12 research outputs found
The Friedrichs Operator and Circular Domains
The Friedrichs operator of a domain (in ) is closely related to
its Bergman projection and encodes crucial information (geometric, quadrature,
potential theoretic etc.) about the domain. We show that the Friedrichs
operator of a domain has rank one if the domain can be covered by a circular
domain via a proper holomorphic map of finite multiplicity whose Jacobian is a
homogeneous polynomial. As an application, we show that the Friedrichs operator
is of rank one on the tetrablock, pentablock, and the symmetrized polydisc -
domains of significance in the study of -synthesis in control theory.Comment: 10 page
Cartan uniqueness theorem on nonopen sets
Cartan's uniqueness theorem does not hold in general for CR mappings, but it
does hold under certain conditions guaranteeing extendibility of CR functions
to a fixed neighborhood. These conditions can be defined naturally for a wide
class of sets such as local real-analytic subvarieties or subanalytic sets, not
just submanifolds. Suppose that is a locally connected and locally closed
subset of such that the hull constructed by contracting
analytic discs close to arbitrarily small neighborhoods of a point always
contains the point in the interior. Then restrictions of holomorphic functions
uniquely extend to a fixed neighborhood of the point. Using this extension, we
obtain a version of Cartan's uniqueness theorem for such sets. When is a
real-analytic subvariety, we can generalize the concept of infinitesimal CR
automorphism and also prove an analogue of the theorem. As an application of
these two results we show that, for circular subvarieties, the only
automorphisms, CR or infinitesimal, are linear.Comment: 10 pages, updated proof of Lemma 2.2, expanded section 3, and added
reference [9