We consider the fractional Laplacian operator (−Δ)s (let s∈(0,1)) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the L2(Rd) scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space H˙s(Rd). More precisely, we focus on functions belonging to some weighted L2 space whose fractional Laplacian belongs to another weighted L2 space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight ρ(x) at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by ρ−1(−Δ)s. The generality of the techniques developed allows us to deal with weighted Lp spaces as well.

The fractional Laplacian in power-weighted Lp spaces: Integration-by-parts formulas and self-adjointness

MURATORI, MATTEO
2016

Abstract

We consider the fractional Laplacian operator (−Δ)s (let s∈(0,1)) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the L2(Rd) scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space H˙s(Rd). More precisely, we focus on functions belonging to some weighted L2 space whose fractional Laplacian belongs to another weighted L2 space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight ρ(x) at infinity. The latter is assumed to be like a power both near the origin and at infinity (the two powers being possibly different). Our results have direct consequences for the self-adjointness of the linear operator formally given by ρ−1(−Δ)s. The generality of the techniques developed allows us to deal with weighted Lp spaces as well.
Degenerate diffusions; Fractional Laplacian; Fractional Sobolev spaces; Self-adjointness; Weights; Analysis
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1015181
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