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Spectral multipliers for the Ornstein-Uhlenbeck semigroup

Research paper by José García-Cuerva, Giancarlo Mauceri, Peter Sjögren, José Luis Torrea

Indexed on: 01 Dec '99Published on: 01 Dec '99Published in: Journal d'Analyse Mathématique



Abstract

The setting of this paper is Euclidean space with the Gaussian measure. We letL be the associated Laplacian, by means of which the Ornstein-Uhlenbeck semigroup is defined. The main result is a multiplier theorem, saying that a function ofL which is of Laplace transform type defines an operator of weak type (1,1) for the Gaussian measure. The (distribution) kernel of this operator is determined, in terms of an integral involving the kernel of the Ornstein-Uhlenbeck semigroup. This applies in particular to the imaginary powers ofL. It is also verified that the weak type constant of these powers increases exponentially with the absolute value of the exponent.