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Generalized Littlewood-Paley characterizations of fractional Sobolev spaces. (English) Zbl 1410.46024

Summary: In this paper, the authors characterize the Sobolev spaces \(W^{\alpha,p}(\mathbb R^n)\) with \(\alpha\in(0,2]\) and \(p\in(\max\{1,\frac{2n}{2\alpha+n}\},\infty)\) via a generalized Lusin area function and its corresponding Littlewood-Paley \(g_\lambda^\ast\)-function. The range \(p\in(\max\{1,\frac{2n}{2\alpha+n}\},\infty)\) is also proved to be nearly sharp in the sense that these new characterizations are not true when \(\frac{2n}{2\alpha+n}>1\) and \(p\in(1,\frac{2n}{2\alpha+n})\). Moreover, in the endpoint case \(p=\frac{2n}{2\alpha+n}\), the authors also obtain some weak type estimates. Since these generalized Littlewood-Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B25 Maximal functions, Littlewood-Paley theory
47G40 Potential operators
31B15 Potentials and capacities, extremal length and related notions in higher dimensions
30L99 Analysis on metric spaces
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