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

On Scales of Sobolev spaces associated to generalized Hardy operators