Holomorphic extensions of functions on submanifolds: A generalization of H, lewy's example

Abstract

In the present paper the differential equation

$$\frac{{\partial u}}{{\partial t}} = \sigma (x,y,t)\frac{{\partial u}}{{\partial x}} + \tau (x,y,t)\frac{{\partial u}}{{\partial y}}$$

is investigated from a new point of view. Instead of the holomorphy of the coefficients, another sufficient condition is obtained which ensures the existence of at least two linearly independent solutions such that any solution may be extended to a holomorphic function.