  We reconsider some estimates the paper "M. Griebel, P. Oswald, On additive
Schwarz preconditioners for sparse grid discretizations. Numer. Math. 66
(1994), 449-463" concerning the hierarchical basis preconditioner for sparse
grid discretizations. The improvement is in three directions: We consider
arbitrary space dimensions d>1, give bounds for generalized sparse grid spaces
with arbitrary monotone index set, and show that the bounds are sharp up to
constants depending only on d, at least for a subclass of generalized sparse
grid spaces containing full grid, standard sparse grid spaces, and energy-norm
optimized sparse grid spaces.
