  We propose a multi-level type operator that can be used in the framework of
operator (or Cald\'{e}ron) preconditioning to construct uniform preconditioners
for negative order operators discretized by piecewise polynomials on a family
of possibly locally refined partitions. The cost of applying this multi-level
operator scales linearly in the number of mesh cells. Therefore, it provides a
uniform preconditioner that can be applied in linear complexity when used
within the preconditioning framework from our earlier work [Math. of Comp.,
322(89) (2020), pp. 645-674].
