Cauchy-Kowalevski extensions, Fueter s theorems and boundary values of special systems in Clifford analysis

Transcription

1 Cauchy-Kowalevski extensions, Fueter s theorems and boundary values of special systems in Clifford analysis by Dixan Peña Peña A thesis submitted to Ghent University for the degree of Doctor of Philosophy in Mathematics Promotors: Prof. Dr. Franciscus Sommen Prof. Dr. Juan Bory Reyes Ghent University - Faculty of Sciences Department of Mathematical Analysis - Clifford Research Group Academic year

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3 A mis padres, mi esposa Barbara, y a mis hijos Aymara y Héctor.

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5 Acknowledgement This thesis would not have been realized without the support, the enthusiasm and the encouragement of many people. That is the reason why I would like to start with some words of thanks. First of all, I want to express my deep gratitude to Prof. Em. Dr. Richard Delanghe for offering me the possibility to do a Ph.D. at Ghent University. It was a very pleasant experience to be a member of the Clifford Research Group. I also appreciate his willingness to contribute to the development of mathematics and the promotion of Clifford analysis in Cuba. Next, I would like to thank my supervisor in Ghent Prof. Dr. Frank Sommen for the wisdom he shared with me and the nice conversations we had. It was a great privilege to be his disciple and he had an important influence on my understanding of mathematics. I am also very grateful to Prof. Dr. Juan Bory Reyes, my supervisor in Cuba, who introduced me in the fascinating world of mathematical research. He gave me very good ideas and he was a professional and personal coach during all these years. Many thanks also to Prof. Dr. Ricardo Abreu Blaya for his contributions to the papers we have written together and for his continuous support. Of course, the University of Ghent gave me the possibility to do my research by offering me a doctoral grant. I am very grateful for that. This grant was not only a great stimulus to my mathematical career, but it changed the whole course of my life since I met my wife in Belgium and by now we have two lovely children. It also offered me the opportunity to travel all around the world. v

6 vi Acknowledgement It is also a great pleasure to thank Prof. Dr. Fred Brackx for the stimulating conversations and his helpful comments as well as Prof. Dr. Hennie De Schepper who read my thesis and gave me excellent suggestions to improve it. They both were always friendly and ready to help. Moreover, I want to thank my colleagues, in particular David for reading my thesis and Bram for being a nice office mate. Additionally, thanks to Samuel and Hans: all kind of administrative problems were solved immediately and in a friendly way, which I appreciate a lot. I would like to thank my parents for their constant support and love and my brother, just for being a good brother. Finally, a special thank-you to my lovely wife Barbara and my little treasures Aymara and Héctor for being my source of strength, love, happiness and inspiration.

7 Contents Introduction ix 1 Some basic elements of Clifford analysis Clifford algebras Monogenic functions The Cauchy type integral Special monogenic series and expressions 1.1 Steering monogenic functions Monogenic power series of axial and biaxial type: toroidal expansions Generalized CK-extensions of codimension Fueter s theorems An alternative proof Generalized Fueter s theorem The jump problem for Hermitean monogenic functions Introduction Integral criterion for h-monogenicity Conservation law for h-monogenic functions vii

8 viii Contents 5 Isotonic Clifford analysis Isotonic functions The isotonic Cauchy type integral Bochner-Martinelli type integrals Holomorphic and biregular extension theorems Holomorphic functions Holomorphic extension for Hölder continuous functions Holomorphic extension for continuous functions Biregular extension for Hölder continuous functions Biregular extension for continuous functions Conclusion 105 Bibliography 107

9 Introduction In the year 1878 William Kingdon Clifford introduced the algebras named after him which may be regarded as a generalization of the complex numbers and Hamilton s quaternions see [33]. They are a type of finite-dimensional associative algebra and have important applications in a variety of fields including geometry and theoretical physics. Clifford analysis is a successful generalization to higher dimension of the theory of holomorphic functions in the complex plane. It involves the study of functions on Euclidean space with values in a Clifford algebra. For a thorough treatment of this function theory we refer the reader to e.g. [6, 34, 45, 61, 64, 65]. The main objects of study in Clifford analysis are the so-called monogenic functions which may be described as null solutions of the Dirac operator, the latter being the higher dimensional analogue of the Cauchy- Riemann operator. Some of the earlier results on Clifford analysis were obtained by Dixon [37], Moisil and Théodoresco [83], Fueter [55, 56], Iftimie [71], Hestenes [69] and Delanghe [39, 40, 41, 4]. The basic theory of Clifford analysis was developed in the book by Brackx, Delanghe and Sommen [6] in 198. This is the first book written on Clifford analysis and it is the basic reference work on the subject. Nowadays Clifford analysis is a well established mathematical discipline as well as an active area of scientific research. The subject of this thesis fits in the framework of Clifford analysis. The first part deals with some techniques to generate monogenic functions and ix

10 x Introduction the second part is devoted to the study of extension theorems for special systems arising in Clifford analysis. In order to make the reader familiar with the concepts used in this thesis, the first chapter contains a review of the definitions and fundamental results concerning Clifford algebras, Clifford analysis, the Cauchy type integral and the singular integral operator. Despite the fact that Clifford analysis generalizes the most important features of classical complex analysis, monogenic functions do not enjoy all properties of holomorphic functions of one complex variable. For instance, due to the non-commutativity of the Clifford algebras, the product of two monogenic functions is in general not monogenic. It is therefore natural to look for specific techniques to construct monogenic functions. There are several techniques available to generate monogenic functions, see [6, 43, 45]. Two of those techniques are considered in this thesis: the Cauchy-Kowalevski extension problem and Fueter s theorem. We also introduce a new technique leading to so-called steering monogenic functions. The first technique mentioned consists in monogenically extending analytic functions defined on a given subset in R m+1 of positive codimension. The second one gives a method to generate monogenic functions starting from a holomorphic function in the upper half of the complex plane. Finally, steering monogenic functions can be roughly described as a class of monogenic functions generated from families of complex-valued functions which are closed under conjugation and under the action of the Cauchy-Riemann operator. In Chapter we introduce the notion of steering monogenic functions and we discuss the Cauchy-Kowalevski extension around special surfaces of codimension two. In Chapter 3 we provide an alternative proof for Fueter s theorem. Using the main idea of this proof, we also establish a new generalization of Fueter s theorem. Some examples of applications are also computed including a closed formula for the Cauchy-Kowalevski extension of the Gaussdistribution in R m.

11 Introduction xi Chapter 4 deals with a recent refinement of the theory of monogenic functions: Hermitean Clifford analysis. It studies so-called Hermitean monogenic functions which are simultaneous null solutions of two mutually related Euclidean Dirac operators see [4, 5, 7, 101, 10]. We derive two criteria providing necessary and sufficient conditions for the existence of a Hermitean monogenic extension of a continuous function defined on a surface in R m, m = n. These characterizations are then used to study the jump problem in this context. In the even dimensional case the Dirac equation may be reduced to the so-called isotonic Dirac system in which different Dirac operators in half the dimension act from both sides on the unknown function. Solutions of this system are called isotonic functions and are closely related with Hermitean monogenic functions. Chapter 5 is devoted to the study of these functions. First, we obtain an integral representation formula. Next, some direct applications of this formula are indicated. The remainder of this chapter is devoted to the study of the isotonic Cauchy type integral and its singular version. Finally, in the last chapter, extension theorems for holomorphic and biregular functions are studied. The latter may be considered as monogenic functions of two higher dimensional variables. As holomorphic and biregular functions are particular cases of isotonic functions, the results obtained in Chapter 5 enable us to get simplified and elegant proofs.

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13 Chapter 1 Some basic elements of Clifford analysis This chapter contains a summary of the Clifford analysis theory we will use. For a thorough treatment we refer the reader to [6, 34, 45, 61, 64, 65]. 1.1 Clifford algebras Clifford algebras, also called geometric algebras, extend the real number system to include vectors and their products. Clifford algebras have important applications in geometry and theoretical physics. They are named after the English geometer and philosopher W. K. Clifford see [33]. We denote by R 0,m m N the real Clifford algebra constructed over the orthonormal basis e 1,..., e m of the Euclidean space R m. The basic axiom of this associative but non-commutative algebra is that the product of a vector with itself equals its squared length up to a minus sign, i.e. for any vector x = m j=1 x je j in R m, we have that x = x = 1 m x j. j=1

14 CK-extensions, Fueter s theorems and boundary values It thus follows that the elements of the basis submit to the multiplication rules e j = 1, j = 1,..., m, e j e k + e k e j = 0, 1 j k m. A basis for the algebra is then given by the elements e A = e j1 e jk, where A = {j 1,..., j k } {1,..., m} is such that j 1 < < j k. For the empty set, we put e = e 0 = 1, the latter being the identity element. It follows that the dimension of R 0,m is m. Any Clifford number a R 0,m may thus be written as a = A a A e A, a A R. For each k {0, 1,..., m}, we call R k 0,m = a R 0,m : a = A =k a A e A the subspace of k-vectors, i.e. the space spanned by the products of k different basis vectors. In particular, the 0-vectors and 1-vectors are simply called scalars and vectors respectively. An important subspace of the real Clifford algebra R 0,m is the so-called space of paravectors R R 1 0,m, being sums of scalars and vectors. Observe that R m+1 may be naturally identified with R R 1 0,m by associating to any element x 0, x = x 0, x 1,..., x m R m+1 the paravector x = x 0 + x. Note that R 0,m = m k=0 R k 0,m

15 Some basic elements of Clifford analysis 3 and hence for any a R 0,m a = m [a] k, k=0 where [a] k is the projection of a on R k 0,m. The product of two Clifford vectors x = m j=1 x je j and y = m j=1 y je j splits into a scalar part and a -vector or so-called bivector part where x y = x y + x y, 1.1 x y = x, y = m x j y j equals, up to a minus sign, the standard Euclidean inner product between x and y, while m m x y = e j e k x j y k x k y j j=1 k=j+1 represents the standard outer or wedge product between them. More generally, for a vector x and a k-vector Y k, the inner and outer product between x and Y k are defined by j=1 x Y k = { [xyk ] k 1 for k 1 0 for k = 0 and x Y k = [xy k ] k+1. In a similar way, Y k x = { [Yk x] k 1 for k 1 0 for k = 0 and Y k x = [Y k x] k+1. We thus have that xy k = x Y k + x Y k, Y k x = Y k x + Y k x,

16 4 CK-extensions, Fueter s theorems and boundary values where also x Y k = 1 k 1 Y k x, x Y k = 1 k Y k x. Two important examples of real Clifford algebras are the field of complex numbers C and the skew field of quaternions H. Indeed, note that R 0,1 is a two-dimensional algebra generated by a single vector e 1 which squares to 1, and therefore R 0,1 is isomorphic to C. On the other hand, R 0, is a fourdimensional algebra spanned by {1, e 1, e, e 1 e }. The latter three elements square to 1 and all anticommute, and so the algebra R 0, is isomorphic to the quaternions H. Three anti-involutions are defined on R 0,m : the main involution, the reversion and the conjugation. The main involution a ã is given by ã = A a A ẽ A, where ẽ A = 1 k e A if A = k. The reversion a a is given by a = A a A e A, where e A = e j k e j1 = 1 k 1k e A if e A = e j1 e jk. Finally, the conjugation a a is a combination of the main involution and the reversion introduced above. It is defined as a = ã = A a A ẽ A. One easily checks that ãb = ã b,

17 Some basic elements of Clifford analysis 5 for any a, b R 0,m. ab = b a, ab = ba, By means of the conjugation, a norm a may be defined for each a R 0,m by putting a = [aa] 0 = a A. A It immediately follows that for any a, b R 0,m a + b a + b and ab m a b. In this thesis we also deal with the complex Clifford algebra C m, which may be defined as C m = C R 0,m = R 0,m i R 0,m. Any complex Clifford number a C m may thus be represented as a = A a A e A, a A C. All concepts introduced above in the context of R 0,m may be reformulated in the case of C m in a very similar way. The major difference lies in the conjugation, where the additional rule i = i has to be included. It is worth pointing out that for m 3 the real Clifford algebra R 0,m has zero divisors. Indeed, it is easily seen that e 13 squares to 1 and hence 1 + e 13 1 e 13 = 1 e e 13 = 0. Thus for m 3 not every Clifford number in R 0,m has a multiplicative inverse. Fortunately, any non-zero paravector x does have a multiplicative inverse given by x 1 = x x. In the case of C m we also have that with ω = x x. 1 + iω 1 iω = 1 iω 1 + iω = 0,

18 6 CK-extensions, Fueter s theorems and boundary values 1. Monogenic functions Monogenic functions are the central object of study in Clifford analysis. The concept of monogenicity of a function may be seen as the higher dimensional counterpart of holomorphy in the complex plane. The functions under consideration are defined on an open subset of R m or R m+1 and take values in the Clifford algebra R 0,m or in its complexification C m. They are of the form f = A f A e A, where the functions f A are R-valued or C-valued. Whenever a property such as continuity, differentiability, etc. is ascribed to f it is clear that in fact all the components f A possess the cited property. Next, we introduce the Dirac operator x = m e j xj j=1 and the generalized Cauchy-Riemann operator x = x0 + x. These operators factorize the Laplace operator in the sense that x = m x j = x 1. j=1 and x = x 0 + x = x x = x x. 1.3 Definition 1.1 A function fx resp. fx defined and continuously differentiable in an open set Ω of R m resp. R m+1 and taking values in R 0,m

19 Some basic elements of Clifford analysis 7 or C m, is called a left monogenic function in Ω if and only if it fulfills in Ω the equation m m x f e j e A xj f A = 0 resp. x f e j e A xj f A = 0. j=1 A Note that in view of the non-commutativity of R 0,m and C m a notion of right monogenicity may be defined in a similar way by letting act the Dirac operator or the generalized Cauchy-Riemann operator from the right. Nevertheless, we will just say f is monogenic in Ω instead of f is left monogenic in Ω. From 1. and 1.3 it follows that any monogenic function in Ω is harmonic in Ω and hence real-analytic in Ω. To fix the ideas let us examine two special cases of monogenic functions. First, if m = 1, then a function f : Ω R R 0,1 is of the form f = f 0 + f 1 e 1 and x = x0 + e 1 x1, so the monogenicity of f reduces to the system { x0 f 0 x1 f 1 = 0 x1 f 0 + x0 f 1 = 0 which is nothing else but the classical Cauchy-Riemann system for holomorphic functions of one complex variable. Next, let f be a vector-valued function in Ω R m, i.e. m fx = f j xe j. Then, from 1.1 we obtain j=1 x f = x f + x f. Claiming that x f = 0 in Ω is thus equivalent to saying that its components f j, j = 1,..., m, satisfy the so-called Riesz system m xj f j = 0, j=1 xj f k xk f j = 0, 1 j k m. j=0 A

20 8 CK-extensions, Fueter s theorems and boundary values It is clear that the set of R 0,m -valued resp. C m -valued monogenic functions in Ω provided with the classical rules for addition and for multiplication with Clifford scalars is a right R 0,m -module resp. C m -module. We emphasize that the product of two monogenic functions is, in general, not monogenic. For a vector-valued differentiable function f = m j=1 f je j and a Clifford algebra-valued differentiable function g, we have the following Leibniz rule the general version will be given in the third chapter Indeed, x fg = x fg = x fg f x g m f j xj g. 1.4 j=1 m e j xj fg + f xj g = x fg + j=1 which results in 1.4 on account of the equality In particular, for f = x we have e j f = fe j f j, j = 1,..., m. m e j f xj g, j=1 x x g = mg x x g E x g, 1.5 E x = m j=1 x j xj being the Euler operator. Using 1.4 we may also prove the following simple but interesting statement: if f is a monogenic function in some open connected set Ω of R m such that e j f is also monogenic in Ω for each j = 1,..., m, then f is a constant in Ω. Indeed, e j f being monogenic, we have, for each j = 1,..., m 0 = x e j fx = xj fx, x Ω. From the above it follows that all first order partial derivatives of f vanish, and consequently f is a constant function in Ω.

21 Some basic elements of Clifford analysis 9 Let Γ x denote the spherical Dirac operator or Gamma operator on the unit sphere S m 1 in R m, i.e. Γ x = x x = m m j=1 k=j+1 e j e k x j xk x k xj. From 1.1 we see that x x = E x Γ x. 1.6 Introducing spherical coordinates x = rω r = x, ω S m 1 and using the fact that E x = r r, we obtain the spherical decomposition of the Dirac operator x = ω r + 1 r Γ x. 1.7 Next, we recall two properties of the spherical Dirac operator Γ x that are frequently used in calculations see [45]. If fr is a function of r, then Γ x fr = = = m m j=1 k=j+1 m m j=1 k=j+1 m m j=1 k=j+1 e j e k xj xk fr x k xj fr e j e k xj r fr xk r x k r fr xj r e j e k x j r fr x k r x k r fr x j. r So that Γ x fr = On account of the above remark and using 1.7, we see that x fr = ω r fr.

22 10 CK-extensions, Fueter s theorems and boundary values From 1.6 and applying 1.5 we also get 1 Γ x ωfx = Γ x xfx + 1 r r Γ xxfx = 1 x x xfx + E x xfx r = 1 mxfx + x x fx + xe x fx r E x xfx xe x fx This gives = ω m 1fx + x x fx + E x fx. Γ x ωfx = m 1ωfx ω Γ x fx. 1.9 By the above and using 1.7, we can assert that x ω = ω r Γ x ω = m 1 ω r 1 = m. r Let us now consider monogenic functions of the form Ax0, r + ω Bx 0, r P k x, 1.10 where Ax 0, r and Bx 0, r are R-valued continuously differentiable functions, and P k x is a homogeneous monogenic polynomial of degree k in R m, i.e. x P k x = 0, x R m, P k tx = t k P k x, t R. Functions 1.10 are called axial monogenic functions of degree k see [77, 111, 116] and they generate monogenic functions in axially symmetric domains. Note that x [ A + ω B Pk x ] = x AP k x + A x P k x + x BωP k x + B x ωp k x = ω r AP k x r BP k x + B x ωp k x,

23 Some basic elements of Clifford analysis 11 where also x ωp k x = x ωp k x ω x P k x r E xp k x = k + m 1 r P k x, which follows from 1.4 and Euler s homogeneous function theorem. We thus get x [ A + ω B Pk x ] = For this reason [ ω r A r B + k + m 1 ] B P k x. r [ x A + ω B Pk x ] [ = x0 A r B k + m 1 ] B + ω x0 B + r A P k x r and so the assumed monogenicity requires the functions A and B to satisfy the Vekua-type system { x0 A r B = k + m 1 B r 1.11 x0 B + r A = 0. We will also deal with another technique to generate monogenic functions in R m+1 : the so-called Cauchy-Kowalevski extension CK-extension problem. The CK-extension problem consists in finding a monogenic extension g of an analytic function g defined on a given subset in R m+1 of positive codimension see e.g. [6, 35, 44, 45, 7, 110, 11, 118]. For analytic functions g on the plane {x 0, x R m+1 : above problem may be stated as follows: find g such that x0 g = x g, in R m+1 g 0, x = gx. x 0 = 0} the

24 1 CK-extensions, Fueter s theorems and boundary values Formally solving this equation we obtain g x 0, x = exp x 0 x gx x 0 k = k k! xgx. k=0 1.1 It may be proved that 1.1 is a monogenic extension of the function g in R m+1 see [6]. Moreover, by the uniqueness theorem for monogenic functions this extension is also unique. More in general, the CK-extension for analytic functions on an analytic m-surface in R m+1 exists and it is also unique see [11]. For the case of surfaces with higher codimension this problem has not yet been solved, except in the flat case see [44, 45]. Before introducing the basic integral formulae of Clifford analysis, we need a few definitions from geometric measure theory. Geometric measure theory can be roughly described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth, and applied to the calculus of variations. For a detailed exposition we refer the reader to [46, 47, 5, 53, 54, 6, 67, 81, 84, 109]. Let A be a subset of R m and let k m be a positive integer. With each δ > 0 we associate the infimum of all numbers of the form k αkdiamb j k, j=1 where A B j, with diamb j < δ, j = 1,, 3,. j=1 Here αk denotes the volume of the unit sphere in R k, and for any nonempty subset B of R m, its diameter is defined as diamb = sup{ x y : x, y B}.

25 Some basic elements of Clifford analysis 13 If δ tends to zero, this infimum is non-decreasing; it approaches the limit H k A, which is by definition the k-dimensional Hausdorff measure of A. In 1918, F. Hausdorff introduced this k-dimensional measure on R m, defined for all subsets, and coinciding for nice subsets, with the usual k-dimensional surface area. When k = m, it equals the Lebesgue measure. The definition of Hausdorff measure extends to any non-negative real number k, taking π k αk = Γ k + 1, where Γ stands for the usual Gamma function. Observe that H 0 equals the counting measure, i.e. H 0 A is the number of elements of A. Throughout the thesis we assume Ω + to be a simply connected bounded and open set in R m, Ω = R m \ Ω +, Σ is the boundary surface of Ω +, and H m 1 Σ <. The open ball of radius δ > 0 centered at a point x in R m will be denoted by Bx, δ and is defined by Bx, δ = {y R m : y x < δ}. Definition 1. A unit vector w is said to be an exterior normal of Ω + at x Σ in the sense of Federer if and only if δ m L m {y : y x, w < 0, y Bx, δ \ Ω + } 0 and δ m L m {y : y x, w > 0, y Bx, δ Ω + } 0 as δ 0+. Here L m denotes the m-dimensional Lebesgue measure over R m. Such a unit vector w, if it exists, is uniquely determined by Ω + and x, and will be denoted by νx. In case no such w exists, νx is the

26 14 CK-extensions, Fueter s theorems and boundary values null vector. This defines for each x Σ a vector νx with components ν 1 x,..., ν m x, i.e. m νx = ν j xe j. j=1 We note that if x is a smooth boundary point of Σ, then νx is the usual exterior normal. In order to work with sets with very general boundaries the following version of the Gauss-Green Theorem provided by H. Federer will be needed see [48, 49, 50, 51]. For other generalizations we refer the reader to e.g. [38, 68, 93, 94, 10]. Theorem 1.1 Gauss-Green Theorem If the vector-valued function F is differentiable in Ω +, continuous on Ω +, and such that Ω + divf x dl m x <, then divf x dl m x = Ω + Σ F x, νx dh m 1 x. We are now ready to introduce the basic integral formulae of Clifford analysis. But first we recall that the fundamental solution of the Dirac operator x is the L loc 1 -function Ex = 1 x ω m x m, x Rm \ {0}, where ω m is the area of the unit sphere S m 1 in R m. It is easily seen that Ex is monogenic in R m \ {0} and vanishes at infinity. Theorem 1. Clifford-Gauss-Green Theorem Let f and g be continuously differentiable functions in Ω +, which are continuous on Ω +, and moreover satisfy Ω + fx x gx + fx x gx dl m x <,

27 Some basic elements of Clifford analysis 15 then fxνxgx dh m 1 x = [fx x gx + fx x gx] dl m x. Σ Ω + Theorem 1.3 Borel-Pompeiu Formula Let f be a continuously differentiable function in Ω +, continuous on Ω +, and such that Ω + x fx dl m x <. Then Σ Ey xνyfy dh m 1 y Ω + Ey x y fy dl m y = { fx for x Ω +, 0 for x Ω. Theorem 1.4 Cauchy s Integral Formula Suppose that f is a continuous function on Ω +. If f is monogenic in Ω +, then fx = Ey xνyfy dh m 1 y, x Ω +. Σ As in classical complex analysis, Cauchy s Integral Formula is an essential tool in Clifford analysis. Applications of this result include the Mean Value Theorem, Liouville s Theorem, the Maximum Modulus Theorem and Weierstrass Convergence Theorem see [6]. In a similar way the integral formulae for the generalized Cauchy- Riemann operator x may be introduced. 1.3 The Cauchy type integral One of the most important tools in the theory of boundary value problems for holomorphic functions is the Cauchy type integral see e.g. [57, 78, 85].

28 16 CK-extensions, Fueter s theorems and boundary values Hence, it is not surprising that this object has also been studied in the context of Clifford analysis see e.g. [1,, 5, 6, 7, 17, 18, 19, 0, 1,, 71, 86, 107, 108, 1, 13]. Definition 1.3 If f is a continuous function defined on the surface Σ, then the Cauchy type integral of f is the function given by C Σ fx = Ey xνyfy dh m 1 y, x R m \ Σ. Σ It immediately follows that C Σ f is a monogenic function in R m \ Σ and vanishes at infinity. In this section, we shall spell out some important properties of the Cauchy type integral provided in [7] that will be useful in the thesis. But first we need some definitions. Put d = diamσ. Let θ z ɛ = H m 1 Σ Bz, ɛ for z Σ and ɛ > 0. Definition 1.4 The surface Σ is called Ahlfors-David-regular AD-regular if there exists a constant C > 0 such that for all z Σ and 0 < ɛ d. C 1 ɛ m 1 θ z ɛ C ɛ m 1 AD-regular surfaces include smooth, Liapunov and Lipschitz surfaces but also many other arbitrary subsets of R m see [36]. Let us denote by SΣ the set of all continuous functions f on Σ such that the following integrals Ey zνyfy fz dh m 1 y Σ Bz,ɛ converge uniformly to zero for z Σ as ɛ 0.

29 Some basic elements of Clifford analysis 17 For f SΣ, we consider the singular version of the Cauchy type integral: the so-called singular integral operator or Hilbert transform S Σ f defined by S Σ fz = lim Ey zνyfy fz dh ɛ 0 Σ\Bz,ɛ m 1 y+fz, z Σ. Note that for any f SΣ, the singular integral operator S Σ f exists for all z Σ and it defines a continuous function on Σ. The modulus of continuity of a continuous function f on Σ will be denoted by ω f and is defined by ω f τ = τ sup δ τ δ 1 sup z 1 z δ fz 1 fz, τ 0, d]. A function ϕ : 0, d] R + with ϕ0+ = 0 is said to be a majorant if ϕτ is non-decreasing and ϕτ/τ is non-increasing for τ 0, d]. Let us denote by H ϕ Σ the set of continuous functions f on Σ satisfying a generalized Hölder condition, i.e. or equivalently fz 1 fz C ϕ z 1 z, z 1, z Σ, ω f τ C ϕτ, τ 0, d], where ϕ is a majorant and C is a positive constant. It is evident that for ϕτ = τ α 0 < α 1, H ϕ Σ is nothing else but the classical set of Hölder continuous functions C 0,α Σ. If α = 1, then the function f satisfies a Lipschitz condition. A norm f Hϕ may be defined for each f H ϕ Σ by putting f Hϕ = max fz + z Σ sup z 1,z Σ fz 1 fz ϕ z 1 z. If moreover for a majorant ϕ there exists a constant C > 0 such that ɛ 0 ϕτ τ dτ + ɛ d ɛ ϕτ τ dτ C ϕɛ, ɛ 0, d],

30 18 CK-extensions, Fueter s theorems and boundary values then ϕ is said to be a regular majorant see [66]. Note that ϕτ = τ α 0 < α < 1 is a regular majorant. It is worth noting that if Σ is an AD-regular surface and ϕ is a regular majorant, then H ϕ Σ SΣ. In fact, if f H ϕ Σ, then Ey zνyfy fz dh m 1 y Σ Bz,ɛ m fy fz Σ Bz,ɛ y z m 1 dh m 1 y ϕ y z C y z m 1 dhm 1 y = C Σ Bz,ɛ ɛ 0 Cϕɛ and from this it follows that f SΣ. ϕτ τ m 1 dθ zτ C ɛ 0 ϕτ τ The following results are extensions to the case of Clifford analysis of those obtained in [103, 104] for complex-valued functions see [7, ]. Theorem 1.5 Plemelj-Sokhotski Formulae Let Σ be an AD-regular surface and let f SΣ. Then C Σ f has continuous limit values on Σ given by C ± Σ fz = lim C Σfx = 1 SΣ fz ± fz, z Σ. Ω ± x z We must remark that if moreover Σ is a rectifiable surface, i.e. Σ is the Lipschitz image of some bounded subset of R m 1, then f SΣ is also a necessary condition for the continuity up to the boundary of the function C Σ f see []. Theorem 1.6 Let Σ be an AD-regular surface. Then the singular integral operator S Σ is an involution on SΣ, i.e. S Σf = f dτ

31 Some basic elements of Clifford analysis 19 for all f SΣ. Theorem 1.7 Plemelj-Privalov Theorem Assume that Σ is an ADregular surface and let ϕ be a regular majorant. Then S Σ is a bounded operator mapping H ϕ Σ into itself. It is worth remarking that if Σ is an AD-regular surface and if ϕ is a regular majorant, then C Σ f f H ϕ Σ has continuous limit values on Σ, which by Theorems 1.5 and 1.7 belong to H ϕ Σ.

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33 Chapter Special monogenic series and expressions In this chapter some special power series expansions related to the CKextension problem for surfaces of codimension and a new class of monogenic functions are introduced see [89, 90, 9]..1 Steering monogenic functions The aim of this section is to present a new collection of special monogenic functions: the so-called steering monogenic functions. Consider the biaxial splitting R m+1 = R R m 1. In this way, for any x R m+1 we may write x = z + y, where m z = x 0 + x 1 e 1 and y = x j e j. j= 1

34 CK-extensions, Fueter s theorems and boundary values By the above, we can also split the generalized Cauchy-Riemann operator x as x = z + y, the operators z and y being given by z = x0 + e 1 x1, y = m e j xj. Our basic assumption is the following: let Φ denote a family of functions fz with values in R 0,1 which is closed under conjugation and under the action of the operator z. That is, for any f Φ, f Φ and z f may be expressed as a linear combination of elements in Φ. j= We shall consider monogenic expressions of the form f j zg j x,.1 j where each f j belongs to Φ and each g j is a R 0,m -valued function. As the elements of Φ steer the functions g j in such a way that.1 is monogenic, we will call the expressions.1 steering monogenic functions. In what follows, the exponential, trigonometric and power functions of z will be regarded as in the complex case by making the identification i e 1. This idea leads to the following special monogenic functions. Exponential steering monogenic functions. Consider with Φ = {expz, expz}. An easy computation shows that expzax + expzbx,. x expz A + expz B = expz z A + y B + expz y A + z B + B.

35 Special monogenic series and expressions 3 Hence, if A and B satisfy the system { z A + y B = 0 y A + z B + B = 0 then. is monogenic. In particular, if A and B only depend on the variable y, then the above system also constitutes a necessary condition for the monogenicity of., and it takes the form y B = 0 B = 1 ya. Substitution of the second equation of the latter system into the first one yields m y A = x j A = 0, i.e. the function Ay is harmonic. j= Note that we have actually proved that if H is a harmonic function of y, then is monogenic. expzhy 1 expz yhy For instance, if Hy = x j j =,..., m, then we get the monogenic function expz x j 1 expz e j. Taking we can also assert that Hy = 1 y m 3, y Rm 1 \ {0}, 1 m 3 expz + expz y m 3 y y m 1

36 4 CK-extensions, Fueter s theorems and boundary values is monogenic. Trigonometric steering monogenic functions. Consider cos z A 1 x + sin z B 1 x + cos z A x + sin z B x,.3 with Φ = {cos z, sin z, cos z, sin z}. A direct computation then yields x cos z A 1 + sin z B 1 + cos z A + sin z B = cos z z A 1 + y A + sin z z B 1 + y B + cos z y A 1 + z A + B + sin z y B 1 + z B A. Consequently, if z A 1 + y A = 0 z B 1 + y B = 0 y A 1 + z A + B = 0 y B 1 + z B A = 0 then.3 is a monogenic function. Similarly to the exponential steering monogenic functions, if A k and B k k = 1, are functions of y, then.3 is monogenic if and only if A 1 and B 1 are harmonic functions of y and A = 1 yb 1, B = 1 ya 1. This gives rise to monogenic functions of the form cos z H 1 y + sin z H y + 1 cos z yh y 1 sin z yh 1 y, where H 1 and H are harmonic functions of y. Power steering monogenic functions.

37 Special monogenic series and expressions 5 Consider A 0 x + with Φ = {z k, z k : k N 0 }. It is easily seen that, if and if for k 1 then.4 is monogenic. k=1 z k A k x + z k B k x,.4 x A 0 + B 1 = 0 { z A k + y B k = 0 y A k + z B k + k + 1B k+1 = 0 In particular, if the coefficients in the series.4 are functions of y only, we conclude that A k k 0 are harmonic functions of y and B k = 1 k ya k 1, k 1. This yields monogenic functions of the form H 0 y + z k H k y 1 k zk y H k 1 y, k=1 the functions H k k 0 being harmonic of the variable y. Mixed steering monogenic functions. We can also consider combinations of the previous cases. For example, z k expza k y + z k expzb k y..5 k=0 It is a simple matter to check that.5 is monogenic if and only if for k 0 { y B k = 0 y A k + k + 1B k+1 + B k = 0.

38 6 CK-extensions, Fueter s theorems and boundary values From the above it follows that each function A k is harmonic and the functions B k satisfy the recurrence relation 1 B k+1 = y A k + B k, k + 1 with B 0 a given monogenic function of y.. Monogenic power series of axial and biaxial type: toroidal expansions We first start with a series expansion around the sphere S m 1 of codimension two in R m+1 in which axial symmetry plays a central role. In what follows, a convergent series of the form Sx = k=0 l=0 will be called a toroidal expansion of axial type. Z k Z l A k,l x, Z = x 0 + r 1 ω,.6 Theorem.1 A sufficient condition for Sx to be monogenic is given by A k,l+1 x = 1 x0 A k,l x + ω r A k,l x + 1 l + 1 r Γ xa l,k x m 1ω + Ak,l x A l,k x, k, l 0..7 r Proof. Let z = x 0 + r 1i. Using the zero divisors 1 + iω and 1 iω we obtain Z k Z l = Z k Z l 1 iω 1 + iω +.8 = z k l 1 iω z + z k l 1 + iω z.

39 Special monogenic series and expressions 7 In the same way we can see that Z k Z l = z k l 1 iω z + z k l 1 + iω z..9 Applying 1.7, 1.8 and 1.9 we get x Z k Z l A k,l = kz k 1 z l iω This gives lz k z l 1 iω 1 iω A k,l k z k 1 z l iω 1 iω A k,l + l z k z l 1 iω 1 + iω A k,l 1 + iω A k,l +z k l 1 iω z ω r A k,l + z k l 1 + iω z ω r A k,l +z k l 1 + iω ω z r Γ xa k,l + z k z + m 1i z k z l A k,l r l 1 iω m 1i z k z l A k,l. r ω r Γ xa k,l x Z k Z l A k,l = kz k 1 Z l A k,l + lz k Z l 1 A k,l + Z k Z l ω r A k,l where also + Z k Z l ω r Γ xa k,l + z k z l z k z l = iω Z k Z l Z k Z l, the latter following from.8 and.9. We thus get x Z k Z l A k,l = kz k 1 Z l A k,l + lz k Z l 1 A k,l +Z k Z l ω r A k,l + m 1ω A k,l + Z k ω Z l r r Γ xa k,l m 1i z k z l z k z l A k,l, r m 1ω A k,l. r

40 8 CK-extensions, Fueter s theorems and boundary values As x0 Z k Z l A k,l = kz k 1 Z l A k,l + lz k Z l 1 A k,l + Z k Z l x0 A k,l, we have x Z k Z l A k,l = lz k Z l 1 A k,l + Z k Z l x0 A k,l + ω r A k,l + + Z k Z l ω r Γ xa k,l So the action of the operator x on S is given by m 1ω A k,l. r m 1ω A k,l r.10 x S = k=0 l=0 Z k Z l l + 1A k,l+1 + x0 A k,l + ω r A k,l + 1r Γ m 1ω xa l,k + Ak,l A l,k. r Hence we may conclude that the recurrence relation.7 is sufficient for Sx to be monogenic. Although the computations are far from trivial, we note that the toroidal expansion.6 generates monogenic functions. All one has to do is to start from the sequence of functions {A k,0 } k 0 initial condition and calculate the functions A k,l via the recurrence formula.7. It is of natural interest to investigate under which conditions on the initial condition {A k,0 } k 0 the corresponding series generated by the recurrence formula.7 is convergent. This question, however, is still open. An interesting particular case of the toroidal expansion is the case where the coefficients do not depend on the variable x 0 and satisfy the symmetric relation A k,l x = A l,k x. With this assumption we can explicitly calculate the coefficients in.6. Indeed, from.7 we see that A k,l x = 1 l xa k,l 1 x.

41 Special monogenic series and expressions 9 It follows that A k,l x = 1k+l k+l k! l! k+l x A 0,0 x. Substituting the above into.6 gives Clearly, Sx = = = k=0 l=0 k=0 k=0 k Z k l Z l A k l,l x 1 k k k! k l=0 k! k l! l! Zk l Z l k xa 0,0 x 1 k k k! Z + Zk k xa 0,0 x. Sx = x 0 k k k! xa 0,0 x, k=0 which is the classical CK-extension 1.1. We can also solve the recurrence formula.7 if the coefficients satisfy the relation A k,l x = 1 k+l A l,k x. In this case, we obtain 1 l xa k,l 1 x for k + l odd, A k,l x = 1 l P xa k,l 1 x for k + l even, where the differential operator P x is defined by P x g = x0 g + ω r g + 1 r Γ xωg. Therefore A k,l x = 1 l k+l k! l! xp x x k+l 1 A 0,0 x for k + l odd, 1 l k+l k! l! P x x k+l A0,0 x for k + l even.

42 30 CK-extensions, Fueter s theorems and boundary values We thus get Sx = = + = + k k=0 l=0 k=0 k=0 k=0 k=0 and, as a result: Sx = Z k l Z l A k l,l x + 1 k k! k l=0 k=0 k+1 l=0 Z k+1 l Z l A k+1 l,l x 1 l k! k l! l! Zk l Z l P x x k A 0,0 x k l k + 1! k+1 k + 1! k + 1 l! l! Zk+1 l Z l x P x x k A 0,0 x l=0 1 k k! Z Zk P x x k A 0,0 x 1 k+1 k + 1! Z Zk+1 x P x x k A 0,0 x k r 1k 1 P x x k A 0,0 x k! + 1 k r 1k+1 ω x P x x k A 0,0 x..11 k + 1! k=0 k=0 This expansion may be considered as a kind of CK-extension for the cylinder r = 1 in R m+1. If moreover the initial function A 0,0 x does not depend on the variable x 0, then.11 may be regarded as the CK-extension for the sphere S m 1 in R m. Let us compute this series for three simple examples. Example.1. Let A 0,0 x = x 0. It may be easily proved by induction that P x x k A 0,0 x = C k ω r k 1, x P x x k A 0,0 x = k mc k r k,

43 Special monogenic series and expressions 31 where the constants C k satisfy the recurrence relation C k+1 = k mk m + 1C k, k 1, C 1 = m 1. Let and We thus get A 1 r 1k k x = 1k P x x k A 0,0 x k! A k x = r 1k 1k+1 ω x P x x k A 0,0 x. k + 1! lim k 1 A k+1 x 1 A k x = lim k A k+1 x A k x = r 1 r. Since r 1 /r < 1 for r > 1/, it follows that the series.11 converges pointwise for r > 1/ and converges uniformly on every compact subset of {x R m+1 : r > 1/}. Example.. Let A 0,0 x = ω. With this choice of initial function, we obtain P x x k A 0,0 x = C k ω r k, x P x x k A 0,0 x = k m + 1C k r k+1, where the constants C k satisfy the recurrence relation C k+1 = k m + 1k m + C k, k 1, C 1 = m 1m. Similar arguments to those above show that the series.11 converges pointwise for r > 1/ and converges uniformly on every compact subset of {x R m : r > 1/}.

44 3 CK-extensions, Fueter s theorems and boundary values Example.3. Let A 0,0 x = P l ω, where P l ω is the restriction of a homogeneous monogenic polynomial P l x of degree l in R m to S m 1. It follows that P x x k A 0,0 x = C k r k P lω, x P x x k A 0,0 x = k + lc kω r k+1 where the constants C k satisfy the recurrence relation P l ω, C k+1 = k + lk + l + 1C k, k 1, C 1 = ll + 1. For this initial function, we also obtain that the series.11 converges pointwise for r > 1/ and converges uniformly on every compact subset of {x R m : r > 1/}. We now investigate the generalization of the previous theorem to the biaxially symmetric case. To that end we split up R m as R m = R p 1 R p, p 1 + p = m, yielding and accordingly x = x 1 + x, x 1 = x j e j, x = p 1 j=1 p 1 p j=1 x p1 +je p1 +j x = x 1 + x, x 1 = e j xj, x = e p1 +j. xp1 +j Introducing spherical coordinates on R p 1 and R p respectively, i.e. we thus have that j=1 p j=1 x k = r k ω k, r k = x k, ω k S p k 1, k = 1, x = ω 1 r1 + 1 r 1 Γ x 1 + ω r + 1 Γ r x

45 Special monogenic series and expressions 33 where Similar to 1.9, we have Γ x k = x k x k, k = 1,. Γ x kω k f = p k 1ω k f ω k Γ x kf, k = 1,. A convergent series around S p 1 1 S p 1 is called a toroidal expansion of biaxial type if it has the form Sx = k=0 l=0 Z k Z l A k,l x, Z = r r 1ω 1 ω..1 We thus obtain the following generalization of Theorem.1. Theorem. A sufficient condition for Sx to be monogenic is given by ω A k+1,l x = 1 ω k r1 A k,l x + 1 Γ r x 1A l,k x 1 +ω r A k,l x + 1 Γ r x A l,k x.13 p1 1ω p 1ω Ak,lx A l,kx, k, l 0. r 1 r Proof. The proof is similar to the one of Theorem.1. In fact, we have that Z k Z l = z k z l 1 iω 1ω + z k z l 1 + iω 1ω, with z = r r 1i. Therefore x Z k Z l A k,l = kz k 1 Z l ω 1 A k,l +Z k Z l ω 1 r1 A k,l + ω r A k,l + p1 1ω 1 + p 1ω A k,l r 1 r

46 34 CK-extensions, Fueter s theorems and boundary values +Z k Z l ω1 Γ r x 1A k,l + ω Γ 1 r x A k,l p1 1ω 1 + p 1ω A k,l r 1 r and the action of the operator x on S is given by.14 x S = k=0 l=0 Z k Z k l + 1ω 1 A k+1,l +ω 1 r1 A k,l + 1 r 1 Γ x 1A l,k + ω r A k,l + 1 r Γ x A l,k p1 1ω p 1ω Ak,l A l,k. r 1 r We thus have that the recurrence relation.13 is sufficient for the function Sx to be monogenic. Note that for the toroidal expansion of biaxial type.1 the sequence of functions {A 0,l x} l 0 is the initial condition. Let P x 1 and P x be the differential operators defined by P x kg = ω k rk g + 1 r k Γ x kω k g, k = 1,. In a completely similar way as in the axial case, using.13, we obtain the following Cauchy-Kowalevski like extensions around S p 1 1 and S p 1 respectively. i A k,l x = A l,k x: Sx = 1 k r 1 1 k k P k! x 1 x x A0,0 x + 1 k r 1 1 k+1 ω k 1 x P k + 1! x 1 x x A0,0 x. k=0 k=0

47 Special monogenic series and expressions 35 ii A k,l x = 1 k+l A l,k x: Sx = r 1 k k=0 k! + k=0 k x 1 P x x A0,0 x r 1 k+1 ω k + 1! x x 1 P x x k A0,0 x..3 Generalized CK-extensions of codimension In this section we focus on the CK-extension around special surfaces of codimension, more specifically: around spheres and products of spheres. Theorem.3 CK-extension theorem for S m 1 Let A k,0 ω k 0 be given functions. Then there exist unique functions A k,l ω, k 0, l > 0, such that the following toroidal expansion of axial type Sx = k=0 l=0 Z k Z l A k,l ω is monogenic. Moreover, those functions can be calculated using the recurrence relation A k,l+1 = 1 l with c n1,n = 1 n k n 1 =0 n =0 m 1 n1 n l c n1 +n,n Γ x A l n,k n 1 ω Ak n1,l n ω A l n,k n 1 ω, k, l 0, ω n1. ω

48 36 CK-extensions, Fueter s theorems and boundary values Proof. Using.10 and the series expansion 1 r = we obtain x = n 1 =0 n 1 1 r n 1 = n 1 =0 n =0 1 n ω n1 Z Z n 1 n 1 =0 n1 n Z k Z l A k,l = lz k Z l 1 A k,l n 1 + It follows that x S = k=0 l=0 n 1 =0 n =0 ω n1 Z n 1 n Z n, 0 < r <, Z k+n 1 n Z l+n m 1 c n1,n A k,l + Z k+n Z l+n 1 n c n1,n Γ x A k,l Z k Z l l + 1A k,l+1 + Γ x A l n,k n 1 + m 1 k l n 1 =0 n =0 c n1 +n,n m 1 A k,l. Ak n1,l n A l n,k n 1, which proves the theorem. Theorem.4 CK-extension theorem for S p 1 1 S p 1 Consider a toroidal expansion of biaxial type of the form Sx = k=0 l=0 Z k Z l A k,l ω 1, ω, where A 0,l ω 1, ω l 0 are given functions. Then there exist unique functions A k,l ω 1, ω, k > 0, l 0, such that the above sum Sx is monogenic. Moreover, those functions can be calculated using the recurrence

49 Special monogenic series and expressions 37 relation A k+1,l ω 1, ω = ω 1 k + 1 +c k l n 1 =0 n =0 c 1 n 1 +n,n Γ x 1A l n,k n 1 ω 1, ω n 1 +n,n Γ x A l n,k n 1 ω 1, ω p1 1 + c 1 n 1 +n,n + p 1 c n 1 +n,n A k n1,l n ω 1, ω A l n,k n 1 ω 1, ω, k, l 0 with c 1 n 1,n = 1 n1 n1 n c n 1,n = 1 n 1+n n1 n ω 1 ω1 ω n1 ω. Proof. Using.14 and the series expansions 1 r 1 = 1 r = n 1 n 1 =0 n =0 n 1 n 1 =0 n =0 1 n1 n1 Z n 1 n Z n, 0 < r 1 < n ω1 1 n 1+n n1 ω n1 n Z 1 n Z n, 0 < r <. n we obtain x Z k Z l A k,l = kz k 1 Z l ω 1 A k,l + n 1 n 1 =0 n =0 Z k+n 1 n Z l+n p1 1 c 1 n 1,n + p 1 c n 1,n A k,l +Z k+n Z l+n 1 n c 1 n 1,n Γ x 1A k,l + c n 1,n Γ x A k,l

50 38 CK-extensions, Fueter s theorems and boundary values p1 1 c 1 n 1,n + p 1 c n 1,n A k,l. The proof now follows easily. Generalized CK-extension theorems may also be obtained for more general surfaces. We end this chapter with the example of a general surface of codimension which intersects the coordinate planes parallel to the x 0, x 1 -plane transversally. Let p 1 = 1 and assume that α x and β x are given R-valued functions. Theorem.5 Consider the convergent series Sx = k=0 l=0 z k z l A k,l x, z = x 0 α x + x 1 β x e 1. Sufficient for Sx to be monogenic is the recurrence relation l + 1A k,l+1 l + 1 k + 1 x α + e 1 x β x α e 1 x β A l+1,k A l,k+1 + x A l,k = 0, k, l Proof. An easy computation shows that x z k z l A k,l = lz k z l 1 A k,l k z k 1 z l x α e 1 x β l z k z l 1 x α + e 1 x β A k,l A k,l + z k z l x A k,l, from which the theorem follows. In particular, if α x = β x = 0 for all x, then.15 takes the form A k,l+1 x = 1 l + 1 x A l,k x, k, l 0.

51 Special monogenic series and expressions 39 Solving this recurrence relation we get l k l! A k,l x 1 4 l k! l! l A x k l,0 x for k l, = 1 k+1 l k 1! 4 k k! l! x k A x l k 1,0 x for k < l,.16 where x = m j= x j. We thus have obtained the following codimension generalization of the CK-extension theorem. Corollary.1 Let A k,0 x k 0 be given functions, and consider the formal series fx = k=0 l=0 x 0 + x 1 e 1 k x 0 x 1 e 1 l A k,l x. Then there exist unique functions A k,l x, k 0, l > 0, such that the above sum f is monogenic. Moreover, those functions can be calculated using.16.

52

53 Chapter 3 Fueter s theorems In this chapter we present an alternative proof for and a generalization of Fueter s theorem for monogenic functions see [87, 88, 91]. 3.1 An alternative proof Fueter s theorem is named after the Swiss mathematician R. Fueter who in his 1935-paper [55] obtained a method to generate monogenic quaternionic functions starting from a holomorphic function in the upper half of the complex plane. More precisely, if fz = ux, y + ivx, y z = x + iy is a holomorphic function in some open subset Ξ C + = {z = x + iy C : y > 0}, then in the corresponding region, the function F q 0, q = uq 0, q + q q vq 0, q is both left and right monogenic with respect to the quaternionic Cauchy- Riemann operator D = q0 + i q1 + j q + k q3, 41

54 4 CK-extensions, Fueter s theorems and boundary values i.e. DF = F D = 0. Here q = q 1 i + q j + q 3 k is a pure quaternion and = q 0 + q 1 + q + q 3 denotes the Laplace operator in four dimensional space. In [105] Sce extended Fueter s theorem to R 0,m for m odd, i.e. under the same assumptions on f, he showed that the function m 1 x ux0, r + ω vx 0, r is monogenic in Ω = {x R m+1 : x 0, r Ξ}. Using Fourier transformation, Qian proved this result for m being even see [96] and also [73]. In [117] Sommen generalized Sce s result as follows: if m is an odd positive integer and P k x is a homogeneous monogenic polynomial of degree k in R m, then m 1 k+ [ ux0, r + ω vx 0, r P k x ] 3.1 is also monogenic in Ω. x His proof was based on the fact that ux0, r + ω vx 0, r P k x may be written locally as x hx0, rp k x for some R-valued harmonic function h of x 0 and r. Thus 3.1 is monogenic if and only if m+1 k+ x hx0, rp k x = 0, which is true for any R-valued harmonic function h in the variables x 0 and r. The aim of this section is to provide an alternative proof of Sommen s generalization. It is a constructive proof, whence it has the advantage of allowing to compute some examples. Let us outline our proof. First, note that this version of Fueter s theorem provides us with the axial monogenic functions of degree k, i.e. m 1 k+ x [ ux0, r + ω vx 0, r P k x ] = Ax 0, r + ω Bx 0, r P k x

55 Fueter s theorems 43 for some R-valued and continuously differentiable functions A and B. Hence the proof consists in showing that A and B satisfy the Vekua-type system It relies on the following two lemmata. Lemma 3.1 Suppose that ft 1,..., t d and gt 1,..., t d are R-valued infinitely differentiable functions on R d and that D tj and D t j are differential operators defined by D tj 0{f} = D t j 0{f} = f and for n 1 Then one has D tj n{f} = 1 n tj {f}, j = 1,..., d, t j D t j n{f} = tj D t j n 1{f} t j, j = 1,..., d. i t j D tj n{f} = D tj n{ t j f} nd tj n + 1{f}, ii tj D tj n 1 {f/t j } = D t j n{f}, iii D t j n{ tj f} = tj D tj n{f}, iv D tj n{ tj f} tj D t j n{f} = n/t j D t j n{f}, v t j D t j n{f} = D t j n{ t j f} nd t j n + 1{f}, vi D tj n{fg} = n s=0 n s Dtj n s{f}d tj s{g}, vii D t j n{fg} = n s=0 n s Dtj n s{f}d t j s{g}. Proof. We prove i by induction. When n = 1, we have t j D tj 1{f} = 3 t j f t j t j f t j + t j f t 3 j = D tj 1 { t j f } D tj {f}

56 44 CK-extensions, Fueter s theorems and boundary values as desired. Now we proceed to show that when i holds for a positive integer n, then it also holds for n + 1. Indeed, t j D tj n + 1{f} = D tj 1 { t j D tj n{f} } D tj { D tj n{f} } { = D tj 1 D tj n { t j f } } n D tj n + 1{f} D tj n + {f} = D tj n + 1 { t j f } n + 1 D tj n + {f}. Statement ii easily follows from the definition of D t j n{f}. Next, using ii, we obtain iii as D t j n{ tj f} = tj D tj n 1{ tj f/t j } = tj D tj n{f}. To obtain iv we use i and ii: D tj n{ tj f} tj D t j n{f} = D tj n{ tj f} t j D tj n 1{f/t j } = D tj n{ tj f} D tj n 1 { t j {f/t j } } + n 1 D tj n{f/t j } = D tj n{ tj f} D tj n 1 { D tj 1{ tj f} D tj 1{f/t j } } +n 1 D tj n{f/t j } = n D tj n{f/t j } = n t j D t j n{f}. From i-iii it follows that t j D t j n{f} = 3 t j D tj n 1{f/t j } = tj D tj n 1{ t j {f/t j }} n 1 tj D tj n{f/t j } = tj D tj n{ tj f} n tj D tj n{f/t j } = D t j n{ t j f} nd t j n + 1{f}. Finally, vi and vii may be easily proved by induction.