We characterize the lack of compactness in the critical embedding of functions spaces X ⊂ Y having similar scaling properties in the following terms: a sequence (un)n≥0 bounded in X has a subsequence that can be expressed as a finite sum of translations and dilations of functions (ϕl)l>0 such that the remainder converges to zero in Y as the number of functions in the sum and n tend to +∞. Such a decomposition was established by Gérard in [13] for the embedding of the homogeneous Sobolev space X = Ḣs into the Y = Lp in d dimensions with 0 < s = d/2 - d/p, and then generalized by Jaffard in [15] to the case where X is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular, we identify two generic properties on the spaces X and Y that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of X and Y satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces.
Hajer Bahouri 1 ; Albert Cohen 1 ; Gabriel Koch 1
@article{CML_2011__3_3_387_0,
author = {Hajer Bahouri and Albert Cohen and Gabriel Koch},
title = {A general wavelet-based profile decomposition in the critical embedding of function spaces},
journal = {Confluentes Mathematici},
pages = {387--411},
publisher = {World Scientific Publishing Co Pte Ltd},
volume = {3},
number = {3},
year = {2011},
doi = {10.1142/S1793744211000370},
language = {en},
url = {https://cml.centre-mersenne.org/articles/10.1142/S1793744211000370/}
}
TY - JOUR AU - Hajer Bahouri AU - Albert Cohen AU - Gabriel Koch TI - A general wavelet-based profile decomposition in the critical embedding of function spaces JO - Confluentes Mathematici PY - 2011 SP - 387 EP - 411 VL - 3 IS - 3 PB - World Scientific Publishing Co Pte Ltd UR - https://cml.centre-mersenne.org/articles/10.1142/S1793744211000370/ DO - 10.1142/S1793744211000370 LA - en ID - CML_2011__3_3_387_0 ER -
%0 Journal Article %A Hajer Bahouri %A Albert Cohen %A Gabriel Koch %T A general wavelet-based profile decomposition in the critical embedding of function spaces %J Confluentes Mathematici %D 2011 %P 387-411 %V 3 %N 3 %I World Scientific Publishing Co Pte Ltd %U https://cml.centre-mersenne.org/articles/10.1142/S1793744211000370/ %R 10.1142/S1793744211000370 %G en %F CML_2011__3_3_387_0
Hajer Bahouri; Albert Cohen; Gabriel Koch. A general wavelet-based profile decomposition in the critical embedding of function spaces. Confluentes Mathematici, Tome 3 (2011) no. 3, pp. 387-411. doi : 10.1142/S1793744211000370. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000370/
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