Adv. appl. Clifford alg. 17 (2007), 145–152 c  2006 Birkh¨auser Verlag Basel/Switzerland 0188-7009/020145-8, published online February 10, 2007 DOI 10.1007/s00006-007-0025-z Advances in Applied Clifford Algebras The Isotonic Cauchy Transform Ricardo Abreu Blaya, Juan Bory Reyes, Dixan Pe˜ na Pe˜ na and Frank Sommen Abstract. Starting with an integral representation for the class of continuously differentiable solutions f : R 2n → C0,n of the system ∂x 1 f + i ˜ f∂x 2 =0, where C0,n is the complex Clifford algebra constructed over R n , x 1 , x 2 are some suitable Clifford vectors and ∂x 1 , ∂x 2 their corresponding Dirac oper- ators, we define the isotonic Cauchy transform and establish the Sokhotski- Plemelj formulae. Some consequences of this result are also derived. Mathematics Subject Classification (2000). 30G35. Keywords. Clifford analysis, isotonic functions, Sokhotski-Plemelj formulae. 1. Preliminaries We will denote by {e 1 ,...,e m } an orthonormal basis of the Euclidean space R m . Let C 0,m be the complex Clifford algebra constructed over R m . The non- commutative multiplication in C 0,m is governed by the rules: e 2 j = -1, j =1,...,m, e j e k + e k e j =0, j = k. The Clifford algebra C 0,m is generated additively by elements of the form e A = e j1 ...e j k , where A = {j 1 ,...,j k }⊂{1,...,m} is such that j 1 < ··· <j k , and so the dimension of C 0,m is 2 m . For A = ∅, e ∅ = 1, is the identity element. Any Clifford number a ∈ C 0,m may thus be written as a =  A a A e A ,a A ∈ C.