tailieunhanh - Đề tài " Van den Ban-SchlichtkrullWallach asymptotic expansions on nonsymmetric domains "

Let X = G/K be a homogeneous Riemannian manifold where G is the identity component of its isometry group. A C ∞ function F on X is harmonic if it is annihilated by every element of DG (X), the algebra of all G-invariant differential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture, which states that on a Riemannian symmetric space of noncompact type, a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G/Po where. | Annals of Mathematics Van den Ban-Schlichtkrull- Wallach asymptotic expansions on nonsymmetric domains By Richard Penney Annals of Mathematics 158 2002 711 768 van den Ban-Schlichtkrull-Wallach asymptotic expansions on nonsymmetric domains By Richard PENNEy Introduction Let X G K be a homogeneous Riemannian manifold where G is the identity component of its isometry group. A C function F on X is harmonic if it is annihilated by every element of Dg X the algebra of all G-invariant differential operators without constant term. One of the most beautiful results in the harmonic analysis of symmetric spaces is the Helgason conjecture which states that on a Riemannian symmetric space of noncompact type a function is harmonic if and only if it is the Poisson integral of a hyperfunction over the Furstenberg boundary G Po where Po is a minimal parabolic subgroup. See 14 17 . One of the more remarkable aspects of this theorem is its generality one obtains a complete description of all solutions to the system of invariant differential operators on X without imposing any boundary or growth conditions. If X is a Hermitian symmetric space then one is typically interested in complex function theory in which case one is interested in functions whose boundary values are supported on the Shilov boundary rather than the Fursten-berg boundary. The Shilov boundary is G P where P is a certain maximal parabolic containing Po. In this case it turns out that the algebra of G invariant differential operators is not necessarily the most appropriate one for defining harmonicity. Johnson and Koranyi 16 generalizing earlier work of Hua 15 Korányi-Stein 19 and Korányi-Malliavin 18 introduced an invariant system of second order differential operators the HJK system defined on any Hermitian symmetric space. In 9 we noted that this system could be defined entirely in terms of the geometric structure of X as HJK - V2 ZiZj R Zi Zj T01 This work was partially supported by NSF grant DMS-9970762. 712 .