DISCRETE AND CONTINUOUS Website: http://math.smsu.edu/journal DYNAMICAL SYSTEMS Volume 7, Number 2, April 2001 pp. 241–246 COMPOSITION IN FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1)(2) AND PETRU MIRONESCU (3) 1. Introduction. A classical result about composition in Sobolev spaces asserts that if u ∈ W k,p (Ω) ∩ L ∞ (Ω) and Φ ∈ C k (R), then Φ ◦ u ∈ W k,p (Ω). Here Ω denotes a smooth bounded domain in R N , k ≥ 1 is an integer and 1 ≤ p< ∞. This result was first proved in [13] with the help of the Gagliardo-Nirenberg inequality [14]. In particular if u ∈ W k,p (Ω) with kp > N and Φ ∈ C k (R) then Φ ◦ u ∈ W k,p since W k,p ⊂ L ∞ by the Sobolev embedding theorem. When kp = N the situation is more delicate since W k,p is not contained in L ∞ . However the following result still holds (see [2],[3]) Theorem 1. Assume u ∈ W k,p (Ω) where k ≥ 1 is an integer, 1 ≤ p< ∞, and kp = N. (1) Let Φ ∈ C k (R) with D j Φ ∈ L ∞ (R) ∀j ≤ k. (2) Then Φ ◦ u ∈ W k,p (Ω) The proof is based on the following Lemma 1. Assume u ∈ W k,p (Ω) ∩ W 1,kp (Ω) where k ≥ 1 is an integer and 1 ≤ p< ∞. Assume Φ ∈ C k (R) satisfies (2). Then Φ ◦ u ∈ W k,p (Ω). Proof of Theorem 1. Since u ∈ W k,p we have ∇u ∈ W k-1,p ⊂ L q by the Sobolev embedding with 1 q = 1 p - k - 1 N . AMS Subject Classification: 46E39 Keywords and phrases: Fractional Sobolev spaces, Besov spaces, composition of functions. Acknowledgment: The first author (H.B.) is partially supported by a European Grant ERB FMRX CT98 0201. He is also a member of the Institut Universitaire de France. Part of this work was done when the second author (P.M.) was visiting Rutgers University; he thanks the Mathematics Department for its invitation and hospitality. We thank S. Klainerman for useful discussions. 241