COMPOSITION IN FRACTIONAL SOBOLEV SPACES HAIM BREZIS (1)() AND PETRU MIRONESCU (3) 1. Introduction A classical result about composition in Sobolev spaces asserts tat if u W k,p (Ω) L (Ω) and Φ C k (R), ten Φ u W k,p (Ω). Here Ω denotes a smoot bounded domain in R N, k 1 is an integer and 1 p <. Tis result was first proved in [13] wit te elp of te Gagliardo-Nirenberg inequality [14]. In particular if u W k,p (Ω) wit kp > N and Φ C k (R) ten Φ u W k,p since W k,p L by te Sobolev embedding teorem. Wen kp = N te situation is more delicate since W k,p is not contained in L. However te following result still olds (see [],[3]) Teorem 1. Assume u W k,p (Ω) were k 1 is an integer, 1 p <, and (1) kp = N. Let Φ C k (R) wit () D j Φ L (R) j k. Ten Φ u W k,p (Ω) Te proof is based on te following Lemma 1. Assume u W k,p (Ω) W 1,kp (Ω) were k 1 is an integer and 1 p <. Assume Φ C k (R) satisfies (). Ten Φ u W k,p (Ω). Acknowledgment: Te first autor (H.B.) is partially supported by a European Grant ERB FMRX CT98 001. He is also a member of te Institut Universitaire de France. Part of tis work was done wen te second autor (P.M.) was visiting Rutgers University; e tanks te Matematics Department for its invitation and ospitality. We tank S. Klainerman for useful discussions. 1 Typeset by AMS-TEX
HAIM BREZIS (1)() AND PETRU MIRONESCU (3) Proof of Teorem 1. Since u W k,p we ave Du W k 1,p L q by te Sobolev embedding wit 1 q = 1 p k 1 N. Applying assumption (1) we find q = N = kp and tus u W 1,kp. Lemma 1 tat Φ u W k,p. We deduce from Proof of Lemma 1. Note tat if u W k,p L wit k 1 integer and 1 p < ten u W 1,kp by te Gagliardo - Nirenberg inequality [14]. Tus, Lemma 1 is a generalization of te standard result about composition. In fact, it is proved exactly in te same way as in te standard case (wen u W k,p L ). Wen k = te conclusion is trivial. Assume, for example tat, k = 3, ten W 3,p W 1,3p W,3p/ by te Gagliardo - Nirenberg inequality. Ten D 3 (Φ u) = Φ (u)d 3 u + 3Φ (u)d udu + Φ (u)(du) 3, and tus Φ u W 3,p since D u p Du p ( D u 3p/ ) /3( Du 3p) 1/3 A simular argument olds for any k 4. C u p/ W 3,p u 3p/ W 1,3p. Starting in te mid-60 s a number of autors considered composition in various classes of Sobolev spaces W s,p, were s > 0 is a real number and 1 p <. Te most commonly used are te Bessel potential spaces L s,p (R N ) = {f = G s g; g L p (R N )} were Ĝs = (1 + ξ ) s/ and te Besov spaces Bp s,p (R N ) (wo s definition is recalled below wen s is not an integer). It is well-known (see e.g. [1],[19] and [0]) tat if k is an integer, L k,p coincides wit te standard Sobolev space W k,p ; also if p =, te Bessel potential spaces L s, and te Besov spaces B s, coincide for every s non-integer and tey are usually denoted by H s. Wen p te spaces L s,p and Bp s,p are distinct. Te first result about composition in fractional Sobolev spaces seems to be due to Mizoata [1] for H s,s > N/. In 1970 Peetre [15] considered Bp s,p L using interpolation
COMPOSITION IN FRACTIONAL SOBOLEV SPACES 3 tecniques; a very simple direct argument for te same class, B s,p p L, was given by M. Escobedo [10] (see te proof of Lemma below). Starting in 1980 tecniques of dyadic analysis and Littlewood-Paley decomposition à la Bony [5] were introduced. For example, Y. Meyer [11] considered composition in L s,p for sp > N; see also [16],[4],[9] for H s wit s > N/ or for H s L, any s > 0. We refer to [17],[6],[7],[18] and teir bibliograpies for oter directions of researc concerning composition in Sobolev spaces. In wat follow we denote by W s,p (Ω) te restriction of Bp s,p (R N ) to Ω wen s is not an integer. Our main result is te following Teorem. Assume u W s,p (Ω) were s > 1 is a real number, 1 < p <, and (3) sp = N. Let Φ C k (R), were k = [s] + 1, be suc tat (4) D j Φ L (R) j k. Ten Φ u W s,p (Ω). Te proof of Teorem relies on a variant of Lemma 1 for fractional Sobolev spaces. Lemma. Let u W s,p (Ω), were s > 1 is a real number and 1 < p <. Assume, in addition, tat u W σ,q for some σ (0, 1) wit (5) q = sp/σ. Let Φ C k (R), were k = [s] + 1, be suc tat (4) olds. Ten Φ u W s,p Proof of Teorem. By te Sobolev embedding teorem we ave W s,p W r,q wit r < s and In view of assumption (3) we find 1 q = 1 (s r) p N. q = N/r.
4 HAIM BREZIS (1)() AND PETRU MIRONESCU (3) In particular, u W σ,q for all σ (0, 1) wit q = N σ = sp σ. Tus we may apply Lemma and conclude tat Φ u W s,p. Remark 1. Teorem is known to be true wen te Sobolev spaces W s,p are replaced by te Bessel potential spaces L s,p wit sp = N; see D. Adams and M. Frazier [3]. Even toug te two results are closely related it does not seem possible to deduce one from te oter. Teir argument relies on a variant of Lemma for Bessel potential spaces: Let u L s,p L 1,sp were s > 1 is a real number and 1 < p <. Let Φ be as in Lemma. Ten Φ u L s,p. Remark. Te assumption in Lemma, u W s,p W σ,q, wit q = sp/σ for some σ (0, 1), is weaker tan te assumption u W s,p L but it is stronger tan te assumption u W 1,sp ; tis is a consequence of Gagliardo - Nirenberg type inequalities (see e.g. te proof of Lemma D.1 in te Appendix D of [8]). It is terefore natural to raise te following: Open Problem. Is te conclusion of Lemma valid if one assumes only u W s,p W 1,sp were s > 1 is a (non-integer) real number? Before giving te proof of Lemma we recall some properties of W s,p wen s is not an integer. Wen 0 < σ < 1 and 1 < p < te standard definition of W σ,p is W σ,p (Ω) = {f L p (Ω); f(x) f(y) p dxdy < }. x y N+σp If s > 1 is not an integer write s = [s] + σ were [s] denotes te integer part of s and 0 < σ < 1. Ten W s,p (Ω) = {f W [s],p (Ω), D α f W σ,p for α = [s]}. Tere is a very useful caracterization of W s,p in terms of finite differences (see Triebel [0], p.110). Here it is more convenient to work wit functions defined on all of R N and to consider teir restrictions to Ω. Set (δ u)(x) = u(x + ) u(x), R N,
COMPOSITION IN FRACTIONAL SOBOLEV SPACES 5 so tat (δ u)(x) = u(x + ) u(x + ) + u(x), etc... Given s > 0 not integer, fix any integer M > s. Ten W s,p = {f L p ; M f(x) p dxd < }. Proof of Lemma. It suffices to consider te case were s is not an integer. For simplicity we treat just te case were 1 < s <. Te same argument extends to general s >, s noninteger, using te same type of computations as in Escobedo [10]. Te key observation is tat δ (Φ u) can be expressed in terms of δ u and δ u. Tis is te purpose of our next computation. Set X = u(x + ) Y = u(x + ) Z = u(x). Since Φ L (R) we ave (6) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ) and since Φ L (R) we also ave (7) Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y ). Combining (6) and (7) we find Φ(X) Φ(Y ) = Φ (Y )(X Y ) + 0( X Y a ) for any 1 a ( we will coose a specific value of a later) Similarly Φ(Z) Φ(Y ) = Φ (Y )(Z Y ) + 0( Z Y a ) Since δ (Φ u)(x) = (Φ(X) Φ(Y )) + (Φ(Z) Φ(Y )), one finds (8) δ (Φ u)(x) C( δ u(x) + δ u(x + ) a + δ u(x) a ).
6 HAIM BREZIS (1)() AND PETRU MIRONESCU (3) Tis yields (9) (Φ u)(x) p dxd C u(x) p u(x) ap dxd + C dxd. Te first integral on te rigt-and side of (9) is finite since u W s,p. To andle te second integral we argue as follows. From te assumption u W s,p W σ,q wit σ (0, 1) and q given by (5) we know tat (10) u(x) p dxd < and u(x) q From (10) and Hölder s inequality we derive tat (11) u(x) r dxd < for all r [p, q], i.e., u W τ,r wit τ = sp/r. We now coose a = min{, s/σ}, so tat a [1, ] and r = ap [p, q]. It follows tat u(x) ap wic is te desired in equality. dxd <, dxd <. Remark 3. Tere could be anoter natural proof of Teorem by induction on [s]. One migt attempt to prove tat D(Φ u) = Φ (u)du W s 1,p. Note tat u W (s 1),N/(s 1) and tus (by induction) we would ave Φ (u) W (s 1),N/(s 1). On te oter and Du W s 1,p. In order to conclude we need a lemma about products, but we are not aware of any suc tool. Remark 4. Wen s (or equivalently p) is a rational number, and Φ C wit D j Φ L j, tere is a simple proof of Teorem based on trace teory and Teorem 1. Assume for simplicity tat Ω = R N. Suppose tat s is not an integer, but tat s 1 = s + 1/p is an integer. Ten u is te trace of some function u 1 W s1,p (R N+1 ). Ten s 1 p = N +1 and by Teorem 1 we deduce tat Φ u 1 W s1,p (R N+1 ). Taking traces we find Φ u W s,p (R N ). If s 1 is not an integer we keep extending u 1 to iger dimensions and stop at te first integer k suc tat s k = s + k/p is an integer ( tis is possible since p is rational and s + k/p = (N + k)/p becomes an integer for some integer k). We ave an extension u k W sk,p (R N+k ) of u. Ten Φ u k W sk,p (R N+k ) by Teorem 1. Taking back traces yields u W s,p.
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