Algebras which satisfy a second order linear partial differential equation

Abstract

Let A be an algebra of complex valued functions satisfying a second order linear partial differential equation in a plane domain. If the equation is hyperbolic or parabolic, the functions of A are locally functions of only one variable. If the equation is elliptic, there exists a unique complex function such that f x = / y for each / in A, and after a change of variables each function in A is analytic. If an algebra of functions satisfies the maximum principle, and one nonconstant function and its square satisfy an elliptic equation, then every function in the algebra satisfies this equation.