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

Estimates for generalized sparse grid hierarchical basis preconditioners