First Order Differential Operators Associated to the Cauchy—Riemann Operator in the Plane

Abstract

The article deals with initial value problems of type δw/δt = Fw, w(0, ·) = φ where t is the time and F is a linear first order operator acting in the z = x ☎ iy-plane. In view of the classical Cauchy-Kovalevkaya Theorem, the initial value problem is solvable provided F has holomorphic coefficients and the initial function is holomorphic. On the other hand, the Lewy example [H. Lewy (1957). An example of a smooth linear partial differential equation without solution. Ann. of Math., 66, 155–158.] shows that there are equations of the above form with infinitely differentiable coefficients not having any solutions. The article in hand constructs, conversely, all linear operators F for which the initial value problem with an arbitrary holomorphic initial function is always solvable. In particular, we shall see that there are equations of that type whose coefficients are only continuous.