Heat and Harmonic Extensions for Function Spaces of Hardy–Sobolev–Besov Type on Symmetric Spaces and Lie Groups

Abstract

Function spaces of Hardy–Sobolev–Besov type on symmetric spaces of noncompact type and unimodular Lie groups are investigated. The spaces were originally defined by uniform localization. In the paper we give a characterization of the spaceFsp,q(X) andBsp,q(X) in terms of heat and Poisson semigroups, for 1⩽p,q⩽∞ ands∈R. The main tool we use, is an atomic decomposition of function spaces on manifolds.