Equivalence of Sobolev norms involving generalized Hardy operators

Abstract

We consider the fractional Schrödinger operator with Hardy potential and critical or subcritical coupling constant. This operator generates a natural scale of homogeneous Sobolev spaces which we compare with the ordinary homogeneous Sobolev spaces. As a byproduct, we obtain generalized and reversed Hardy inequalities for this operator. Our results extend those obtained recently for ordinary (non-fractional) Schrödinger operators and have an important application in the treatment of large relativistic atoms.

21 pages; v3 contains a new appendix, which extends Theorem 1.1 to all functions in the domains of $(|p{|}^{\alpha }+a|x{|}^{-\alpha }{)}^{s/2}$ and $|p{|}^{\alpha s/2}$, respectively

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• Lower bounds on the moduli of three-dimensional Coulomb-Dirac operators via fractional Laplacians with applications
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Cited by in corpus (11)

• Proof of the Strong Scott Conjecture for Chandrasekhar Atoms
• On Scales of Sobolev spaces associated to generalized Hardy operators
• The Scott conjecture for large Coulomb systems: a review
• On Sobolev norms involving Hardy operators in a half-space
• On the excess charge of a relativistic statistical model of molecules with an inhomogeneity correction
• Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels
• On complex-time heat kernels of fractional Schrödinger operators via Phragmén-Lindelöf principle
• Proof of the Strong Scott Conjecture for Heavy Atoms: the Furry Picture
• On the number and sums of eigenvalues of Schrödinger-type operators with degenerate kinetic energy
• On the fractional powers of a Schrödinger operator with a Hardy-type potential
• The Ground State Energy of Heavy Atoms: Leading and Subleading Asymptotics