Borel-Carathéodory Type Theorem for monogenic functions
aa r X i v : . [ m a t h . C V ] O c t Borel-Carath´eodory Type Theorem formonogenic functions
K. G¨urlebeck, J. Morais ∗ P. Cerejeiras † November 21, 2018
Abstract
In this paper we give a generalization of the classical Borel-Carath´eodoryTheorem in complex analysis to higher dimensions in the framework ofQuaternionic Analysis.
MSC 2000 : 30G35
Keywords : spherical monogenics, homogeneous monogenic polynomials,Borel-Carath´eodory Theorem.
The Borel-Carath´eodory Theorem is a well known theorem about analytic func-tions on the unit disc in the complex plane. It states that an analytic functionis essentially bounded by its real part, the proof being based on the maximummodulus principle.
Theorem 1.1. (Borel-Carath´eodory)
Let a function f be analytic on aclosed disk of radius R centered at the origin. Suppose that r < R . Then,we have the following inequality k f k r ≤ rR − r sup | z |≤ R ℜ f ( z ) + R + rR − r | f (0) | . We recall that the norm on the left-hand side is defined as the maximumvalue of | f ( z ) | in the closed disk, that is k f k r = max | z |≤ r | f ( z ) | = max | z | = r | f ( z ) | . Since the concept of an analytic, or holomorphic, function in the complexplane is replaced, in higher dimensions, by the one of monogenic function, it isnatural to ask whether this theorem can be generalized to monogenic functionson a ball in the Euclidean space R n . In this paper we present a generalizationof this theorem to monogenic quaternionic functions. ∗ Institute of Mathematics / Physics, Bauhaus-University, Weimar, Germany. † Research (partially) supported by
Unidade de Investiga¸c˜ao Matem´atica e Aplica¸c˜oes ofthe University of Aveiro. Preliminaries
Let { e , e , e , e } be an orthonormal basis of the Euclidean vector space R .We consider e to be the real scalar unit and e , e , e the imaginary units.We introduce a multiplication of the basis vectors e i subject to the followingmultiplication rules e i e j + e j e i = − δ i,j e , i, j = 1 , , e e i = e i e = e i , i = 0 , , , . This non-commutative product, together with the extra condition e e = e , generates the algebra of real quaternions denoted by H . The real vector space R will be embedded in H by means of the identification a := ( a , a , a , a ) ∈ R with a = a e + a e + a e + a e ∈ H , where a i ( i = 0 , , ,
3) are real numbers. Remark that the vector e is themultiplicative unit element of H . From now on, we will identify e with 1 . We denote by ℜ ( a ) := a the scalar part of a and by Vec a := a e + a e + a e the vector part of a . As in the complex case, the conjugate element of a is the quaternion a := a − a e − a e − a e . The norm of a is given by | a | = √ aa and coincides with the corresponding Euclidean norm of a , as vectorin R .Let us consider the subset A := span R { , e , e } of H . The real vectorspace R is to be embedded in A via the identification of each element x =( x , x , x ) ∈ R with the reduced quaternion x = x + x e + x e ∈ A . As a consequence, no distintion will be made between x as point in R or itscorrespondent reduced quaternion. Also, we emphasize that A is a real vectorialsubspace, but not a sub-algebra, of H .Let now Ω be an open subset of R with piecewise smooth boundary. Aquaternion-valued function on Ω is a mapping f : Ω ⊂ R −→ H , with f ( x ) = P i =0 f i ( x ) e i , where the coordinate-functions f i are real-valued functions in Ω ,i = 0 , , , . Properties such as continuity, differentiability or integrability areascribed coordinate-wisely.We introduce the first order operator D = ∂ x + e ∂ x + e ∂ x (1)acting on C functions. This operator will be denoted as generalized Cauchy-Riemann operator on R . The corresponding conjugate generalized Cauchy-Riemann operator is defined ad D = ∂ x − e ∂ x − e ∂ x . (2)2 function f : Ω ⊂ R −→ H of class C is said to be left (resp. right ) monogenic in Ω if it verifies Df = 0 in Ω (resp ., f D = 0 in Ω) . The generalized Cauchy-Riemann operator (1) and its conjugate (2) factorizethe Laplace operator in R . In fact, it holds∆ = DD = DD and it implies that every monogenic function is also harmonic.At this point we would like to remark that, in general, left (resp. right)monogenic functions are not right (resp. left) monogenic. From now on, werefer only to left monogenic functions. For simplicity, we will call them mono-genic. However, all results achieved to left monogenic functions can also beadapted to right monogenic functions.Trough the remaining of this paper, we will consider the following notations: B := B (0) will denote the unit ball in R centered at the origin, S = ∂B itsboundary and dσ the Lebesgue measure on S . In what follows, we will denoteby L ( S ; X ; F ) (resp. L ( B ; X ; F )) the F -linear Hilbert space of square integrablefuntions on S (resp. B ) with values in X ( X = R or A or H ), where F = H or R . For any f, g ∈ L ( S ; A ; R ) the real-valued inner product is given by h f, g i L ( S ) = Z S ℜ ( f g ) dσ. (3)Each homogeneous harmonic polynomial P n of order n can be written inspherical coordinates as P n ( x ) = r n P n ( ω ) , ω ∈ S, (4)its restriction, P n ( ω ), to the boundary of the unit ball is called sphericalharmonic of degree n . From (4), it is clear that a homogeneous polynomialis determined by its restriction to S . Denoting by H n ( S ) the space of real-valued spherical harmonics of degree n in S , it is well-known (see [3] and [16])that dim H n ( S ) = 2 n + 1 . It is also known (see [3] and [16]) that if n = m , the spaces H n ( S ) and H m ( S )are orthogonal in L ( S ; R ; R ).Homogeneous monogenic polynomial of degree n will be denoted in generalby H n . In an analogously way to the spherical harmonics, the restriction of H n to the boundary of the unit ball is called spherical monogenic of degree n . Wedenote by M n ( H ; F ) the subspace of L ( B ; H ; F ) ∩ ker D ( B ) of all homogeneousmonogenic polynomials of degree n . Sudbery proved in [17] that the dimensionof M n ( H ; H ) is n + 1. In [5], it is proved that the dimension of M n ( H ; R ) is4 n + 4. 3 Homogeneous Monogenic Polynomials
In [5] and [6], R -linear and H -linear complete orthonormal systems of H -valuedhomogeneous monogenic polynomials in the unit ball of R are constructed. Themain idea of these constructions is based on the factorization of the Laplace op-erator. We take a system of real-valued homogeneous harmonic polynomials andwe apply the D operator in order to obtain systems of H -valued homogeneousmonogenic polynomials. For an easier description, we introduce the sphericalcoordinates x = r cos θ, x = r sin θ cos ϕ, x = r sin θ sin ϕ, where 0 < r < ∞ , 0 < θ ≤ π , 0 < ϕ ≤ π . Each point x = ( x , x , x ) ∈ R admits a unique representation x = r w , where r = | x | and | w | = 1. Therefore, w i = x i r for i = 0 , , . We will apply the operator D to each homogeneousharmonic polynomial of the family { r n +1 U n +1 , r n +1 U mn +1 , r n +1 V mn +1 , m = 1 , ..., n + 1 } n ∈ N , (5)in order to obtain a system of spherical monogenic polynomials.The elements of the previous family (5) are homogenous extensions to theball of the spherical harmonics (see e.g. [18]), U n +1 ( θ, ϕ ) = P n +1 (cos θ ) U mn +1 ( θ, ϕ ) = P mn +1 (cos θ ) cos mϕ (6) V mn +1 ( θ, ϕ ) = P mn +1 (cos θ ) sin mϕ, m = 1 , ..., n + 1 . Here, P n +1 stands for the standard Legendre polynomial of degree n + 1,while the functions P mn +1 are its associated Legendre functions, P mn +1 ( t ) := (1 − t ) m/ d m dt m P n +1 ( t ) , m = 1 , ..., n + 1 . (7)Notice that the Legendre polynomials together with the associated Legendrefunctions satisfy several recurrence formulae. We point out only the ones nec-essary for what follows in the next section. Following [2], Legendre polynomialsand its associated Legendre functions satisfy the recurrence formulae(1 − t )( P mn +1 ( t )) ′ = ( n + m + 1) P mn ( t ) − ( n + 1) tP mn +1 ( t ) , m = 0 , ..., n + 1 , (8)and P mm ( t ) = (2 m − − t ) m/ , m = 1 , ..., n + 1 . (9)Finally, these functions are mutually orthogonal in L ([ − , Z − P mn +1 ( t ) P mk +1 ( t ) dt = 0 , n = k L -norms are given by Z − ( P mn +1 ( t )) dt = 22 n + 3 ( n + 1 + m )!( n + 1 − m )! , m = 0 , ..., n + 1 . (10)For a detailed study of Legendre polynomials and associated Legendre func-tions we refer, for example, [2] and [18].Restricting the functions of the set (5) to the sphere, we obtain the sphericalmonogenics X n , X mn , Y mn , m = 1 , ..., n + 1 , (11)given by X n := (cid:18) D (cid:19) ( r n +1 U n +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 = A ,n + B ,n cosϕ e + B ,n sinϕ e , (12)where A ,n = 12 (cid:18) sin θ ddt [ P n +1 ( t )] t = cosθ + ( n + 1) cosθP n +1 ( cosθ ) (cid:19) (13) B ,n = 12 (cid:18) sinθcosθ ddt [ P n +1 ( t )] t = cosθ − ( n + 1) sinθP n +1 ( cosθ ) (cid:19) , (14)while for the remaining polynomials we have X mn := (cid:18) D (cid:19) ( r n +1 U mn +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 = A m,n cos( mϕ )+ ( B m,n cos ϕ cos mϕ − C m,n sin ϕ sin mϕ ) e + ( B m,n sin ϕ cos mϕ + C m,n cos ϕ sin mϕ ) e (15) Y mn := (cid:18) D (cid:19) ( r n +1 V mn +1 ) (cid:12)(cid:12)(cid:12)(cid:12) r =1 = A m,n sin( mϕ )+ ( B m,n cos ϕ sin mϕ + C m,n sin ϕ cos mϕ ) e + ( B m,n sin ϕ sin mϕ − C m,n cos ϕ cos mϕ ) e (16)with A m,n = 12 (cid:18) sin θ ddt (cid:2) P mn +1 ( t ) (cid:3) t = cosθ + ( n + 1) cosθP mn +1 ( cosθ ) (cid:19) B m,n = 12 (cid:18) sinθcosθ ddt (cid:2) P mn +1 ( t ) (cid:3) t = cosθ − ( n + 1) sinθP mn +1 ( cosθ ) (cid:19) C m,n = 12 m sinθ P mn +1 ( cosθ ) , m = 1 , ..., n + 1 . For each fixed n ∈ N , we obtain the set of homogeneous monogenic poly-nomials { r n X n , r n X mn , r n Y mn : m = 1 , ..., n + 1 } (17)by taking the homogeneous monogenic extension of the previous spherical mono-genics into the ball.For future use in this paper we will need norm estimates of the sphericalmonogenics described in (11) and of its real part. Proposition 3.1. (see [15]) For n ∈ N the homogeneous monogenic polynomialssatisfy the following inequalities: | r n X n ( x ) | ≤ r n ( n + 1)2 n r π ( n + 1)2 n + 3 | r n X mn ( x ) | ≤ r n ( n + 1)2 n s π n + 1)(2 n + 3) ( n + 1 + m )!( n + 1 − m )! | r n Y mn ( x ) | ≤ r n ( n + 1)2 n s π n + 1)(2 n + 3) ( n + 1 + m )!( n + 1 − m )! , whit m = 1 , ..., n + 1 . Proposition 3.2.
Given a fixed n ∈ N , the norms of the spherical harmonics ℜ ( X n ) , ℜ ( X mn ) and ℜ ( Y mn ) are given by kℜ ( X n ) k L ( S ) = ( n + 1) r π n + 1 and kℜ ( X mn ) k L ( S ) = kℜ ( Y mn ) k L ( S ) = s π n + 1 + m )(2 n + 1) ( n + 1 + m )!( n − m )! , for m = 1 , ..., n + 1 . Proof.
For simplicity sake, we just present the proof for the case of ℜ ( X n ), theproof for ℜ ( X mn ) and ℜ ( Y mn ) being similar.Using the definition of real-valued inner product (3) and (12)-(14), we get kℜ ( X n ) k L ( S ) = π Z π " sin θ (cid:18) ddt [ P n +1 ( t )] t =cos θ (cid:19) + ( n + 1) cos θ ( P n +1 (cos θ )) + 2 sin θ ( n + 1) cos θ ddt [ P n +1 ( t )] t =cos θ P n +1 (cos θ ) (cid:21) sin θdθ. t = cos θ and using the recurrence formula (8) , the last expression becomes kℜ ( X n ) k L ( S ) = π Z − (1 − t ) ( P ′ n +1 ( t )) dt − ( n + 1) Z − t ( P n +1 ( t )) dt +2( n + 1) Z − tP n ( t ) P n +1 ( t ) dt = π n + 1) Z − ( P n ( t )) dt. Due to (10) we get kℜ ( X n ) k L ( S ) = π ( n + 1) n + 1 . Proposition 3.3.
Given a fixed n ∈ N , the spherical harmonics ℜ ( X n +1 n e ) and ℜ ( Y n +1 n e ) are orthogonal to each other (w. r. t. (3)). Moreover, theirnorms satisfy kℜ ( X n +1 n e ) k L ( S ) = kℜ ( Y n +1 n e ) k L ( S ) = 12 p π ( n + 1)(2 n + 2)! . Proof.
Again, we only present the proof for the spherical harmonics ℜ{ X n +1 n e } ,the one for ℜ{ Y n +1 n e } being similar. Using (see [15]) ℜ{ X n +1 n e } = n + 12 1 sinθ P n +1 n +1 (cos θ ) cos nϕ, the definition of real-valued inner product and (12) and (14) , we obtain kℜ ( X n e ) }k L ( S ) = π (cid:18) n + 12 (cid:19) Z π θ (cid:0) P n +1 n +1 (cos θ ) (cid:1) dθ. We make the change of variable t = cos θ and, by (9), we get kℜ ( X n e ) }k L ( S ) = π (cid:18) n + 12 (cid:19) Z − (1 − t ) − ( P n +1 n +1 ( t )) dt. Now, due to the equality (10) we finally get kℜ ( X n e ) k L ( S ) = π n + 1)(2 n + 2)! . Borel-Carath´eodory’s Theorem
We will denote by X , ∗ n , X m, ∗ n , Y m, ∗ n the normalized basis functions in L ( S ; H ; H ). Theorem 4.1. (see [5]) Let M n ( R ; A ) be the space of A -valued homogeneousmonogenic polynomials of degree n in R . For each n , the set of n + 3 homo-geneous monogenic polynomials (cid:8) √ n + 3 r n X , ∗ n , √ n + 3 r n X m, ∗ n , √ n + 3 r n Y m, ∗ n , m = 1 , ..., n + 1 (cid:9) (18) forms an orthonormal basis in M n ( R ; A ) . According to this theorem, a monogenic L -function f : Ω ⊂ R −→ A canbe decomposed into f = f (0) + f + f (19)where the components f and f have Fourier series f ( x ) = ∞ X n =1 √ n + 3 r n X , ∗ n ( x ) α n + n X m =1 [ X m, ∗ n ( x ) α mn + Y m, ∗ n ( x ) β mn ] ! f ( x ) = ∞ X n =1 √ n + 3 r n (cid:2) X n +1 , ∗ n ( x ) α n +1 n + Y n +1 , ∗ n ( x ) β n +1 n (cid:3) . Moreover, we remark that the associated Fourier coefficients are real-valued.In what follows, we proof that a monogenic L -function f : Ω ⊂ R −→ A function can be bounded by its real part. For this purpose, we must find rela-tions between the Fourier coefficients and the real part of f . Lemma 4.1.
Given a fixed n ∈ N , the spherical harmonics (cid:8) ℜ ( X n ) , ℜ ( X mn ) , ℜ ( Y mn ) : m = 1 , ..., n (cid:9) are orthogonal to each other with respect to the inner product (3). The proof is immediate if one takes in consideration (12), (15) and (16).
Lemma 4.2.
Given a fixed n ∈ N , the set of spherical harmonics (cid:8) ℜ ( X n e ) , ℜ ( X mn e ) , ℜ ( Y mn e ) : m = 1 , ..., n (cid:9) is orthogonal to the set (cid:8) ℜ ( X n +1 n e ) , ℜ ( Y n +1 n e ) (cid:9) with respect to the inner product (3). emma 4.3. Let f be a square integrable A -valued monogenic function. Then,the Fourier coefficients are given by √ n + 3 α n = k X n k L ( S ) kℜ ( X n ) k L ( S ) Z S ℜ ( f ) ℜ ( X n ) dσ √ n + 3 α pk = k X mn k L ( S ) kℜ ( X mn ) k L ( S ) Z S ℜ ( f ) ℜ ( X mn ) dσ √ n + 3 β mn = k Y mn k L ( S ) kℜ ( Y mn ) k L ( S ) Z S ℜ ( f ) ℜ ( Y mn ) dσ, m = 1 , ..., n √ n + 3 α n +1 n = k X n +1 n e k L ( S ) kℜ ( X n +1 n e ) k L ( S ) Z S ℜ ( f e ) ℜ ( X n +1 n e ) dσ √ n + 3 β n +1 n = k Y n +1 n e k L ( S ) kℜ ( Y n +1 n e ) k L ( S ) Z S ℜ ( f e ) ℜ ( Y n +1 n e ) dσ. Proof.
According to Theorem 4.1, a monogenic L -function f : Ω ⊂ R −→ A can be written as Fourier series f ( x ) = f (0) + ∞ X n =1 √ n + 3 r n X , ∗ n ( x ) α n + n +1 X m =1 [ X m, ∗ n ( x ) α mn + Y m, ∗ n ( x ) β mn ] ! . We will present the proof for the coefficients α n of f , the remaining coeffi-cients α mn and β mn ( m = 1 , ..., n + 1) being obtain in a similar way.We aim to compare each Fourier coefficient α n with ℜ ( f ). In fact, multiply-ing both sides of the expression ℜ ( f ) = ∞ X n =0 √ n + 3 r n ( ℜ ( X , ∗ n ) α n + n X m =1 [ ℜ ( X m, ∗ n ) α mn + ℜ ( Y m, ∗ n ) β mn ] ) (20)by the real part of the homogeneous monogenic polynomials described in (17)and integrating over the sphere, we get the desired relations. In particular,multiplying both sides of (20) by Sc { X k } k = 1 , ... and integrating over thesphere, we obtain √ k + 3 α k = k X k k L ( S ) kℜ ( X k ) k L ( S ) Z S ℜ ( f ) ℜ ( X k ) dσ. We now study the coefficients α n +1 n and β n +1 n . Multiplying f at right by e we get˜ f := f e = ∞ X n =0 √ n + 3 r n " ℜ ( X , ∗ n e ) α n + n X m =1 [ ℜ ( X m, ∗ n e ) α mn + ℜ ( Y m, ∗ n e ) β mn ] . α n +1 n and β n +1 n , with ℜ ( ˜ f ).Multiplying ℜ ( ˜ f ) = ∞ X n =0 √ n + 3 r n " ℜ ( X , ∗ n e ) α n + n +1 X m =1 [ ℜ ( X m, ∗ n e ) α mn + ℜ ( Y m, ∗ n e ) β mn ] (21)by the homogeneous harmonic polynomials ℜ ( X k +1 k e ) ( resp. ℜ ( Y k +1 k e )),using Lemma 4.2 and integrating over the sphere carries our results √ k + 3 α k +1 k = k X k +1 k e k L ( S ) kℜ ( X k +1 k e ) k L ( S ) Z S ℜ ( f e ) ℜ ( X k +1 k e ) dσ √ k + 3 β k +1 k = k Y k +1 k e k L ( S ) kℜ ( Y k +1 k e ) k L ( S ) Z S ℜ ( f e ) ℜ ( Y k +1 k e ) dσ. Corollary 4.1.
Let f be a square integrable A -valued monogenic function.Then, the Fourier coefficients satisfy the following inequalities: √ n + 3 | α n | ≤ k X n k L ( S ) kℜ ( X n ) k L ( S ) kℜ ( f ) k L ( S ) √ n + 3 | α mn | ≤ k X mn k L ( S ) kℜ ( X mn ) k L ( S ) kℜ ( f ) k L ( S ) √ n + 3 | β mn | ≤ k X mn k L ( S ) kℜ ( X mn ) k L ( S ) kℜ ( f ) k L ( S ) , m = 1 , ..., n √ n + 3 | α n +1 n | ≤ k X n +1 n e k L ( S ) kℜ ( X n +1 n e ) k L ( S ) kℜ ( f e ) k L ( S ) √ n + 3 | β n +1 n | ≤ k Y n +1 n e k L ( S ) kℜ ( Y n +1 n e ) k L ( S ) kℜ ( f e ) k L ( S ) . The proof follows directly from Lemma 4.3 and Schwarz inequality.
Theorem 4.2.
Let f be a square integrable A -valued monogenic function in B .Then, for ≤ r < we have the following inequality: | f | ≤ | f (0) | + 4 r (2 r − (cid:0) kℜ ( f ) k L ( S ) A ( r ) + kℜ ( f e ) k L ( S ) A ( r ) (cid:1) where A ( r ) = 3(3 − r ) + 8 r (2 − r )(2 r − A ( r ) = 3(1 − r ) . roof. Considering f written as in (19) we have | f | ≤ | f (0) | + | f | + | f | . We start now to study the function f . Using the previous corollary it followsthat | f | = kℜ ( f ) k L ( S ) ∞ X n =1 (cid:20) | X , ∗ n | k X n k L ( S ) kℜ ( X n ) k L ( S ) + n X m =1 (cid:18) | X m, ∗ n | k X mn k L ( S ) kℜ ( X mn ) k L ( S ) + | Y m, ∗ n | k Y mn k L ( S ) kℜ ( Y mn ) k L ( S ) (cid:19) and, due to the Proposition 3.1 | f | ≤ kℜ ( f ) k L ( S ) ∞ X n =1 r n ( n + 1)2 n ( k X n k L ( S ) kℜ ( X n ) k L ( S ) + 2 n X m =1 k X mn k L ( S ) kℜ ( X mn ) k L ( S ) ) Now, using the estimates given by Proposition 3.2 | f | ≤ kℜ ( f ) k L ( S ) ∞ X n =1 (2 r ) n ( n + 1)( n + 2)(2 n + 1) . Note that the previous inequality is also based on [5] where the following rela-tions are proved k X n k L ( S ) = p π ( n + 1) k X mn k L ( S ) = k Y mn k L ( S ) = s π n + 1) ( n + 1 + m )!( n + 1 − m )! , m = 1 , ..., n + 1 . In the same way, we can study the function f . In fact it follows | f | ≤ kℜ ( f e ) k L ( S ) ∞ X n =1 (2 r ) n ( n + 1) . Finally | f | ≤ | f (0) | + 3 kℜ ( f e ) k L ( S ) ∞ X n =1 (2 r ) n ( n + 1)+ 12 kℜ ( f ) k L ( S ) ∞ X n =1 (2 r ) n ( n + 1)( n + 2)(2 n + 1) . Now, note that the last series are convergent for 0 ≤ r < .As a immediate consequence of the previous theorem we can state a type ofSchwartz Lemma as follows: 11 orollary 4.2. Let f be a square integrable A -valued monogenic function in B . If f (0) = 0 and kℜ ( f ) k L ( S ) A ( r ) + kℜ ( f e ) k L ( S ) A ( r ) ≤ r − , then | f | ≤ r, f or ≤ r < . The proof follows directly from the previous theorem.
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