On scales of Sobolev spaces associated to generalized Hardy operators

We consider the fractional Laplacian with Hardy potential and compare the scale of homogeneous $$L^p$$ Sobolev spaces generated by this operator with the ordinary homogeneous Sobolev spaces. The proof relies on a generalized Hardy inequality, a reversed Hardy inequality expressed in terms of square functions, and a Hormander multiplier theorem which is proven for positive coupling constants. The latter is crucial to obtain Bernstein and square function estimates associated to this operator. 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.