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On holomorphic extendability and the strong maximum principle for CR functions. (English) Zbl 1472.32017

The paper investigates necessary and sufficient conditions for a \(\mathcal{C}^\infty\) surface \(\mathcal{M}\) in \(\mathbb{C}^n\) to satisfy a strong maximum principle for CR-functions.
Definition. \(\mathcal{M}\) satisfies the weak maximum principle if \(\forall U\subset \mathcal{M}\) connected open set and \(\forall h\) CR-function in \(U\), either \(h\) is constant or \(|h|\) has no weak local maximum in \(U\).
Definition. \(\mathcal{M}\) satisfies the strong maximum principle if \(\forall U\subset \mathcal{M}\) connected open set and \(\forall h\) CR-function in \(U\), \(|h|\) has no strict maximum in \(U\).
While the weak maximum principle is well understood, the same is not true for the strong one. In the paper various results (for the general case or specific cases) are proven:
If \(\mathcal{M}\) satisfies the extendability property (i.e., all CR-functions on \(\mathcal{M}\) extend to holomorphic functions), then \(\mathcal{M}\) satisfies the strong maximum principle.
(nearly a viceversa) If \(\mathcal{M}\) satisfies the strong maximum principle, then there is a dense subset \(\Sigma\subset\mathcal{M}\) such that CR-functions on \(\mathcal{M}\) extend to a neighbourhood of \(\Sigma\).
For real analytic tubes, the strong maximum principle is equivalent to analytic hypoellipticity.
If \(\mathcal{M}\) is of hypersurface type and every locally integrable CR distribution defined on an open subset of \(\mathcal M\) is smooth, then \(\mathcal M\) satisfies the strong maximum principle.

The paper also devotes some space to counterexamples, e.g., an example of a hypersurface \(\mathcal{M}\subset\mathbb{C}^6\) satisfying the strong maximum principle for the restrictions of holomorphic functions, but not for CR-functions. An example like this was already known [the author, Contemp. Math. 205, 1–13 (1997; Zbl 0901.32015)], but the example here is simpler.

MSC:

32V40 Real submanifolds in complex manifolds
32D15 Continuation of analytic objects in several complex variables

Citations:

Zbl 0901.32015
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References:

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