Fractional Laplacian on the torus

By Luz Roncal, Pablo Raúl Stinga. 2012

109 citations recorded by the source corpus.

We study the fractional Laplacian $(-\Delta)^{\sigma/2}$ on the $n$-dimensional torus $\mathbb{T}^n$, $n\geq1$. First, we present a general extension problem that describes \textit{any} fractional power $L^\gamma$, $\gamma>0$, where $L$ is a general nonnegative selfadjoint operator defined in an $L^2$-space. This generalizes to all $\gamma>0$ and to a large class of operators the previous known results by Caffarelli and Silvestre. In particular it applies to the fractional Laplacian on the torus. The extension problem is used to prove interior and boundary Harnack's inequalities for $(-\Delta)^{\sigma/2}$, when $0<\sigma<2$. We deduce regularity estimates on H\"older, Lipschitz and Zygmund spaces. Finally, we obtain the pointwise integro-differential formula for the operator. Our method is based on the semigroup language approach.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/50e1a24c1d0258bd925fdb4f7109b85ecedcc0b1

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In Applied Mathematics

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Citations in the corpus

Some constructions for the fractional Laplacian on noncompact manifolds (2012, methodology): “1) has been considered in [64], using a double Fourier series expansion.”
User’s guide to the fractional Laplacian and the method of semigroups (2018, methodology): “s using the extension technique for fractional powers of operators in divergence form were systematically developed in [95]. For applications of the extension method to other fractional operators see [14, 29, 39, 46, 70, 76, 90, 91, 92, 93, 94]. Harnack inequalities for degenerate elliptic equations”
Fractional elliptic equations, Caccioppoli estimates and regularity (2014, background): “under the norm ‖u‖Λ∗(Rn) := ‖u‖L∞(Rn) + [u]Λ∗(Rn), see [32] (also [26, Chapter V] or [21]).”
Nonlocal discrete diffusion equations and the fractional discrete Laplacian, regularity and applications (2016, background): “…has been a substantial revision of a big amount of problems in diﬀerential equations where the Laplacian is replaced by the fractional Laplacian or more general integro-diﬀerential operators, see for example [3, 5, 9, 21, 22, 23, 24, 25, 28, 29] and references therein for models and techniques.”
What is the fractional Laplacian? A comparative review with new results (2018, background): “, [60], and for discussions of extension characterizations as well as well-posedness and regularity, see [61,62].”
Sharp commutator estimates via harmonic extensions (2016, methodology): “operator (1 − ∆)s/2, and thus for which our arguments might be useful. We treat only differential orders < 2, sometimes even < 1. Since there are higher order extensions to local operators, see [57, 41], it should, in principle, be also possible to obtain higher order commutator estimates. Fur”