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 \(f_I\) of an eigenfunction \(f\) are parametrized by subsets \(I\) of the set \(S\) of spherical roots which determine the fine geometry of \(Z\) at infinity. Constant terms are transitive i.e. \((f_J)_I=f_I\) for \(I\subset J\), and our main result is a quantitative bound of the difference \(f-f_I\), which is uniform in the parameter of the eigenfunction.