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A Fourier approach to the profile decomposition in Orlicz spaces. (English) Zbl 1311.46030

Summary: This paper is devoted to the characterization of the lack of compactness of the Sobolev embedding of \(H^N(\mathbb R^{2N})\) into the Orlicz space using Fourier analysis. The approach adopted in this paper is strikingly different from the one used in 2D, which consists in tracking the large values of the sequences considered. The analysis we employ in this work is inspired by the strategy of P. Gérard [ESAIM, Control Optim. Calc. Var. 3, 213–233 (1998; Zbl 0907.46027)] and is based on the notion introduced in [H. Bahouri, Trends in Mathematics, 1–15 (2013; Zbl 1291.46030)] of being log-oscillating with respect to a scale.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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