  We consider the fractional Laplacian with Hardy potential and study the scale
of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides
generalized and reversed Hardy inequalities, the analysis relies on a
H\"ormander multiplier theorem which is crucial to construct a basic
Littlewood--Paley theory. The results extend those obtained recently in $L^2$
but do not cover negative coupling constants in general due to the slow decay
of the associated heat kernel.
