Resumen de Morera type problems in Clifford analysis

Emilio Marmolejo Olea

• The Pompeiu and the Morera problems have been studied in many contexts and generality. For example in different spaces, with different groups, locally, without an invariant measure, etc. The variations obtained exhibit the fascination of these problems.

  In this paper we present a new aspect: we study the case in which the functions have values over a Clifford Algebra. We show that in this context it is completely natural to consider the Morera problem and its variations. Specifically, we show the equivalence between the Morera problem in Clifford analysis and Pompeiu problem for surfaces in Rn. We also show an invariance theorem. The non-commutativity of the Clifford algebras brings in some peculiarities.

  Our main result is a theorem showing that the vanishing of the first moments of a Clifford valued function implies Clifford analyticity. The proof depends on results which show that a particular matrix system of convolution equations admits spectral synthesis.