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Erschienen in:
2014 | OriginalPaper | Buchkapitel
Transference of Fractional Laplacian Regularity
verfasst von : Luz Roncal, Pablo Raúl Stinga
Erschienen in: Special Functions, Partial Differential Equations, and Harmonic Analysis
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Summary
In this note we show how to obtain regularity estimates for the fractional Laplacian on the multidimensional torus \(\mathbb{T}^{n}\) from the fractional Laplacian on \(\mathbb{R}^{n}\). Though at first glance this may seem quite natural, it must be carefully precised. A reason for that is the simple fact that L
2 functions on the torus cannot be identified with L
2 functions on \(\mathbb{R}^{n}\). The transference is achieved through a formula that holds in the distributional sense. Such an identity allows us to transfer Harnack inequalities, to relate the extension problems, and to obtain pointwise formulas and Hölder regularity estimates.