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Maximum modulus sets in pseudoconvex boundaries. (English) Zbl 0772.32012
We quote the author’s abstract: “Let $$D$$ be a strictly pseudoconvex domain in $$\mathbb{C}^ n$$ with $$C^ \infty$$ boundary. We denote by $$A^ \infty(D)$$ the set of holomorphic functions in $$D$$ that have a $$C^ \infty$$ extension to $$\overline D$$. A closed subset $$E$$ of $$\partial D$$ is locally a maximum modulus set of $$A^ \infty(D)$$ if for every $$p\in E$$ there exists a neighborhood $$U$$ of $$p$$ and $$f\in A^ \infty(D\cap U)$$ such that $$| f|=1$$ on $$E\cap U$$ and $$| f|<1$$ on $$\overline D\cap U\backslash E$$. A submanifold $$M$$ of $$\partial D$$ is an interpolation manifold if $$T_ p(M)\subset T^ c_ p(\partial D)$$ for every $$p\in M$$, where $$T^ c_ p(\partial D)$$ is the maximal complex subspace of the tangent space $$T_ p(\partial D)$$. We prove that a local maximum modulus set for $$A^ \infty(D)$$ is locally contained in totally real $$n$$-dimensional submanifolds of $$\partial D$$ that admit a unique foliation by $$(n-1)$$-dimensional interpolation submanifolds. Let $$D=D_ 1\times\cdots\times D_ r\subset\mathbb{C}^ n$$ where $$D_ i$$ is a strictly pseudoconvex domain with $$C^ \infty$$ boundary in $$\mathbb{C}^{n_ i}$$, $$i=1,\dots,r$$. A submanifold $$M$$ of $$\partial D_ 1\times\cdots\times\partial D_ r$$ verifies the cone condition if $$\Pi_ p(T_ p(M))\cap\overline C[Jn_ 1(p),\dots,Jn_ r(p)]=\{0\}$$ for every $$p\in M$$, where $$n_ i(p)$$ is the outer normal to $$D_ i$$ at $$p$$, $$J$$ is the complex structure of $$\mathbb{C}^ n$$, $$\overline C[Jn_ 1(p),\dots,Jn_ r(p)]$$ is the closed positive cone of the real space $$V_ p$$ generated by $$Jn_ 1(p),\dots,Jn_ r(p)$$, and $$\Pi_ p$$ is the orthogonal projection of $$T_ p(\partial D)$$ on $$V_ p$$. We prove that a closed subset $$E$$ of $$\partial D_ 1\times\cdots\times\partial D_ r$$ which is locally a maximum modulus set for $$A^ \infty(D)$$ is locally contained in $$n$$-dimensional totally real submanifolds of $$\partial D_ 1\times\cdots\times\partial D_ r$$ that admit a foliation by $$(n-1)$$-dimensional submanifolds such that each leaf verifies the cone condition at every point of $$E$$. A characterization of the local peak subsets of $$\partial D_ 1\times\cdots\times\partial D_ r$$ is also given”.

##### MSC:
 32A38 Algebras of holomorphic functions of several complex variables 32T99 Pseudoconvex domains 32V40 Real submanifolds in complex manifolds
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##### References:
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