FOURIER MULTIPLIERS FOR LOCAL HARDY SPACES ON CH EBLI-TRIMÈCHE ´ HYPERGROUPS

ABSTRACT. In this paper we consider Fourier multipliers on local Hardy spaces hp (0 Ú p 1) for Chébli-Trimèche hypergroups. The molecular characterization is investigated which allows us to prove a version of Hörmander’s multiplier theorem. The theory of Fourier multipliers is well developed on euclidean spaces, with various results having been established to give sufficient conditions for a multiplier operator to be bounded on the Lebesgue spaces Lp (p Ù 1) or Hardy spaces Hp (0 Ú p 1). Among these are Hörmander’s multiplier theorem and its variants. Over the past twenty years considerable effort has been made to extend the classical Fourier multiplier theory to groups and hypergroups (see [W], [FS], [Al], [FX] on Lie groups, [CS], [ST], [K], [An] on noncompact symmetric spaces and [S] on Bessel-Kingman hypergroups). In the consideration of this problem a dichotomy is emerging, based on the growth of the volume of balls centered at the identity as their radii become large (polynomial or exponential growth). While in the case of polynomial growth the condition on a multiplier is similar to that on euclidean spaces (see [Al], [FX], [FS] and [W]), some holomorphy of the multiplier is necessary for the operator to be bounded on Lp (p Ù 1) when the volume