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Computational Methods and Function Theory Volume 5 (2005), No. 1, 111–134

Cauchy Integral Decomposition of Multi-Vector Valued Functions on Hypersurfaces Ricardo Abreu Blaya, Juan Bory Reyes, Richard Delanghe and Frank Sommen (Communicated by Stephan Ruscheweyh) Abstract. Let Ω be a bounded open and connected subset of Rm which has a C∞ -boundary Σ and let Fk ∈ C∞ (Σ) be a k-multi-vector valued function on Σ. Under which conditions can Fk be decomposed as Fk = Fk+ + Fk− where Fk± are extendable to harmonic k-multi-vector fields in Ω± with Ω+ = Ω and Ω− = Rm \ Ω? This question is answered by proving a set of equivalent assertions, including a conservation law on Fk and conditions on the Cauchy transform CΣ Fk and on the Hilbert transform HΣ Fk of Fk . Keywords. Clifford analysis, multi-vector valued functions, Cauchy transform, Hilbert transform. 2000 MSC. 30G35, 45B20.

1. Introduction Let Ω be an open bounded subset of the plane having as boundary a C1 -Jordan curve. Furthermore, let f ∈ C 0,α (Γ), 0 < α < 1, and let CΓ be the Cauchy transform of f , i.e. Z 1 CΓ f (z) = n(t)f (t) ds(t), Γ t−z where n(t) is the outward pointing unit normal at t ∈ Γ. Then a classical result tells us that, if Ω+ = Ω and Ω− = R2 \ Ω, the function CΓ f which is holomorphic in Ω+ ∪ Ω− with CΓ f (∞) = 0 admits the boundary values CΓ± f (t) =

lim CΓ f (z),

Ω± 3z→t

t ∈ Γ,

where CΓ± f ∈ C 0,α (Γ). Moreover, on Γ we have f = CΓ+ f − CΓ− f. Received August 16, 2004. The first two authors were supported by the FWO Research Network WO. 003. 01N and the last author was supported by the FWO “Krediet aan Navorsers: 1.5.065.04, 1.5.106.02”. c 2005 Heldermann Verlag ISSN 1617-9447/$ 2.50

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The question of looking at a natural multidimensional analogue of such a decomposition is closely connected with the problem of generalizing analytic function theory in the plane to higher dimensions. So far, several ways of possible generalizations have been considered. A first analogue was studied within the theory of several complex variables (see [2] and [3]). However, complex analysis in Cm , m > 1, does not seem to have direct applications to harmonic forms in Rm . A second analogue, essentially worked out by E. Dyn’kin in [6] and [7], was based on studying the decomposition of a continuous k-form ηk on a C1 -smooth hypersurface Σ in Rm as a sum of continuous k-forms ηk± on Σ (1)

ηk = ηk+ + ηk− ,

where (i) Σ is the boundary of an open domain G in Rm ; (ii) ηk± are extendable to harmonic k-forms of class H s , s > 0, in, respectively, G+ = G and G− = Rm \ G with ηk− (∞) = 0. Dyn’kin gave necessary and sufficient conditions under which ηk can be represented by (1), his so-called Condition (A) and Condition (B). Let us recall that a smooth k-form ηk in an open domain G of Rm is said to be harmonic if it satisfies in G the Hodge-de Rham system  dηk = 0, (2) d∗ ηk = 0. P j In the particular case of 1-forms ξ = m j=1 ξj dx , (2) also reads  ∂ξi ∂ξj   − = 0, i 6= j,   ∂xj ∂xi m (3) X ∂ξj   = 0.   ∂xj j=1

In terms of smooth vector fields ~ξ = (ξ1 , . . . , ξm ), this means that ~ξ satisfies the so-called Riesz system ( div ~ξ = 0, (4) curl ~ξ = 0. The systems (3) and (4) clearly generalize the classical Cauchy-Riemann system in the plane to the m-dimensional Euclidean space Rm . A solution in G to the system (3) was called (in [15]) a system of conjugate harmonic functions in G. By considering 1-vector valued null-solutions of the Dirac operator ∂x in Rm M. Riesz derived in [13] the system (4).

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As such, we touch upon a third possible approach to generalizing complex analysis in the plane to higher dimensions, namely Clifford analysis. For more detailed information about Clifford analysis and in particular about its relation to harmonic analysis in Rm , we refer the reader to for e.g. [5] and [8]. Let us recall that if ∂x is the Dirac operator in Rm , and R0,m is the Clifford algebra constructed over the vector space Rm , the latter being provided with a quadratic form of signature (0, m), then an R0,m -valued C1 -function f in an (k) open domain G of Rm is said to be left monogenic in G if ∂x f = 0 in G. If R0,m (k) denotes the space of k-multi-vectors in R0,m , then a left monogenic R0,m -valued function Fk in G is called a harmonic k-multi-vector field in G. As in open domains G of Rm , there is a one-one correspondence between harmonic k-forms and harmonic k-multi-vector fields in G (see e.g. [1]), the question naturally arises of considering the following problem: given a continuous k-multivector valued function Fk on a hypersurface Σ in Rm , under which conditions can one decompose Fk on Σ as a sum (5)

Fk = Fk+ + Fk−

where (i) Σ is the boundary of an open bounded domain Ω in Rm ; (ii) Fk± are extendable to harmonic k-multi-vector fields Fk± in, respectively Ω+ = Ω and Ω− = Rm \ Ω with Fk− (∞) = 0? A first approach to this problem was made by M. Shapiro who considered in [14] the case of a H¨older continuous 1-vector valued function on a Liapunov surface Σ. In [1], the case of decomposing a H¨older continuous k-multi-vector valued function Fk on an Ahlfors-David regular hypersurface Σ in Rm was studied. A set of equivalent assertions was established (see [1, Theorem 4.1]) which are of a pure function theoretic nature and which, in some sense, replace Dyn’kin’s Condition (B) for harmonic k-forms. Notice also that, as pointed out in [7], for an open domain in Rm and 0 < s < 1, the class H s of harmonic k-forms appearing in Dyn’kin’s Condition (B) simply consists of all harmonic k-forms in that domain satisfying a H¨older condition of order s in the closed domain. As to Dyn’kin’s Condition (A), namely   dηk Σ = 0,  d∗ η = 0 k Σ — a natural condition at least in the sense of currents for harmonic k-forms which are continuously extendable to the boundary of the domain — no counterpart in the framework of Clifford analysis was derived in [1].

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One of the main aims of this paper is to show that Dyn’kin’s Condition (A) for harmonic k-forms is equivalent to a so-called conservation law (CL) for harmonic k-multi-vector fields (see Section 5). In fact, such a conservation law is already established in Section 4 for two-sided monogenic functions f in Ω (i.e. ∂x f = f ∂x = 0 in Ω) which are extendable to C∞ (Ω). We also suppose that Σ is a C∞ -hypersurface. We assume such type of smoothness conditions on both Σ and the trace f |Σ in order to describe how a decent calculus for boundary values of harmonic k-multi-vector fields may be established. Less restrictive conditions on Σ and f |Σ could as well be envisaged and could therefore be the subject of further study. Within this framework, our main result Theorem 4.3 states a list of equivalent assertions concerning the possibility of decomposing a given Fk ∈ C∞ (Σ) into a sum of the type (5). For the convenience of the reader, we have recalled in Section 2 some preliminaries concerning Clifford algebras and Clifford analysis, while in Section 3, results are listed about the Cauchy transform CΣ and the Hilbert transform HΣ on C∞ (Σ).

2. The tangential Dirac operator In this section an expression is derived for the Dirac operator in an -normal open neighborhood Σ of a smooth hypersurface Σ in Rm . 2.1. Clifford algebras and multivectors. Let R0,m be the vector space Rm equipped with a quadratic form of signature (0, m) and let R0,m be the universal Clifford algebra constructed over R0,m . If e = (e1 , . . . , em ) is an orthonormal basis of R0,m , then the multiplication rules in R0,m are governed by (see [5]) ei 2 = −1,

i = 1, . . . , m

and ei ej + ej ei = 0, i 6= j. For any subset A = {i1 , . . . , ik } ⊂ {1, . . . m} with 1 ≤ i1 < i2 < · · · < ik ≤ m, define the element eA as follows eA := ei1 ei2 · · · eik . Putting e∅ = 1, the identity element of R0,m , a basis for R0,m is then given by (eA : A ⊂ {1, . . . , m}), whence an arbitrary element a ∈ R0,m may be written as X a= aA eA , aA ∈ R. A

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(k)

For 0 ≤ k ≤ m fixed, the space R0,m of k-vectors or k-grade multivectors in R0,m is defined by (k) R0,m = spanR (eA : |A| = k). Clearly m X

R0,m =

(k)

⊕R0,m

k=0

and any element a ∈ R0,m may thus also be written as (6)

a=

m X

[a]k ,

k=0 (k)

(k)

where [·]k : R0,m → R0,m denotes the projection of R0,m onto R0,m . (0)

It is customary to identify R with R0,m , the so-called set of scalars in R0,m , and (1) (2) Rm with R0,m ∼ = R0,m , the so-called set of vectors in R0,m . The elements of R0,m (m) are also called bivectors while the elements of R0,m = ReM , where eM = e1 · · · em , are called pseudo-scalars. Notice that for any two vectors x and y, their product is given by xy = x • y + x ∧ y where


1 x y + y x = −hx, yi, 2 P x y being the standard inner product between x and y, while hx, yi = m j=1 j j
X 1 x∧y = xy − yx = ei ej (xi yj − xj yi ) 2 i 0 sufficiently small, the -normal open neighborhood Σ of Σ is determined by Σ = {x = n(x ⊥ )ν(x) + x ⊥ ∈ Rm : x ⊥ ∈ Σ, |ν(x)| < } where for x ⊥ ∈ Σ, the outward pointing unit normal at x ⊥ is denoted by n(x ⊥ ) and ν(x) = hn(x ⊥ ), xi = ±|x − x ⊥ |. If (ε1 , . . . , εm−1 ) is an orthonormal frame for the tangent space at x ⊥ ∈ Σ and n = n(x ⊥ ), then for x ∈ Σ x = nhn, xi +

m−1 X

εj hεj , xi.

j=1

Putting ν = hn, xi, (ν, ω1 , . . . , ωm−1 ) determines a coordinate system at x. As   ∂x ∂x ,..., ∂ω1 ∂ωm−1 is also a basis for the tangent space at x ⊥ , for each j = 1, . . . , m − 1, there ought to exist C∞ -functions Qji (ν, ω1 , . . . , ωm−1 ) such that εj =

m−1 X i=1

Qji (ν, ω1 , . . . , ωm−1 )

∂x . ∂ωi

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Now, let ∂x be the Dirac operator in Rm , i.e. ∂x =

m X

ei ∂xi .

i=1

Then in terms of the orthonormal basis (n, ε1 , . . . , εm−1 ) (7)

∂x = nhn, ∂x i + ∂kx

where ∂kx =

m−1 X

εj hεj , ∂x i

j=1

is the tangential part of the Dirac operator, also called the tangential Dirac operator. Clearly n = ∂x ν = ∂ν x, where ∂ν = hn, ∂x i, X ∂kx = εj Qji ∂ωi , j,i

whence ∂x = n∂ν + ∂kx = n∂x +

X

εj Qji ∂ωi .

j,i

Moreover, as n∂x = n • ∂x + n ∧ ∂x , ∂x n = ∂x • n + ∂x ∧ n, we have that nhn, ∂x i = −n(n • ∂x ) and that ∂kx = −n(n ∧ ∂x ) when acting from the left, respectively ∂kx = −(n ∧ ∂x )n when acting from the right, on smooth R0,m -valued functions. Finally, let f be an R0,m -valued C∞ -function in Σ and put g(ω) = f (0, ω), the restriction of f to Σ. Then ∂kx f |Σ = ∂ω g, where ∂ω =

X j,i

εj Qji (0, ω)∂ωi .

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Example. An important example is the case of the unit sphere S m−1 in Rm . Let us describe ∂kx as an operator acting from the left. Using polar coordinates x = rξ with r = |x| and ξ ∈ S m−1 , we may write ∂x as (see [5])   Γξ ∂x = ξ ∂r + r where Γξ = x ∧ ∂x . As at ξ ∈ S m−1 , n(ξ) = ξ, we thus have in terms of polar coordinates that in Σ = S m−1 ×] − , [,  sufficiently small, ξΓξ ∂kx = , r its restriction ∂ω to S m−1 being given by ∂ω = ξΓξ . In the case m = 2, i.e. the case of the unit circle S 1 in the plane, straightforward computations then lead to T (ξ)∂θ ∂kx = , r ∂ω = T (ξ)∂θ , where

π  π  T (ξ) = e1 cos + θ + e2 sin +θ 2 2 is the unit tangent vector at ξ ∈ S 1 . 2.3. Some elements of Clifford analysis. Let G be an open subset of Rm and let f : G → R0,m be a C1 -function. Then f is said to be left (resp. right) monogenic in G if ∂x f = 0 in G (resp. f ∂x = 0 in G). If f is both left and right monogenic in G, i.e. ∂x f = f ∂x = 0 in G, then f is called two-sided monogenic in G. Monogenic functions in G belong to C∞ (G); even more, they are real analytic in G (see e.g. [5]). The space of left monogenic functions and of twosided monogenic functions in G is denoted, respectively, by M(G) and M(G). An important example of a two-sided monogenic function in G = Rm \ {0} is given by the fundamental solution E of ∂x , namely 1 x E(x) = . Am |x|m Here Am stands for the surface area of the unit sphere S m−1 in Rm . (k)

Notice that if Fk is an R0,m -valued C1 -function in G (Fk is also called k-multivector valued, 0 ≤ k ≤ m), then in G ∂x Fk = 0

⇐⇒

Fk ∂x = 0.

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This follows from ∂x Fk = Fk ∂x k(k+1)/2

with ∂x = −∂x and Fk = (−1)

Fk .

(k) R0,m -valued

It thus follows that an left monogenic function Fk in G is automatically two-sided monogenic in G. Such functions are also called harmonic k-multi-vector fields in G (see also Section 5). P Of particular interest is the case k = 1: a harmonic 1-vector field u = m j=1 ej uj in G satisfies  ∂ui ∂uj   − = 0, i 6= j,   ∂xj ∂xi m (8) X ∂uj   = 0,   ∂xj j=1

i.e. if ~u = (u1 , . . . , um ), ~u satisfies the Riesz system (8).

3. Plemelj-Sokhotzki formulae In this section we suppose that Ω is a connected bounded open domain in Rm with C∞ -boundary Σ such that Rm \ Ω is also connected. For f ∈ C∞ (Σ), its left and right Cauchy transforms CΣ f and f CΣ and its left and right Hilbert transforms HΣ f and f HΣ are defined respectively by Z CΣ f (x) = E(y − x)n(y)f (y) dS(y), x ∈ Rm \ Σ, Σ

Z E(y − x)n(y)f (y) dS(y),

HΣ f (x) = 2 lim

→0+

Z f CΣ (x) =

x ∈ Σ,

{y∈Σ: |x−y|>}

f (y)n(y)E(y − x) dS(y),

x ∈ Rm \ Σ,

Σ

Z f (y)n(y)E(y − x) dS(y),

f HΣ (x) = 2 lim

→0+

x ∈ Σ.

{y∈Σ: |x−y|>}

Here n(y) is the outward pointing unit normal at y ∈ Σ. Clearly CΣ f is left monogenic while f CΣ is right monogenic in Rm \ Σ with CΣ f (∞) = f CΣ (∞) = 0. Let us now formulate some important properties of CΣ f and HΣ f , which, mutatis mutandis, may be translated in similar results for f CΣ and f HΣ . For their proofs, we refer to [4].

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(i) CΣ f extends to C∞ (Ω), i.e. CΣ f ∈ M(Ω) ∩ C∞ (Ω). (ii) If CΣ+ f denotes the trace of CΣ f to Σ, i.e. for x ∈ Σ, if CΣ+ f (x) = lim CΣ f (˜ x), Ω3˜ x→x

then 1 (f (x) + HΣ f (x)) . 2 (iii) HΣ f ∈ C∞ (Σ) and CΣ+ (CΣ+ f ) = CΣ+ f . CΣ+ f (x) =

Now let R > 0 be such that Ω ⊂ B(R) and put Ω∗ = B(R) \ Ω. Then Ω∗ is a compact, connected manifold with C∞ -boundary Σ∗ = ∂B(R) ∪ Σ, whence for F ∈ C∞ (Σ∗ ), Calderbank’s results [4, §§6–9] remain valid. Putting for f ∈ C∞ (Σ), ( f (x), x ∈ Σ F (x) = 0, x ∈ ∂B(R), we thus have for the Cauchy and Hilbert transforms CΣ∗ F and HΣ∗ F that CΣ∗ F ∈ M(Ω∗ ) ∩ C∞ (Ω∗ ). Moreover, straightforward arguments lead to the following results. (iv) For x ∈ Ω∗ , CΣ∗ F (x) = −CΣ f (x). (v) For x ∈ Σ

∗

1 CΣ+∗ F (x) = (F (x) + HΣ∗ F (x)) = 2

( −CΣ f (x), x ∈ ∂B(R) 1 (f (x) − HΣ f (x)), x ∈ Σ. 2

It thus follows from (iv) and (v) that for x ∈ Σ, CΣ− f (x) =

lim x→x Rm \Ω3˜

CΣ f (˜ x) =

1 (−f (x) + HΣ f (x)) . 2

Putting Ω+ = Ω and Ω− = Rm \Ω, we obtain the following theorem by combining the foregoing results. Theorem 3.1. Let Ω ⊂ Rm be an open bounded and connected domain with C∞ -boundary Σ such that Rm \ Ω is connected. Furthermore, let for f ∈ C∞ (Σ), CΣ f and HΣ f be, respectively, the Cauchy and Hilbert transforms of f . Then (i) CΣ f ∈ M(Rm \ Σ) with CΣ f (∞) = 0; (ii) HΣ f ∈ C∞ (Σ);

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(iii) for x ∈ Σ, CΣ± f (x) =

lim CΣ f (˜ x)

Ω± 3˜ x→x

exist and belong to C∞ (Σ); (iv) (Plemelj-Sokhotski Formulae) for x ∈ Σ 1 CΣ± f (x) = (±f (x) + HΣ f (x)) . 2 Obviously, similar results hold for the right monogenic function f CΣ and its boundary values f CΣ± . Now we prove a remarkable result connecting two-sided monogenicity of a function f in a domain Ω and the Hilbert transform HΣ f + and f + HΣ of its trace on the boundary Σ of Ω. Let M{ l } (Σ) denote the set of functions on Σ which are the boundary values of r

functions in M{ l } (Ω) ∩ C∞ (Ω), where M{ l } (Ω) denotes the space of left, respecr r tively right, monogenic functions in Ω. The Hardy space H{2 l } (Σ) is defined to r

be the closure in L2 (Σ) of M{ l } (Σ). It is well known that (see [4]) HΣ extends r from C∞ (Σ) to a bounded operator on L2 (Σ) and that 2 f ∈ H{l} (Σ)

⇐⇒

HΣ f = f,

2 f ∈ H{r} (Σ)

⇐⇒

f HΣ = f.

respectively Theorem 3.2. Let Ω be an open bounded and connected domain in Rm with C∞ -boundary Σ and let f ∈ M(Ω) ∩ C∞ (Ω) have trace f + = f |Σ on Σ. Then the following are equivalent: (i) f is two-sided monogenic in Ω; (ii) HΣ f + = f + HΣ . Proof. Suppose that f ∈ M(Ω) ∩ C∞ (Ω). Then clearly f + ∈ H{2 l g} (Σ) whence r

+

+

+

HΣ f = f = f HΣ . Conversely, suppose that HΣ f + = f + HΣ . As f ∈ M(Ω) ∩ C∞ (Ω), we have that f = CΣ f + and that for x ∈ Σ,
1 + BV+ f (x) = lim f (˜ x) = CΣ+ f + (x) = f + HΣ f + (x). Ω3˜ x→x 2 In view of the assumption made
1 + BV+ f = f + f + HΣ 2 whence BV+ f = f + CΣ+ .

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Now put g = (BV+ f )CΣ . Then g is right monogenic in Ω and belongs to C∞ (Ω), with x) = (f + CΣ+ )CΣ+ = f + CΣ+ = BV+ f. BV+ g = lim g(˜ Ω3˜ x→x

Put h = f −g. Then h is harmonic in Ω and it belongs to C∞ (Ω) with BV+ h = 0 on Σ. The classical Dirichlet problem then implies that h ≡ 0 in Ω or f = g in Ω. Consequently f is also right monogenic in Ω. Remark. Plemelj-Sokhotski type formulae 1 (9) CΣ± f (x) = (±f (x) + HΣ f (x)) 2 may also be obtained in the following cases. (i) Σ ⊂ Rm is the graph of a Lipschitz function, Ω± are domains in Rm which lie, respectively, above and below Σ and f ∈ Lp (Σ), 1 < p < +∞ (see [11, 12]). The relations (9) then hold for a.e. x ∈ Σ. (ii) Σ is the boundary of a bounded Lipschitz domain and f belongs to the class Lp (Σ), 1 < p < +∞ (see [8]). Again the relations (9) then hold for a.e. x ∈ Σ. (iii) Ω is a bounded open domain in Rm such that its boundary Σ is a Liapunov hypersurface and f ∈ C 0,α (Σ), 0 < α < 1 (see [10]). The relations (9) hold for all x ∈ Σ. For f ∈ C 0,α (Σ), the relations (9) remain valid in the case where Σ is an Ahlfors-David regular hypersurface, although a change in the definition of HΣ has to be carried out (see [1]). (iv) Since in the case of f ∈ C 0,α (Σ) (0 < α < 1), Σ being a Liapunov hypersurface, (9) holds and (CΣ+ )2 = CΣ+ on C 0,α (Σ), Theorem 3.2 remains valid for f ∈ M(Ω) ∩ C 0,α (Ω).

4. A conservation law for two-sided monogenic functions In this section a condition is established on the boundary value of a function f which is two-sided monogenic in Ω and belongs to C∞ (Ω). This condition gives rise to a set of equivalent assertions concerning the Cauchy integral decomposition of a k-multi-vector field Fk ∈ C∞ (Σ). 4.1. The general case. Suppose that Ω is a connected open bounded domain in Rm with C∞ -boundary Σ. Furthermore, following Section 2, assume that in a small -normal open neighborhood Σ of Σ, x ∈ Σ has local coordinates x = (ν, ω1 , . . . , ωm−1 ) = (ν, ω) with respect to the orthonormal frame (n, ε1 , . . . , εm−1 ) where (i) x = n(x ⊥ )ν(x) + x ⊥ with x ⊥ ∈ Σ, n = n(x ⊥ ) the unit normal at x ⊥ and ν(x) =P hn, xi; (ii) x ⊥ = m−1 j=1 εj hεj , xi, (ε1 , . . . , εm−1 ) being an orthonormal basis of the tangent space T (x ⊥ ) at x ⊥ ;

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(iii) ω = (ω1 , . . . , ωm−1 ) is a local C∞ -coordinate system at x ⊥ on Σ. As we have seen, the Dirac operator ∂x may be expressed as ∂x = nhn, ∂x i + ∂kx where ∂kx is the tangential Dirac operator with X ∂kx = εj Qji ∂ωi . j,i

For f ∈ C∞ (Ω) we have that (10) (11)

∂x f = nhn, ∂x if + ∂kx f, f ∂x = f hn, ∂x in + f ∂kx .

Furthermore, if f + = f |Σ , then ∂kx f |Σ = ∂ω f + where X ∂ω = εj Qji (0, ω)∂ωi . j,i

Now suppose that f ∈ M(Ω) ∩ C∞ (Ω). From ∂x f = 0 and f ∂x = 0, it follows respectively, that (12) (13)

hn, ∂x if − n∂kx f = 0, hn, ∂x if − (f ∂kx )n = 0

whence in Ω, n∂kx f = (f ∂kx )n or equivalently, (14)

n(∂kx f )n + f ∂kx = 0.

We thus obtain from (14) that for f ∈ M(Ω) ∩ C∞ (Ω), on Σ (15)

n(∂ω f + )n + f + ∂ω = 0.

Relation (15) is called conservation law or (CL)-condition for f ∈ M(Ω)∩C∞ (Ω). Remark. The term “conservation law” is borrowed from electrodynamics. The free field Maxwell equations correspond to a bivector field f (x, t) in spacetime satisfying the Hodge system or, equivalently, the two-sided monogenic system. Assume that one would have such a Maxwell field f (x, t) in a domain of the form Ω × R say, Ω ⊂ R3 and R the time axis, then the boundary value f (ω, t), ω ∈ ∂Ω has the form nJ(ω, t) where n is the unit normal to ∂Ω and J(ω, t) is the current generating the field f (ω, t). The conservation laws in our paper correspond in this case to the conservation of electromagnetic charge i.e. divJ = 0. Conversely, suppose that f ∈ M(Ω) ∩ C∞ (Ω) satisfies the (CL)-condition (15). Put g = f ∂x . Then combining (11) and (12) we have for g + , the restriction of g to Σ, that (16)

g + = n(∂ω f + )n + f + ∂ω .

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But, as f + satisfies the (CL)-condition (15), (16) yields g + = −f + ∂ω + f + ∂ω = 0. Since g ∈ M(Ω) ∩ C∞ (Ω) with g + = g|Σ = 0 we have by the classical Dirichlet problem that g = 0 in Ω and hence f ∂x = 0 in Ω or f is right monogenic as well. The foregoing results lead to the following theorem. Theorem 4.1. Let f ∈ M(Ω) ∩ C∞ (Ω). Then the following are equivalent: (i) f is two-sided monogenic in Ω; (ii) f + = f |Σ satisfies the (CL)-condition. Now assume that Ω as well as Rm \ Ω are connected and that f ∈ C∞ (Rm \ Ω), i.e. f ∈ C∞ (Rm \ Ω) may be extended to C∞ (Σ). Denoting for x ∈ Σ, f − (x) = f (x) = lim f (˜ x), Σ

x→x Rm \Ω3˜

similar arguments as before give for f ∈ M(Rm \ Ω) ∩ C∞ (Rm \ Ω) the boundary condition ((CL)-condition) n∂ω f − + f − ∂ω n = 0.

(17)

Consequently, if f ∈ M(Rm \ Σ) ∩ C∞ (Ω) ∩ C∞ (Rm \ Ω) has restrictions f + and f − on Σ, its jump BV f , defined by BV f = f + − f − , also satisfies the (CL)-condition. Conversely, suppose that F ∈ C∞ (Σ) satisfies the (CL)-condition n(∂ω F )n + F ∂ω = 0 and put f (x) = CΣ F (x), x ∈ Rm \ Σ. Then we know (see Section 3) that f ∈ M(Rm \ Σ) ∩ C∞ (Ω) ∩ C∞ (Rm \ Ω) with f (∞) = 0 and that on Σ, BV f = f + − f − = F. We claim that f is also right monogenic in Rm \ Σ. Indeed, put g = f ∂x . Then clearly g is left monogenic in Rm \Σ with g(∞) = 0 and g ∈ C∞ (Ω)∩C∞ (Rm \Ω). Furthermore, as in Rm \ Σ g = f ∂x = n(∂kx f )n + f ∂kx , on Σ we have that However, as F = f + − f −

g ± = n(∂ω f ± )n + f ± ∂ω . satisfies the (CL)-condition,

g + − g − = n(∂ω F )n + F ∂ω = 0,

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whence on Σ g+ = g−. By virtue of the Painlev´e Theorem (see [1]), g is left monogenic in Rm . But, as g(∞) = 0, by means of the Liouville Theorem (see [1]), g ≡ 0 in Rm \ Σ and so f ∈ M(Rm \ Σ). This gives us the following result. Theorem 4.2. Let Ω ⊂ Rm be an open bounded and connected set with C∞ -boundary Σ such that Rm \ Ω is connected. Furthermore, let F ∈ C∞ (Σ) satisfy the (CL)-condition n(∂ω F )n + F ∂ω = 0. If f ∈ M(Rm \ Σ) ∩ C∞ (Ω) ∩ C∞ (Rm \ Ω) with f (∞) = 0 is such that its jump BV f = f + − f − = F , then f = CΣ F . Proof. We have already shown that CΣ F satisfies all properties of the given function f . The Painlev´e and Liouville Theorems then imply that f = CΣ F . 4.2. The multi-vector case. Let Ω be again a bounded open and connected (k) subset of Rm such that Rm \Ω is connected. Furthermore, let Fk be an R0,m -valued C∞ -function on Σ. If Fk satisfies the (CL)-condition n(∂ω Fk )n + Fk ∂ω = 0, then we know by virtue of Theorem 4.2 that f = CΣ Fk is two-sided monogenic in Rm \ Σ. Similar arguments as used in proving the equivalence of (ii), (iii) and (iv) in [1, Theorem 4.1] then imply that f is a harmonic k-multi-vector field in Rm \ Σ (k) with f (∞) = 0. Consequently, f ± are also R0,m -valued on Σ and by virtue of Theorem 3.1, they belong to C∞ (Σ) with Fk = f + − f − . By means of the Plemelj-Sokhotski formula 1 f + = (Fk + HΣ Fk ) , 2 (k)

whence HΣ Fk is also R0,m -valued on Σ with HΣ Fk ∈ C∞ (Σ). (k)

Conversely, suppose that for an R0,m -valued C∞ -function Fk on Σ, the function (k) HΣ Fk is also R0,m -valued. Again put f = CΣ Fk and split f into its l-vector parts fl = [CΣ Fk ]l , l = k − 2, k, k + 2, i.e. (18)

f = fk−2 + fk + fk+2 .

As f is left monogenic in Ω, ∆f = 0 in Ω and hence also ∆fl = 0, l = k−2, k, k+2. Furthermore, as by the Plemelj-Sokhotski formulae 1 f ± = (±Fk + HΣ Fk ) , 2

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(k)

we obtain that f ± are both R0,m -valued C∞ -functions on Σ. Taking restrictions in (18) to Σ, we have that ± ± f ± = fk−2 + fk± + fk+2 ± whence fk±2 = 0 on Σ. By virtue of the classical Dirichlet problem fk±2 = 0 in Ω. It thus follows that f ≡ fk in Ω. Analogously, in Rm \ Ω, ∆fk±2 = 0, − fk±2 (∞) = 0 and fk±2 = 0 on Σ, whence fk±2 = 0 in Rm \ Ω and so f = fk in Rm \ Ω. Consequently, f = CΣ Fk is two-sided monogenic in Rm \ Σ and so

Fk = f + − f − satisfies the (CL)-condition. Under the assumptions made upon Ω and taking into account that (i) [1, Theorem 4.1] remains valid if Fk ∈ C∞ (Σ); (ii) CΣ Fk = CΣ Fk+ in Ω and CΣ Fk = CΣ Fk− in Rm \ Ω with (CΣ− )2 = CΣ− on C∞ (Σ); (iii) Theorem 3.2 remains valid for functions f ∈ M(Rm \ Ω) ∩ C∞ (Rm \ Ω) satisfying f (∞) = 0, we obtain the following result. Theorem 4.3. Let Ω ⊂ Rm be open, bounded and connected with smooth boun(k) dary Σ such that Rm \ Ω is connected. Furthermore let Fk be an R0,m -valued C∞ -function on Σ. Then the following are equivalent: (i) (i) Fk admits on Σ a decomposition Fk = Fk+ + Fk− ,

(ii) (iii) (iv) (v) (vi) (vii)

where Fk± ∈ C∞ (Σ) are the restrictions to Σ of harmonic k-multi-vector fields in Ω± ; the Cauchy transform CΣ Fk is two-sided monogenic in Rm \ Σ; [CΣ Fk ]k is a harmonic k-multi-vector field in Rm \ Σ; [CΣ Fk ]k−2 and [CΣ Fk ]k+2 vanish in Rm ; Fk satisfies the (CL)-condition; (k) HΣ Fk is R0,m -valued on Σ; HΣ Fk = Fk HΣ .

Remarks. • In fact, in (iv) it suffices to suppose that [CΣ Fk ]k−2 = [CΣ (Fk ]k+2 = 0 in Rm \ Σ.

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• If Fk ∈ (C∞ (Σ), then for x ∈ Rm \ Σ (see (also [1]): Z E(y − x) • (n(y) • Fk (y)) dS(y), [CΣ Fk ]k−2 (x) = Σ Z E(y − x) ∧ (n(y) ∧ Fk (y)) dS(y). [CΣ Fk ]k+2 (x) = Σ

Consequently, requiring that [CΣ Fk ]k±2 = 0 in Rm \ Σ is equivalent to imposing the conditions Z E(y − x) • (n(y) • Fk (y)) dS(y) = 0, (19) Σ Z (20) E(y − x) ∧ (n(y) ∧ Fk (y)) dS(y) = 0. Σ (k)

Call Nk (C∞ (Σ)) the subspace of C∞ (Σ) consisting of those R0,m -valued C∞ -functions Fk on Σ for which the conditions (19) and (20) are satisfied and call Mk (Ω+ ∪ Ω− ) the space of harmonic k-multi-vector fields Fk in Ω+ ∪ Ω− with Fk (∞) = 0. Then Theorem 4.3 tells us that CΣ maps Nk (C∞ (Σ)) into Mk (Ω+ ∪ Ω− ). In the special case where k = 1, for any 1-valued C∞ -function F1 on Σ, the condition (19) is automatically satisfied since on Σ E(y − x) • (n(y) • F1 (y)) ≡ 0. Consequently N1 (C∞ (Σ)) consists of those F1 ∈ C∞ (Σ) for which (20) holds. Theorem 4.3 implies that CΣ maps N1 (C∞ (Σ)) into M1 (Ω+ ∪Ω− ), the space of harmonic vector fields in Ω+ ∪ Ω− which vanish at ∞. • In [1, Theorem 4.1] , the equivalent conditions (i)-(iv) from Theorem 4.3 were obtained in the case where Fk ∈ C 0,α (Σ), Σ being an Ahlfors-David regular hypersurface in Rm . Denote by N1 (C 0,α (Σ)) the space of 1-vector valued functions on Σ which belong to C 0,α (Σ), and which satisfy the condition (20). Then CΣ maps N1 (C 0,α (Σ)) into M1 (Ω+ ∪ Ω− ) ∩ C 0,α (Ω+ ∪ Ω− ). A similar result, although formulated in different terms, was obtained by M. Shapiro in [14] and this in the case where Σ is a Liapunov hypersurface. As is well known, any Liapunov hypersurface is an Ahlfors-David regular hypersurface. • In [7, Theorem 3] Dyn’kin gave an explicit representation formula for the harmonic k-forms ηk± in Ω± which lead to the decomposition ηk = ηk+ + ηk− of a k-form on Σ satisfying the conditions (A) and (B). A careful analysis of this formula shows that, by means of the isomorphism Θ (see Lemma 5.1), ηk = Θ[CΣ Fk ]k , where Fk satisfies the condition (i) of Theorem 4.3.

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Consequently, Dyn’kin’s integral representation formulae for harmonic kforms can be obtained from the Cauchy integral formula in Clifford analysis, by projecting the latter formula onto its k-vector part. Notice in this regard that a Cauchy integral formula specific to harmonic k-forms was also derived directly from the Clifford analysis setting in [9].

5. Dyn’kin’s condition (A) in Clifford analysis In this section it is shown that Dyn’kin’s Condition (A) on the boundary value of a harmonic k-form is equivalent to the conservation law for the boundary value of the corresponding k-multi-vector field. 5.1. Harmonic k-multivector fields and k-forms. If G is an open subset of Rm , then there is a natural vectorial isomorphism Θ between the spaces Ek (G) and Λk (G) of, Prespectively, smooth k-multi-vector fields and smooth k-forms in G: for Fk = |A|=k FA eA , put X ΘFk = ηk = ηA dxA |A|=k

where for all A = {i1 , . . . , ik } ⊂ {1, . . . , m}, ηA = FA and dxA = dxi1 ∧ · · · ∧ dxik . This isomorphism Θ may also be obtained as follows: consider the form m X dx = dxj ej . j=1

Then clearly for each k ∈ {1, . . . , m}, dxk = k!

X

eA dxA .

|A|=k

Using the classical inner product h·, ·i on k-multi-vectors, we thus obtain the following lemma. Lemma 5.1. If Fk ∈ Ek (G), then ΘFk = ηk =

1 hFk , dxk i. k!

Now suppose that Fk is left-and hence two-sided- monogenic in G, i.e. ∂x Fk = 0 in G. Then, as ∂x Fk = ∂x • Fk + ∂x ∧ Fk , we obtain that ( ∂x ∧ Fk = 0, (21) ∂x Fk = 0 ⇐⇒ ∂x • Fk = 0.

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Putting ηk = ΘFk , it is easily verified that (21) is equivalent to  dηk = 0, (22) , d∗ ηk = 0 or to  (23)

dηk = 0, d ∗ ηk = 0.

In turn (23) is equivalent to  (24)

∂x ∧ Fk = 0, , ∂x ∧ eM Fk = 0

since 0 = dηk = d(ΘFk ), 0 = d ∗ ηk = d(Θ(eM Fk )). As already mentioned in the introduction, a k-multi-vector field Fk which is monogenic in G — or equivalently a k-form ηk which satisfies the Hodge-de Rham system (22) in G — is called harmonic in G. 5.2. Dyn’kin’s condition (A) for harmonic k-forms versus the (CL)condition for harmonic k-multi-vector fields. Let Ω be a bounded open and connected subset of Rm having as boundary the C∞ -surface Σ and let for  > 0, Σ be an -normal open neighborhood of Σ (see Section 2). In terms of the coordinate system (ν, ω1 , . . . , ωm−1 ) we obtain that in Ω ∩ Σ , m−1

m−1

X ∂x X X ∂x dx = dν + Ki,j εj dωi = n dν + εj φj dωj = ndν + ∂ν ∂ω j j=1 i,j j=1 where the Ki,j ’s are C∞ -functions in (ν, ω) and the φj ’s are 1-forms depending only on dω1 , . . . , dωm−1 . For k ∈ {1, . . . , m} fixed, we thus get !k !k m−1 m−1 X X dxk = n dν + εj φj = εj φj + kn dν j=1

j=1

Here we used the commutation relation ! m−1 X (n dν) εj φj = j=1

m−1 X

!k−1 εj φj

.

j=1

m−1 X

! εj φj

(n dν) .

j=1

k

Now let Fk ∈ E (Ω) ∩ C∞ (Ω), i.e. Fk is a smooth k-multi-vector field in Ω which is extendable to C∞ (Σ). In terms of the orthonormal frame (n, ε1 , . . . , εm−1 ), the function Fk may be written in Ω ∩ Σ as (25)

Fk = F (k) + nF (k−1)

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where F (l) , l = k, k − 1, are l-multi-vector fields over (ε1 , . . . , εm−1 ). Applying Lemma 5.1 to (25) we get ηk = ΘFk = ΘF (k) + Θ(nF (k−1) ) * !k + * m−1 X kdν 1 F (k) , + nF (k−1) , n = εj φj k! k! j=1

m−1 X

!k−1 + εj φj

j=1

= η (k) + dν η (k−1) . Here η (l) , l = k, k − 1, are l-forms over (dω1 , . . . , dωm−1 ). Now suppose ηk = ΘFk is harmonic in Ω, i.e. it satisfies the Hodge-de Rham system (23). As ηk is assumed to be extendable to C∞ (Σ), by restricting to the boundary, we immediately obtain Dyn’kin’s Condition (A) (see [7]), namely  dηk = 0 (A1) . d ∗ ηk = 0 (A2) We now claim that in Clifford analysis terms, Condition (A) is equivalent to the (CL)-condition for the boundary value Fk+ = Fk |Σ of the harmonic k-multi-vector field Fk (see (15)) n(∂ω Fk+ )n + Fk+ ∂ω = 0. Let us prove this statement in the case where k is odd, the proof of the case k even being similar. In order to avoid tedious calculations, we only give a sketch of the proof. Let us first show how the (CL)-condition may be derived from Condition (A). In fact, we shall establish separately conditions upon Fk+ which are equivalent to, respectively, the Conditions (A1) and (A2). Let us start by translating Condition (A1). As we have seen (see (24)), Condition (A1) also reads: d(ΘFk ) = (Θ(∂x ∧ Fk )) = 0. Σ

Σ

As k is odd, 1 1 (∂x Fk − Fk ∂x ) = [∂x , Fk ]. 2 2 Moreover, as [∂x , Fk ] is a (k + 1)-multi-vector field, we may write it in the form (see (25)) ∂x ∧ Fk =

(26)

[∂x , Fk ] = F (k+1) + nF (k)

where F (k+1) and F (k) are constructed over (ε1 , . . . , εm−1 ). By virtue of (26)

(Θ[∂x , Fk ]) Σ = ΘF (k+1) + dνΘF (k) Σ = ΘF (k+1) Σ = Θ(F (k+1) Σ ). But, k being odd, we have that nF (k+1) = F (k+1) n, nF (k) = −F (k) n,

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whence from (26) we get (27)

F





(k+1)

Σ

=

 −n {n, [∂x , Fk ]} . 2 Σ

Condition (A1), namely d(ΘFk ) Σ = 0 is thus equivalent to Θ(F (k+1) |Σ ) = 0 or to F (k+1) |Σ = 0. In view of (27) it is hence also equivalent to {n, [∂x , Fk ]} Σ = 0. Now writing Fk in the form (see also (25)) Fk = F˜ (k) + nF˜ (k−1) , by using the relation (7), straightforward calculations lead to (A1) ⇐⇒ (CL1) where (CL1) is given by 0 = {n, [∂x , Fk ]} Σ = {n, [∂kx , Fk ]} Σ = {n, [∂ω , Fk+ ]}.

(28)

Analogously it may be shown that (A2) ⇐⇒ (CL2) where (CL2) is given by 0 = [n, {∂x , Fk }] Σ = [n, {∂kx , Fk }] Σ = [n, {∂ω , Fk+ }].

(29)

Obviously, (28) and (29) imply the (CL)-condition n(∂ω Fk+ )n + Fk+ ∂ω = 0. Conversely, suppose that Fk is a harmonic k-multi-vector field in Ω which extends to C∞ (Σ) and which is such that its trace on the boundary Fk+ = Fk |Σ satisfies the (CL)-condition n(∂ω Fk+ )n + Fk+ ∂ω = 0 or equivalently, n(∂ω Fk+ ) = (Fk+ ∂ω )n.

(30) (k)

As Fk+ is R0,m -valued, ∂ω Fk+ , which splits into a (k+1)- and a (k−1)-multivector, may be written as (31)

∂ω Fk+ = A + nB,

where A = A(k+1) + A(k−1) , B = B (k) + B (k−2) .

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Here the A(l) ’s, l = k − 1, k + 1, and B (j) ’s, j = k, k − 2, are l- and j-multi-vector fields constructed over ε1 , . . . , εm−1 . Put σ = (−1)k(k+1)/2 . On the one hand, we obtain that ∂ω Fk+ = −σFk+ ∂ω

(32) while on the other hand

∂ω Fk+ = A − Bn = A

(k+1)

+A

(k−1)

−B

(k)

n−B

(k−2)

n.

But now, for k odd, A A

(k+1)

= σA(k+1) ,

(k−1)

= −σA(k−1)

while B B

(k)

(k−2)

= σB (k) , = −σB (k−2) .

Taking also into account that A(k+1) n = nA(k+1) , it follows from (30), (31) and (32) that A(k+1) = 0 and B (k−2) = 0. Now the (k + 1)-vector A(k+1) is the tangent part of [∂ω Fk+ ]k+1 . But 1 [∂ω Fk+ ]k+1 = ∂ω ∧ Fk+ = [∂ω , Fk+ ]. 2 Hence n A(k+1) = − {n, [∂ω , Fk+ ]} 2 (k+1) and so A = 0 leads to {n, [∂ω , F + ]} = {n, [∂kx , Fk ]} = 0, k

Σ

thus giving condition (CL1) (see (29)). Similarly, from B (k−2) = 0, it follows that [n, {∂ω , Fk+ }] = [n, {∂kx , Fk }] Σ = 0 thus implying condition (CL2) (see (30)). Consequently, the (CL)-condition implies condition (A). We may thus conclude with the following theorem. Theorem 5.1. Let Ω be an open, bounded and connected subset of Rm having the C∞ - surface Σ as boundary. Furthermore, let Fk be a harmonic k-multi-vector field in Ω which is extendable to C∞ (Σ) and has the trace Fk+ = Fk |Σ on Σ. If ηk = ΘFk is the corresponding harmonic k-form, then  dηk Σ = 0 ⇐⇒ n(∂ω Fk+ )n + Fk+ ∂ω = 0 d∗ ηk Σ = 0 i.e. Dyn’kin’s Condition (A) and the (CL)-condition are equivalent.

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Remarks. • For smooth differential forms in Ω which are extendable to C∞ (Σ), Dyn’kin’s Conditions (A1) and (A2) are natural, since closedness of differential forms in Ω is inherited when restricting to the boundary Σ. For a k-multi-vector field Fk ∈ M(Ω) ∩ C∞ (Ω), the conservation law (CL) for the trace Fk+ = Fk |Σ indicates the possibility of eliminating the normal derivative at the boundary. • Let again Fk ∈ Ek (Ω) ∩ C∞ (Σ) and let ηk = ΘFk . Then Θ(∂x ∧ Fk ) = dηk = d(ΘFk ). It should however be stressed that (Θ(∂x ∧ Fk )) Σ 6= Θ((∂x ∧ Fk ) Σ ). That is the reason why, in deriving the (CL)-condition from Condition (A), we wrote out ∂x ∧ Fk in Ω ∩ Σ as (see (26)) ∂x ∧ Fk = F (k+1) + nF (k) . This observation indicates an essential difference between the calculus of differential forms and the calculus of multi-vector fields when taking traces on the boundary of a domain. Acknowledgement. This paper was written while the first two authors were guests at the Department of Mathematical Analysis of Ghent University. They wish to thank all members of this Department for their kind hospitality. The authors would like to thank the referees for their valuable suggestions.

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9. J. Gilbert, J. Hogan and J. Lakey, Frame decompositions of form-valued Hardy spaces, in: J. Ryan (ed.), Clifford algebras in analysis and related topics, CRC Press, Boca Raton, 1996, 239–259. 10. V. Iftimi´e, Fonctions hypercomplexes, Bull. Math. Soc. Sci. Math. R´epub. Soc. Roum. Nouv. S´er. 9 (1965), 279–332. 11. A. McIntosh, Clifford algebras, Fourier theory, singular integrals, and harmonic functions on Lipschitz domains, in: J. Ryan (ed.), Clifford algebras in analysis and related topics, CRC Press, Boca Raton, 1996, 33–87. 12. M. Mitrea, Clifford Wavelets, Singular Integrals, and Hardy Spaces, Lecture Notes in Mathematics 1575, Springer-Verlag, Berlin, 1994. 13. M. Riesz, Clifford Numbers and Spinors (Chapters I–IV) Lecture Series, No. 38, Institute for Physical Science and Technology, Maryland, 1958. 14. M. Shapiro, On the conjugate harmonic functions of M. Riesz–E. Stein–G. Weiss, in: S. Dimiev and K. Sekigawa (Eds), Topics in omplex analysis, differential geometry and mathematical physics, World Sci. Publishing, 1997, 8-32. 15. E. Stein and G. Weiss, On the theory of harmonic functions of several variables, I: The theory of H p -spaces, Acta Math. 103 (1960), 25–62. Ricardo Abreu Blaya E-mail: [email protected] Address: Faculty of Mathematics and Informatics, University of Holgu´ın, Holgu´ın 80100, Cuba. Juan Bory Reyes E-mail: [email protected] Address: Department of Mathematics, University of Oriente, Santiago de Cuba 90500, Cuba. Richard Delanghe E-mail: [email protected] Address: Department of Mathematical Analysis, University of Ghent, Galglaan 2, B-9000 Gent, Belgium. Frank Sommen E-mail: [email protected] Address: Department of Mathematical Analysis, University of Ghent, Galglaan 2, B-9000 Gent, Belgium.