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

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• 作者：Matteo Muratori ^{matteo.muratori@unipv.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional Laplacian ; Fractional Sobolev spaces ; Self-adjointness ; Degenerate diffusions ; Weights
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：271
• 期：12
• 页码：3662-3694
• 全文大小：611 K

文摘

We consider the fractional Laplacian operator (−Δ)^s${(-\mathrm{\Delta })}^s$ (let s∈(0,1)$s\in (0,1)$) on Euclidean space and investigate the validity of the classical integration-by-parts formula that connects the L²(R^d)${\mathrm{L}}^2({\mathbb{R}}^d)$ scalar product between a function and its fractional Laplacian to the nonlocal norm of the fractional Sobolev space ${\dot{\mathrm{H}}}^s({\mathbb{R}}^d)$. More precisely, we focus on functions belonging to some weighted L²${\mathrm{L}}^2$ space whose fractional Laplacian belongs to another weighted L²${\mathrm{L}}^2$ space: we prove and disprove the validity of the integration-by-parts formula depending on the behaviour of the weight ρ(x)$\rho (x)$ 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 ρ⁻¹(−Δ)^s${\rho }^{-1}{(-\mathrm{\Delta })}^s$. The generality of the techniques developed allows us to deal with weighted L^p${\mathrm{L}}^p$ spaces as well.