On Bergman spaces induced by a v-Laplacian vector fields theory

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5,8,15,28] and the references therein, which contain a broad summary and historical notes of the subject, that frees us from referring to missing details. In their recent works González-Cervantes, Luna-Elizarrarás and Shapiro [11,12], laid the foundations for the generalization of the theory of Bergman spaces induced by Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields by taking advantage on the intimate connections between harmonic vector fields theory and quaternionic analysis for the Moisil-Theodorescu operator (MT-operator for short). A deeper discussion of the last mentioned relation can be found in [1]. On the setting of general bounded domains in R³, we extend the aforementioned study in a very natural way to the case of an introduced v-MT-operator for v∈R³, proving several properties of induced Bergman spaces and some relative results about Stokes and Borel-Pompieu formulas for v-MT-hyperholomorphic functions, i.e., functions which belong to kernel of the v-MT-operator. In addition, we show that this v-MT operator satisfies a conformal co-variant property. Application of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory, which represents the main goal of this paper.