A General Wavelet-Based Profile Decomposition in the Critical Embedding of Function Spaces

Abstract

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 (φ l ) 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 =Ḣ s into the Y = L p 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.

Minimax Critical Points in Ginzburg-Landau Problems with Semi-Stiff Boundary Conditions: Existence and Bubbling

“…This contrasts with the case of thin doubly connected domains, in which minimizers of E ε with prescribed degrees one and one do exist [28], [6]. 2 Next natural question is existence of critical points. Our main result is the following.…”

“…Let us simply mention the pioneering work of Sacks and Uhlenbeck [39] about minimal 2-spheres, the analysis of Brezis and Coron [17] of constant mean curvature surfaces, or the one of Struwe [41] of equations involving the critical Sobolev exponent. There are also abstract approaches to bubbling as in the work of Lions [33] about concentration-compactness or the characterization of lack of compactness of critical embeddings in Gérard [26], Jaffard [31] or Bahouri, Cohen and Koch [2].…”

Description of the lack of compactness of some critical Sobolev embedding

“…Under these assumptions, we proved in [5] that as in the previous works [9] and [12] translational and scaling invariance are the sole responsible for the defect of compactness of the embedding of X → Y . More precisely, we established that the lack of compactness in this embedding can be described in terms of an asymptotic decomposition in the following terms: a sequence (u n ) n≥0 bounded in X can be decomposed up to a subsequence extraction according to…”

“…Recently in [5], the wavelet-based profile decomposition introduced by S. Jaffard in [12] is revisited in order to treat a larger range of examples of critical embedding of functions spaces X → Y including Sobolev, Besov, Triebel-Lizorkin, Lorentz, Hölder and BMO spaces. (One can consult [4] and the references therein for an introduction to these spaces).…”

Logarithmic Littlewood-Paley Decomposition and Applications to Orlicz Spaces

“…Contrary to the case of the elementary concentrations involved in the framework studied by P. Gérard in [18] (see also [3,15,19]) concerning the critical Sobolev embedding…”