The C-property for slice regular functions and applications to the Bergman space

Abstract

This article has a twofold purpose: on one hand, we deepen the study of slice regular functions by studying their behaviour with respect to the so-called C-property and anti-C-property. We show that, for any fixed basis of the algebra of quaternions ℍ any slice regular function decomposes into the sum of four slice regular components, each of them satisfying the C-property. Then, we will use these results to show a reproducing property of the Bergman kernels of the second kind.