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arXiv · 1503.02716

Sobolev spaces adapted to the Schr\"odinger operator with inverse-square potential

R. Killip·C. Miao·M. Visan·J. Zhang·J. Zheng
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Abstract

We study the $L^p$-theory for the Schr\"odinger 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 $\mathcal L_a$.

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R. Killip, C. Miao, M. Visan, J. Zhang, J. Zheng. 2015-03-09. Sobolev spaces adapted to the Schr\"odinger operator with inverse-square potential. https://arxiv.org/abs/1503.02716

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