A Boxing inequality for the fractional perimeter
(2020) Scuola Normale Superiore di Pisa. Annali. Classe di Scienze — Vol. 20, p. 107-141 (2020)
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- Abstract
- We prove a fractional version of the Boxing inequality that involves the Hausdorff content \(\mathcal{H}^{d-\alpha}_\infty\). We then show how this estimate implies a trace inequality in the fractional Sobolev space \(W^{\alpha,1} (\mathbb{R}^d)\) that includes Sobolev’s \(L^{\frac{d}{d-\alpha}}\) embedding, its Lorentz-space improvement, and Hardy’s inequality. All these estimates are thus obtained with the appropriate asymptotics as \(\alpha\) tends to 0 and 1, recovering in particular the classical inequalities of first order. Their counterparts in the full range \(\alpha\in(0,d)\) are also investigated.
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Ponce, A., & Daniel Spector. (2020). A Boxing inequality for the fractional perimeter. Scuola Normale Superiore di Pisa. Annali. Classe di Scienze, 20, 107-141. https://doi.org/10.2422/2036-2145.201711_012 (Original work published 2020)