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End date: 2020 × Start date: 2019 × End date: 2014 × Start date: 2013 × Type: Text × Publisher: Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach × Subject: Harmonic manifolds ×

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    Noncompact harmonic manifolds
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Knieper, Gerhard; Peyerimhoff, Norbert
    The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szab´o [Sz] for harmonic manifolds with compact universal cover. E. Damek and F. Ricci [DR] provided examples showing that in the noncompact case the conjecture is wrong. However, such manifolds do not admit a compact quotient. The classification of all noncompact harmonic spaces is still a very difficult open problem. In this paper we provide a survey on recent results on noncompact simply connected harmonic manifolds, and we also prove many new results, both for general noncompact harmonic manifolds and for noncompact harmonic manifolds with purely exponential volume growth.
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    Solid extensions of the Cesàro operator on the Hardy space H2(D)
    (Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach, 2013) Curbera, Guillermo P.; Ricker, Werner J.
    We introduce and study the largest Banach space of analytic functions on the unit disc which is solid for the coefficient- wise order and to which the classical Ces`aro operator C : H2 → H2 can be continuously extended, while still maintaining its values in H2. Properties of this Banach space H(ces2) are presented as well as a characterization of individual analytic functions which belong to H(ces2). In addition, both the multiplier space of H(ces2) and the spectrum of C : H(ces2) → H(ces2) are determined.