On Some Complex Differential and Singular Integral Operators

Abstract

The Pompeiu operator in complex analysis as the right inverse of the Cauchy-Riemann operator provides a particular solution to the inhomogeneous Cauchy-Riemann equation. In case of the entire plane ℂ or the whole space ℂⁿ under proper decay conditions on the solution it gives even the unique solution. Taking the z-derivative then relates this derivative to the $\bar z$ -derivative of the same function via the Ahlfors-Beurling operator. This area integral operator is singular of Calderon-Zygmund type. This situation is reflected to any higher order partial differential operator of fixed order. All n-th order derivatives are expressible by just one particular one through proper singular integral operators of Calderon-Zygmund type emerging from higher order Pompeiu operators within a hierarchy of integral operators through proper differentiation.

The situation is found true also in bounded domains. If the kernels of the higher order Pompeiu operators are altered by replacing them through proper derivatives of higher order Green functions then these operators turn out to be projections on L₂-subspaces orthogonal to the kernel of the related higher order partial differential operator. The unique solution to the related inhomogeneous partial differential equation is provided by this projective operator. All other derivatives of the same order of the solution are then expressed by the given one through singular integral operators. The situation is considered in particular for the unit disc in ℂ, the unit ball and the unit polydisc in ℂⁿ. In ℂ² also the Fueter system is treated.