Codomains for the Cauchy-Riemann and Laplace operators in ℝ2

Abstract

A codomain for a nonzero constant-coefficient linear partial differential operator P(∂) with fundamental solution E is a space of distributions T for which it is possible to define the convolution E*T and thus solving the equation P(∂)S=T. We identify codomains for the Cauchy-Riemann operator in ℝ2 and Laplace operator in ℝ2 . The convolution is understood in the sense of the S′-convolution.