Rarita-Schwinger Type operators on Cylinders
aa r X i v : . [ m a t h . A P ] N ov Rarita Schwinger Type Operators on Cylinders
Junxia Li and John Ryan
Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, USA.
Carmen J. Vanegas
Departamento de Matem´aticas, Universidad Sim´on Bol´ıvar, Caracas, Venezuela.
Abstract
Here we define Rarita-Schwinger operators on cylinders and construct their fun-damental solutions. Further the fundamental solutions to the cylindrical Rarita-Schwinger type operators are achieved by applying translation groups. In turn,a Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform arepresented for the cylinders. Moreover we show a construction of a number of con-formally inequivalent spinor bundles on these cylinders. Again we construct Rarita-Schwinger operators and their fundamental solutions in this setting. Finally westudy the remaining Rarita-Schwinger type operators on cylinders.
This paper is dedicated to the memory of Jaime Keller
The Rarita-Schwinger operators are generalizations of the Dirac operator which in turnis a natural generalization of the Cauchy-Riemann operator. They have been studied inEuclidean space in [BSSV, BSSV1, DLRV, Va, Va1] and on spheres and real projectivespaces in [LRV].Conformally flat manifolds are manifolds with atlases whose transition functions areM¨obius transformations. They can be constructed by factoring out a subdomain U ofeither the sphere S n or R n by a Kleinian subgroup Γ of the M¨obius group where Γ actsstrongly discontinuously on U . This gives rise to the conformally flat manifold U \ Γ. Realprojective spaces are examples of conformally flat manifolds of type S n \ {± } . Othersimple examples are R n \ Z l , where Z l is an integer lattice and 1 ≤ l ≤ n . We callthese manifolds cylinders. In case l = n we have the n -torus. For cylinders and tori theconformal structure is given by translations.In this paper we define Rarita-Schwinger operators on cylinders and construct theirfundamental solutions. Further the fundamental solutions to the cylindrical Rarita-Schwinger type operators are achieved by applying translation groups. In turn, Borel-Pompeiu Formula, Cauchy Integral Formula and a Cauchy Transform are presented for1he cylinders. We also show a construction of a number of conformally inequivalent spinorbundles on these cylinders. Again we construct Rarita-Schwinger operators and their fun-damental solutions in this setting. The Rarita Schwinger type operators on tori are leftfor future research.In [KR] R. Krausshar and J. Ryan introduce Clifford analysis on cylinders and torimaking use of the fact that the universal covering space of all of these manifolds is R n .So provided the functions and kernels are l -periodic for some l ∈ { , . . . , n } then R.Krausshar and J. Ryan use the projection map to obtain the equivalent function or kernelon those manifolds. Following them, we take the Rarita-Schwinger kernel in R n and usethe translation group to construct new kernels that are l -fold periodic, which are projectedon the cylinders. In order to prove that the new kernels are well defined we adapt theEisenstein series argument developed in [K]. We also show a construction of a number ofconformally inequivalent spinor bundles on these cylinders. In the last section, we discussthe remaining Rarita-Schwinger operators studied in [LR] and determine their kernels oncylinders. We finally establish some basic integral formulas associated with the remainingRarita-Schwinger operators on cylinders. A Clifford algebra, Cl n , can be generated from R n by considering the relationship x = −k x k for each x ∈ R n . We have R n ⊆ Cl n . If e , . . . , e n is an orthonormal basis for R n , then x = −k x k tells us that e i e j + e j e i = − δ ij . Let A = { j , · · · , j r } ⊂ { , , · · · , n } and1 ≤ j < j < · · · < j r ≤ n . An arbitrary element of the basis of the Clifford algebracan be written as e A = e j · · · e j r . Hence for any element a ∈ Cl n , we have a = P A a A e A , where a A ∈ R . For a ∈ Cl n , we will need the anti-involution called Clifford conjugation:¯ a = X A ( − | A | ( | A | +1) / a A e A , where | A | is the cardinality of A. In particular, we have e j · · · e j r = ( − r e j r · · · e j and ab = ¯ b ¯ a for a, b ∈ Cl n . For each a = a + · · · + a ··· n e · · · e n ∈ Cl n the scalar part of ¯ aa gives the square ofthe norm of a, namely a + · · · + a ··· n .We recall that if y ∈ S n − ⊆ R n and x ∈ R n , then yxy gives a reflection of x in the y direction, because yxy = yx k y y + yx ⊥ y y = − x k y + x ⊥ y where x k y is the projection of x onto y and x ⊥ y is perpendicular to y .The Dirac Operator in R n is defined to be D := n X j =1 e j ∂∂x j . M k denote the space of Cl n − valued polynomials, homogeneous of degree k andsuch that if p k ∈ M k then Dp k = 0 . Such a polynomial is called a left monogenicpolynomial homogeneous of degree k . Note if h k ∈ H k , the space of Cl n − valued harmonicpolynomials homogeneous of degree k , then Dh k ∈ M k − . But Dup k − ( u ) = ( − n − k +2) p k − ( u ) , so H k = M k M u M k − , h k = p k + up k − . This is the so-called Almansi-Fischer decomposition of H k . See [BDS].Suppose U is a domain in R n . Consider a function of two variables f : U × R n −→ Cl n such that for each x ∈ U, f ( x, u ) is a left monogenic polynomial homogeneous of degree k in u . Consider the action of the Dirac operator: D x f ( x, u ) . As Cl n is not commutative then D x f ( x, u ) is no longer monogenic in u but it is stillharmonic and homogeneous of degree k in u . So by the Almansi-Fischer decomposition, D x f ( x, u ) = f ,k ( x, u ) + uf ,k − ( x, u ) where f ,k ( x, u ) is a left monogenic polynomialhomogeneous of degree k in u and f ,k − ( x, u ) is a left monogenic polynomial homogeneousof degree k − u . Let P k be the left projection map P k : H k → M k , then R k f ( x, u ) is defined to be P k D x f ( x, u ). The left Rarita-Schwinger equation is definedto be (see [BSSV]) R k f ( x, u ) = 0 . For an integer l , 1 ≤ l ≤ n we define the l -cylinder C l to be the n -dimensional manifold R n / Z l , where Z l denote the l -dimensional lattice defined by Z l := Z e + · · · + Z e l . Wedenote its members m e + · · · + m l e l for each m , · · · , m l ∈ Z by a bold letter m . When l = n , C l is the n -torus, T n . For each l the space R n is the universal covering of thecylinder C l . Hence there is a projection map π l : R n → C l . An open subset U of the space R n is called l − fold periodic if for each x ∈ U the point x + m ∈ U . So π l ( U ) = U ′ is an open subset of the l -cylinder C l . Suppose that U ⊂ R n is a l − fold periodic open set. Let f ( x, u ) be a function definedon U × R n with values in Cl n , and such that f is a monogenic polynomial homogeneousof degree k in u . Then we say that f ( x, u ) is a l -fold periodic function if for each x ∈ U we have that f ( x, u ) = f ( x + m , u ).Now if f : U × R n → Cl n is a l − fold periodic function then the projection π l inducesa well defined function f ′ : U ′ × R n → Cl n , where f ′ ( x ′ , u ) = f ( π − l ( x ′ ) , u ) for each x ′ = π l ( x ) ∈ U ′ . Moreover, any function f ′ : U ′ × R n → Cl n lifts to a l − fold periodicfunction f : U × R n → Cl n , where U = π − l ( U ′ ) . π l induces a projection of the Rarita-Schwinger operator R k toan operator R C l k acting on domains on C l × R n which is defined by P k D ′ , where D ′ is theprojection of the Dirac operator D . That is R C l k f ′ ( x ′ , u ) = P k D ′ x ′ f ′ ( x ′ , u ) . We call the operator R C l k a l -cylindrical Rarita-Schwinger type operator and the solutionsof the equation R C l k f ′ ( x ′ , u ) = 0 (1) l -cylindrical Rarita-Schwinger functions.As I − P k : H k → u M k − , where I is the identity map, then we can define the remaining Rarita-Schwinger operators Q k := ( I − P k ) D x : u M k − → u M k − ug ( x, u ) : → ( I − P k ) D x ug ( x, u ) . See[BSSV, LR].The remaining Rarita-Schwinger equation is defined to be ( I − P k ) D x ug ( x, u ) = 0or Q k ug ( x, u ) = 0 , for each x and ( x, u ) ∈ U × R n , where U is a domain in R n and g ( x, u ) ∈ M k − . Suppose that g : U × R n → Cl n is a l − fold periodic function then the projection π l induces a well defined function g ′ : U ′ × R n → Cl n , where g ′ ( x ′ , u ) = g ( π − l ( x ′ ) , u ) foreach x ′ = π l ( x ) ∈ U ′ . The projection map π l also induces a projection of the remainingRarita-Schwinger operator Q k to an operator Q C l k acting on domains on C l × R n which isdefined by ( I − P k ) D ′ . That is Q C l k ug ′ ( x ′ , u ) = P k D ′ x ′ ug ′ ( x ′ , u ) . We call the operator Q C l k a l -cylindrical remaining Rarita-Schwinger type operator andthe solutions of the equation Q C l k ug ′ ( x ′ , u ) = 0 (2) l -cylindrical remaining Rarita-Schwinger functions. R C l k Let U a domain in R n . We recall the fundamental solution of the Rarita-Schwingeroperator R k in R n , see [DLRV]: E k ( x, u, v ) = 1 ω n c k x k x k n Z k ( xux k x k , v ) , where c k = n − n + 2 k − , ω n is the surface area of the unit sphere in R n ,Z k ( u, v ) := X σ P σ ( u ) V σ ( v ) v , P σ ( u ) = 1 k ! X σ ( u i − u e − e i ) . . . ( u i k − u e − e i k ) , V σ ( v ) = ∂ k G ( v ) ∂v k i ...∂v knik , k + . . . + k n = k, i k ∈ { , · · · , n } and the summation is taken over all permutations of the mono-mials without repetition. See [BDS].Now we construct the following functionscot l,k ( x, u, v ) = X ( m , ··· ,m l ) ∈ Z l E k ( x + m e + · · · + m l e l , u, v ) , for 1 ≤ l ≤ n − . (3)These functions are defined on the l -fold periodic domain R n / Z l for fixed u and v in R n and are Cl n -valued. It is easy to see that they are l -fold periodic functions.Now, we will prove the locally uniform convergence of the series cot l,k ( x, u, v ). First,we give a detailed proof for the locally normal convergence of the series X m ∈ Z l G ( x + m ),where G ( x + m ) = x + m || x + m || n . We will need the following proposition whose proof canbe found on page 42 in [K]. Proposition 1.
Let N be the set of non-negative integers and A k +1 be the space ofparavectors z = x + x with Sc ( z ) = x and V ec ( z ) = x ∈ R k . Let s ∈ { , . . . , k } . For allmulti-indices α ∈ N k +10 , | α | ≥ , α = (0 , α , . . . , α k ) , the following estimate holds for all z ∈ A k +1 \ { } : || ∂ | α | ∂z α q ( s ) ( z ) || ≤ ( k + 1 − s )( k + 2 − s ) . . . ( k + | α | − s ) || z || k + | α | +1 − s , where q ( z ) := ¯ z || z || k +1 ∂ | α | ∂z α q ( z ) := ∂ α + ... + α k ∂x α . . . ∂x α k k q ( z ) and q ( s ) ( z ) is the kernel ofthe s th-power of the Dirac operator. Remark 1.
In the vector formalism in R n , q ( x ) := − x || x || n . From now on we are working only in the case s = 1.Now we have a particular case of Proposition 2.2 appearing in [K]. Proposition 2.
Let p ∈ N with ≤ p ≤ n − . Let Z p be the p -dimensional lattice. Thenthe series X m ∈ Z p q ( x + m ) (4) converges normally in R n \ Z p .Proof. We consider an arbitrary compact subset K ⊂ R n and a real number R > B (0 , R ) covers K completely. Let x ∈ ¯ B (0 , R ) , x = ( x , . . . , x n ). WLOG wewill study the convergence of the series summing only over those lattice m that satisfy || m || > nR ≥ || x || . 5he function q ( x + m ) is left monogenic in 0 ≤ || x || < nR . Hence it is real analytic in¯ B (0 , nR ) and therefore can be represented in the interior of this ball by its Taylor series,i.e., q ( x + m ) = ∞ X ν =0 (cid:16) X l + ... + l n = ν l ! x l . . . x l n n q l ( m ) (cid:17) , where l = ( l , · · · , l n ) and q l ( m ) = ∂ | l | ∂ m l q ( m ) . Using Proposition 1 and observing that || x || ν ≤ R ν , we obtain: || q ( x + m ) || ≤ ∞ X ν =0 (cid:16) X l + ... + l n = ν l ! || x || ν || q l ( m ) || (cid:17) ≤ ∞ X ν =0 (cid:16) X l + ... + l n = ν l ! . . . l n ! R ν ν ! n − Y γ =1 ( ν + γ ) 1 || m || n − ν (cid:17) . Using the multinomial formula at X l + ... + l n = ν ν ! l ! . . . l n ! 1 l · · · l n = n ν , we can write the former estimate as || q ( x + m ) || ≤ ∞ X ν =0 n − Y γ =1 ( ν + γ ) (cid:16) nR || m || (cid:17) ν || m || n − . Since the series ∞ X k = − s r k + s converges absolutely for | r | < − r , we can consider its s th-derivative which also converges absolutely for | r | < d s dr s ( 11 − r ). So we obtain ∞ X k =0 ( k + 1)( k + 2) · · · ( k + s ) r k = s !(1 − r ) s +1 . Taking this into account and observing that nR || m || <
1, we get ∞ X ν =0 n − Y γ =1 ( ν + γ ) (cid:16) nR || m || (cid:17) ν = ( n − − nR || m || ) n − . Therefore we have || q ( x + m ) || ≤ ( n − − nR || m || ) n − · || m || n − . Due to Eisenstein’s Lemma (see [E]) a series of the form X ( m ,...,m p ) ∈ Z p \{ } || m ω + . . . + m p ω p || − ( p + α )
6s convergent if and only if α ≥
1, where ω i , i = 1 , . . . , p, are R -linear independentparavectors in A n . In our case, 1 ≤ p ≤ n − A || m || n − = A || m || p + α for α ≥ A = ( n − (cid:16) − nR || m || (cid:17) n − , and therefore observing that also e j , j = 1 , . . . , p, are R -linearindependent paravectors in A n and using the comparison test, we obtain that the series(4) converges normally in R n \ Z p . (cid:4) Returning to the series defined by (3)cot l,k ( x, u, v ) = X m ∈ Z l E k ( x + m e + · · · + m l e l , u, v )= X m ∈ Z l G ( x + m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) , ≤ l ≤ n − , we observe that Z k ( ( x + m ) u ( x + m ) k x + m k , v ) is a bounded function on a bounded domain in R n , because its first variable, ( x + m ) u ( x + m ) k x + m k , is a reflection in the direction x + m || x + m || for each m , and hence is a linear transformation which is a continuous function. Onthe other hand, we would get with respect to the second variable, bounded homogeneousfunctions of degree k .Consequently, applying Proposition 2 the series (3) is a uniformly convergent seriesand represents a kernel for the Rarita-Schwinger operators under translations by m ∈ Z l ,with 1 ≤ l ≤ n − n − Z l into three parts: the origin { } and a positive and anegative parts. The last two parts are equal and disjoint:Λ l = { m e : m ∈ N } ∪ { m e + m e : m , m ∈ Z , m > }∪ · · · ∪ { m e + · · · + m l e l : m , · · · , m l ∈ Z , m l > }− Λ l = ( Z l \ { } ) \ Λ l . For l = n − , we definecot n − ,k ( x, u, v ) = E k ( x, u, v ) + X m ∈ Λ n − [ E k ( x + m , u, v ) + E k ( x − m , u, v )] . (5)To show the uniform convergence of the above series, we need the following propositionwhich is a special case in [K]. Proposition 3.
Let Z n − be the ( n − -dimensional lattice. Then the series q ( x ) + X m ∈ Z n − \{ } (cid:0) q ( x + m ) − q ( m ) (cid:1) , (6)7 onverges normally in R n \ Z n − .Proof. Following the proof of Proposition 2, again we suppose that x ∈ ¯ B (0 , R ) andconsider only those lattice points with || m || > nR ≥ || x || . We consider the function q ( x + m ) − q ( m ) and expand it into a Taylor series in B (0 , R ′ ) where 0 < R ′ < R . Sowe have q ( x + m ) − q ( m ) = ∞ X ν =1 (cid:16) X l + ... + l n = ν l ! x l . . . x l n n q l ( m ) (cid:17) . Using similar arguments to those of proof of Proposition 2, we obtain || q ( x + m ) − q ( m ) || ≤ ∞ X ν =1 n − Y γ =1 ( ν + γ ) (cid:16) nR || m || (cid:17) ν || m || n − = ∞ X ν =0 n − Y γ =2 ( ν + γ ) (cid:16) nR || m || (cid:17) ν nR || m || n . Since || q ( x + m ) − q ( m ) || ≤ KR || m || n = KR || m || p +1 , p = n − , where K is a positive real constant and applying the Eisenstein’s Lemma, the series (6)turns into a normally convergent series in R n \ Z n − . (cid:4) Remark 2.
Because Λ n − and − Λ n − belong to Z n − \ { } , we have also the normalconvergence of the series X m ∈ Λ n − (cid:0) q ( x + m ) − q ( m ) (cid:1) , X m ∈− Λ n − (cid:0) q ( x + m ) − q ( m ) (cid:1) Now we consider the expression G ( x + m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + G ( x − m ) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17) , where the function Z k ( · , · ) has the hypotheses stated formerly. Rewriting the formerexpression as( G ( x + m ) + G ( x − m )) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + ( − G ( x − m ) − G ( m )) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + ( G ( x − m ) + G ( m )) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17) + G ( m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) − G ( m ) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17) Z n − \ { } we obtain X m ∈ Z n − \{ } (cid:16) G ( x + m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + G ( x − m ) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17)(cid:17) = X m ∈ Z n − \{ } ( G ( x + m ) + G ( x − m )) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + X m ∈ Z n − \{ } ( − G ( x − m ) − G ( m )) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) + X m ∈ Z n − \{ } ( G ( x − m ) + G ( m )) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17) + X m ∈ Z n − \{ } (cid:16) G ( m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) − G ( m ) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17)(cid:17) . (7)The last sum in (7) vanishes because the terms G ( m ) Z k (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) for each m ∈ Λ n − and m ∈ − Λ n − cancel with the terms G ( m ) Z k (cid:16) ( x − m ) u ( x − m ) k x − m k , v (cid:17) for each m ∈ − Λ n − and m ∈ Λ n − , respectively. On the other side, since || G ( x + m ) + G ( x − m ) || ≤ || G ( x + m ) − G ( m ) || + || G ( x + m ) − G ( m ) || . and observing that Z k ( · , · ) is bounded, we can apply Proposition 3 to obtain the normalconvergence for the series (7). Consequently the series defined by (5) is uniformly con-vergent and it is a kernel for the Rarita-Schwinger type operator under translations by m ∈ Λ n − .For x, y ∈ R n \ Z l , 1 ≤ l ≤ n −
1, the functions cot l,k ( x − y, u, v ) induce functionscot ′ l,k ( x ′ , y ′ , u, v ) = cot l,k ( π − l ( x ′ ) − π − l ( y ′ ) , u, v ) . These functions are defined on ( C l × C l ) \ diag ( C l × C l ) for each fixed u, v ∈ R n , where diag ( C l × C l ) = { ( x ′ , x ′ ) : x ′ ∈ C l } and they are l -cylindrical Rarita-Schwinger func-tions, i.e, the equation R C l k cot ′ l,k ( x ′ , y ′ , u, v ) = 0 is satisfied. Furthermore for each l theyrepresent a kernel for the operator R C l k . Definition 1.
For any Cl n -valued polynomials P, Q , the inner product ( P ( u ) , Q ( u )) u withrespect to u is given by ( P ( u ) , Q ( u )) u = Z S n − P ( u ) Q ( u ) dS ( u ) , where S n − is the unit sphere in R n . p k ∈ M k , one obtains p k ( u ) = ( Z k ( u, v ) , p k ( v )) v = Z S n − Z k ( u, v ) p k ( v ) dS ( v ) . See [BDS].
Theorem 1. [DLRV] (Stokes’ Theorem for R k ) Let Ω and Ω ′ be domains in R n andsuppose the closure of Ω lies in Ω ′ . Further suppose the closure of Ω is compact and ∂ Ω is piecewise smooth. Then for f, g ∈ C (Ω ′ , M k ) , we have Z Ω [( g ( x, u ) R k , f ( x, u )) u + ( g ( x, u ) , R k f ( x, u )) u ] dx n = Z ∂ Ω ( g ( x, u ) , P k dσ x f ( x, u )) u , where dx n = dx ∧· · ·∧ dx n , dσ x = n X j =1 ( − j − e j d ˆ x j , and d ˆ x j = dx ∧· · · dx j − ∧ dx j +1 · · ·∧ dx n . Theorem 2. [DLRV] (Borel-Pompeiu Theorem) Let Ω ′ and Ω be as in Theorem 1. Thenfor f ∈ C (Ω ′ , M k ) f ( y, v ) = Z ∂ Ω ( E k ( x − y, u, v ) , P k dσ x f ( x, u )) u − Z Ω ( E k ( x − y, u, v ) , R k f ( x, u )) u dx n . Now by Stokes’ Theorem and Borel-Pompeiu Theorem for the Rarita-Schwinger op-erator R k in R n , we may easily obtain: Theorem 3.
Let V be a bounded domain in R n and V be the closure of V . For each x ∈ V , the shifted lattice x + Z l intersected with V only contains the points x. Supposethat the boundary of V , ∂V , is piecewise smooth and V is compact. Further suppose f ( x, u ) : V × R n → Cl n is a monogenic homogeneous polynomial of degree k in u and withrespect to x is C . Then for ≤ l ≤ n − and each y ∈ V,f ( y, v ) = Z ∂V (cot l,k ( x − y, u, v ) , P k dσ x f ( x, u )) u − Z V (cot l,k ( x − y, u, v ) , R k f ( x, u )) u dx n . Proof. . When m = 0 , this is the Borel-Pompeiu Formula given by Theorem 2 f ( y, v ) = Z ∂V ( E k ( x − y, u, v ) , P k dσ x f ( x, u )) u − Z V ( E k ( x − y, u, v ) , R k f ( x, u )) u dx n . (8)When m = 0 , for each m ∈ Z l , by the hypothesis that the shifted lattice x + Z l intersected with V only contains the points x , we have x + m / ∈ V but y ∈ V , so there10s no singularity in E k ( x − y + m , u, v ) = E k ( x + m − y, u, v ) . Hence by Stokes’ Theoremwe obtain Z ∂V ( E k ( x − y + m , u, v ) , P k dσ x f ( x, u )) u − Z V ( E k ( x − y + m , u, v ) , R k f ( x, u )) u dx n = Z V ( E k ( x − y + m , u, v ) R k , f ( x, u )) u dx n = 0 . (9)Now for all m ∈ Z l , by adding the equations (8) and (9), we obtain f ( y, v ) = Z ∂V X m ∈ Z l E k ( x − y + m , u, v ) , P k dσ x f ( x, u ) ! u − Z V ( X m ∈ Z l E k ( x − y + m , u, v ) , R k f ( x, u )) u dx n . (cid:4) The case we are most interested in here is the one that V is l − fold periodic and f is l − fold periodic. If we use the projection map π l at this moment, we obtain theBorel-Pompeiu Theorem on the cylinder C l . Theorem 4. (Borel-Pompeiu Theorem for R C l k ) Suppose V ′ is a domain in C l withcompact closure and smooth boundary. Suppose f ( x, v ) is defined as in Theorem 3. Thenfor ≤ l ≤ n − and each y ′ ∈ V ′ ,f ′ ( y ′ , v ) = Z ∂V ′ (cid:0) cot ′ l,k ( x ′ , y ′ , u, v ) , P k dσ ′ x ′ f ′ ( x ′ , u ) (cid:1) u − Z V ′ (cot ′ l,k ( x ′ , y ′ , u, v ) , R C l k f ′ ( x ′ , v )) v dµ ( x ′ ) , where x ′ = π l ( x ) , dσ ′ x ′ = ∂ x π l dσ x , ∂ x π l is the derivative of π l at x , µ is the projectionof Lebesgue measure on R n onto C l , dσ x = n ( x ) dσ ( x ) , n ( x ) is the unit exterior normalvector at x = π − l ( x ′ ) and dσ ( x ) is the surface measure element. Theorem 5. (Cauchy integral formula) Suppose the hypotheses as in Theorem 11 and f ′ ( x ′ , u ) is annihilated by the operator R C l k , then for ≤ l ≤ n − and each y ′ ∈ V ′ , wehave f ′ ( y ′ , u ) = Z ∂V ′ (cid:0) cot ′ l,k ( x ′ , y ′ , u, v ) , P k dσ ′ x ′ f ′ ( x ′ , u ) (cid:1) u . If the function given in the Borel-Pompeiu Theorem has its support with respect to x ′ in V ′ ⊂ C l , then by the same theorem, we have the following: Theorem 6. Z V ′ − (cot ′ l,k ( x ′ , y ′ , u, v ) , R C l k ψ ( x ′ , u )) u dx ′ n = ψ ( y ′ , v ) , for ψ ∈ C ∞ ( C l ) . We also can introduce a Cauchy transform for the Rarita-Schwinger operator R C l k :11 efinition 2. For a subdomain V ′ of cylinder C l and a function f ′ : V ′ −→ Cl n , theCauchy transform of f ′ is formally defined to be ( T f ′ )( y ′ , v ) = − Z V ′ (cid:0) cot ′ l,k ( x ′ , y ′ , u, v ) , f ′ ( x ′ , u ) (cid:1) u dx ′ n , y ′ ∈ V ′ . Consequently, one may obtain a right inverse for R C l k Theorem 7. R C l k Z V ′ (cot ′ l,k ( x ′ , y ′ , u, v ) , ψ ( x ′ , u )) u dx ′ n = ψ ( y ′ , v ) , for ψ ∈ C ∞ ( C l ) . C l We shall now show a construction of a number of conformally inequivalent spinor bundleson C l . In the previous sections the spinor bundle over C l is chosen to be the trivial one C l × Cl n , however we can construct 2 l spinor bundles on C l .Different spin structures on a spin manifold M are detected by the number of differenthomomorphisms from the fundamental group Π ( M ) to the group Z = { , } . In ourcase we have Π ( C l ) = Z l . Because there are two homomorphisms of Z to Z , we have 2 l distinct spin structures on C l , see [MP, KR1].The following construction is for some of spinor bundles over C l but all the othersare constructed similarly. First let p be an integer in the set { , . . . , l } and consider thelattice Z p := Z e + · · · + Z e p . We also consider the lattice Z l − p := Z e p +1 + · · · + Z e l . Inthis case Z l = { m + n : m ∈ Z p and n ∈ Z l − p } . Suppose that m = m e + · · · + m p e p .Let us make the identification ( x, X ) with ( x + m + n , ( − m + ··· + m p X ) where x ∈ R n and X ∈ Cl n . This identification gives rise to a spinor bundle E p over C l .The Rarita-Schwinger operator over R n induces a Rarita-Schwinger operator acting onsections of the bundles E p over C l . We will denote this operator by R C l k,p . The projection π l maps U ⊂ R n to a domain U ′ ⊂ C l . Now if f : U × R n → Cl n is a l − fold periodicfunction then the projection π l induces a well defined function f ′ p : U ′ × R n → E p , where f ′ p ( x ′ + m + n , u ) = ( − m + ··· + m p f ( π − l ( x ′ )+ m + n , u ) for each x ′ + m + n = π l ( x )+ m + n ∈ U ′ . If R C l k,p ( f ′ p ) = 0 then f ′ p is called an E p left Rarita-Schwinger section. Moreover, any E p left Rarita-Schwinger section f ′ p : U ′ × R n → E p lifts to a l − fold periodic function f : U × R n → Cl n , where U = π − l ( U ′ ), and R C l k f = 0.By considering the seriescot l,k ( x, u, v ) = X m ∈ Z l E k ( x + m , u, v ) , for 1 ≤ l ≤ n − R n \ Z l , we can obtain the kernel (see section 3)cot l,k ( x, y, u, v ) = X m ∈ Z l E k ( x − y + m , u, v ) , for 1 ≤ l ≤ n − . π l : R n → C l to these kernels induce kernelscot ′ l,k ( x ′ , y ′ , u, v )defined on ( C l × C l ) \ diag( C l × C l ), where diag( C l × C l ) = { ( x ′ , x ′ ) : x ′ ∈ C l } . We canadapt these functions as follows. For 1 ≤ l ≤ n − l,k,p ( x, u, v ) = X m ∈ Z p , n ∈ Z l − p ( − m + ··· + m p E k ( x + m + n , u, v ) . These are well defined functions on R n \ Z l . Therefore we obtain from these functions thecotangent kernelscot l,k,p ( x, y, u, v ) = X m ∈ Z p , n ∈ Z l − p ( − m + ··· + m p E k ( x − y + m + n , u, v ) . Again applying the projection map π l these kernels give rise to the kernelscot ′ l,k,p ( x ′ , y ′ , u, v ) . In the case l = n −
1, by considering the seriescot n − ,k ( x, u, v ) = E k ( x, u, v ) + X m ∈ Z n − [ E k ( x + m , u, v ) + E k ( x − m , u, v )]we obtain the kernelcot n − ,k ( x, y, u, v ) = E k ( x − y, u, v ) + X m ∈ Z n − [ E k ( x − y + m , u, v )+ E k ( x − y − m , u, v )]which in turn using the projection map induces kernels cot ′ n − ,k ( x ′ , y ′ , u, v ). Definingcot n − ,k,p ( x, u, v ) = E k ( x + m + n , u, v )+ X m ∈ Z p , n ∈ Z n − − p ( − m + ··· + m p h E k ( x + m + n , u, v ) + E k ( x − m − n , u, v ) i we obtain the cotangent kernelscot n − ,k,p ( x, y, u, v ) = E k ( x − y + m + n , u, v )+ X m ∈ Z p , n ∈ Z n − − p ( − m + ··· + m p h E k ( x − y + m + n , u, v ) + E k ( x − y − m − n , u, v ) i and by π l the kernels cot ′ n − ,k,p ( x ′ , y ′ , u, v ). 13 Remaining Rarita-Schwinger type operators oncylinders
Consider the fundamental solution of the remaining Rarita-Schwinger operator Q k in R n ,see [LR]: H k ( x, u, v ) := − ω n c k u x k x k n Z k − ( xux k x k , v ) v, where c k = n − n − k . Now we construct functions
Cot l,k ( x, u, v ) = X m ∈ Z l H k ( x + m e + · · · + m l e l , u, v )= X m ∈ Z l uG ( x + m ) Z k − (cid:16) ( x + m ) u ( x + m ) k x + m k , v (cid:17) v, ≤ l ≤ n − . These functions are defined on the l -fold periodic domain R n / Z l for fixed u and v in R n and are Cl n -valued. We can observe that they are l -fold periodic functions. Usingthe similar arguments in Section 3 , we may easily obtain that the series Cot l,k ( x, u, v ) isnormally convergent over R n \ Z l .For l = n − , we define Cot n − ,k ( x, u, v ) = H k ( x, u, v ) + X m ∈ Z n − [ H k ( x + m , u, v ) + H k ( x − m , u, v )] . We can establish that the previous series converges uniformally following the proof inSection 3 for the case l = n − . For x, y ∈ R n \ Z l , 1 ≤ l ≤ n −
1, the functions
Cot l,k ( x − y, u, v ) induce functions Cot ′ l,k ( x ′ , y ′ , u, v ) = Cot l,k ( π − l ( x ′ ) − π − l ( y ′ ) , u, v ) . These functions are defined on ( C l × C l ) \ diag ( C l × C l ) for each fixed u, v ∈ R n , where diag ( C l × C l ) = { ( x ′ , x ′ ) : x ′ ∈ C l } and they are l -cylindrical remaining Rarita-Schwingerfunctions. Consequently, Q C l k Cot ′ l,k ( x ′ , y ′ , u, v ) = 0 . Furthermore for each l they representa kernel for the operator Q C l k .Now we will establish some integral formulas associated with the remaining Rarita-Schwinger operators on cylinders. Theorem 8. [LR](Stokes’ Theorem for Q k operators) Let Ω ′ and Ω be domains in R n and suppose the closure of Ω lies in Ω ′ . Further suppose the closure of Ω is compact and he boundary of Ω , ∂ Ω , is piecewise smooth. Then for f, g ∈ C (Ω ′ , M k − ) , we have Z Ω [( g ( x, u ) uQ k,r ) , uf ( x, u )) u + ( g ( x, u ) u, Q k uf ( x, u )) u ] dx n = Z ∂ Ω ( g ( x, u ) u, ( I − P k ) dσ x uf ( x, u )) u = Z ∂ Ω ( g ( x, u ) udσ x ( I − P k,r ) , uf ( x, u )) u . Where Q k,r is the right remaining Rarita-Schwinger operator. Theorem 9. [LR](Borel-Pompeiu Theorem for Q k operators)Let Ω ′ and Ω be as in theprevious Theorem. Then for f ∈ C (Ω ′ , M k − ) and y ∈ Ω , we obtain uf ( y, u ) = Z Ω ( H k ( x − y, u, v ) , Q k vf ( x, v )) v dx n − Z ∂ Ω ( H k ( x − y, u, v ) , ( I − P k ) dσ x vf ( x, v )) v . Applying Stokes’ Theorem and Borel-Pompeiu Theorem for the Q k operator in R n ,we may have: Theorem 10.
Let V be a bounded domain in R n and V be the closure of V . For each x ∈ V , the shifted lattice x + Z l intersected with V only contains the points x. Supposethat the boundary of V , ∂V , is piecewise smooth and V is compact. Further suppose g ( x, u ) : V × R n → Cl n is a monogenic homogeneous polynomial of degree k − in u andwith respect to x is C . Then for ≤ l ≤ n − and each y ∈ V,vg ( y, v ) = Z ∂V Z V ( Cot l,k ( x − y, u, v ) , Q k uf ( x, u )) u − ( Cot l,k ( x − y, u, v ) , ( I − P k ) dσ x ug ( x, u )) u dx n . Now using the projection map π l , we obtain the Borel-Pompeiu Theorem for the Q k operators on the cylinder C l . Theorem 11. (Borel-Pompeiu Theorem for Q C l k ) Suppose V ′ is a domain in C l withcompact closure and smooth boundary. Suppose g ( x, u ) is defined as in Theorem 10. Thenfor ≤ l ≤ n − and each y ′ ∈ V ′ ,vg ′ ( y ′ , v ) = Z ∂V ′ (cid:0) Cot ′ l,k ( x ′ , y ′ , u, v ) , ( I − P k ) dσ ′ x ′ ug ′ ( x ′ , u ) (cid:1) u − Z V ′ ( Cot ′ l,k ( x ′ , y ′ , u, v ) , Q C l k ug ′ ( x ′ , u )) u dµ ( x ′ ) , where x ′ = π l ( x ) , dσ ′ x ′ = ∂ x π l dσ x , ∂ x π l is the derivative of π l at x , µ is the projectionof Lebesgue measure on R n onto C l , dσ x = n ( x ) dσ ( x ) , n ( x ) is the unit exterior normalvector at x = π − l ( x ′ ) and dσ ( x ) is the surface measure element. heorem 12. (Cauchy integral formula for Q C l k ) Suppose the hypotheses as in Theorem10 and ug ′ ( x ′ , u ) is annihilated by the operator Q C l k , then for ≤ l ≤ n − and each y ′ ∈ V ′ , we have vg ′ ( y ′ , v ) = Z ∂V ′ (cid:0) Cot ′ l,k ( x ′ , y ′ , u, v ) , ( I − P k ) dσ ′ x ′ ug ′ ( x ′ , u ) (cid:1) u . If the function given in Borel-Pompeiu Theorem has its support with respect to x ′ in V ′ ⊂ C l , then we have the following: Theorem 13. Z V ′ − ( Cot ′ l,k ( x ′ , y ′ , u, v ) , Q C l k uψ ( x ′ , u )) u dx ′ n = vψ ( y ′ , v ) , for ψ ∈ C ∞ ( C l × R n ) . We may introduce a Cauchy transform for the remaining Rarita-Schwinger operator Q C l k : Definition 3.
For a subdomain V ′ of cylinder C l and a function g ′ : V ′ × R n −→ Cl n , which is monogenic in u with degree k − , the Cauchy transform of f ′ is formally definedto be ( T vg ′ )( y ′ , v ) = − Z V ′ (cid:0) Cot ′ l,k ( x ′ , y ′ , u, v ) , ug ′ ( x ′ , u ) (cid:1) u dx ′ n , y ′ ∈ V ′ . Consequently, one may obtain:
Theorem 14. Q C l k Z V ′ ( Cot ′ l,k ( x ′ , y ′ , u, v ) , uψ ( x ′ , u )) u dx ′ n = vψ ( y ′ , v ) , for ψ ∈ C ∞ ( C l × R n ) . Similarly, we can carry on the theory of the remaining Rarita-Schwinger operators tothe setting of conformally spinor bundles over the cylinders.
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