Uniqueness of analytic continuation: Necessary and sufficient conditions

AbstractNecessary and sufficient conditions for uniqueness of analytic continuation are investigated for a system of m ⩾ 1 first-order linear homogeneous partial differential equations in one unknown, with complex-valued b∞ coefficients, in some connected open subset of Rk, k ⩾ 2. The type of system considered is one for which there exists a real k-dimensional, b∞, connected C-R submanifold Mk of Cn, for k, n ⩾ 2, such that the system may be identified with the induced Cauchy-Riemann operators on Mk. The question of uniqueness of analytic continuation for a system of partial differential equations is thus transformed to the question of uniqueness of analytic continuation for C-R functions on the manifold Mk ⊂ Cn. Under the assumption that the Levi algebra of Mk has constant dimension, it is shown that if the excess dimension of this algebra is maximal at every point, then Mk has the property of uniqueness of analytic continuation for its C-R functions. Conversely, under certain mild conditions, it is shown that if Mk has the property of uniqueness of analytic continuation for all b∞ C-R functions, and if the Levi algebra has constant dimension on all of Mk, then the excess dimension must be maximal at every point of Mk