Applications on Bergman spaces induced by a 𝐯-Laplacian vector fields theory

J. Oscar González-Cervantes,∗, J. Bory-Reyes [29] in their recent works Gonzalez-Cervantes, Luna-Elizarraras and Shapiro [11,12], laid the foundations for the generalization of the theory of Bergman spaces induced by the foundations for the generalization of the theory of Bergman spaces induced byLaplacian (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 ℝ 3 , we extend the aforementioned study in a very natural way to the case of an introduced 𝒗-MToperator for 𝒗 ∈ ℝ 3 , 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. They show that this v-MT operator satisfies a conformal co-variant property. Following [29] we improved the applications of all the above allows to study of Bergman type spaces induced by v-Laplacian vector fields theory.