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

By Rowan Killip, Changxing Miao, Monica Visan, Junyong Zhang, Jiqiang Zheng. 2015

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We study the $$L^p$$Lp-theory for the Schrödinger operator $$\mathcal L_a$$La with inverse-square potential $$a|x|^{-2}$$a|x|-2. Our main result describes when $$L^p$$Lp-based Sobolev spaces defined in terms of the operator $$(\mathcal L_a)^{s/2}$$(La)s/2 agree with those defined via $$(-\Delta )^{s/2}$$(-Δ)s/2. We consider all regularities $$0<s<2$$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 $$\mathcal L_a$$La.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/092949ebb6f2448953841a5f46a9541852737038