  We present quasi-diagonal preconditioners for piecewise polynomial
discretizations of pseudodifferential operators of order minus two in any space
dimension. Here, quasi-diagonal means diagonal up to a sparse transformation.
Considering shape regular simplicial meshes and arbitrary fixed polynomial
degrees, we prove, for dimensions larger than one, that our preconditioners are
asymptotically optimal. Numerical experiments in two, three and four dimensions
confirm our results. For each dimension, we report on condition numbers for
piecewise constant and piecewise linear polynomials.
