L P {l Q {estimates for Functions of the Laplace{beltrami Operator on Noncompact Symmetric Spaces, Ii *
نویسندگان
چکیده
In this paper we continue the study of functional calculus for the Laplace{ Beltrami operator on symmetric spaces of the noncompact type begun in 3]; this paper is dedicated to a study of the Poisson semigroup, which we deene shortly. Let G and K be a connected noncompact semisimple Lie group with nite center and a maximal compact subgroup thereof, and consider the symmetric space G=K; also denoted by X: We denote by n the dimension of X; byìts real rank, and by the \pseudo{dimension" 2 + 0 + `; where + 0 is the cardinality of the set of the positive indivisible (restricted) roots. There is a canonical invariant Riemannian metric on X; denote by ?L 0 the associated Laplace{Beltrami operator. By general nonsense, L 0 is positive and essentially self-adjoint on C 1 c (X); let L be the unique self-adjoint extension of L 0 and fP g the spectral resolution of the identity for which
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