Hadamard's real part theorem for monogenic functions

Hadamard’s real part theorem for monogenic functions K. Gürlebeck and J. Morais Bauhaus-Universität Weimar, Institut für Mathematik/Physik, Coudraystr. 13B, D-99421 Weimar, Germany Abstract. The purpose of this paper is to generalize the classical Hadamard’s real part theorem from complex onedimensional analysis to the practically important 3-dimensional case in the framework of Quaternionic Analysis. Keywords: spherical monogenics, solid spherical monogenics, Hadamard’s real part theorem. PACS: 02.30.-f,02.30.Mv,02.50.Sk

INTRODUCTION In the context of complex analytic functions, Hadamard’s real part theorem contains only the modulus of the function in the left-hand side of an inequality and bounds the growth of a function by the growth of its real part. More precisely, the inequality | f (z) − f (0)| ≤

Cr sup |Re ( f (ξ ) − f (0)) |, R − r |ξ |≤R

holds for analytic functions in the disk D = {z : |z| = r < R}. Such an inequality appeared first in Hadamard’s work in 1892 (see [3]) with C = 4. Later, Borel and Carathéodory found the sharp constant C = 2. As a matter of this fact, a more general estimate for | f (z)| with f (0) 6= 0 was noticed by Carathéodory (see Landau [7, 8]) | f (z)| ≤

R+r 2r sup |Re f (ξ )| + | f (0)|. R − r |ξ |≤R R−r

In higher dimensions, the class of holomorphic functions is replaced by the set of null solutions of a generalized Cauchy-Riemann system, the class of monogenic functions. We will discuss several attempts to generalize Hadamard’s real part theorem to monogenic functions on a ball in the Euclidean space R3 . Having in mind the type of inequalities we want to prove, we will restrict ourselves to the three-dimensional case. Notice that in the four-dimensional case (quaternion-valued functions) there are a lot of non-trivial monogenic functions with vanishing scalar part. For such functions we cannot get a theorem as desired.

PRELIMINARIES Let {1, e1 , e2 , e3 } be an orthonormal basis of the Euclidean vector space R4 with the (quaternionic) product given according to the multiplication rules ei e j + e j ei = −2δi, j , i, j = 1, 2, 3, where δi, j stands for the Kronecker symbol. This non-commutative, associative product, together with the extra condition e1 e2 = e3 , generates the algebra of real quaternions denoted by H. The real vector space R4 will be embedded in H by identifying a := (a0 , a1 , a2 , a3 ) ∈ R4 with the element a = a0 + a1 e1 + a2 e2 + a3 e3 ∈ H. The real number Sc(a) := a0 is called the scalar part of a and Vec(a) := a1 e1 + a2 e2 + a3 e3 is the vector part of a. Analogously to the complex case, the conjugate of a is the quaternion a := a0 − a1 e1 − a2 e2 − a3 e3 . A norm of
1/2 a is given by |a| = a20 + a21 + a22 + a23 and coincides with the corresponding Euclidean norm of a, as a vector in R4 . Considering the subset A := spanR {1, e1 , e2 } of H, the real vector space R3 can be embedded in A by the identification of each element x = (x0 , x1 , x2 ) ∈ R3 with the paravector (sometimes also called reduced quaternion) x = x0 + x1 e1 + x2 e2 ∈ A .

As a consequence, we will often use the same symbol x to represent a point in R3 as well as to represent the corresponding reduced quaternion. Let us consider an open set Ω ⊂ R3 with a piecewise smooth boundary. An Hvalued function is a mapping f : Ω −→ H such that f (x) = f0 (x) + f1 (x)e1 + f2 (x)e2 + f3 (x)e3 , where the coordinates fi are real-valued functions defined in Ω. For continuously real-differentiable functions f : Ω −→ H, the operator D = ∂x0 + e1 ∂x1 + e2 ∂x2

(1)

is called generalized Cauchy-Riemann operator. We define the conjugate generalized Cauchy-Riemann operator by D = ∂x0 − e1 ∂x1 − e2 ∂x2 .

(2)

Solutions of the differential equations D f = 0 (resp. f D = 0) are called left-monogenic (resp. right-monogenic) functions in the domain Ω. From now on we only use left monogenic functions and call them for simplicity monogenic. In what follows, we will use the following notations: B := B1 (0) is the unit ball in R3 centered at the origin, S = ∂ B its boundary and dσ is the Lebesgue measure on S. We will denote by L2 (S; H; R) the R-linear Hilbert space of square integrable functions on S with values in H. For any f , g ∈ L2 (S; H; R) a real-valued inner product is defined by h f , giL2 (S) =

Z

Sc( f g)dσ ,

(3)

S

and leads to the usual L2 -norm (in S).

HOMOGENEOUS MONOGENIC POLYNOMIALS In [1] and [2], R-linear and H-linear complete orthonormal systems of H-valued homogeneous monogenic polynomials in the unit ball of R3 are explicitly constructed. The main idea of these constructions is based on the factorization of the Laplace operator, ∆3 = DD = DD. The authors have taken a system of real-valued homogeneous harmonic polynomials and applied the D operator in order to obtain systems of H-valued homogeneous monogenic polynomials. To be precise, we introduce the spherical coordinates, x0 = r cos θ , x1 = r sin θ cos ϕ, x2 = r sin θ sin ϕ, where 0 ≤ r < ∞, 0 ≤ θ ≤ π, 0 < ϕ ≤ 2π. Each point x = (x0 , x1 , x2 ) ∈ R3 \ {0} admits a unique representation x = rω, where r = |x| and |ω| = 1. Therefore, ωi = xri for i = 0, 1, 2. As described, the homogeneous monogenic polynomials rn Xn0 , rn Xnm , rn Ynm , m = 1, . . . , n + 1

(4)

are obtained by applying the operator 12 D to the system of homogeneous harmonic polynomials 0 m m rn+1Un+1 , rn+1Un+1 , rn+1Vn+1 , m = 1, . . . , n + 1. 0 ,U m ,V m : m = 1, . . . , n + 1} stands for the standard orthogonal basis of spherical harmonics Hereby the set {Un+1 n+1 n+1 3 in R (considered, e.g., in [9]).

In order to obtain estimates for the basis polynomials in (4), we begin by considering the following norm estimates already established in [1]. Proposition 1 For a given fixed n ∈ N0 , the spherical monogenics Xn0 , Xnm and Ynm (m = 1, . . . , n + 1) are orthogonal to each other with respect to the inner product (3) and their norms are given by p k Xn0 kL2 (S) = π(n + 1) s π (n + 1 + m)! m m k Xn kL2 (S) = k Yn kL2 (S) = (n + 1) . 2 (n + 1 − m)! Moreover, for future use in this paper we will also need pointwise estimates of the spherical monogenics Xn0 , Xnm and Ynm (m = 1, . . . , n + 1) and norm estimates of their real part. We can use results from [4] and [5].

Proposition 2 For n ∈ N0 , the spherical monogenics Xn0 , Xnm and Ynm (m = 1, . . . , n + 1) satisfy the inequalities |Xn0 (x)| ≤ |Xnm (x)|, |Ynm (x)| ≤

1 √ (n + 1)2n k Xn0 kL2 (S) 2 π 1 √ (n + 1)2n k Xnm kL2 (S) . 2 π

Proposition 3 Given a fixed n ∈ N0 , the spherical harmonics Sc(Xn0 ), Sc(Xnm ) and Sc(Ynm ) (m = 1, . . . , n) are orthogonal to each other with respect to the inner product (3) and their norms are given by r π 0 kSc(Xn )kL2 (S) = (n + 1) 2n + 1 s π 1 (n + m)! kSc(Xnm )kL2 (S) = kSc(Ynm )kL2 (S) = (n + 1 + m) . 2 (2n + 1) (n − m)! Proposition 4 Given a fixed n ∈ N0 , the spherical harmonics Sc(Xnn+1 e1 ) and Sc(Ynn+1 e1 ) are orthogonal to each other with respect to the inner product (3) and their norms are given by kSc(Xnn+1 e1 )kL2 (S) = kSc(Ynn+1 e1 )kL2 (S) =

1p π(n + 1)(2n + 2)!. 2

The important fact is here that the scalar parts of the constructed solid monogenics build again an orthogonal system.

HADAMARD’S REAL PART THEOREM FOR MONOGENIC FUNCTIONS Let Mn (R3 ; A ) be the space of A -valued homogeneous monogenic polynomials of degree n in R3 . Based on the homogeneous monogenic polynomials described in (4), an orthonormal basis in the space Mn (R3 ; A ) has been constructed in [1]. This system is given by the set of 2n + 3 homogeneous monogenic polynomials o n√ √ √ 2n + 3rn Xn0,∗ , 2n + 3rn Xnm,∗ , 2n + 3rn Ynm,∗ : m = 1, ..., n + 1 . The previous result makes it possible to work with the Fourier expansion of a square integrable A -valued monogenic function. Moreover, each monogenic function can be decomposed in an orthogonal sum of a monogenic "main part" (g) of the function and a monogenic constant (h). More precisely, it holds: Lemma 1 A monogenic L2 -function f : Ω ⊂ R3 −→ A can be decomposed into f = f (0) + g + h, where the functions g and h have Fourier series ! n ∞ √ n 0,∗ 0 m,∗ m m,∗ m g(x) = ∑ 2n + 3 r Xn (x)αn + ∑ [Xn (x)αn +Yn (x)βn ] h(x) =

n=1 ∞ √

∑

m=1

  2n + 3 rn Xnn+1,∗ (x)αnn+1 +Ynn+1,∗ (x)βnn+1 .

n=1

We remark that the associated Fourier coefficients αn0 , αnm , βnm (m = 1, ..., n + 1) are real-valued. Of course, f (0) belongs also to the subspace of constants. Having in mind the desired theorem it is natural to write f (0) as an extra summand. Originally, the Fourier coefficients are given by the inner product of the function f and elements of the space Mn (R3 ; A ). However, taking into account the orthogonality of the spherical harmonics Sc(Xn0 ), Sc(Xnm ) and Sc(Ynm ) (m = 1, . . . , n) (resp. Sc(Xnn+1 e1 ) and Sc(Ynn+1 e1 )), the Fourier coefficients, up to a factor, can also be expressed as inner products of the real part of f (resp. the real part of he1 ) and Sc(Xn0 ), Sc(Xnm ) and Sc(Ynm ) (see [5]). For our purpose we need also relations between the Fourier coefficients and the C-norm of the functions. Such an estimate is given by the following lemma.

Lemma 2 Let f be a square integrable A -valued monogenic function. Given a fixed n ∈ N0 , the Fourier coefficients satisfy the inequalities √ √ 2n + 3 |αn0 | ≤ 2 π

kXn0 kL2 (S) sup |Sc ( f (x) − f (0)) | kSc(Xn0 )kL2 (S) |x|