Morera type problems in Clifford analysis

Abstract

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. Specically we show the equivalence between the Morera problem in Clifford analysis and Pompeiu problem for surfaces in ${\mathbb{R}}^n$. We also show an invariance theorem. The noncommutativity 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.