Convolution equations and mean-value theorems for solutions of linear elliptic equations with constant coefficients in the complex plane

Abstract

In terms of the Bessel functions, we characterize smooth solutions of some convolution equations in the complex plane and prove a two-radius theorem for solutions of homogeneous linear elliptic equations with constant coefficients whose left-hand sides are representable in the form of a product of some non-negative integer powers of the complex differentiation operators ∂ and \( \overline{\partial} \).