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. Keywords. Author Keywords:Pseudodifferential operator of negative orderDiagonal scalingAdditive Schwarz methodPreconditionerNegative order Sobolev spacesRegular 2015FONDECYTFONDECYT