The Constant Term of Tempered Functions on a Real Spherical Space

Let Z be a unimodular real spherical space. We develop a theory of constant terms for tempered functions on Z, which parallels the work of Harish-Chandra. The constant terms fI of an eigenfunction f are parametrized by subsets I of the set S of spherical roots that determine the fine geometry of Z at infinity. Constant terms are transitive i.e., (fJ)I=fI for I⊂ J, and our main result is a quantitative bound of the difference f-fI, which is uniform in the parameter of the eigenfunction.