, xandy∈Rⁿ.

Now applying the Cayley transformation to the above kernel, we obtain E_k^S(x_s, y_s, u, v) : = 1

ωnck

J(C⁻¹, x_s)J(C⁻¹, x_s)⁻¹ x_s−y_s kxs−yskⁿ

×J(C⁻¹, y_s)⁻¹Z_k(au˜a, v)

= 1

ωnck

xs−ys

kxs−yskⁿJ(C⁻¹, ys)⁻¹Zk(au˜a, v), (8)

wherea=a(x_s, y_s) = J(C⁻¹, x_s)⁻¹(x_s−y_s)J(C⁻¹, y_s)⁻¹ kJ(C⁻¹, xs)⁻¹k k(xs−ys)k kJ(C⁻¹, ys)⁻¹k.

E_k^S(x_s, y_s, u, v) is the fundamental solution to R^S_kf(x_s, u) = 0 on Sⁿ. This function is left monogenic inuand it is also right monogenic inv.

In the same way we obtain that

(9) 1

ωnck

Z_k(u,ava)J˜ (C⁻¹, y_s)⁻¹ x_s−y_s kxs−yskⁿ

is a non trivial solution tog(xs, v)R^S_k,r= 0. In fact, this function isE_k^S(xs, ys, 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 ωnck

Z_k(u,˜ava)J(C⁻¹, y_s)⁻¹ x_s−y_s kxs−yskⁿ

=− 1

ω_nc_k˜aZ_k(au˜a, v)aJ(C⁻¹, y_s)⁻¹ xs−ys

kxs−y_skⁿ

=− 1 ωnck

J(C⁻¹, ys)⁻¹ xs−ys

kxs−yskⁿ

J(C⁻¹, xs)⁻¹

kJ(C⁻¹, xs)⁻¹kZk(au˜a, v) J(C⁻¹, xs)⁻¹ kJ(C⁻¹, xs)⁻¹k

=− J(C⁻¹, x_s)⁻¹ kJ(C⁻¹, xs)⁻¹k

1 ωnck

x_s−y_s

kxs−yskⁿJ(C⁻¹, ys)⁻¹Zk(au˜a, v) J(C⁻¹, x_s)⁻¹ kJ(C⁻¹, xs)⁻¹k. Theorem 6 (Stokes’ Theorem for then-spherical Dirac operatorDs[8]). Suppose Usis a domain onSⁿ andf, g:Us×Rⁿ→Cln+1areC¹, then for∂Vsa sufficiently smooth hypersurface inUsbounding a subdomain Vsof Us, we have

Z

∂Vs

g(xs, u)n(xs)f(xs, u)dΣ(xs)

= Z

V_s

g(x_s, u)D_s,xs

f(x_s, u) +g(x_s, u) D_s,xsf(x_s, u)

dS(x_s), where dS(x_s)is then-dimensional area measure onV_s,dΣ(x_s)is then−1-dimen- sional scalar Lebesgue measure on ∂V_s andn(x_s)is the normal vector tangent to the sphere at xs, orthogonal to∂Vsand pointing outward.

Definition 1 ([6]). For any Cln+1-valued polynomials P(u), Q(u), the inner product(P(u),Q(u))u with respect tou∈Rⁿ is given by

(P(u), Q(u))_u= Z

Sⁿ⁻¹

P(u)Q(u)ds(u), whereSⁿ⁻¹ is the unit sphere inRⁿ.

For anyp_k∈ Mk, one obtains

p_k(u) = (Z_k(u, v), p_k(v))_v= Z

Sⁿ⁻¹

Z_k(u, v)p_k(v)ds(v). See [2].

Theorem 7(Stokes’ Theorem for then-spherical Rarita-Schwinger type operator R^S_k). LetU_s,V_s,∂V_s be as in Theorem 6. Then forf,g∈C¹(U_s×Rⁿ,Mk), we have

Z

V_s

g(xs, u)R^S_k, f(xs, u)

u+ g(xs, u), R^S_kf(xs, u)

u

dS(xs)

= Z

∂Vs

g(xs, u), Pkn(xs)f(xs, u)

udΣ(xs)

= Z

∂V_s

g(x_s, u)n(x_s)P_k,r, f(x_s, u)

udΣ(x_s)

= Z

∂Vs

g(xs, u)n(xs)f(xs, u)

udΣ(xs)

where dS(x_s)is then-dimensional area measure on V_s,n(x_s)anddΣ(x_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 Stokes’ Theorem

Z

∂Vs

(g(xs, u)n(xs)f(xs, u))udΣ(xs)

= Z

V_s

(g(x_s, u)D_s,xs, f(x_s, u))_u+ (g(x_s, u), D_s,xsf(x_s, u))_u

dS(x_s). By applying the Almansi-Fischer decomposition tog(xs, u)Ds,xs andDs,xsf(xs, u) and Definition 1 the right side of the previous equation becomes

Z

V_s

g(xs, u)R^S_k, f(xs, u)

u+ g(xs, u), R^S_kf(xs, u)

u

dS(xs).

The other identities follow from arguments given in the proof of Theorem 6 in

[6].

Corollary 1 (Cauchy’s Theorem). If R^S_kf(xs, u) = 0and g(xs, u)R^S_k = 0 forf, g∈C¹(Us×Rⁿ,Mk), then we have

Z

∂Vs

(g(xs, u), Pkn(xs)f(xs, u))udΣ(xs) = 0,

where∂V_sis a sufficiently smooth hypersurface in U_s bounding a subdomainV_s of U_s.

Now let us look at Stokes’ Theorem for Rarita-Schwinger operatorsR_k inRⁿ. SupposeU is a domain onRⁿ andf_?,g_?:U×Rⁿ→Cl_n+1 areC¹, then for∂V a sufficiently smooth hypersurface inU bounding a relatively compact subdomain V ofU, we have

Z

V

[(g?(x, u)Rk, f?(x, u))u+ (g?(x, u), Rkf?(x, u))u]dxⁿ

= Z

∂V

(g?(x, u), Pkn(x)f?(x, u))_udσ(x),

wheredσ(x) is the scalar Lebesgue measure on∂V. Now consider the integral on the right hand side

Z

∂V

(g_?(x, u), P_kn(x)f_?(x, u))_udσ(x)

= Z

∂V

Z

Sⁿ⁻¹

g_?(x, u)P_kn(x)f_?(x, u)ds(u)dσ(x)

= Z

C(∂V)

Z

Sⁿ⁻¹

g?(C⁻¹(xs), u)Pk,uJ(C⁻¹, xs)n(xs)

×J(C⁻¹, x_s)f_?(C⁻¹(x_s), u)ds(u)dΣ(x_s),

where x_s = C(x), C(∂V) bounds a domain C(V) in Sⁿ, dΣ(x_s) is the scalar Lebesgue measure on C(∂V) and J(C⁻¹, x_s) = x_s−e_n+1

kxs−en+1kⁿ. SinceP_k,u inter- changes with J(C⁻¹, xs), the previous integral becomes

Z

C(∂V)

Z

Sⁿ⁻¹

g?(C⁻¹(xs),(xs−en+1)w(xs−en+1)

kxs−en+1k² )J(C⁻¹, xs)Pk,wn(xs)J(C⁻¹, xs)

×f_?(C⁻¹(x_s),(xs−en+1)w(xs−en+1)

kxs−e_n+1k² )ds(w)dΣ(x_s) (10)

whereu=(xs−en+1)w(xs−en+1) kxs−e_n+1k² .

Consider the integral on the left hand side (11)

Z

V

g_?(x, u)R_k, f_?(x, u)

u+ g_?(x, u), R_kf_?(x, u)

u

dxⁿ

= Z

V

Z

Sⁿ⁻¹

[g_?(x, u)R_k,r,uf_?(x, u) +g_?(x, u)R_k,uf_?(x, u)]ds(u)dxⁿ Applying Theorem 3, the integral now is equal to

Z

C(V)

Z

Sⁿ⁻¹

h g?

C⁻¹(xs),(xs−en+1)w(xs−en+1) kxs−en+1k²

J(C⁻¹, xs)R^S_k,r,wJ(C⁻¹, xs)

×f_?

C⁻¹(x_s),(xs−en+1)w(xs−en+1) kxs−e_n+1k²

+g_?

C⁻¹(x_s),(x_s−e_n+1)w(x_s−e_n+1) kxs−en+1k²

J(C⁻¹, x_s)R^S_k,wJ(C⁻¹, x_s)

×f?

C⁻¹(xs),(xs−en+1)w(xs−en+1) kxs−en+1k²

i

ds(w)dS(xs) (12)

whereC(V) =Vsis a domain inSⁿ.

Stokes’ Theorem for Rarita-Schwinger operatorsRk inRⁿ tells us that (10) is equal to (12). Therefore Stokes’ Theorem for Rarita-Schwinger type operators is conformally invariant under the Cayley transformation.

Now let us consider Stokes’ Theorem forR^S_k in Sⁿ. Z

Vs

g(x_s, u)R^S_k, f(x_s, u)

u+ g(x_s, u), R^S_kf(x_s, u)

u

dS(x_s)

= Z

∂V_s

g(xs, u), Pkn(xs)f(xs, u)

udΣ(xs), whereVs,∂Vs,dS(xs) anddΣ(xs) are stated as in Theorem 7.

First look at Z

∂V_s

g(xs, u), Pkn(xs)f(xs, u)

udΣ(xs)

= Z

∂Vs

Z

Sⁿ⁻¹

g(xs, u), Pkn(xs)f(xs, u)ds(u)Σ(xs)

= Z

C⁻¹(∂V_s)

Z

Sⁿ⁻¹

g C(x), u

P_k,uJ C(x)n(x)

J(C, x)f C(x), u

ds(u)dσ(x),

where J(C, x) = x+en+1

kx+e_n+1kⁿ, x = C⁻¹(xs) and C⁻¹(∂Vs) bounds a domain C⁻¹(Vs) inRⁿ. Since we can interchangePk,u withJ(C, x), the previous integral is equal to

(13) Z

C⁻¹(∂Vs)

Z

Sⁿ⁻¹

g

C(x),(x+en+1)w(x+en+1) kx+e_n+1k²

J C(x)Pk,wn(x) J(C, x)

×f

C(x),(x+en+1)w(x+en+1) kx+en+1k²

ds(w)dσ(x),

whereu=(x+en+1)w(x+en+1) kx+en+1k² . Second we look at

Z

Vs

g(xs, u)R^S_k, f(xs, u)

u+ g(xs, u), R^S_kf(xs, u))u dS(xs)

= Z

V_s

Z

Sⁿ⁻¹

g(x_s, u)R^S_k,r,u

f(x_s, u) +g(x_s, u) R_k,u^S f(x_s, u)

ds(u)dS(x_s). Applying Theorem 4, the integral becomes

Z

C⁻¹(V_s)

Z

Sⁿ⁻¹

g

C(x),(x+e_n+1)w(x+e_n+1) kx+en+1k²

J(C, x)R^S_k,r,w

×J(C, x)f

C(x),(x+en+1)w(x+en+1) kx+en+1k²

+g

C(x),(x+en+1)w(x+en+1) kx+e_n+1k²

J(C, x)

×R^S_k,wJ(C, x)f

C(x),(x+e_n+1)w(x+e_n+1) kx+en+1k²

ds(w)dS(x_s). (14)

Stokes’ Theorem forR^S_k on the sphere shows that (13) is equal to (14). Thus Stokes’

Theorem for Rarita-Schwinger operators is also conformally invariant under the inverse of the Cayley transformation.

Theorem 8(Borel-Pompeiu Theorem). SupposeU_s,V_s and∂V_s are as in Theo- rem 6 andys∈Vs. Then for f ∈C¹(Us×Rⁿ,Mk)we have

f(ys, u⁰) =J(C⁻¹, ys) Z

∂Vs

E_k^S(xs, ys, u, v), Pkn(xs)f(xs, v)

vdΣ(xs)

−J(C⁻¹, y_s) Z

V_s

E_k^S(x_s, y_s, u, v), R^S_kf(x_s, v)

vdS(x_s) whereu⁰= J(C⁻¹, y_s)⁻¹uJ(C⁻¹, y_s)⁻¹

kJ(C⁻¹, ys)⁻¹k² , dS(x_s)is then-dimensional area measure on Vs⊂Sⁿ,n(xs)anddΣ(xs)are as in Theorem 6.

Proof. In this proof we will use the representation ofE_k^S(xs, ys, u, v) given by (9).

LetBs(ys, ) be the ball centered atys∈Sⁿ with radius.

We denote C⁻¹(Bs(ys, )) by B(y, r), and C⁻¹(∂Bs(ys, )) by ∂B(y, r), where y =C⁻¹(ys)∈Rⁿ andris the radius ofB(y, r) inRⁿ. Consider Bs(ys, )⊂Vs, then we have

Z

Vs

E_k^S(xs, ys, u, v), R^S_kf(xs, v)

vdS(xs)

= Z

V_s\Bs(y_s,)

E_k^S(x_s, y_s, u, v), R^S_kf(x_s, v)

vdS(x_s) +

Z

Bs(ys,)

E_k^S(xs, ys, u, v), R_k^Sf(xs, v)

vdS(xs).

Because of the degree of homogeneity ofx_s−y_sinE_k^S, the second integral on the right hand goes to zero asgoes to zero. Applying Theorem 7 to the first integral on the right hand we obtain

Z

V_s\B_s(y_s,)

E_k^S(x_s, y_s, u, v), R_k^Sf(x_s, v)

vdS(x_s)

= Z

∂Vs

E_k^S(xs, ys, u, v), Pkn(xs)f(xs, v)

vdΣ(xs)

− Z

∂B_s(y_s,)

E_k^S(x_s, y_s, u, v), P_kn(x_s)f(x_s, v)

vdΣ(x_s). Since f(x_s, v) = (f(x_s, v)−f(y_s, v)) +f(y_s, v) and taking into account the degree of homogeneity ofx_s−y_sinE^S_k and the continuity off, we can replace the second integral on the right hand by

Z

∂Bs(ys,)

E_k^S(xs, ys, u, v), Pkn(xs)f(ys, v)

vdΣ(xs).