Ψ-Weighted Cauchy-Riemann operators and some associated integral representation

Abstract

In the present work we introduce a weighted Cauchy-Riemann type operator in the complex plane, where the weights are complex non-constant  functions. We construct a fundamental solution for this operator where the weights are complex constant functions and orthogonal functions inspired by the idea for the construction of the Levy function proposed by Miranda (see [23]). Therefore we obtain a Cauchy-Pompeiu integral representation formula. We also present some examples of such representations when we take some particular weights.

Key words: Weighted Cauchy-Riemann operator, Cauchy integral formula, Cauchy-Pompeiu integral formula, elliptic operator.