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In this paper we define Besov-Lipschitz and Triebel-Lizorkin spaces in the context of Gaussian harmonic analysis, the harmonic analysis of Hermite polynomial expansions. We study inclusion relations among them, some interpolation results and continuity results of some important operators (the Ornstein-Uhlenbeck and the Poisson-Hermite semigroups and the Bessel potentials) on them. We also prove that the Gaussian Sobolev spaces L p α (γ d ) are contained in them. The proofs are general enough to allow extensions of these results to the case of Laguerre or Jacobi expansions and even further in the general framework of diffusion semigroups.

In this paper we consider an alternative orthogonal decomposition of the space L 2 associated to the d-dimensional Jacobi measure in order to obtain an analogous result to P.A. Meyer's Multipliers Theorem for d-dimensional Jacobi expansions. Then we define and study the Fractional Integral, the Fractional Derivative and the Bessel potentials induced by the Jacobi operator. We also obtain a characterization of the Sobolev or potential spaces and a version of Calderón's reproduction formula for the d-dimensional Jacobi measure. R ÉSUM É. Dans cet article nous considérons une décomposition orthogonale alternative de l'espace L 2 associée à la mesure de Jacobi d-dimensionelle afin d'obtenir de résultat analogues au Théorème des Multiplicateurs de P.A. Meyer pour les développements d-dimensionnels de Jacobi. Nous définissons et étudions l'integral Fractionnaire, la dérivée Fractionnaire et les potentiels de Bessel induits par l'operateur de Jacobi. Nous obtenons ègalement une charactérisation des espaces

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