A note on the fractional perimeter and interpolation. (Une note sur le périmètre fractionnaire et l’interpolation.) (English. French summary) Zbl 1385.46024
Summary: We present the fractional perimeter as a set-function interpolation between the Lebesgue measure and the perimeter in the sense of E. De Giorgi [in: Equazioni differenziali e calcolo delle variazioni. Bologna: Pitagora Editrice. 237–250 (1995; Zbl 0942.49503)]. Our motivation comes from a new fractional boxing inequality that relates the fractional perimeter and the Hausdorff content and implies several known inequalities involving the Gagliardo seminorm of the Sobolev spaces \(W^{\alpha, 1}\) of order \(0 < \alpha < 1\).
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 46B70 | Interpolation between normed linear spaces |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 28A78 | Hausdorff and packing measures |
Citations:
Zbl 0942.49503References:
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