Holmgren's Uniqueness Theorem
A simple consequence of the Bros and Iagolnitzer characterisation of local analyticity is the following regularity result of Lars Hörmander and Mikio Sato (Sjöstrand (1982)).
Theorem. Let P be an elliptic partial differential operator with analytic coefficients defined on an open subset X of Rn. If Pf is analytic in X, then so too is f.
When "analytic" is replaced by "smooth" in this theorem, the result is just Hermann Weyl's classical lemma on elliptic regularity, usually proved using Sobolev spaces (Warner 1983). It is a special case of more general results involving the analytic wave front set (see below), which imply Holmgren's classical strengthening of the Cauchy–Kowalevski theorem on linear partial differential equations with real analytic coefficients. In modern language, Holmgren's uniquess theorem states that any distributional solution of such a system of equations must be analytic and therefore unique, by the Cauchy–Kowalevski theorem.
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