Complex Variables, Vol. 48, No. 12, pp. 1023–1039 December 2003
Conjugate Hyperharmonic Functions and CauchyType Integrals in Douglis Analysis RICARDO ABREU BLAYAa,*, DIXAN PEN‹A PEN‹Aa,y and JUAN BORY REYESb,z a
Faculty of Mathematics and Informatic, University of Holgu|¤ n, Holgu|¤ n, 80100, Cuba; Faculty of Mathematics and Computer Science, University of Oriente, Santiago of Cuba, 90500, Cuba
b
Communicated by H. Begehr (Received 2 March 2003) In the recent years the so-called hyperanalytic functions theory, i.e., null solutions of the Douglis operator in R2 , has emerged as an increasingly important area of activity for mathematicians. The definition of conjugate hyperharmonic Douglis algebra-valued functions presented in this work is shown to be a generalization of the classical conjugate harmonic functions in the Complex analysis case. An approach to the problem of constructing a conjugate hyperharmonic function to a given hyperharmonic function is considered. The most interesting feature of the article is contained in the following results: It is proved that singular Cauchy integral operator in the sense of Douglis analysis defines a bounded operator on generalized Ho¨lder spaces. A complete characterization of those classes of Douglis algebra-valued functions such that for any regular curve the Cauchy transform (in the Douglis analysis sense) has continuous limit values everywhere on the curve is also obtained. Keywords: Hyperanalytic function; Conjugate hyperharmonic function; Cauchy type integrals Classification Categories: 30G35
1 INTRODUCTION The function theory associated with the Douglis operator in R2 (identifying R2 with C in the usual way) has received much recent attention, both because it offers a natural generalization of classical complex analysis in the plane and also because of its practical use. For greater details the reader is directed to the Gilbert and Buchanan’s book [8], and Begehr and Gilbert’s book [3].
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[email protected] ISSN 0278-1077 print: ISSN 1563-5066 online ß 2003 Taylor & Francis Ltd DOI: 10.1080/02781070310001634548
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R.A. BLAYA et al.
The well-known Douglis system, that is, an elliptic system of first order in two independent variables, can be represented by a single ‘‘hypercomplex’’ equation. Solutions of such equations (null solutions of the Douglis operator) are termed hyperanalytic functions. In [5] Douglis presented a complete study of the hyperanalytic function theory. To get an overview of these topics we recommend the monograph by Wendland [18], and the references therein. In more recent times hyperanalytic function theory has been developed for solving problems of mathematical physics such as plate and shell problems, a wealth of applications are discussed in the monograph by Wendland [18]. For an introductory information regarding the applications of the complex analytic methods for partial differential equations refer to [1]. The title of this article may be somewhat misleading: we have to give some explanation about the meaning of the name Douglis analysis, because it should not be misunderstood. To proclaim this subject one should remember and admire the contribution of Richard Delanghe [4] in the study of mathematical analysis with Clifford algebras. The authors of this article suggested the name of Douglis analysis to give an explicit analogy to the case of Clifford analysis. Concerning boundary behavior properties of the Cauchy transform and singular Cauchy integral operator in Douglis algebras context were investigated in [8,10]. Special singular integral equations using Douglis analytical techniques were treated in [11–13]. More related results to generalize hypercomplex function theory are developed in [7,9]. This article is organized as follows. First we briefly review some basic definitions and results concerning Douglis algebras, associated function theory and related function spaces. Then we give the definition of conjugate hyperharmonic Douglis algebravalued function and consider the problem of constructing a conjugate hyperharmonic function to a given hyperharmonic function. Finally we study boundary behavior of Cauchy transform and singular Cauchy integral operator. 2
PRELIMINARIES
We consider the Douglis algebra generated by i and e, subjected to the multiplication rules i2 ¼ 1,
e0 ¼ 1,
ie ¼ ei,
er ¼ 0,
where r is a positive integer. The elements of this algebra are linear combinations with real coefficients of the 2r independent elements ek ,
iek ,
k ¼ 0, 1, . . . , r 1:
Therefore, an arbitrary element a of the algebra is a hypercomplex number of the form a¼
r1 X
ak e k ,
k¼0
P k where each ak is a complex number; a0 is called the complex part of a and r1 k¼1 ak e the nilpotent part. In particular, if each ak is a real number we say that a is a real Douglis number.
DOUGLIS ANALYSIS
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It is possible to introduce the conjugate element of a which is defined as a :¼
r1 X
ak e k :
k¼0
Note P that the Douglis algebra is commutative. The norm of a is defined by jaj :¼ r1 k¼0 jak j, then for any hypercomplex numbers a and b and
jabj jajjbj
ja þ bj jaj þ jbj:
The multiplicative inverse a1 of a with complex part a0 6¼ 0 is given by a1
or
k r1 1 1X A ¼ ð1Þk , a a0 k¼0 a0
where A is the nilpotent part of a. Observe that if a0 ¼ 0, then a does not have a multiplicative inverse and is called nilpotent. Observe that any hypercomplex number a may be uniquely decomposed as a¼
r1 X
Reðak Þek þ i
k¼0
r1 X
Imðak Þek :
k¼0
Using this decomposition we define the following mappings
