A general wavelet-based profile decomposition in the critical embedding of function spaces

Details A general wavelet-based profile decomposition in the critical embedding of function spaces A general wavelet-based profile decomposition in the critical embedding of function spaces

2011-03-12

arxiv.org/abs/1103.2468v1

Mathematics

Functional Analysis

24 pages

Scientific paper

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0

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