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A partial generalization of the Helgason Conjecture to general bounded homogeneous domains by Richard C. Penney is a Mathematics article available to read on EtoBox.

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Let X = G/K be a Riemannian symmetric space and I ⊂ D G (X ) a co-finite ideal. A function F on X is I-harmonic if I F = 0. A result of Oshima and Sekiguchi [12] says such a function is the Poisson integral of a distribution over the Furstenberg boundary G/P if and only if it has moderate growth. We prove a partial generalization of this result to general, non-symmetric, bounded homogeneous domains in C n . Instead of D G (X ), we use an algebra of geometrically defined differential operators, D geo (X ) which, in the symmetric case, is a subalgebra of D G (X ). We prove the existence of an asymptotic expansion for I-harmonic functions that reduces in the symmetric space case to the expansions due to Wallach [16] and van den Ban and Schlichtkrull [1]. We prove a convergence theorem for these expansions that seems to be new even in the symmetric space case. These expansions are used to define boundary values for F which uniquely determine F. An algorithm constructing F from its boundary values is given.

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Author

Richard C. Penney

Publisher

Walter de Gruyter GmbH

Published

2009

Language

EN

Field

Mathematics (Physical Sciences)