Spherical Π -type Operators in Clifford Analysis and Applications
aa r X i v : . [ m a t h . C V ] S e p Spherical Π-type Operators in Clifford Analysis andApplications
Wanqing Cheng, John Ryan and Uwe K¨ahler
Abstract
The Π-operator (Ahlfors-Beurling transform) plays an important role in solvingthe Beltrami equation. In this paper we define two Π-operators on the n-sphere.The first spherical Π-operator is shown to be an L isometry up to isomorphism.To improve this, with the help of the spectrum of the spherical Dirac operator,the second spherical Π operator is constructed as an isometric L operator over thesphere. Some analogous properties for both Π-operators are also developed. We alsostudy the applications of both spherical Π-operators to the solution of the sphericalBeltrami equations. Keywords:
Singular integral operator, Π-operator, Spectrum, Beltrami equation.
The Π-operator is one of the tools used to study smoothness of functions over Sobolevspaces and to solve some first order partial differential equations such as the Beltramiequation which describes quasi-conformal mappings. In one dimensional complex analysis,the Beltrami equation is the partial differential equation: ∂w∂z = µ ∂w∂z where µ = µ ( z ) is a given complex function, and z = x + iy ∈ C , ∂ z = ∂∂x − i ∂∂y , ∂ ¯ z = ∂∂x + i ∂∂y . It can be transformed to a fixed-point equation h = q ( z )( I + Π Ω h )where Π Ω h ( z ) = − πi Z Ω h ( ξ )( ξ − z ) dξ dξ is the complex Π-operator. This singular integral operator acts as an isometry from L ( C )to L ( C ) with the L p -norm being a long standing conjecture by Ivaniec.1ith the help of Clifford algebras, the classical Beltrami equation and Π-operator withsome well known results can be generalized to higher dimensions. Abundant results inEuclidean space have been found( see [4, 9, 8]). In order to generate results in Euclideanspace to the unit sphere, we define two Π-operators related to the conformally invariantspherical Dirac operator. The idea to consider the n -sphere is not only motivated by be-ing the classic example of a manifold and being invariant under the conformal group, butalso by the fact that in the case of n = 3 due to the recently proved Poincar´e conjecturethere is a wide class of manifolds which are homeomorphic to the 3-sphere. This makesour results much more general and valid for any simply connected closed 3-manifold. Inparticular, results on local and global homeomorphic solutions of the sperical Beltramiequation carry over to such manifolds.This paper is organized as follows: In section 2, we briefly introduce Clifford algebras,Clifford analysis, the Euclidean Dirac operator, and some well known integral formulas.In Section 3, we review the construction and some properties for the Π-operator in Eu-clidean space. In section 4, we construct the Π-operator in a generalized spherical spaceand solve the Beltrami equation with a singular integral operator Π s, . In the last section,we will investigate the spectra of several spherical Dirac type operators and the sphericalLaplacian, and construct the isometric spherical Π-operator Π s, . Dedication : This paper is dedicated to Franciscus Sommen on the occasion of his 60thbirthday.
Let e , · · · , e n be an orthonormal basis of R n +1 . The Clifford algebra C l n is the algebraover R n generated by the relation x = −|| x || e where e is the identity of C l n . These algebras were introduced by Clifford in 1878 in [6].Each element of the algebra C l n can be represented in the form x = X A ⊂{ , ··· ,n } x A e A where x A are real numbers. The norm of a Clifford number x is defined as k x k = X A ⊂{ , ··· ,n } x A . If the set A contains k elements, then we call e A a k-vector . Likewise, we call each linearcombination of k -vectors a k -vector. The vector space of all k -vectors is denoted by Λ k R n .Obviously, C l n is the direct sum of all Λ k R n for k ≤ n . The following anti-involutions arewell known: 2 Reversion: ˜ a = X A ( − | A | ( | A |− / a A e A , where | A | is the cardinality of A . In particular, ^ e j · · · e j r = e j r · · · e j . Also e ab = ˜ b ˜ a for a, b ∈ C l n . • Clifford conjugation: a † = X A ( − | A | ( | A | +1) / a A e A , satisfying e j · · · e j r † = ( − r e j r · · · e j and ( ab ) † = b † a † for a, b ∈ C l n . • Clifford involution: ¯ a = ˜ a † = e a † . In the following we identify the Euclidean space R n +1 with the direct sum Λ R n ⊕ Λ R n .For all that follows let Ω ⊂ R n +1 be a domain with a sufficiently smooth boundary Γ = ∂ Ω.Then functions f defined in Ω with values in C l n are considered. These functions may bewritten as f ( x ) = X A ⊆{ e ,e ,...e n } e A f A ( x ) , ( x ∈ Ω) . Properties such as continuity, differentiability, integrability, and so on, which are ascribedto f have to be possessed by all components f A ( x ) , ( A ⊆ { e , e , ...e n } ). The spaces C k (Ω , C l n ) , L p (Ω , C l n ) are defined as right Banach modules with the corresponding tradi-tional norms. The space L (Ω , C l n ) is a right Hilbert module equipped with a C l n -valuedsesquilinear form ( u, v ) = Z Ω u ( η ) v ( η ) d Ω η . Furthermore, W kp (Ω , C l n ) , k ∈ N ∪ { } , ≤ p < ∞ denotes the Sobolev spaces as the rightmodule of all functionals whose derivatives belong to L p (Ω , C l n ), with norm k f k W kp (Ω , C l n ) := X A X k α k≤ k k D αw f A k pL p (Ω , C l n ) /p . The closure of the space of test functions C ∞ (Ω , C l n ) in the W kp -norm will be denoted by ◦ W kp (Ω , C l n ). 3he Euclidean Dirac operators D x and D arise as generalizations of the Cauchy-Riemann operator of one complex variable. As homogenous linear differential operators, D x := n X i =1 e i ∂ x i ,D := e ∂ x + n X i =1 e i ∂ x i = e ∂ x + D x . Note D x = − ∆ x , where ∆ x is the Laplacian in R n +1 , and ∆ n +1 = D D , where D is theClifford conjugate of D . Definition 1. A C l n -valued function f ( x ) defined on a domain Ω in R n +1 is called leftmonogenic if D x f ( x ) = n X i =1 e i ∂ x i f ( x ) = 0 . Similarly, f is called a right monogenic function if it satisfies f ( x ) D x = n X i =1 ∂ x i f ( x ) e i = 0Let f ∈ C (Ω , C l n ), G ( x − y ) = x − y k x − y k n +1 beign the fundamental solution of D .Hence, the Cauchy transform is defined as T Ω f ( x ) = Z Ω G ( x − y ) f ( y ) dy, where T is the generalization of the Cauchy transform in the complex plane to Euclideanspace, and it is the right inverse of D , that is D T = I . Also, the non-singular boundaryintegral operator is given by F ∂ Ω f ( x ) = Z ∂ Ω G ( x − y ) n ( y ) f ( y ) dσ ( y ) . We have the Borel-Pompeiu Theorem as follows.
Theorem 1. ([8]) For f ∈ C (Ω , C l n ) ∩ C ( ¯Ω) , we have f ( x ) = Z ∂ Ω G ( x − y ) n ( y ) f ( y ) dσ ( y ) + Z Ω G ( x − y ) D f ( y ) dy, In particular, if f ∈ ◦ W (Ω , C l n ) , then f ( x ) = Z Ω G ( x − y ) D f ( y ) dy. Π -operator in Euclidean space It is well known that in complex analysis, the Π-operator can be realized as the com-position of ∂ ¯ z and the Cauchy transform. As the generalization to higher dimension inClifford algebra, we have the Π-operator in R n +1 defined as follows. Definition 2.
The Π -operator in Euclidean space R n +1 is defined as Π = D T. The following are some well known properties for the Π-operator.
Theorem 2. ([8]) Suppose f ∈ ◦ W kp (Ω)(1 < p < ∞ , k ≥ , then1. D Π f = D f, Π D f = D f − D F ∂ Ω f, F ∂ Ω Π f = (Π − T D ) f, D Π f − Π D f = D F ∂ Ω f. The following decomposition of L (Ω , C l n ) helps us to observe that the Π-operatoractually maps L (Ω , C l n ) to L (Ω , C l n ). Theorem 3. ([8]) ( L (Ω , C l n ) Decomposition) L (Ω , C l n ) = L (Ω , C l n ) \ KerD M D ( ◦ W (Ω , C l n )) , and L (Ω , C l n ) = L (Ω , C l n ) \ KerD M D ( ◦ W (Ω , C l n )) . Notice that, since Π( L (Ω , C l n ) \ KerD ) = L (Ω , C l n ) \ KerD , Π( D ( ◦ W (Ω , C l n ))) = D ( ◦ W (Ω , C l n )) , hence, this Π-operator is from L (Ω , C l n ) to L (Ω , C l n ).One key property of the Π-operator is that it is an L isometry, in other words, Theorem 4. ([4]) For functions in L (Ω , C l n ) , we have Π ∗ Π = I.
5o complete this section, we give the classic example of the Π-operator solving theBeltrami equation. Let Ω ⊆ R n +1 , q : Ω → C l n a bounded measurable function and ω : Ω → C l n be a sufficiently smooth function. The generalized Beltrami equation D ω = qD ω could be transformed into an integral equation h = q ( D φ + Π h )where ω = φ + T h , which could have a unique solution if k q k ≤ q < k Π k , see [8], with q being a constant. This tells us that the existence of a unique solution to the Beltramiequation depends on the norm estimate for the Π-operator. Π -typeoperator with generalized spherical Dirac operator Recall that in one dimensional complex analysis, the Π-operator is defined asΠ f ( z ) := ∂ ¯ z T f ( z ) = ∂ ¯ z Z Ω f ( z ) η − z dz, where z = x + iy ∈ C and ∂ ¯ z = ∂∂x + i ∂∂y . This suggests us to generalize the Π-operator, weneed to consider a variable z with “real” and “imaginary” parts, so we can take conjugateof ∂ z to define the Π-operator. Π -type operator with generalized spherical Diracoperator Let S n be the n-unit sphere. The spherical Dirac operator D s, on S n is defined as follows. xD = n X j =1 e e j ( x ∂ x j − x j ∂ x ) − n X i =1 ,j>i e i e j ( x i ∂ x j − x j ∂ x i ) + n X j =0 ( x j ∂ x j ) . Denote Γ = n X j =1 e e j (( x ∂ x j − x j ∂ x )) − n X i =1 ,j>i e i e j (( x i ∂ x j − x j ∂ x i )) . Hence, D s, = x − xD s, = x k x k ( E r + Γ ) = ξ ( D r + Γ r ) , where rD r = E r , r = k x k and ξ ∈ S n .In particular, we have the conformally invariant spherical Dirac operator as follows, D s, = w (Γ − n . D s = ξ ( D r + Γ r ), and since D s is also conformally invariant, wehave D s = w (Γ − n ), whereΓ = − n X j =1 e e j ( x ∂ x j − x j ∂ x ) − n X i =1 ,j>i e i e j ( x i ∂ x j − x j ∂ x i ) . Here D s is the Clifford involution of D s . Lemma 1. Γ w = nw − w Γ ;Γ w = nw − w Γ . Proof.
The proof is similar to Theorem 3 in [10].
Theorem 5. D s w = − wD s , D s w = − wD s . Proof.
Applying the last Lemma, a straight forward calculation completes the proof.
Theorem 6.
Since D s and D s are both conformally invariant, we have their fundamentalsolutions as follows: D s G s ( w − v ) = D s w − v k w − v k n = δ ( v ) ,D s G s ( w − v ) = D s w − v k w − v k n = δ ( v ) , where w, v ∈ S n .Proof. The proof is similar to Proposition 4 in [10].Let Ω be a bounded smooth domain in S n and f ∈ C (Ω , C l n ), we have the Cauchytransforms for both D s and D s , T Ω f ( w ) = Z Ω G s ( w − v ) f ( v ) dv = Z Ω w − v k w − v k n f ( v ) dv,T Ω f ( w ) = Z Ω G s ( w − v ) f ( v ) dv = Z Ω w − v k w − v k n f ( v ) dv. Also, the non-singular boundary integral operators are given by F ∂ Ω f ( w ) = Z ∂ Ω G s ( w − v ) n ( v ) f ( v ) dσ ( v ) ,F ∂ Ω f ( w ) = Z ∂ Ω G s ( w − v ) n ( v ) f ( v ) dσ ( v )Then we have Borel-Pompeiu Theorem as follows.7 heorem 7. ([10]) (Borel-Pompeiu Theorem) For f ∈ C (Ω) ∩ C ( ¯Ω) , we have f ( w ) = Z ∂ Ω G s ( w − v ) n ( v ) f ( v ) dσ ( v ) + Z Ω G s ( w − v ) D s f ( v ) dv, in other words, f = F ∂ Ω f + T Ω D s f . Similarly, f = F ∂ Ω f + T Ω D s ff ( w ) = Z ∂ Ω G s ( w − v ) n ( v ) f ( v ) dσ ( v ) + Z Ω G s ( w − v ) D s f ( v ) dv, If f is a function with compact support, then T D s = T D s = I . Since the conformally invariant spherical Laplace operator ∆ s has the fundamentalsolution H s ( w − v ) = − n − k w − v k n − , see [10]. We have factorizations of ∆ s asfollows. Theorem 8. ∆ s = D s ( D s + w ) = D s ( D s + w ) .Proof. The proof is similar to Proposition 5 in [10].We also have the dual of D s as follows. Theorem 9. D ∗ s = − D s . Proof.
Let f, g : Ω → C l n both have compact supports, < D s f, g > = < w (Γ − n f, g > = < (Γ − n f, wg > = < Γ f, wg > − n < f, wg > = < f, Γ wg > − n < f, wg > = < f, ( nω − ω Γ ) g > − n < f, wg > = < f, − w (Γ − n g > = < f, − D s g > . Definition 3.
Define the generalized spherical Π -type operator as Π s, f = ( D s + w ) T f.
We have some properties of Π s, as follows. Proposition 1. D s Π s, = D s − w, Π s, D s = D s + w. roof. D s Π s, = D s ( D s + w ) T = ( D s − w ) D s T = D s − w, Π s, D s = ( D s + w ) T D s = D s + w. From the proposition above, we can have decompositions of L (Ω , C l n ) as follows. Theorem 10. L (Ω , C l n ) = L (Ω , C l n ) \ Ker ( D s − w ) M D s ( ◦ W (Ω , C l n )) ,L (Ω , C l n ) = L (Ω , C l n ) \ KerD s M ( D s + w )( ◦ W (Ω , C l n )) . Notice that Π s, ( L (Ω , C l n ) \ Ker ( D s − w ) = L (Ω , C l n ) \ KerD s , Π s, D s ( ◦ W (Ω , C l n )) = ( D s + w )( ◦ W (Ω , C l n )) . Hence, Π s, operator is from L (Ω , C l n ) to L (Ω , C l n ). The proof is similar to Theorem 1in [8]. Definition 4.
We define the Π + s operator as Π + s f = D s T + f, where T + f = Z Ω G + ( w − v ) f ( v ) dv , G + ( w − v ) = G s ( w − v ) + wH s ( w − v ) − G (3) s ( w − v ) , and G (3) s ( w − v ) = 1( n − n − w − v k w − v k n − . Notice that G (3) s ( w − v ) is actually the reproducing kernel of D (3) s = ( D s − w ) D s ( D s + w )and the proof is similar to a proof in [10]. Proposition 2. Π s, ( L (Ω , C l n ) \ KerD s ) = L (Ω , C l n ) \ Ker ( D s − w ) , Π s, ( D s + w )( ◦ W (Ω , C l n )) = D s ( ◦ W (Ω , C l n )) . heorem 11. Π s is an isometry on ◦ W (Ω , C l n ) up to isomorphism.Proof. Let f ∈ L (Ω , C l n ), then h Π s f, Π + s g i = h ( D s + w ) T f, D s T + g i = h T f, ( − D s + w ) D s T + g i = −h T f, ( D s − w ) D s T + g i = −h T f, D s ( D s + w ) T + g i = h D s T f, ( D s + w ) T + g i = h f, g i . Π s, to the solution of a Beltrami equation We have a Beltrami equation related to Π s, as follows. Let Ω ⊆ S n − be a bounded,simply connected domain with sufficiently smooth boundary, q : Ω −→ C l n a measurablefunction. Let f : Ω −→ C l n be a sufficiently smooth function. The spherical Beltramiequation is as follows: D s f = q ( D s + w ) f. It has a unique solution f = φ + T h where φ ia an arbitrary left-monogenic function suchthat D s f = 0 and h is the solution of an integral equation h = q (cid:0) ( D s + w ) φ + Π s, h (cid:1) . By the Banach fixed point theorem, the previous integral equation has a unique solutionin the case where k q k ≤ q < k Π s, k , with q being a constant. Hence, for the rest of this section, we will estimate the L p normof Π s, with p > D s = w (Γ − n ) = w ( wD − E r − n ) = D − wE r − n w , thenΠ s, f ( w ) = ( D s + w ) T f ( w ) = ( DT + w (1 − E w ) T − n T ) f ( w ) . it is easy to see that ∂∂w j Z S n w − v k w − v k n f ( v ) dv = Z S n e j − n ( w j − v j ) w − v k w − v k k w − v k n f ( v ) dv + ω n e j n f ( v ) , since ∂∂w j w − v k w − v k n = e j − n ( w j − v j ) w − v k w − v k k w − v k n , § Z S w − v k w − v k cos( r, w j ) dS = ω n e j n , where S is a sufficiently small neighborhood of w on S n .Hence, we have DT f ( w ) = 1 ω n Z S n P e j − n P ( w j − v j ) e j w − v k w − v k k w − v k n f ( v ) dv + P e j n f ( v )= 1 ω n Z S n (1 − n ) − n w − v k w − v k k w − v k n f ( v ) dv + 1 − nn f ( v ) E w T f ( w ) = 1 ω n Z S n P w j e j − n P w j ( w j − v j ) w − v k w − v k k w − v k n f ( v ) dv + P w j e j n f ( v )= 1 ω n Z S n w − n < w, w − v > w − v k w − v k k w − v k n f ( v ) dv + wn f ( v ) . Therefore, we have an integral expression of Π s, as follows. Theorem 12. Π s, f ( w ) = ( DT + w (1 − E w ) T − n T ) f ( w )= 1 ω n Z S n − n − w k w − v k n f ( v ) dv + nω n Z S n v − h w, v i w k w − v k n +1 · w − v k w − v k f ( v ) dv + (1 − n wω n Z S n w − v k w − v k n f ( v ) dv + 1 − nn f ( v ) . Since Π s, = ( D s + w ) T = ( w (Γ − n w ) T = w Γ T + (1 − n wT, where Γ = − n X j =1 e e j ( x ∂ x j − x j ∂ x ) − n X i =1 ,j>i e i e j ( x i ∂ x j − x j ∂ x i ). To estimate the L p normof Π s, , we need the following result. Theorem 13.
Suppose p is a positive integer and p > , then k T k L p ≤ ω n − . roof. Since k T f k pL p = ( 1 ω n ) p Z Ω k Z Ω G s ( w − v ) f ( v ) dv n k p dw n = ( 1 ω n ) p Z Ω k Z Ω G s ( w − v ) q G s ( w − v ) p f ( v ) dv n k p dw n ≤ ( 1 ω n ) p Z Ω (cid:0) ( Z Ω k G s ( w − v ) k dv n ) pq · Z Ω k G s ( w − v ) kk f ( v ) k p dv n (cid:1) dw n ≤ ( 1 ω n ) p C pq Z Ω Z Ω k G s ( w − v ) kk f ( v ) k p dv n dw n = ( 1 ω n ) p C pq Z Ω k f ( v ) k p ( Z Ω k G s ( w − v ) k dw n ) dv n ≤ ( 1 ω n ) p C pq +11 Z Ω k f ( v ) k p ( Z Ω k G s ( w − v ) k dw n ) dv n = ( 1 ω n ) p C p · Z Ω k f ( v ) k p dv n = ( 1 ω n ) p C p · k f k pL p where p, q > p + 1 q = 1, where C ≤ (cid:12)(cid:12) Z S n k G s ( w − v ) k dv n (cid:12)(cid:12) = (cid:12)(cid:12) Z S n k w − v k n − dv n (cid:12)(cid:12) . Due to the symmetry we can choose any fixed point w , hence we choose w = (1 , , , ..., v = ( x , x , · · · , x n ) ∈ S n , i.e. n X i =0 k x i k = 1. Let v = cos θe + sin θζ , where ζ is avector on n − dv n = sin n − θdθ , Z S n − x )] n − dv n = 2 − n − Z π − cos θ ) n − sin n − θdθ = 2 − n − Z π (2 sin θ − n − (2 sin θ θ n − dθ = Z π cos n − θ dθ = 2 · · Γ( )Γ( n )Γ( n − + 1)= √ π Γ( n )Γ( n +12 ) . ω n = 2 π ( n +1) / Γ( n +12 ) , we have k T k L p ≤ ω n − G be the operator defined by G g ( w ) = − n − ω n Z S n k w − v k n − g ( v ) dv, n ≥ , and R s = Γ ◦ G is a Riesz transformation of gradient type (see [1]). Then we have, Proposition 3. [1], The operator R s is a L p operator and the L p norm is bounded by π / √ pp − / B p , where B p = C M,p + C p , C M,p is the L p norm of the maximal truncated Hilbert transforma-tion on S , and C p = cot π p ∗ , p + p ∗ = 1 . Hence, k Γ ω n Z Ω k w − v k n − · w − v k w − v k f ( v ) dv k L p ≤ ( n − π / √ pp − / B p k f ( v ) k L p = ( n − π / √ pp − / B p k f ( v ) k L p . (1)Recall that Π s, f = ( D s + w ) T f = ( w (Γ − n ) + w ) T f = w Γ T f + (1 − n ) wT f ,and by Theorem 13, || (1 − n wT f || L p = k (1 − n wω n Z Ω w − v k w − v k n f ( v ) dv k L p ≤ ( n − ω n − k f k L p . (2)By inequalities (1) and (2), we show that Π s, is a bounded operator mapping from L p space to itself, and k Π s, k L p ≤ ( n − π / √ pp − / B p + ( n − ω n − . Remark:
The spherical Π-type operator Π s, preserves most properties of the Πoperator in Euclidean space and more importantly, it is a singular integral operator whichhelps to solve the corresponding Beltrami equation. Unfortunately, it is also only an L isometry up to isomorphism as shown in Theorem 11 . In the next section, we will usethe spectrum theory of differential operators to claim that there is a spherical Π-typeoperator which is also an L isometry. 13 Eigenvectors of spherical Dirac type operators
In this section, we will investigate the spectrums of several spherical Dirac type operatorsand the spherical Laplacian. During the investigation, we will point out there is a spher-ical Π-type operator which is an L isometry.Since Γ = xD − E r , it is easy to verify the fact that if p m is a monogenic poly-nomial and is homogeneous with degree m , that is D f m = 0 and E r f m = mf m , thenΓ f m = − mf m , so f m is an eigenvector of Γ with eigenvalue − m . Similarly, if D g m = 0, g m is an eigenvector of Γ with eigenvalue − m .Let H k be the space of C l n -valued harmonic polynomials homogeneous of degree k and M k be the C l n -valued monogenic polynomials homogeneous of degree k, M k is the cliffordinvolution of M k . By an Almansi-Fischer decomposition [7], H k = M k L ¯ x M k − . Hence,for for all harmonic functions with homogeneity of degree k , there exist p k ∈ KerD , and p k − ∈ KerD such that h k = p k + ¯ xp k − . Then, it is easy to get that Γ p k = − kp k andΓ ¯ xp k − = ( n + k )¯ xp k − .Let H m denote the restriction to S n of the space of C l n -valued harmonic polynomialswith homogeneity of degree m . P m is the space of spherical C l n -valued left monogenicpolynomials with homogeneity of degree − m and Q m is the space of spherical C l n -valuedleft monogenic polynomials with homogeneity of degree n + m , m = 0 , , , ... .Then wehave H m = P m L Q m ([3]). It is well known that L ( S n ) = ∞ X m =0 H m ([2]), it follows L ( S n ) = ∞ X m =0 P m M Q m . If p m ∈ P m , since Γ p m = − mp m , it is an eigenvector of Γ with eigenvalue − m , and for q m ∈ Q m , it is an eigenvector of Γ with eigenvalue n + m .Therefore, the spectrum of Γ is σ (Γ ) = {− m, m = 1 , , ... } ∪ { m + n, m = 0 , , , ... } , .Since D s = w (Γ − n D s is σ ( D s ) = σ (Γ ) − n , which is {− m − n , m =0 , , , ... } ∪ { m + n , m = 0 , , , ... } .As mentioned in the previous section that D s T = T D s = I , and we know that D s : P m −→ Q m ([3]). Hence,we have T : Q m −→ P m and the spectrum of T is the reciprocalof the spectrum of D s , which is σ ( T ) = { m + n , m = 0 , , , ... } S { − m − n , m = 0 , , , ... } .Similar arguments apply for D s and T , in fact σ ( D s ) = σ ( D s ) and σ ( T ) = σ ( T ).Now with similar strategy as in [3], we consider the operator D s T which maps L ( S n )to L ( S n ). If u ∈ C ( S n ) then u ∈ L ( S n ). It follows that u = ∞ X m =0 X p m ∈ P m p m + −∞ X m =0 X q m ∈ Q m q m , p m and q m are eigenvectors of Γ . Further the eigenvectors p m and q m can bechosen so that within P m they are mutually orthogonal. The same can be done for theeigenvectors q m . Moreover, as u ∈ C ( S n ) then D s T u ∈ C ( S n ) and so D s T u ∈ L ( S n ).Consequently, D s T u = ∞ X m =0 X p m ∈ P m D s T p m + ∞ X m =0 X q m ∈ Q m D s T q m = ∞ X m =0 X q m ∈ Q m D s m + n q m + ∞ X m =0 X p m ∈ P m D s − m − n p m and || D s T u || L = ∞ X m =0 ( 1 m + n ) X q m ∈ Q m k D s q m k L + ∞ X m =0 ( 1 − m − n ) X p m ∈ P m k D s p m || L = ∞ X m =0 ( 1 m + n ) ( m + n X p m ∈ P m k p m || L + ∞ X m =0 ( 1 − m − n ) ( − m − n X q m ∈ Q m k q m || L = ∞ X m =0 X p m ∈ P m || p m || L + ∞ X m =0 X q m ∈ Q m || q m || L = || u || L . This shows D s T is an L ( S n ) isometry.By the help of the spectrum of T , we have the L norm estimate of the Π s, , that is k Π s, u k L ≤ k D s T u k L + k w k L k T u k L = k u k L + ( 1 m + n ) ( ∞ X m =0 X p m ∈ P m k p m k L + ∞ X m =0 X q m ∈ Q m k q m k L ) ≤ (1 + 4 n ) k u k L . Hence we have k Π s, k L ≤ n .By Theorem 13, ∆ s = D s ( D s + w ) = ( D s − w ) D s = D s ( D s + w ) = ( D s − w ) D s .Since D s = w (Γ − n , D s = w (Γ − n s = − (Γ − n − ww (Γ − n − Γ + ( n − − ( n − n − (Γ − n − ww (Γ − n − Γ + ( n − − ( n − n . < r <
1, any harmonic function h m ∈ B (0 , r ) = { x ∈ R n : || x || < r } withhomogeneity degree m, we have h m = f m + g m , where f m ∈ KerD and g m ∈ D , theyare both homogeneous with degree m (see Lemma 3 [13]). Consequently,∆ s f m = ( − Γ + ( n − − ( n − n f m = ( − m − m ( n − − ( n − n f m , and∆ s g m = ( − Γ + ( n − − ( n − n g m = ( − m − m ( n − − ( n − n g m . Hence ∆ s h m = ∆ s ( f m + g m ) = ( − m − m ( n − − ( n − n f m + g m )= ( − m − m ( n − − ( n − n h m . Since for any function u ∈ L ( S n ) : Ω
7→ C l n , u = ∞ X m =0 h m , where h m ∈ H m , it follows that∆ s has spectrum σ (∆ s ) = {− m − m ( n − − ( n − n ) : m = 0 , , , ... } .In order to preserve the property of isometry of the Π-operator on the sphere, wedefine the isometric spherical Π-operator as Π s, as Π s, = D s T , which is isometry in L space. We can solve the Beltrami equation related to Π s, as follows.Let Ω ⊆ S n − be a bounded, simply connected domain with sufficiently smooth bound-ary, and q, f : Ω −→ C l n , q is a measurable function, and f is sufficiently smooth. Thespherical Beltrami equation is as follows: D s f = qD s f. It has a unique solution f = φ + T h where φ ia an arbitrary left-monogenic function suchthat D s f = 0 and h is the solution of an integral equation h = q ( D s φ + Π s, h ) . By the Banach fixed point theorem, the previous integral equation has a unique solutionin the case of k q k ≤ q < k Π s, k with q being a constant. Hence, we can use the estimate of the L p norm of Π s, with p >
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