  We present weighted covariant derivatives and wave operators for
perturbations of certain algebraically special Einstein spacetimes in arbitrary
dimensions, under which the Teukolsky and related equations become weighted
wave equations. We show that the higher dimensional generalization of the
principal null directions are weighted conformal Killing vectors with respect
to the modified covariant derivative. We also introduce a modified Laplace-de
Rham-like operator acting on tensor-valued differential forms, and show that
the wave-like equations are, at the linear level, appropriate projections {\em
off shell} of this operator acting on the curvature tensor; the projection
tensors being made out of weighted conformal Killing-Yano tensors. We give off
shell operator identities that map the Einstein and Maxwell equations into
weighted scalar equations, and using adjoint operators we construct solutions
of the original field equations in a compact form from solutions of the
wave-like equations. We study the extreme and zero boost weight cases; extreme
boost corresponding to perturbations of Kundt spacetimes (which includes Near
Horizon Geometries of extreme black holes), and zero boost to static black
holes in arbitrary dimensions. In 4 dimensions our results apply to Einstein
spacetimes of Petrov type D and make use of weighted Killing spinors.
