  We consider multilevel decompositions of piecewise constants on simplicial
meshes that are stable in $H^{-s}$ for $s\in (0,1)$. Proofs are given in the
case of uniformly and locally refined meshes. Our findings can be applied to
define local multilevel diagonal preconditioners that lead to bounded condition
numbers (independent of the mesh-sizes and levels) and have optimal
computational complexity. Furthermore, we discuss multilevel norms based on
local (quasi-)projection operators that allow the efficient evaluation of
negative order Sobolev norms. Numerical examples and a discussion on several
extensions and applications conclude this article.
