The fractional Laplacian in power-weighted $L^p$ spaces: integration-by-parts formulas and self-adjointness

Abstract

We consider the fractional Laplacian operator $(-\Delta {)}^s$ (let $s\in (0,1)$) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the $L^2({\mathbb{R}}^d)$ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space ${\dot{H}}^s({\mathbb{R}}^d)$. More precisely, we focus on functions belonging to some weighted $L^2$ space whose fractional Laplacian belongs to another weighted $L^2$ space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight $\rho (x)$ 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 ${\rho }^{-1}(-\Delta {)}^s$. The generality of the techniques developed allows us to deal with weighted $L^p$ spaces as well.

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• From the long jump random walk to the fractional Laplacian
• Existence, Uniqueness and Asymptotic behaviour for fractional porous medium equations on bounded domains