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

TL;DR

This paper develops a unified wavelet-based method to decompose bounded sequences in function spaces with critical embeddings, generalizing previous results to a broader class of spaces like Sobolev, Besov, and BMO.

Contribution

It introduces two generic properties for spaces that enable simplified construction of profile decompositions across various critical embeddings.

Findings

Unified wavelet-based profile decomposition framework

Applicable to Sobolev, Besov, Triebel-Lizorkin, Lorentz, H"older, and BMO spaces

Simplifies previous approaches and broadens applicability.

Abstract

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 $0<s=d/2-d/p$, and then generalized by Jaffard 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, H\"older and BMO spaces.