Uhlenbeck’s Decomposition in Sobolev and Morrey–Sobolev Spaces

Decomposition of S1-valued maps in Sobolev spaces

Let n ≥ 2, s > 0, p ≥ 1 be such that 1 ≤ sp < 2. We prove that for each map u ∈W s,p(Sn;S1) one can find φ ∈ W s,p(Sn;R) and v ∈ W sp,1(Sn;S1) such that u = ve. This yields a decomposition of u into a part that has a lifting in W , e, and a map "smoother" than u but without lifting, namely v. Our result generalizes a previous one of Bourgain and Brezis (which corresponds to the case s = 1/2, p ...

Improved Sobolev Embeddings, Profile Decomposition, and Concentration-compactness for Fractional Sobolev Spaces

We obtain an improved Sobolev inequality in Ḣ spaces involving Morrey norms. This refinement yields a direct proof of the existence of optimizers and the compactness up to symmetry of optimizing sequences for the usual Sobolev embedding. More generally, it allows to derive an alternative, more transparent proof of the profile decomposition in Ḣ obtained in [19] using the abstract approach of di...

Sobolev Spaces

Lifting in Sobolev Spaces

for some function φ : Ω → R. The objective is to find a lifting φ “as regular as u permits.” For example, if u is continuous one may choose φ to be continuous and if u ∈ C one may also choose φ to be C. A more delicate result asserts that if u ∈ VMO (= vanishing means oscillation), then one may choose φ to be also VMO (see R. Coifman and Y. Meyer [1] and H. Brezis and L. Nirenberg [1]). In this...

Commuatators and Sobolev Spaces

In this paper, we consider the commutators of fractional integrals ∫ Rd b(x)− b(y) |x− y| f(y)dy on the Sobolev spaces . H s p where b is a locally integrable function and r ∈ (0, d). We establish the equivalence between the boundedness of the commutators and the paraproducts of J.M. Bony. Then, we establish necessary and sufficient conditions for the boundedness of the paraproduit operator fro...