Sobolev spaces adapted to the Schrödinger operator with inverse-square potential

Abstract

We study the $L^p$-theory for the Schrödinger operator ${\mathcal{L}}_a$ with inverse-square potential $a|x{|}^{-2}$. Our main result describes when $L^p$-based Sobolev spaces defined in terms of the operator $({\mathcal{L}}_a{)}^{s/2}$ agree with those defined via $(-\Delta {)}^{s/2}$. We consider all regularities $0<s<2$. In order to make the paper self-contained, we also review (with proofs) multiplier theorems, Littlewood-Paley theory, and Hardy-type inequalities associated to the operator ${\mathcal{L}}_a$.

25 pages

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