Journal of Lie Theory Volume 8 (1998) 67–82 ◯^{C}1998 Heldermann Verlag

Dirac Operators, Conformal Transformations and Aspects of Classical Harmonic Analysis

John Ryan

Communicated by M. Cowling

Abstract. The main thrust of this paper is to investigate the intimate link between the conformal group and singular integral operators, in particular, but not exclusively, operators of Calderón–Zygmund type, together with associated commutators acting on the L^{2} spaces of surfaces. Clifford analysis and Dirac operators are the basic tools used to help to unify these themes. These surfaces lie in euclidean space, the sphere or the hyperbola. We illustrate how these results extend to a general class of submanifolds with arbitrary codimension in euclidean space, the sphere or the hyperbola.

1. Introduction

Clifford analysis has been extensively applied to clarify and solve various problems arising in classical harmonic analysis, see for instance [AuT, GaLQ, LMcQ, LMcS, Mc, Mi, S]. In these papers Dirac operators and Cauchy kernels are used to study properties of singular integral operators acting on the L^{p} spaces associated to various types of surfaces in R^{n}. Usually these surfaces are Lipschitz surfaces, or some minimal smoothness criterion is assumed of the surface. In most cases the Clifford algebra is used to show that results known for singular integral operators acting on L^{p} spaces of minimally smooth curves in the complex plane can be adapted to higher dimensions, see for instance [10, 9]. In this setting Clifford analysis has proved to be a powerful tool in extending complex variable techniques to solving problems in higher dimensions.

Links between Möbius transformations and Dirac operators have been developed in a number of papers, see for instance [5, 8, 19, 21, 22]. All of these papers make use of the Vahlen matrices arising from work in [1, 28] and elsewhere.

In particular, in [8, 21] it is shown that if two surfaces in R^{n} are conformally equivalent then their L^{2} spaces are isometric, and there is a simple association between the operators acting on these L^{2} spaces. In [21] it is pointed out that the associated Hardy spaces H^{2} of solutions to the Dirac equation for the domains complementing the surfaces are also isometric. This follows from the conformal invariance of the Clifford analysis analogues of the Plemelj operators.

ISSN 0949–5932 / $2.50 ◯^{C} Heldermann Verlag

The main thrust here is to show that by introducing Clifford algebras and, in particular, Vahlen matrices one can easily show that many important results known in classical harmonic analysis on euclidean space possess a conformal covariance. In other words once one extends from euclidean space and includes the Clifford algebra generated from that space many traditional results associated with classical harmonic analysis described in [25] and elsewhere have a rather natural association with the conformal group, and seem to be intimately related to that structure. For instance, in [21] it is shown how certain key results on convolution operators acting on the L^{2} spaces of Lipschitz graphs, described in [15, 16], can be carried over to some other surfaces via conformal transformations. In this paper we develop this theme further. We demonstrate that convolution operators of Calderón–Zygmund type acting on the L^{2} space of R^{n−1} transform under Möbius transformations acting on R^{n−1} to other operators of Calderón–Zygmund type. We are also able to show that the commutators of such operators with multiplier operators are preserved in the sense that the resulting operators are also the commutators of operators of Calderón–Zygmund type with multiplier operators. This analysis leads us to show that a considerable part of the key results in [15, 16] naturally lend themselves to invariance under the action of the Lie group V (n − 1), the group of Vahlen matrices over R^{n−1}. In particular, when viewed as operators the left and right monogenic functions defined on sector domains in [15, 16] possess an extremely simple and elegant conformal invariance which we describe here on the sphere and hyperbola. This makes use of various results and constructions made in [22] for Dirac operators on spheres and hyperbolae.

We also consider the intimate link between the conformal group and the Riesz potentials described in [25] and elsewhere.

We conclude the paper by illustrating that many of the results appearing in the earlier part of the paper also hold for a wide class of submanifolds of euclidean space, the sphere or hyperbola. With this objective in mind we establish Plemelj formulae for such manifolds. Plemelj formulae for general manifolds are described in [7]. Here we explicitly use vector space structures to set up the formulae.

Most of the proofs in this paper follow by similar arguments to those used in [19, 21, 22]. For this reason we primarily indicate how the results can be deduced rather than repeat details from those references.

2. Background on Clifford Algebras

In this section we set up most of the background material on Clifford algebras that we will need for the rest of the paper. We shall use the real Clifford algebra, Cl_{n} and its complexification Cl_{n}(C). We will assume that R^{n} is embedded in Cl_{n}, and that Cl_{n} is the 2^{n}-dimensional algebra with basis

1, e_{1}, …, e_{n}, …, e_{j_{1}} …, e_{j_{r}}, …, e_{1} …, e_{n},

where e_{1}, …, e_{n} is an orthonormal basis for R^{n}, 1 ≤ r ≤ n, and the basis vectors e_{1}, …, e_{n} satisfy the anti-commutation relationship e_{i}e_{j} + e_{j}e_{i} = −2δ_{i,j}, where δ_{i,j} is the Kronecker delta function. From the anti-commutation relationship it follows that each non-zero vector x ∈ R^{n} has a multiplicative inverse x^{−1} = \frac{−x}{∥x∥^{2}} .

Up to the minus sign this inverse corresponds to the Kelvin inverse of a vector. It follows that the unit sphere, S^{n−1}, of R^{n}, is the generator of a group in Cl_{n}. This group is called the pin group and it is denoted by Pin(n). We will need to use the anti-automorphism

: Cl_{n} → Cl_{n} : (e_{j_{1}} …, e_{j_{r}}) = e_{j_{r}} …, e_{j_{1}}.

For each a ∈ Cl_{n} we denote a by ã. When a ∈ Pin(n) the action axã corresponds to an orthogonal transformation on R^{n}, where x is a variable in R^{n}. This can be seen by considering the action yxy where y ∈ R^{n}, and decomposing x into vectors parallel to y and perpendicular to y. By using the anti-commutation relationship it may be observed that this action gives a reflection of x in the direction of y. It may be observed that the set {a ∈ Pin(n) : a = y_{1} …, y_{m} with m even} is a subgroup of Pin(n). This is the spin group, which is denoted by Spin(n). In [20] and elsewhere it is observed that Pin(n) is a double covering of the orthogonal group O(n) while Spin(n) is a double covering of the special orthogonal group SO(n).

The conformal group of the one point compactification, R^{n} ∪ {∞}, of R^{n} is the group of diffeomorphisms generated by orthogonal transformations, translations, dilations and Kelvin inversion. Under the first three types of transformations that we just mentioned, the point at infinity remains fixed, while under Kelvin inversion the origin and the point at infinity are interchanged.

Following [1, 28] we note that each conformal transformation can be expressed as (ax + b)(cx + d)^{−1}, where a, b, c and d ∈ Cl_{n} and satisfy:

(i) a, b, c and d are all products of vectors from R^{n}. (ii) ac̃, cd̃, db̃, and bã ∈ R^{n}. (iii) ad̃− bc̃ = ±1.

A matrix \binom{a b}{c d}, with a, b, c and d satisfying (i)–(iii) is called a Vahlen matrix. The set of all such matrices form a group under matrix multiplication. This group is a covering group of the conformal group and it is denoted by V (n). If in condition (iii) we insist that ad̃− bc̃ = 1 then we obtain a subgroup of V (n) which we denote by V_{+}(n).

For each natural number m we can consider the Clifford algebra Cl_{m}, and the Vahlen groups V (m) and V_{+}(m). When m = n−1 the Möbius transformation (ax + b)(cx + d)^{−1} preserves upper half space, R^{n,+} = {x = x_{1}e_{1} + … + x_{n}e_{n} : x_{n} > 0} and lower half space, R^{n,−} = {x = x_{1}e_{1} + … x_{n}e_{n} : x_{n} < 0}, whenever \binom{a b}{c d} ∈ V_{+}(n − 1). V_{+}(n − 1) also preserves the boundary R^{n−1} ∪ \{∞\} of R^{n,±}.

When m = n + 1 the Cayley transformation

K_{1}(x) = (x − e_{n+1})(−e_{n+1}x + 1)^{−1}

transforms R^{n} onto the punctured sphere, S^{n}\{e_{n+1}}.

Besides the anti-automorphism we shall also need the anti-automorphism

: Cl_{n} → Cl_{n} : (e_{j_{1}} … e_{j_{r}}) = (−1)^{r}e_{j_{r}} … e_{j_{1}}.

For a ∈ Cl_{n} we denote (a) by \overline{a}. This anti-automorphism is the analogue of conjugation on the complex and quaternionic algebras. In particular, if a = a_{0} + … + a_{1...n}e_{1} … e_{n}, then the real part of a\overline{a} gives (a_{0}^{2} + … + a_{1...n}^{2}), the square of the norm ∥a∥ of a.

Besides the Clifford algebra Cl_{m} we shall also need the Clifford algebra Cl_{n,1} generated from the Minkowski, or Krein, space R^{n,1} = {x_{1}e_{1}+…+x_{n}e_{n}+y_{n+1}f_{n+1} : x_{1}, …, x_{n}, y_{n+1} ∈ R}. On identifying f_{n+1} with ie_{n+1} one can see that Cl_{n,1} is a real 2^{n+1}-dimensional subalgebra of Cl_{n+1}(C), that the vector f_{n+1} anticommutes with each e_{j} for 1 ≤ j ≤ n, and f_{n+1}^{2} = 1.

It may be observed that there is a Cayley transformation

K_{2} : R^{n}\S^{n−1} → H_{n}\{−f_{n+1}} : K_{2}(x) = (−x + f_{n+1})(f_{n+1}x + 1)^{−1},

where H_{n} comprises the hyperbola {y ∈ R^{n,1} : y^{2} = 1}. H_{n} has two components H_{n}^{+} and H_{n}^{−}. The hyperbola H_{n}^{+} contains the vector f_{n+1} while the hyperbola H_{n}^{−} contains the vector −f_{n+1}. Also, K_{2}^{−1}(H_{n}^{+}) = D(0, 1), the open unit disc in R^{n}, while K_{2}^{−1}(H_{n}^{−}\{−f_{n+1}}) = {x ∈ R^{n} : ∥x∥ > 1}.

Further results on Clifford algebras can be found in [20] and elsewhere.

3. Some Clifford Analysis

We begin by developing some background on Clifford analysis. The Dirac operator that we shall use here is the differential operator D = ∑_{j=1}^{n} e_{j} \frac{∂}{∂x_{j}} . One simple but important feature of this operator is that D^{2} = −△_{n}, where △_{n} is the Laplacian on R^{n}. For U a domain in R^{n} and f a Cl_{n} valued C^{1} function defined on U we say that f is left monogenic if Df = 0. Similarly if g : U → Cl_{n} is a C^{1} function satisfying gD = 0 then g is said to be right monogenic. Here gD = ∑_{j=1}^{n} \frac{∂g}{∂x_{j}} e_{j}. It should be noted that f is left monogenic if and only if \overline{f} is right monogenic.

Many results on monogenic function theory can be found in [6, 14] and in many other texts. In particular, there are the following analogues of Cauchy’s theorem and Cauchy’s integral formula.

Theorem 3.1. Suppose that f and g are respectively left and right monogenic functions on U and that S is a compact Lipschitz surface bounding a subregion V of U, and such that cl(V), the closure of V , is contained in U. Then ∫_{S} g(x)n(x)f(x)dσ(x) = 0, where n(x) is the outward normal vector to S at x, and σ is the Borel measure on S.

Theorem 3.2. Suppose that f, V and S are as in the previous theorem, and that y ∈ V . Then f(y) = \frac{1}{ω_{n}} ∫_{S} G(x − y)n(x)f(x)dσ(x), where ω_{n} is the surface area of the unit sphere in R^{n} and G(x) = \frac{−x}{∥x∥^{n}}.

It should be noted that as the surface S is a Lipschitz surface then, as a consequence of a theorem of Rademacher, the outward normal vector function on S is defined almost everywhere on S and is an L^{∞} function. It should also be noted that the function G is both left and right monogenic.

Using the local arguments presented in [13] one also has the following analogues of the Plemelj formulae, see also [15, 16].

Theorem 3.3. Suppose that S is a Lipschitz surface in R^{n} and η ∈ L^{2}(S). Then:

(i) If S^{+} is the component of R^{n}\S into which the normal vector function of S points, y(t) ∈ S^{+} for t ∈ [0, 1), and y(t) approaches y ∈ S non-tangentially as t tends to 1, then

lim_{t→1} ∫_{S} G(x − y(t))n(x)η(x)dσ(x) = \frac{1}{2}η(y) + pv \frac{1}{ω_{n}} ∫_{S} G(x − y)n(x)η(x)dσ(x).

(ii) If S^{−} is the other component of R^{n}\S, w(t) ∈ S^{−} for t ∈ [0, 1), and w(t) approaches w ∈ S non-tangentially as t approaches 1, then

lim_{t→1} ∫_{S} G(x − w(t))n(x)η(x)dσ(x) =

\frac{−1}{2}η(w) + pv \frac{1}{ω_{n}} ∫_{S} G(x − w)n(x)η(x)dσ(x).

In Theorem 3.3 we use the L^{2} space of S, which is defined to be the set of all Cl_{n} valued functions, f, defined on S such that ∥ ∫_{S} \overline{f}(x)f(x)dσ(x)∥ < +∞.

The bounded operators

\frac{1}{2}I ± C_{S} : L^{2}(S) → L^{2}(S) : (\frac{1}{2}I ± C_{S})η = \frac{1}{2}η ± pv \frac{1}{ω_{n}} ∫_{S} G(x−)n(x)η(x)dσ(x)

are projection operators onto the Hardy spaces H^{2}(S^{±}) of left monogenic functions defined on S^{±} respectively and which extend continuously in the L^{2} sense to S. See for instance [15, 16] and elsewhere.

By using the generators of the group V (n) and arguments presented in [22, 26] it may be observed that if y = ψ(x) = (ax + b)(cx + d)^{−1} is a Möbius transformation, and f(y) is a left monogenic function, then J(ψ, x)f(ψ(x)) is left monogenic in the variable x, where J(ψ, x) = ˜ (cx + d)∥cx + d∥^{−n}. Moreover, in [8, 21] it is observed that if S_{1} and S_{2} are surfaces and ψ(S_{1}) = S_{2} in R^{n} ∪ {∞}, then for each pair of functions η, ν ∈ L^{2}(S_{2}) we have that

∫_{S_{2}} \overline{η}(y)ν(y)dσ(y) = ∫_{S_{1}} \overline{η}(ψ(x))ν(ψ(x))\frac{1}{∥cx + d∥^{2n−2}} dσ(x).

As \frac{1}{∥cx+d∥^{2n−2}} = \overline{J}(ψ, x)J(ψ, x), it follows that the right Cl_{n} linear transform

Is_{2,1} : L^{2}(S_{2}) → L^{2}(S_{1}) : Is_{2,1}η(y) = J(ψ, x)η(ψ(x))

is an isometry. It is also a consequence of the previous remarks that Is_{2,1}H^{2}(S_{2}^{+}) = H^{2}(S_{1}^{±}) while Is_{2,1}H^{2}(S_{2}^{−}) = H^{2}(S_{1}^{∓}).

As G(ψ(x)−ψ(w)) = J(ψ, w)^{−1}G(x−w)J̃(ψ, x)^{−1} and the volume element n(y)dσ(y) on S_{2} pulls back to J̃(ψ, x)n(x)J(ψ, x)dσ(x) on S_{1}, then

J(ψ, x)(\frac{1}{2}I ± C_{S_{2}})η(ψ(y))) = (\frac{1}{2}I ± C_{S_{1}})J(ψ, x)η(ψ(x)),

where, as usual, y = ψ(x). Consequently the Plemelj projection operators are preserved via conformal transformations, when acting on L^{2} spaces of surfaces.

We shall denote the operators \frac{1}{2}I ± C_{S} by C_{S}^{±} respectively. Besides the operators C_{S}^{±} we shall need the operators C_{S}^{±⋆}, where

C_{S}^{±⋆}(ν)(w) = \frac{1}{2}ν(w) ± n(w)\frac{1}{ω_{n}} ∫_{S} G(x − w)ν(x)dσ(x),

where ν ∈ L^{2}(S).

From the previous remarks it may be noted that the orthogonal projection operators

P_{S}^{+} : L^{2}(S) → H^{2}(S^{+}),

and

P_{S}^{−} : L^{2}(S) → H^{2}(S^{−})

are conformally covariant.

Lemma 3.4. Suppose that S_{1} and S_{2} are surfaces and that ψ(S_{1}) = S_{2} for some Möbius transformation y = ψ(x) = (ax + b)(cx + d)^{−1}. Then

J(ψ, x)P_{S_{2}}^{±}ν(y) = P_{S_{1}}^{±}J(ψ, x)ν(ψ(x))

for each ν ∈ L^{2}(S_{2}).

The orthogonal projection operators P_{S}^{±} are usually referred to as Szegö projections. Lemma 3.4 demonstrates the conformal covariance of the Szegö projection operators.

Strictly speaking the operators C_{S}^{±⋆} do not have the same conformal covariance as the operators C_{S}^{±} and P_{S}^{±}. However, for each pair ν, μ ∈ L^{2}(S_{2}) it may be observed that

⟨μ(y), C_{S_{2}}^{±⋆}(ν)(y)⟩_{S_{2}} = ⟨J(ψ, x)μ(ψ(x)), C_{S_{1}}^{±⋆}(J(ψ, x)ν)(ψ(x))⟩_{S_{1}},

where ⟨ν, μ⟩_{S} = ∫_{S} \overline{ν}(x)μ(x)dσ(x) for any pair ν, μ ∈ L^{2}(S). Consequently, the bilinear form ⟨ν, C_{S}^{±⋆}μ⟩_{S} is covariant under Möbius transformations. It follows that the Kerzman–Stein kernels C_{S}^{±} − C_{S}^{±⋆} have a bilinear form covariance given by

⟨ν(y), (C_{S_{2}}^{±} − C_{S_{2}}^{±⋆})(μ)(y)⟩_{S_{2}} =
⟨J(ψ, x)ν(ψ(x)), (C_{S_{1}}^{±} − C_{S_{1}}^{±⋆})(J(ψ, x)μ)(ψ(x))⟩_{S_{1}}.
In [3, 23] it is noted that P_{S}^{±} = C_{S}^{±}P_{S}^{±}, P_{S}^{±} = P_{S}^{±⋆} = P_{S}^{±⋆}C_{S}^{±⋆} = P_{S}^{±}C_{S}^{±⋆},

and C_{S}^{±} = P_{S}^{±}C_{S}^{±}, where P_{S}^{±⋆} is the adjoint of P_{S}^{±}. Following [3, 23] it may be observed that P_{S}^{±} − C_{S}^{±} = P_{S}^{±}(C_{S}^{±⋆} − C_{S}^{±}). Consequently

P_{S}^{±}(I + (C_{S}^{±⋆} − C_{S}^{±})) = C_{S}^{±}.  (1)

The conformal covariance of this formula is an immediate consequence of our preceding remarks.

So far we have tacitly assumed that we have been working solely in euclidean space. However, using the Cayley transformations K_{1} and K_{2} and the results in [22] one can transfer all results so far mentioned in this section to surfaces in the sphere S^{n} and on the hyperbola H_{n}. In these cases one uses the Cayley transformations as chart maps to construct vector bundles over S^{n} and H_{n} where each fiber is isomorphic as a vector space to Cl_{n}. We denote the bundle over S^{n} by (S^{n}, Cl_{n}) and the bundle over H_{n} by (H_{n}, Cl_{n}). If U is a domain in either S^{n} or H_{n} then we denote the restriction of the appropriate bundle to U by (U, Cl_{n}). Similarly if S is a surface in S^{n} or H_{n} then the restriction of the appropriate bundle to S is denoted by (S, Cl_{n}). We shall denote the bundle of square integrable sections on (S, Cl_{n}) simply by L^{2}(S), and the corresponding bundle of sections of solutions to the Dirac equation over the complementary domains S^{±} which extend continuously in the L^{2} sense to the boundary, S, by H^{2}(S^{±}) respectively. The Dirac operator D_{S^{n}} over S^{n} is set up in [22], see also [27]. The Dirac operator D_{H_{n}} over H_{n} is set up in [22]. It should be noted that H_{n} has two components. So one does need to clarify what is meant by the domains S^{±} in the context of H_{n}.

Definition 3.5. An open subset U of H_{n} is called a domain if there is a domain V ⊂ R^{n} such that K_{2}(V \(V ∩ S^{n−1})) = U. It follows that U can have two components.

From [24, 26] it may be observed that the H^{2} decomposition of L^{2}(S^{n−1}) is an orthogonal decomposition. It follows that if S is a surface that is conformally equivalent to S^{n−1} ⊂ R^{n} then the H^{2} decomposition of L^{2} is also an orthogonal decomposition. This happens when S = R^{n−1}, or when S is a sphere in S^{n} of codimension 1. It also happens when S = S_{α} = H_{n} ∩ (R^{n} + αf_{n+1}), where either α ∈ (1, ∞) or α ∈ (−∞, −1). When S = aS_{α}ã the decomposition is also orthogonal, where a = a_{1} … a_{m} with a_{j} ∈ H_{n} and 1 ≤ j ≤ m for some arbitrary positive integer m. The orthogonal decomposition also happens when S = H_{n−1}, where H_{n−1} is the restriction of H_{n} to R^{n−1,1}. In this last case S has two components. The decomposition is also orthogonal when S = aH_{n−1}ã where a as before is equal to a_{1} … a_{m} with a_{j} ∈ H_{n} for 1 ≤ j ≤ m. Again in these circumstances the surface has two components.

Via Möbius transformations we may now obtain orthogonal bases or orthonormal bases for L^{2}(S) and H^{2}(S^{±}) for each one of these surfaces. They are simply obtained from the orthogonal and orthonormal bases for L^{2}(S^{n−1}) and H^{2}(S^{n−1,±}) respectively. Details for the special case where S = R^{n−1} are worked out in [21], the other cases follow similarly.

4. Operators of Calderón–Zygmund Type

Consider a Lipschitz continuous function λ : R^{n−1} → R where R^{n−1} is spanned by e_{1}, …, e_{n−1}. Now consider the graph Σ′ = {x + λ(x)e_{n} : x ∈ R^{n−1}} of λ. Besides the Lipschitz graph Σ′ we also shall consider the Lipschitz graph Σ = aΣ′ã for any a ∈ Spin(n). Σ is just an orthogonally transformed copy of Σ′. If the Lipschitz continuous function λ has Lipschitz constant c ∈ (0, ∞), so that ∥λ(u)−λ(v)∥ ≤ c∥u−v∥, then there is a θ ∈ (0, \frac{π}{2}) such that c = tan θ. Following [15, 16, 17] we may introduce the open cone N′(Σ′) = {x + x_{n}e_{n} : x ∈ R^{n−1} and x_{n} > tan ∥x∥}. We shall denote the open cone aN′(Σ′)ã by N(Σ). One important property of N(Σ) is that for each u ∈ Σ the cone u+N(Σ) has empty intersection with Σ\{u}.

Definition 4.1. An L^{2} bounded operator T_{K,k} : L^{2}(Σ) → L^{2}(Σ) is said to be of Calderón–Zygmund type if

T_{K,k}(ϕ)(y) = lim_{ϵ→0}(∫_{Σ(x,ϵ)} K(x, y)n(x)ϕ(x)dσ(x) + k(ϵn(y)ϕ(y))

for almost all y ∈ Σ, where K(x, y) is a smooth Cl_{n}(C) valued function defined on (Σ × Σ)\diag(Σ × Σ) and satisfying

∥K(x, y)∥ ≤ C∥x − y∥^{−n+1},

for some C ∈ R^{+}. Moreover, k is a Cl_{n}(C) valued L^{∞} function defined on N(Σ), while Σ(x, ϵ) = {y ∈ Σ : ∥y − x∥ ≥ ϵ}.

Definition 4.1 gives a minor modification of various types of singular integral operators considered in [16] and elsewhere. In [16] the function K(x, y) is considered to be equal to be a function K′(x − y). The formulation given here is more amenable under conformal transformation than that given in [16] and elsewhere.

Theorem 4.2. Suppose that T_{K,k} : L^{2}(Σ) → L^{2}(Σ) is as in definition 4.1, and that Σ = ψ(Ψ), where y = ψ(u) = (au + b)(cu + d)^{−1}. Then the operator

S_{H,h} : L^{2}(Ψ) → L^{2}(Ψ)

is L^{2} bounded, where

H(v, u) = J(ψ, v)K(ψ(v), ψ(u))J̃(ψ, u),

h(v) = \frac{(cv + d)k(ψ(v)) ˜ (cv + d)}{∥cv + d∥^{2}}

and

S_{H,h}(η)(v) = lim_{δ→0} ∫_{Ψ(v,δ)} H(v, u)n(u)η(u)dσ(u) + h(δn(v))η(v))

for almost all v ∈ Ψ, with Ψ(v, δ) = {w ∈ Ψ : ∥w − v∥ ≥ δ}.

It should be noted that ∥H(v, u)∥ ≤ C′∥v − u∥^{n−1} for some C′ ∈ R^{+} and that h is an L^{∞} function on ψ^{−1}(N(Σ)) = N(Ψ).

We shall also call the operator S_{H,h} appearing in Theorem 4.2 an operator of Calderón–Zygmund type. One important reason for considering the function K(x, y) to not necessarily be equal to K′′(x − y) for some function K′′ is that the difference x − y is not in general preserved under Möbius transformations. An interesting case to consider is when the Lipschitz surface Σ is R^{n−1}. In this case L^{2}(R^{n−1}) is transformed isometrically to itself, and so:

Proposition 4.3. The family of operators of Calderón–Zygmund type described in Definition 4.1 and acting on L^{2}(R^{n−1}) is transformed to itself under the action of V_{+}(n − 1). This gives another reason for not considering K(x, y) to be equal to K′′(x − y).

Let us now turn to look at commutators of multiplier operators with operators of Calderón–Zygmund type. We begin with:

Theorem 4.4. Suppose that g : Σ → Cl_{n}(C) is such that the commutator

[T_{K,k}, M_{g}] : L^{2}(Σ) → L^{2}(Σ)

is L^{2} bounded for each operator T_{K,k} of Calderón–Zygmund type, where M_{g} is the multiplier operator M_{g}(μ) = g(x)μ(x). Then for each Möbius transformation y = ψ(u) = (au + b)(cu + d)^{−1} the operator [T_{K,k}, g] is conformally pulled back to the L^{2} bounded operator

[S_{H,h}, M_{p}] : L^{2}(Ψ) → L^{2}(Ψ),

where S_{H,h} is as in Theorem 4.2 and

p(v) = \frac{˜ (cv + d)g(ψ(v))(cv + d)}{∥cv + d∥^{2}}

Theorem 4.4 shows us that commutators of operators of Calderón–Zygmund type with multiplier operators are preserved under conformal transformations. For the special case where the surface Σ is R^{n−1} we have:

Proposition 4.5. The class of commutators

[T_{K,k}, M_{g}] : L^{2}(R^{n−1}) → L^{2}(R^{n−1})  (2)

is preserved under the action of V_{+}(n − 1).

In [11] it is noted that the operator given in (2) is bounded if and only if g ∈ BMO(R^{n−1}). It follows that in this case p ∈ BMO(R^{n−1}).

It should be pointed out that Propositions 4.3 and 4.5 hold equally well if R^{n−1} is replaced by the unit sphere S^{n−1} in R^{n}, or the unit sphere K_{1}(R^{n−1}) ∪ {e_{n+1}} in R^{n+1} or the hyperbola H_{n−1} = K_{2}(R^{n−1}\S^{n−2}), where S^{n−2} is the unit sphere in R^{n−1}. In these cases the group V_{+}(n − 1) is replaced by the groups KV_{+}(n − 1)K^{−1}, K_{1}V_{+}(n − 1)K_{1}^{−1}, and K_{2}V_{+}(n − 1)K_{2}^{−1} respectively.

By similar observations to those used to derive Theorem 4.4 and the use of induction we may also arrive at:

Proposition 4.6. Suppose that T_{K,k}, S_{H,h}, g and p are as in Theorem 4.4 and the n-fold commutator

[… [T_{K,k}, M_{g}], …, M_{g}] : L^{2}(Σ) → L^{2}(Σ)

is bounded. Then under the Möbius transformation y = ψ(u) = (au + b)(cu + d)^{−1} this operator pulls back to the bounded operator

[… [S_{H,h}, M_{p}], …, M_{p}] : L^{2}(Ψ) → L^{2}(Ψ).

Within Propositions 4.5 and 4.6 we have a description of the conformal covariance of the multiplier operator M_{g}. This conformal covariance can easily be applied to describe the conformal covariance of the following Hankel operators on conformally equivalent surfaces S_{1} and S_{2}.

Lemma 4.7. Suppose that y = ψ(v) = (av + b)(cv + d)^{−1} and ψ(S_{1}) = S_{2}. Suppose also that g ∈ L^{∞}(S_{2}). Then the Hankel operator

(I − P_{S_{2}}^{±})M_{g} : L^{2}(S_{2}) → L^{2}(S_{2})

pulls back to the operator

(I − P_{S_{1}}^{±})M_{p} : L^{2}(S_{1}) → L^{2}(S_{1}),

where p(v) = \frac{_{(cv+d)}g(ψ(v))(cv+d)}{∥cv+d∥^{2}} ∈ L^{∞}(S_{1}).

In all the preceding work the surface Ψ is assumed to lie in R^{n}, S^{n} or H_{n}. In order to adequately cope with the situations where Ψ is in H_{n} we shall assume that Σ ∩ S^{n−1}, seen as a subset of S^{n−1}, is a set of measure zero.

One important reason why in [15, 16, 17] and elsewhere the function K(x, y) is seen to be equal to K′′(x − y) for some function K′′(x) is because it is assumed that K′′ is a right monogenic function on the sector domain S_{θ} = {x + x_{n}e_{n} : x ∈ R^{n−1} and |x_{n}| < ∥x∥ tan θ}. Moreover, ∥K′′(x)∥ ≤ C∥x∥^{−n+1} on S_{θ}.

In [21] it is observed that sector domains are transformed by Möbius transformations on R^{n−1} to sector domains. For each x ∈ R^{n} and each Möbius transformation ψ we shall denote the sector domain ψ^{−1}(S_{θ} + x) by Q_{θ}(v) where ψ(v) = x. Each Q_{θ}(v) is a domain in R^{n}, S^{n} or H_{n}. So for each v ∈ Ψ the function H(u, v) = J(ψ, v)K′′(ψ(u) − ψ(v))J̃(ψ, u) is right monogenic on Q_{θ}(v) and ∥H(u, v)∥ ≤ C′∥u − v∥^{−n+1} for some C′ ∈ R^{+}. In [16] it is shown that K′′ = K′′_{+} + K′′_{−} where K′′_{±} is a right monogenic function on S_{θ}^{±} = S_{θ} ∪ R^{n,±} and ∥K′′_{±}(x)∥ ≤ C_{±}∥x∥^{−n+1}. Consequently, H(u, v) = H_{+}(u, v) + H_{−}(u, v) where H_{±} is a right monogenic function on Q_{θ}^{±}(v) = ψ^{−1}(S_{θ}^{±} + x), for each v = ψ^{−1}(x) ∈ Ψ. Moreover, H_{±}(u, v) = J(ψ, v)K′′_{±}(ψ(u) − ψ(v))J̃(ψ, u), and ∥H_{±}(u, v)∥ ≤ C′_{±}∥u − v∥^{−n+1}.

In [16, 15] a specific construction is given for the L^{∞} function k defined on N_{θ}(Σ), and associated to the right monogenic function K′′. Via this construction k is defined uniquely up to a constant. In [15, 16] it is shown that k(x) = k_{+}(x) + k_{−}(x) where k_{±} ∈ L^{∞}(N_{θ}(Σ)). Moreover explicit constructions of these functions are given in [15, 16]. Under the Möbius transformation ψ these functions are pulled back to the functions h_{±}(v) = \frac{_{(cv+d)}k_{±}(ψ(v))(cv+d)}{∥cv+d∥^{2}} . Furthermore it is easily seen that h(v) = h_{+}(v) + h_{−}(v). In [15, 16] conditions are given for the functions K′′_{±} to be left monogenic on Σ_{θ}^{±} as well as right monogenic, under the assumption that K′′ is also left monogenic on Σ_{θ}. Under these assumptions it is shown that T_{K′′,k} = T_{K′′_{+},k_{+}} + T_{K′′_{−},k_{−}} and that T_{K′′_{±},k_{±}} : H^{2}(Σ^{±}) → H^{2}(Σ^{±}), is a bounded operator and T_{K′′_{±},k_{±}}(H^{2}(Σ^{∓})) = {0}. Consequently we have

Theorem 4.8. Suppose that T_{K′′,k}, T_{K′′_{±},k_{±}} are as in the previous paragraph. Then the operators

T_{K′′_{±},k_{±}} : H^{2}(Σ^{±}) → H^{2}(Σ^{±})

pull back via the Möbius transformation ψ to the bounded operators

S_{H_{±},h_{±}} : H^{2}(Ψ^{±}) → H^{2}(Ψ^{±}).

Moreover, S_{H_{±},h_{±}}(H^{2}(Ψ^{∓})) = {0}.

When Ψ ⊂ R^{n}, Theorem 4.8 has previously been described in [21].

5. Other Conformally Covariant Operators

For U a domain in R^{n}, S^{n} or H_{n}, we will denote by S(U) the space of Cl_{n}(C) valued C^{∞} functions on U with compact support. In the special cases where U ⊂ S^{n} or U ⊂ H_{n} then each ϕ ∈ S(U) is a section on (U, Cl_{n}(C)).

For each γ = α + iβ ∈ C and for each ϕ ∈ S(U) we may at least formally consider the convolutions

G_{γ}(ϕ)(y) = ∫_{U}(x − y)∥x − y∥^{γ}ϕ(x)dx^{n}

and

I_{γ}(ϕ)(y) = ∫_{U} ∥x − y∥^{γ+1}ϕ(x)dx^{n}.

We shall only consider the cases where these operators are well defined on S(U) and when cl(U), the closure of U, is compact. This happens when the real part of γ belongs to (−n + 1, ∞).

Let us denote the image space G_{γ}(S(U)) by P_{γ}(U) and the image space I_{γ}(S(U)) by Q_{γ}(U).

Theorem 5.1. Suppose that y = ψ(v) = (av + b)(cv + d)^{−1} is a Möbius transformation. Then

∫_{U}(x − y)∥x − y∥^{γ}ϕ(x)dx^{n} =
J_{γ}(ψ, v)^{−1} ∫_{ψ^{−1}(U)}(u − v)∥u − v∥^{γ}J̃_{γ+2n}(ψ, u)^{−1}ϕ(ψ(u))du^{n},

and

∫_{U} ∥x − y∥^{γ+1}ϕ(x)dx^{n} =
K_{γ}(ψ, v)^{−1} ∫_{ψ^{−1}(U)} ∥u − v∥^{γ+1}K_{γ+1+2n}(ψ, u)^{−1}ϕ(ψ(u))du^{n},

where J_{γ}(ψ, v) = ˜ (cv + d)∥cv + d∥^{γ} and K_{γ}(ψ, v) = ∥cv + d∥^{−γ−1}. Outline of Proof: The result follows from noting that

(ψ(u) − ψ(v))∥ψ(u) − ψ(v)∥^{γ} = J_{γ}(ψ, v)^{−1}(u − v)∥u − v∥^{γ}J̃(ψ, u)^{−1},

that

∥ψ(u) − ψ(v)∥^{γ+1} = K_{γ}(ψ, v)^{−1}∥u − v∥^{γ+1}K_{γ}(ψ, u)^{−1},

and the Jacobian of the transformation is ∥cv + d∥^{−2n}.

We shall denote the function space {J̃_{γ+2n}(ψ, v)^{−1}ϕ(ψ(v)) : ϕ ∈ S(U)} by S_{γ,J}(ψ^{−1}(U)). We denote the function space {K_{γ+1+2n}(ψ, v)^{−1}ϕ(ψ(v)) : ϕ ∈ S(U)} by S_{γ,K}(ψ^{−1}(U)). Moreover the space G_{γ}S_{γ,J}(ψ^{−1}(U)) will be denoted by R_{γ}(ψ^{−1}(U)) while the space I_{γ}S_{γ,K}(ψ^{−1}(U)) will be denoted by W_{γ}(ψ^{−1}(U)).

Theorem 5.1 tells us that the multiplier operators J_{γ}(ψ,) and J̃_{γ+2n}(ψ,)^{−1} intertwine the convolution operator G_{γ} and the multiplier operators K_{γ}(ψ,) and K_{γ+1+2n}(ψ,)^{−1} intertwine the convolution operator I_{γ}. So the transforms

J_{γ}(ψ,)G_{γ} : S(U) → R_{γ}(ψ^{−1}(U))  (3)

and

G_{γ}J̃_{γ+2n}(ψ,)^{−1} : S(U) → R_{γ}(ψ^{−1}(U))  (4)

are the same, and the transforms

K_{γ}(ψ,)I_{γ} : S(U) → W_{γ}(ψ^{−1}(U))  (5)

and

I_{γ}K_{γ+1+2n}(ψ,)^{−1} : S(U) → W_{γ}(ψ^{−1}(U))  (6)

are the same.

For −n < γ < 0 and U a subset of R^{n} the operators I_{γ} correspond to the Riesz potentials described in [25] and elsewhere. Following the Fourier analysis arguments given for R^{n} in [25] it may be observed that for these values of γ the operator I_{γ} is invertible. From Theorem 5.1 it follows that for these values of γ the operators I_{γ} acting on the function spaces set up in Theorem 5.1 for the domain ψ^{−1}(U) are invertible. When γ = −n + 2 the Riesz potential on R^{n} is the fundamental solution to the Laplacian. This operator has △_{n} as its inverse. When γ = −n + 1 then as observed in [25] the inverse of I_{γ} is △n^{\frac{1}{2}} = ∥D∥. It follows from Theorem 5.1 that each one of these inverse operators have analogues on the appropriate function spaces on the domain ψ^{−1}(U).

From Theorem 5.1 and equations 3–6 it follows that when the operators G_{γ} : S(U) → P_{γ}(U) and I_{γ} : S(U) → Q_{γ}(U) are invertible then

G_{γ}^{−1}J_{γ}(ψ,)^{−1} : R_{γ}(ψ^{−1}(U)) → S(U)

and

J̃_{γ+2n}G_{γ}^{−1} : R_{γ}(ψ^{−1}(U)) → S(U)

are the same, and the transforms

I_{γ}^{−1}K_{γ}(ψ,)^{−1} : W_{γ}(ψ^{−1}(U)) → S(U)

and

K_{γ+1+2n}I_{γ}(ψ,)^{−1} : W_{γ}(ψ^{−1}(U)) → S(U)

are the same.

As a consequence of the preceding remarks one may introduce analogues of the pseudo-differential operator △n^{\frac{1}{2}} , acting on function spaces S(U) where U is a domain in S^{n} or H_{n}.

In [25] and elsewhere it is shown that when −n + 1 < γ < 0 then

I_{γ} : L^{p}(R^{n−1}) → L^{q}(R^{n−1})

is a bounded operator if and only if \frac{1}{q} = \frac{1}{p} − \frac{α}{n}, where α = n − γ, and 1 < p < ∞. First let us note:

Lemma 5.2. The function ϕ(y) ∈ L^{p}(R^{n−1}) if and only if

K_{\frac{−2n+2}{p} −1}(ψ, v)ϕ(ψ(v)) ∈ L^{p}(R^{n−1})

for each ψ ∈ V_{+}(n − 1), and 1 < p < ∞.

Similarly ϕ(y) ∈ L^{p}(R^{n−1}) if and only if J_{\frac{−2n+2}{p}} (ψ, v)ϕ(ψ(v)) ∈ L^{p}(R^{n−1}) for each ψ ∈ V_{+}(n − 1) and 1 < p < ∞.

Consequently one may deduce:

Theorem 5.3. Suppose that 1 < p < ∞ and \frac{1}{p} + \frac{1}{q} = 1 then the multiplier operators K_{( \frac{1}{p} −1)(2n−2)+1}(ψ,) and K_{( \frac{1}{p} +1)(2n−2)+1}(ψ,) intertwine the operator

I_{\frac{2−2n}{p} +1} : L^{p}(R^{n−1}) → L^{q}(R^{n−1})

while the multiplier operators J_{( \frac{1}{p} −1)(2n−2)}(ψ,) and J_{( \frac{1}{p} +1)(2n−2)}(ψ,) intertwine the operator

G_{\frac{2−2n}{p}} : L^{p}(R^{n−1}) → L^{q}(R^{n−1})

for each ψ ∈ V_{+}(n − 1).

In fact most of the previous material from this section can be reformulated for arbitrary smooth orientable manifolds in R^{n}, S^{n} or H_{n}. Consider a pdimensional smooth and orientable manifold M_{p} in R^{n}, S^{n} or H_{n}, and M_{p} is such that cl(M_{p}) is compact. Then we may introduce the space S(M_{p}) of smooth sections on M_{p} with compact support. These sections will take their values in the bundle (M_{p}, Cl_{n}(C)), where (M_{p}, Cl_{n}(C)) is the appropriate sub-bundle of (S^{n}, Cl_{n}(C)) when M_{p} ⊂ S^{n}. It is a sub-bundle of (H_{n}, Cl_{n}(C)) when M_{p} ⊂ H_{n} and it is the bundle M_{p} × Cl_{n}(C) when M_{p} ⊂ R^{n}.

In these circumstances we may formally introduce the convolution operators G_{γ,M_{p}}ϕ(y) = ∫_{M_{p}}(x − y)∥x − y∥^{γ}ϕ(x)dν(x) and I_{γ,M_{p}}ϕ(y) = ∫_{M_{p}} ∥x − y∥^{γ+1}ϕ(x)dν(x), where ϕ ∈ S(M_{p}) and ν is the Borel measure on M_{p}. These operators are well defined and bounded when the real part of γ belongs to (−p, ∞). We shall denote the image space G_{γ,M_{p}}S(U) by P_{γ}(M_{p}) and the image space I_{γ,M_{p}}S(U) by Q_{γ}(M_{p}).

Theorem 5.4. Suppose that y = ψ(v) = (av + b)(cv + d)^{−1} is a Möbius transformation then for each ϕ ∈ S(U)

∫_{M_{p}}(x − y)∥x − y∥^{γ}ϕ(x)dν(x) =

J_{γ}(ψ, v) ∫_{ψ^{−1}(M_{p}}(u − v)∥u − v∥^{γ}J̃_{γ+2p}(ψ, u)^{−1}ϕ(ψ(u))dν(u)

and

∫_{M_{p}} ∥x − y∥^{γ+1}ϕ(x)dν(x) =

K_{γ}(ψ, v)^{−1} ∫_{ψ^{−1}(M_{p})} ∥u − v∥^{γ+1}K_{γ+2p}(ψ, u)^{−1}ϕ(ψ(u))dν(u).

The result follows from noting that the Jacobian of this transformation is ∥cv + d∥^{−2p} and by applying the same arguments used to deduce Theorem 5.1.

Theorem 5.4 tells us that the multiplier operators J_{γ}(ψ,) and J̃_{γ+2p}(ψ,)^{−1} intertwine the operator G_{γ,M_{p}} and also its inverse when that exists, and that K_{γ}(ψ,) and K_{γ+1+2p}(ψ,)^{−1} intertwine the operator I_{γ,M_{p}} and also its inverse when that exists.

6. Generalised Plemelj Formulae

When regarded as a principal value integral, the convolution operator G_{−p+1,M_{p}} is also well defined on S(M_{p}). We shall consider the case where M_{p} is compact and is the boundary of a smooth orientable manifold M. We shall also assume that ∂M_{p} is Liapunov, so that M_{p} is C^{1} and has Hölder continuous first derivative. So the function n : M_{p} → S^{n−1} is Hölder continuous, where the function n assigns to each x ∈ M_{p} the unit vector n(x) in T M_{x} normal to T M_{p}(x) and pointing out from the manifold M, where T M(x) is the tangent space to M at x and T M_{p}(x) is the tangent space to M_{p} at x.

Let L^{2}(M_{p}) denote the class of L^{2} integrable functions defined on M_{p} and with values in the bundle (M_{p}, Cl_{n}(C)). It is a simple matter to follow the classical arguments presented in [4] and elsewhere on compact Liapunov surfaces in R^{n} to deduce

Theorem 6.1. The convolution operator

G_{−p+1,M_{p},M} : L^{2}(M_{p}) → L^{2}(M_{p}) :

\frac{1}{ω_{p}} G_{−p+1,M_{p},M}ϕ(y) = pv \frac{1}{ω_{p}} ∫_{M_{p}} \frac{(x − y)}{∥x − y∥^{p}} n(x)ϕ(x)dν(x)

is an L^{2} bounded operator, where ω_{p} is the surface area of the unit sphere in R^{p}.

It should be noted that the the normal function n(x) to M_{p} depends on the choice of M. Consequently the operator G_{−p+1,M_{p},M} also depends on M. Also, when p = n the operator G_{−p+1,M_{p},M} is the same as the Cauchy kernel G.

Using the Dirac operator over each tangent space T M_{x} with x ∈ M_{p} we obtain the following Plemelj formula, which also follows by the same reasoning as that presented for surfaces in R^{n} in [4].

Theorem 6.2. For each y ∈ M_{p} and each y(t) ∈ M such that y(t) approaches y non-tangentially as t approaches 1 then

lim_{t→1} \frac{1}{ω_{p}} ∫_{M_{p}} \frac{(x − y)}{∥x − y∥^{p}} n(x)ϕ(x)dν(x) =

\frac{1}{2}ϕ(y) + pv \frac{1}{ω_{p}} ∫_{M_{p}} \frac{(x − y)}{∥x − y∥^{p}} n(x)ϕ(x)dν(x),

for almost all y ∈ M_{p} and each ϕ ∈ L^{2}(M_{p}).

We may denote the Hilbert space (\frac{1}{2} + G_{−p+1,M_{p}})L^{2}(M_{p}, M) by H^{2}(M). It is an easy matter to mimic the arguments used earlier here to show

that if ψ is a Möbius transformation and for two manifolds M_{p,1} and M_{p,2} with ψ(M_{p,1}) = M_{p,2} then the Hilbert spaces L^{2}(M_{p,1}) and L^{2}(M_{p,2}) are isometric. In particular, each ϕ ∈ L^{2}(M_{p,2}) is transformed to J_{−p+1}(ψ,)ϕ ∈ L^{2}(M_{p,1}). Via this isometry it may be deduced that H^{2}(M_{p,1}) is isometric to H^{2}(M_{p,2}). Also the multiplier operator J_{−p+1}(ψ,) intertwine the operator G_{−p+1} and also the operator \frac{1}{2} + G_{−p+1}. Consequently, a significant part of the results mentioned earlier for L^{2} spaces of surfaces in R^{n}, S^{n} and H_{n} go through automatically to the setting described in this section. Though we have assumed the manifold M_{p,2} is compact and Liapunov it does not follow that the manifold M_{p,2} is either compact or Liapunov.

References

[1] Ahlfors, L. V., Möbius transformations in R^{n} expressed through 2 × 2 matrices of Clifford numbers, Complex Variables: Theory and Application 5 (1986), 215–224.

[2] Auscher, P and Ph. Tchamitchian, Bases d’ondelettes sur des courbes corde-arc, noyau de Cauchy et espaces de Hardy associés, Rev. Mat. Iberoamericana, 5 (1989), 139–170.

[3] Bell, S., “The Cauchy Transform, Potential Theory, and Conformal Mapping,” CRC Press, 1992.

[4] Bernstein, S., The left-linear Riemann problems in higher dimensions, to appear.

[5] Bojarski, B., Conformally covariant differential operators, in “Proceedings, XXth Iranian Math Congress”, Tehran, 1989.

[6] Brackx, F., R. Delanghe and F. Sommen, “Clifford Analysis, ” Pitman Research Notes in Mathematics 76, 1982.

[7] Calderbank, D., Clifford analysis for Dirac operators of manifolds with boundary, to appear.

[8] Cnops, J., “Hurwitz pairs and applications of Möbius transformations, ” Ph. D. Dissertation, University of Gent, Belgium, 1994.

[9] Coifman, R., P. Jones and S. Semmes, Two elementary proofs of the L^{2}- boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), 350–358.

[10] Coifman, R., A. McIntosh and Y. Meyer, L’intégrale de Cauchy définit un opérateur borné sur L^{2} pour les courbes lipschitziennes, Ann. Math. 116 (1982), 361–387.

[11] Coifman, R., R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. Math. 103 (1976), 611–635.

[12] Gaudry, G., R. Long and T. Qian, A martingale proof of L^{2} boundedness of Clifford-valued singular integrals, Ann. Mat. Pura Appl., 165 (1993), 369–394.

[13] Gilbert, J. and M. A. M. Murray, Clifford Algebras and Dirac Operators in Harmonic Analysis, CUP, 1991.

[14] Gürlebeck, K. and W. Sprössig, “Quaternionic Analysis and Elliptic Boundary Value Problems,” Birkhäuser Verlag, 1990.

[15] Li, C., A. McIntosh and T. Qian, Clifford algebras, Fourier transforms and singular convolution operators on Lipschitz surfaces, Rev. Mat. Iberoamericana, 10(1994), 665–721.

[16] Li, C., A. McIntosh and S. Semmes, Convolution singular integrals on Lipschitz surfaces, J. Amer. Math. Soc. 5(1992), 455–481.

[17] McIntosh, A., Clifford algebras, singular integrals, and partial differential equations on Lipschitz domains, in “Clifford Algebras in Analysis and Related Topics,” J. Ryan, Ed., CRC Press, 1996, 33–87.

[18] Mitrea, M., “Clifford Wavelets, Singular Integrals and Hardy Spaces, ” Lecture Notes in Mathematics 1575, Springer Verlag, 1994.

[19] Peetre, J. and T. Qian, Möbius covariance of iterated Dirac operators, J. Austral. Math. Soc. (Series A) 56 (1994), 1–12.

[20] Porteous, I., Clifford Algebras and the Classical Groups, CUP, 1995.

[21] Qian, T. and J. Ryan, Conformal transformations and Hardy spaces arising in Clifford analysis, J. Operator Theory 35 (1996), 349–372.

[22] Ryan, J., Dirac Operators on Spheres and Hyperbolae, Boletin de la Sociedad Matematica Mexicana, to appear.

[23] Semmes, S., Chord-arc surfaces with small constant, 1, Adv. in Math. 85 (1991), 198–223.

[24] Sommen, F., Spherical monogenics and analytic functionals on the unit sphere, Tokyo J. Math., 4(1981), 427–456.

[25] Stein, E. M., “Singular Integrals and Differentiability Properties of Functions,” Princeton University Press, 1970.

[26] Sudbery, A., Quaternionic analysis, Math. Proc. Cambridge Philos. Soc. 85 (1979), 199–225.

[27] van Lancker, P., Clifford analysis on the sphere, to appear.

[28] Vahlen, K. Th., Über Bewegungen und Complexe Zahlen, Math. Ann. 55 (1902), 585–593.

Department of Mathematics University of Arkansas Fayetteville, Arkansas 72701 jryancomp.uark.edu

Received March 27, 1997 and in final form September 16, 1997