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In Section 2 we recall the classical result on the continuity of Tf in Lp spaces. We take this opportunity to correct some erroneous statement in the literature. In Section 3, we recall the characterization of composition operators acting in inhomogenous and homogeneous Adams-Frazier spaces, and in Sobolev algebras. In Section 4 we explain the specific difficulties concerning the continuity of Tf in homogeneous spaces, which can be par- tially overcome by using realizations. Section 5 is devoted to the proof of the continuity of Tf .

We denote by N the of all positive integers, including 0. All functions occurring in the n paper are assumed to be real valued. We denote by Pk the set of on R , of n degree less or equal to k. If f is a on R , we denote by [f]k its equivalence class n modulo Pk. We consider a classical mollifiers θν(x) := ν θ(νx), ν ≥ 1, where Rn N N R θ ∈D( ) and Rn θ(x) dx = 1. For all N ∈ , we denote by Cb ( ) the space of functions f : R → R, with continuous bounded derivatives up to order N. In all the paper, m is R m Rn ˙ m Rn an integer > 1 and the real number p satisfies 1 ≤ p < +∞. Wp ( ) and Wp ( ) are the classical inhomogeneous and homogeneous Sobolev spaces, endowed with the norms and seminorms (α) (α) kgk m := kg k , kgk ˙ m := kg k , Wp p Wp p |αX|≤m |αX|=m :.respectively. For topological spaces E, F , the symbol E ֒→ F means an imbedding, i.e E ⊆ F and the natural mapping E → F is continuous.

The authors are grateful to Alano Ancona, Mihai Brancovan and Thierry Jeulin for fruitful discussions during the preparation of the paper.

2 Thecaseof Lp In a recent survey paper on , the first author said that all the com- n position operators acting in Lp(R ) are continuous (see [16], in particular the first line of n the tabular at page 123). This assertion is erroneous. Indeed Tf takes Lp(R ) to itself iff |f(t)| ≤ c |t| for some constant c, see [6, thm. 3.1]. Clearly this property does not imply the continuity of f outside of 0. Instead we have the following:

Proposition 1 Let (X,µ) be a measure space. Let f : R → R be s.t. Tf takes Lp(X,µ) to itself. Then Tf is continuous from Lp(X,µ) to itself iff f is continuous.

Proof. Let us assume that Tf is continuous on Lp. Without loss of generality, assume that f(0) = 0. Let A be a measurable set in X s.t. 0 < µ(A) < +∞. For all real numbers u, v, it holds 1/p kf ◦ uχA − f ◦ vχAkp = |f(u) − f(v)| µ(A) . (1)

Clearly v → u in R implies vχA → uχA in Lp. By the continuity of Tf , and by (1), we obtain the continuity of f. For the reverse implication, we refer to [6, thm. 3.7]. We can also use the following statement: Proposition 2 Assume q ∈ [1, +∞[. Let (X,µ) be a measure space. Let f : R → R be a s.t. for some constant c > 0, it holds |f(t)| ≤ c |t|p/q, for all t ∈ R. Then Tf is a continuous mapping from Lp(X,µ) to Lq(X,µ).

2 Proof. We follow [4, thm. 2.2]. Let (gν) be a sequence converging to g in Lp(X,µ). By 1 a classical measure theoretic result , there exists a subsequence (gνk ) and a function h ∈ Lp(X,µ) s.t.

gνk → g a.e., |gνk | ≤ h .

By the continuity of f, it holds f ◦ gνk → f ◦ g a.e.. By assumption on f, it holds

q q p |f ◦ gνk − f ◦ g| ≤ (2c) h .

By Dominated Convergence Theorem, we conclude that kf ◦ gνk − f ◦ gkq tends to 0.

3 Adams-Frazier spaces and related spaces

3.1 Function spaces The inhomogeneous and homogeneous Adams-Frazier spaces are defined as follows:

m Rn m ˙ 1 Rn ˙ m Rn ˙ m ˙ 1 Rn Ap ( ) := Wp ∩ Wmp( ) , Ap ( ) := Wp ∩ Wmp( ) .

Both spaces are endowed with their natural and seminorm:

kfk m := kfk m + kfk ˙ 1 , kfk ˙ m := kfk ˙ m + kfk ˙ 1 . Ap Wp Wmp Ap Wp Wmp

The pertinency of those spaces w.r.t. composition operators was first noticed in [1], see m Rn 1 Rn also the Introduction of [11]. By Sobolev imbedding, it holds Wp ( ) ֒→ Wmp( ), in case m ≥ n/p, hence m Rn m Rn m ≥ n/p ⇒ Ap ( )= Wp ( ) . (2) m Rn In particular the critical Sobolev spaces Wp ( ), m = n/p, are Adams-Frazier spaces.

˙ m Rn m Rn Rn The intersections Wp ∩ L∞( ) and Wp ∩ L∞( ) are subalgebras of L∞( ) for the usual product, see Remark 2 below. We call them the homogeneous and inhomogeneous Sobolev algebras, and we endow them with their natural norms. Let us notice that the second is a proper subspace of the first, since the nonzero constant functions do not belong m Rn to Wp ( ). By the Gagliardo-Nirenberg inequalities, see e.g. [11, (6), p. 6108], we have the imbeddings ˙ m Rn ˙ m Rn m Rn m Rn (Wp ∩ L∞( ) ֒→ Ap ( ) , Wp ∩ L∞( ) ֒→ Ap ( ) . (3 m Rn In particular Wp ( ) coincides with the corresponding Sobolev algebra if m > n/p, or m = n and p = 1. The following statement characterizes the Adams-Frazier spaces which coincide with the corresponding Sobolev algebras:

m Rn Rn Proposition 3 1- The inclusion Ap ( ) ⊂ L∞( ) holds iff m > n/p, or m = n and p =1. ˙ m Rn Rn 2- The inclusion Ap ( ) ⊂ L∞( ) holds iff m = n and p =1.

1 See the proof of the completeness of Lp, e.g. [25, thm. 3.11, p. 67]

3 Proof. 1- In case m ≥ n/p, it suffices to (2) and the known properties of Sobolev λ n spaces. Now assume m < n/p. Define fλ(x) := |x| ϕ(x), where ϕ ∈ D(R ) and ϕ(x)=1 m Rn near 0. Then fλ ∈ Wp ( ) iff λ > m − (n/p). If we take n 1 − <λ< 0 , mp m Rn Rn we obtain fλ ∈ Ap ( ) but fλ ∈/ L∞( ). ˙ m Rn Rn 2- In case m < n/p, or m = n/p and p> 1, the first part implies a fortiori Ap ( ) 6⊂ L∞( ). λ Now assume m > n/p. Define fλ(x) := |x| (1 − ϕ(x)), with the same function ϕ as above. ˙ m Rn Then fλ ∈ Wp ( ) iff λ < m − (n/p). If we take n 0 <λ< 1 − , mp ˙ m Rn Rn we obtain fλ ∈ Ap ( ) but fλ ∈/ L∞( ). ˙ n Rn Rn If f ∈ W1 ( ), there exists g ∈ C0( ) s.t. f − g ∈ Pn−1, see [12, thm. 3]. If moreover ˙ 1 Rn f ∈ Wn ( ), one proves easily that f − g is a constant. Thus we obtain the inclusion ˙ n Rn Rn A1 ( ) ⊂ Cb( ). 2 Rn 2 Rn Remark 1 The above proof shows also that W1 ( ) does not coincide with A1( ) in case n ≥ 3: if we consider the function fλ of the above first part, with n 2 − n<λ< 1 − , 2 2 Rn ˙ 1 Rn then fλ ∈ W1 ( ), but fλ ∈/ W2 ( ).

3.2 Uniform localization Let us introduce a function ψ ∈D(R), positive, s.t.

ψ(x − ℓ) = 1 for all x ∈ R . Z Xℓ∈ If f is any distribution in R, we set fℓ(x) := f(x)ψ(x − ℓ), and we call the decomposition f = ℓ∈Z fℓ a locally uniform decomposition (abbreviated as LU-decomposition) of f. If E is a normed space of distributions in R, we denote by Elu the set of distributions f such that P

kfkElu := sup kfℓkE < +∞ . ℓ∈Z

Under standard assumptions on E, it is known that the space Elu does not depend on the choice of the function ψ. We refer to [9, 2, 3] for details. In case E = Lp(R), Elu is denoted R N ˙ N R by Lp,lu( ). For N ∈ , let us denote by WLp,lu ( ) the homogeneous based (N) upon Lp,lu(R), i.e. the set of functions f s.t. f ∈ Lp,lu(R). The inhomogeneous Sobolev N R space WLp,lu ( ) is defined similarly. Both spaces are endowed with their natural norm and seminorm: N (N) (j) N kfkW˙ N := kf kLp,lu , kfkW := kf kLp,lu . Lp,lu Lp,lu j=0 X N R N R Let us recall that WLp,lu ( ) coincides with (Wp ( ))lu, with equivalent norms. As a conse- quence, we have the following property:

4 Proposition 4 Let N ∈ N. For all compact I of R, there exists a constant c = c(N,p,I) > 0 s.t.

N gℓ ≤ c sup kgℓkWp , ℓ∈Z ℓ∈Z N WL X p,lu

for all sequence (gℓ)ℓ∈Z s.t. the function gℓ is supported by the translated interval I + ℓ, for all ℓ ∈ Z.

Proof. See [9, lem.1].

Proposition 5 N ∞ N R N R For any N ∈ , C ∩ WLp,lu ( ) is a dense subspace of WLp,lu ( ).

Proof. We use the mollifiers sequence (θν)ν≥1 introduced in Notation, with n = 1. Let N R f ∈ WLp,lu ( ) and ε> 0. Let f = ℓ∈Z fℓ be the LU-decomposition of f. Let us set P v(x) := θjℓ ∗ fℓ , Z Xℓ∈ for some convenient sequence (jℓ)ℓ∈Z. First we observe that the sum which defines the function v is locally finite. Indeed θjℓ ∗ fℓ is supported by I + ℓ, for some fixed compact interval I. As a consequence, v is a C∞ function. By condition p < ∞ and by classical

N properties of Sobolev spaces, we can choose jℓ s.t. kθjℓ ∗ fℓ − fℓkWp ≤ ε. By Proposition 4, we deduce

N N kv − fkW ≤ c sup kθjℓ ∗ fℓ − fℓkWp ≤ c ε . Lp,lu ℓ∈Z This ends up the proof.

Proposition 6 N R ˙ N R N−1 R .( ) For any integer N > 0, it holds WLp,lu ( )= WLp,lu ∩ L∞( ) ֒→ Cb

Proof. N R 1- Let f = ℓ∈Z fℓ be the LU-decomposition of f ∈ WLp,lu ( ). By condition N ≥ 1, we have the Sobolev imbedding W N (R) ֒→ C (R), hence P p b

N N sup kfℓk∞ ≤ c1 sup kfℓkWp ≤ c2 kfkW . ℓ∈Z ℓ∈Z Lp,lu

By considering the of fℓ, we conclude that f ∈ Cb(R). ˙ N R N−1 R 2- Let f ∈ WLp,lu ∩ L∞( ). By [10, lem. 1], it holds f ∈ W∞ ( ). A fortiori, it holds (j) R N R f ∈ Lp,lu( ) for all j =0,...,N − 1. Hence f ∈ WLp,lu ( ). N R N R N−1 R 3- If f ∈ WLp,lu ( ), then f ∈ Wp ( )loc. By Sobolev imbedding, it follows that f ∈ Cb ( ).

3.3 Composition operators in Adams-Frazier spaces Proposition 7 The composition operator T takes Am(Rn) to itself if f ′ ∈ W m−1(R) and f p Lp,lu f(0) = 0. Moreover, for some constant c = c(n, m, p) > 0, it holds

′ m m m−1 m kf ◦ gkAp ≤ c kf kW 1+ kgkAp , (4) Lp,lu  m Rn for all g ∈ Ap ( ).

5 Proposition 8 The composition operator T takes A˙ m(Rn) to itself if f ′ ∈ W m−1(R). More- f p Lp,lu over, the estimation (4) holds true with A replaced by A˙.

We refer to [11, thm. 1, p. 6107], [16, thm. 25] and to Proposition 6. The condition f ′ ∈ W m−1(R) is not only sufficient, but also necessary, for all Adams-Frazier spaces which are not Lp,lu n included into L∞(R ). For such spaces, Propositions 7 and 8 constitute characterizations of n the acting composition operators. For spaces imbedded into L∞(R ), we have the following alternative result:

˙ m Rn m R m Theorem 2 The operator Tf takes Wp ∩ L∞( ) to itself iff f ∈ Wp ( )loc, and Wp ∩ Rn m R L∞( ) to itself iff f ∈ Wp ( )loc and f(0) = 0.

Proof. This statement is classical, see [10] and [16, thm. 2]. We recall here the part of the proof which will be useful to prove the continuity of Tf . For every r > 0, we denote by Br ˙ m Rn the ball of center 0 and radius r in Wp ∩ L∞( ). It will suffice to prove that Tf takes Br ˙ m Rn R to Wp ∩ L∞( ), for all r> 0. We introduce a family of auxiliary functions ωr ∈D( ) s.t. ωr(t)=1 for |t| ≤ r. Then

∀g ∈ Br , Tf (g)= Tfωr (g) . (5)

If f ∈ W m(R) , it is clear that (fω )′ ∈ W m−1(R). We can apply Proposition 8 to fω . p loc r Lp,lu r ˙ m Rn ˙ m Rn By (3), we conclude that Tfωr takes Wp ∩ L∞( ) to Wp ( ). By condition m ≥ 1, the R ˙ m Rn Sobolev imbedding yields fωr ∈ Cb( ), and we obtain that Tfωr takes Wp ∩ L∞( ) to n L∞(R ). This ends up the proof.

m ˙ m Rn Remark 2 In particular, any function of class C acts on Wp ∩ L∞( ) by composition. Applying this to the function f(t) := t2, we derive immediately the algebra property.

2 Rn For n = 1, 2, the space W1 ( ) is a Sobolev algebra, for which the acting composition operators are described in Theorem 2. In the other cases, we have the following result (see [10]):

2 Rn ′′ Theorem 3 Assume that n ≥ 3. Then Tf takes W1 ( ) to itself iff f(0) = 0 and f ∈ L1(R). Moreover there exists c = c(n) > 0 s.t.

2 ′ ′′ 2 kf ◦ gkW1 ≤ c (|f (0)| + kf k1) kgkW1 , (6)

2 Rn for all such f’s and all g ∈ W1 ( ).

4 Homogeneous spaces and their realizations

˙ m Rn Usually, an homogeneous function space F , such as Wp ( ), is only a seminormed space, with kfk = 0 iff f ∈ Pk, for some k ∈ N depending on F . The presence of polynomials, with a seminorm equal to 0, has some pathological effects on composition operators. Recall, for instance, the following (see [11, prop. 11]):

˙ m Rn Proposition 9 If m > 1 and n > 1, the only functions f, for which Tf takes Wp ( ) to itself, are the affine ones.

6 This degeneracy phenomenon does not occur in homogeneous Adams-Frazier spaces, see Proposition 8. However, the continuity of Tf is a tricky question. The statement: “ Tf is ˙ m Rn continuous as a mapping of the seminormed space Ap ( ) to itself ” makes sense, but it ˙ m Rn has no chance to be true. Assume that, for a sequence (gν) tending to g in Ap ( ), the ˙ m Rn sequence (f ◦ gν) tends to f ◦ g in Ap ( ). Then, for a sequence (cν) of real numbers, the ˙ m Rn sequence (gν + cν) tends also to g in Ap ( ). But the sequence (f ◦ (gν + cν) − f ◦ g) cannot ˙ m Rn tend to 0 in Ap ( ), whatever be the sequence (cν). If, for instance, f(t) := sin t, g is a n nonzero function in D(R ), and gν := g for all ν ∈ N, then f ◦ (g + π) − f ◦ g = −2f ◦ g, a function which is not constant.

In order to avoid the disturbing effect of polynomials, two ideas seems available. The first one would be to consider the factor space F/Pk. But that does not work. Indeed, if g1 and g2 differ by a , the same does not hold for f ◦ g1 and f ◦ g2, hence we cannot extend the operator Tf to the factor space. The second one consists in restricting Tf to a vector subspace E s.t. F = E ⊕Pk. We will exploit this idea in case of Adams-Frazier spaces ˙ m Rn ˙ 2 Rn Ap ( ), and the space W1 ( ).

4.1 Realizations of homogeneous Adams-Frazier spaces Let us begin with a remark:

m n Proposition 10 1. The factor space A˙ (R )/P , endowed with the norm k−k ˙m , is a p 0 Ap .

˙ m Rn 2. Any subspace E of Ap ( ) s.t.

˙ m Rn Ap ( )= E ⊕ P0 (7)

is a Banach space for the norm k−k ˙ m . Ap

˙ m Rn Proof. 1- It is well known that Wp ( )/Pm−1 is a Banach space if endowed with the norm m n k−k ˙ m , see [23, 1.1.13, thm. 1]. If ([g ] ) is a Cauchy sequence in A˙ (R )/P , then there Wp ν 0 p 0 ˙ m Rn ˙ 1 Rn ˙ m Rn ˙ 1 Rn exist u ∈ Wp ( ) and v ∈ Wmp( ) s.t. (gν) tends to u in Wp ( ) and to v in Wmp( ). By [23, 1.1.13, thm. 2], there exist a sequence (rν) in Pm−1 and a sequence (cν) in P0, s.t. n (gν − rν) tends to u, and (gν − cν) to v, in L1(R )loc. Thus (rν − cν) is a sequence in Pm−1, n which tends to v − u in L1(R )loc. We conclude that v − u ∈ Pm−1, and that ([gν]0) tends to ˙ m Rn [v]0 in Ap ( )/P0. ˙ m Rn 2- If E is a subspace of Ap ( ) s.t. (7), the f 7→ [f]0 is an isometry from ˙ m Rn E onto Ap ( )/P0. The completeness of E follows by the first part. This ends up the proof.

A subspace E satisfying (7) will be of interest only if it is a Banach space of distributions. This motivates the following definition:

˙ m Rn ˙ m Rn Definition 1 A subspace E of Ap ( ) s.t. (7) is called a realization of Ap ( ) if one of the following equivalent properties holds:

1. the inclusion mapping E → S′(Rn) is continuous;

7 n 2. the inclusion mapping E → L1(R )loc is continuous;

3. for every sequence (gν) tending to g in E, there exists a subsequence (gνk ) s.t. gνk → g a.e..

The equivalence between the three properties follows easily by the Closed Graph Theorem.

˙ m Rn Remark 3 In [11], we used a slightly weaker definition for a realization of Wp ( ). We ˙ m Rn said that a subspace E of Wp ( ) is a realization if ˙ m Rn Wp ( )= E ⊕ Pm−1 . (8)

˙ m Rn ′ Rn If (8) holds we obtain a linear mapping σ : Wp ( )/Pm−1 → S ( ) s.t. ˙ m Rn ∀u ∈ Wp ( )/Pm−1 [σ(u)]m−1 = u , and whose range is E. Then σ is a realization, in the sense of [8, 12, 13], if σ is a continuous ˙ m Rn ′ Rn mapping from Wp ( )/Pm−1 to S ( ): this is precisely what means Definition 1.

˙ m Rn Now we turn to the description of the usual realizations of Ap ( ). Except in case m = n, ˙ 1 Rn ˙ m Rn p = 1, it will suffice to realize Wmp( ), then restrict to Ap ( ). The most natural ˙ m Rn realizations are those which retain the invariance properties of Ap ( ) w.r.t. translations or dilations. It is classically known that such realizations do not always exist, see [8, 12, 13].

1- Case m < n/p. Let us set 1 1 1 := − . q mp n ˙ 1 Rn ˙ 1 Rn ˙ m Rn Then Lq ∩ Wmp( ) is a realization of Wmp( ), see [11, prop. 14]. Hence Lq ∩ Ap ( ) is a ˙ m Rn realization of Ap ( ). Clearly it is invariant w.r.t. translations and dilations.

n ˙ 1 Rn Rn 2- Case m > n/p. By condition 1 > mp , Wmp( ) is a of C( ). Then the subspace ˙ m Rn ˙ m Rn {f ∈ Ap ( ): f(0) = 0} is a dilation invariant realization of Ap ( ).

˙ n Rn 3- Case m = n and p = 1. As observed in the proof of Proposition 3, C0 ∩ A1 ( ) is a ˙ n Rn realization of A1 ( ), clearly invariant w.r.t. translations and dilations.

˙ m Rn 4- Case m = n/p and p> 1. In such a case, Ap ( ) does not admit invariant realizations. This can be deduced from [13, thms. 5.4, 5.7].

˙ m Rn In all cases we can use “rough” realizations described as follows. Let us recall that Ap ( ) n is a subset of Lpm(R )loc, see [23, 1.1.2]. Let us define q by 1 1 := 1 − . q mp

n Let ϕ be a compactly supported measurable function in Lq(R ), s.t.

ϕ(x) dx =1 . Rn Z 8 ˙ m Rn We can define a linear on Ap ( ) by setting

Λ(g) := ϕ(x)g(x) dx . Rn Z ˙ m Rn Then the of Λ is a realization of Ap ( ), with no invariance property.

˙ 2 Rn 4.2 Realizations of W1 ( ) ˙ 2 Rn According to Proposition 9, there is no nontrivial composition operator which takes W1 ( ) to itself if n > 1. In such a case, we are forced to introduce realizations, i.e. subspaces E ˙ 2 Rn s.t. W1 ( )= E ⊕ P1, and satisfying the equivalent properties of Definition 1. Let us recall the known results concerning invariant realizations, and composition operators acting on them, see [12] and [11, prop. 18] for details.

˙ 2 R ˙ 2 R 1- Case n =1. Tf takes W1 ( ) to itself iff f ∈ W1 ( ), and, for a such f, it holds ′′ ′′ ′ ′′ ′ k(f ◦ g) k1 ≤ c (kf k1 + |f (0)|) (kg k1 + |g (0)|) , (9) ˙ 2 R for every g ∈ W1 ( ).

3 Lemma 1 For all α := (α1,α2,α3) ∈ R s.t. α1 + α2 + α3 =1, the subspace ˙ 2 R ′ ′ ′ Eα = {g ∈ W1 ( ): g(0) = 0 , α1g (−∞)+ α2g (0) + α3g (+∞)=0} (10) ˙ 2 R is a dilation invariant realization of W1 ( ). ˙ 1 R Proof. Let us recall that every function in W1 ( ) is continuous, with finite limits at ±∞. ˙ 2 R ˙ 2 R Thus the definition of Eα makes sense. It is easily seen that W1 ( )= Eα ⊕P1. If g ∈ W1 ( ) ′ ′ ′′ ′ ′′ and g(0) = 0, it holds kg k∞ ≤|g (−∞)| + kg k1, hence |g(x)|≤|x| (|g (−∞)| + kg k1) for n all x ∈ R. It follows that Eα is imbedded into L1(R )loc. ˙ 2 R Remark 4 It can be proved that any dilation invariant realization of W1 ( ) is necessarily equal to Eα for some α, see [12, prop. 11]. 2- Case n =2. According to [12, thm. 3], ˙ 2 R2 E := C0 ∩ W1 ( ) (11) ˙ 2 R2 is a realization of W1 ( ). Indeed, it is the unique translation invariant realization, see [12, ˙ 2 R2 2 R thm. 6]. By Theorem 2, Tf takes E to W1 ( ) iff f ∈ W1 ( )loc.

1 2 3- Case n ≥ 3. According to [12, thm. 2], if q := 1 − n , then ˙ 2 Rn E := Lq ∩ W1 ( ) (12) ˙ 2 Rn is a realization of W1 ( ). Indeed, it is the unique translation invariant realization, and the ˙ 2 Rn unique dilation invariant realization, of W1 ( ), see [12, thm. 6, prop. 11]. Tf takes E to ˙ 2 R ˙ 2 R W1 ( ) iff f ∈ W1 ( ). The estimation ′ 2 2 2 kf ◦ gkW˙ 1 ≤ c (|f (0)| + kfkW˙ 1 ) kgkW˙ 1 (13) ˙ 2 R holds for all f ∈ W1 ( ) and all g ∈ E.

9 5 Continuity theorems

Theorem 4 Let f : R → R be s.t. f ′ ∈ W m−1(R). Let E be a realization of A˙ m(Rn). Lp,lu p ˙ m Rn Then Tf is continuous from E to Ap ( ). If moreover f(0) = 0, then Tf is continuous from m Rn Ap ( ) to itself.

′ m−1 R m Rn Under the stronger assumption f ∈ Cb ( ), the continuity of Tf on Ap ( ) is a classical result, seemingly with no reference in the literature; in their article on composition operators in fractional Sobolev spaces [17], Brezis and Mironescu said only that the proof is “very easy via the standard Gagliardo-Nirenberg inequality”. This “folkloric” proof will be recalled below, see the proof of Lemma 3: what we do for terms with s < m works as well for s = m, ′ m−1 R in case f ∈ Cb ( ).

m R ˙ m Rn Theorem 5 Let f ∈ Wp ( )loc. Then Tf is continuous from Wp ∩ L∞( ) to itself. If m Rn moreover f(0) = 0, then Tf is continuous from Wp ∩ L∞( ) to itself.

In case p> 1, Theorem 5 has been proved in [14, cor. 2], as a particular case of a continuity m Rn theorem for composition in Lizorkin-Triebel spaces. F.Isaia has also proved it for Wp ( ), m R in case m > n/p and p ≥ 1, but with a stronger condition on f, namely f ∈ W∞ ( )loc, see [20, thm. 2.1, (iii)].

˙ 2 Rn Theorem 6 Let E be the realization of W1 ( ) defined by (10) or (11) or (12) according R R ˙ 2 Rn to the value of n. Let f : → be s.t. Tf takes E to W1 ( ). Then Tf is continuous from ˙ 2 Rn 2 Rn E to W1 ( ). If moreover f(0) = 0, then Tf is continuous from W1 ( ) to itself.

Let us notice that Theorem 6 is less general than Theorem 4 since we do not consider all the realizations, but only the invariant ones.

5.1 Main tools Four propositions will be useful, where the first is elementary, the second follows by H¨older inequality and the third is classical, see e.g. [7, I §8.2, prop. 2 , p. 100].

Proposition 11 If α> 1, the functions t 7→ |t|α and t 7→ sgn(t) |t|α are of class C1 on R.

Proposition 12 Let p1,...,ps ∈ [1, +∞[ s.t.

s 1 1 = . p p j=1 j X

Then the mapping (ϕ1,...,ϕs) 7→ ϕ1 ··· ϕs is continuous from Lp1 ×···× Lps to Lp.

m Proposition 13 Let us denote I := {1,...,m}. Then, for every (x1,...,xm) ∈ R , it holds

m (−1)m x x ··· x = (−1)|H| x , 1 2 m m! k H⊆I ! X kX∈H where the sum runs over all the of I, and |H| denotes the cardinal of H.

10 n n .Proposition 14 Let E be a Banach space of distributions in R s.t. E ֒→ L1(R )loc n Let T be a continuous mapping from E to Lp(R ). Let Φ ∈ Cb(R). Define the mapping n n V : E → Lp(R ) by V (g) := (Φ ◦ g) T (g). Then V is continuous from E to Lp(R ).

n Proof. Let (gν) be a sequence s.t. lim gν = g in E. By the imbedding E ֒→ L1(R )loc, we can assume that lim gν = g a.e., up to replacement by some subsequence. It holds

1/p p p kV (gν) − V (g)kp ≤kΦk∞ kT (gν) − T (g)kp + |Φ ◦ gν − Φ ◦ g| |T (g)| dx . Rn Z  The second term of the above r.h.s. tends to 0 by Dominated Convergence Theorem (Convergence a.e. follows by continuity of Φ, domination by boundedness of Φ and the fact n that T (g) ∈ Lp(R )).

5.2 Proof of Theorem 4 m Rn ˙ m Rn In all the proof, we denote by E the space Ap ( ), or a realization of Ap ( ). Without loss of generality, we assume that f is a smooth function s.t. f ′ ∈ W m−1(R), see Proposition Lp,lu 5, and the estimation (4).

˙ 1 5.2.1 Continuity of Tf from E to Wmp and to Lp ′ By assumption m ≥ 2, and by Proposition 6, we have f ∈ Cb(R). For j =1,...,n, it holds ′ ∂j(f ◦ g)=(f ◦ g) ∂jg. Thus we can apply Proposition 14 and conclude that g 7→ ∂j(f ◦ g) is n ′ a continuous mapping from E to Lmp(R ). We have also kf ◦g1 −f ◦g2kp ≤kf k∞kg1 −g2kp, m Rn Rn hence the continuity from Ap ( ) to Lp( ), in case f(0) = 0.

˙ m 5.2.2 Continuity to Wp : heart of the proof Let us consider the nonlinear operator

(m) m SD(g) := (f ◦ g)(Dg) , (14)

n where D is any first order differential operator with constant coefficients, say D := j=1 cj∂j, for some real numbers c . The fact that S takes A˙ m(Rn) to L (Rn) is the heart of the proof j D p p P of Proposition 8, see [11, par. 2.3]. Thus, we can expect that the following result will be the main argument for proving Theorem 4:

n Lemma 2 The nonlinear operator SD is continuous from E to Lp(R ).

Proof. Let u ∈D(R) be such that u ≥ 0 and

∀y ∈ R , u2(y − ℓ)=1 . Z Xℓ∈ (m) p (m) ∞ Let us define Φℓ(y) := u(y − ℓ) |f (y)| , for all y ∈ R. Since f ∈ Lp,lu ∩ C (R), it holds

Φℓ ∈ C(R) and sup kΦℓk1 < +∞ . ℓ∈Z

11 +∞ We define Ψℓ(y) := Φℓ(t) dt. Then Zy 1 Ψℓ ∈ C (R) and sup kΨℓk∞ < +∞ . (15) ℓ∈Z

Now we assume that mp > 2 (the exceptional case, m = 2 and p = 1, needs a change, see below). We can use Proposition 11 with α := mp − 1. It holds

p (m) p mp 2 kSD(g)kp = |f ◦ g| |Dg| (u ◦ (g − ℓ)) dx Z Rn Xℓ∈ Z

mp−1 = (Φℓ ◦ g)(Dg) sgn(Dg) |Dg| (u ◦ (g − ℓ)) dx , Rn ℓ∈Z Z X  hence n p kSD(g)kp = cj Vj,ℓ , j=1 Z X Xℓ∈ with mp−1 Vj,ℓ := (Φℓ ◦ g) ∂jg sgn(Dg) |Dg| (u ◦ (g − ℓ)) dx . Rn Z  The computation of Vj,ℓ relies upon an integration by parts w.r.t the j-th coordinate. This 1 Rn I.P. is justified by a classical theorem. Any function which belongs locally to Wp ( ) can be modified on a negligible set of Rn, in such a way that the resulting function has the following property: its restriction, on almost every line parallel to a coordinate axis, is locally absolutely continuous. This theorem originates to Calkin [18]; see [23, 1.1.3, thm.1] for the precise statement. Here, this theorem can be applied to the function Ψℓ ◦ g, which belongs locally 1 Rn to Wmp( ), and to the function

w := sgn(Dg) |Dg|mp−1 (u ◦ (g − ℓ)) ,

which satisfies mp w ∈ W˙ 1 ∩ L (Rn) , q := . (16) 1 q mp − 1 Proof of (16). Let k =1,...,n. By the assumption g ∈ E, and by the Gagliardo-Nirenberg inequality m−2 1 m−1 m−1 2 kgkW˙ ≤ ckgk ˙ 1 kgk ˙ m mp/2 Wmp Wp n (see [11, (7), p. 6108]), it holds ∂kDg ∈ Lmp/2(R ). On the other hand, since

mp−2 mp−1 |∂kw| ≤ c1|∂kDg||Dg| + c2|Dg| |∂kg|,

mp mp the H¨older inequality applied to the r.h.s. (with exponents 2 and mp−2 in the first term, mp Rn mp−1 and mp in the second) gives ∂kw ∈ L1( ).

The property (16) implies the following:

12 n−1 For almost every (x1,...,xj−1, xj+1,...,xn) ∈ R , the function

t 7→ w(x1,...,xj−1, t, xj+1,...,xn) is absolutely continuous on R, with limit 0 at ±∞.

Thus we obtain

mp−1 Vj,ℓ = (Ψℓ ◦ g) ∂j sgn(Dg) |Dg| (u ◦ (g − ℓ)) dx , Rn Z hence 

p kSD(g)kp = (F ◦ g) T1(g) dx + (G ◦ g) T2(g) dx , (17) Rn Rn Z Z where ′ F (y) := Ψℓ(y) u(y − ℓ) , G(y) := Ψℓ(y) u (y − ℓ) , Z Z Xℓ∈ Xℓ∈ and 2 mp−2 mp T1(g) := (mp − 1)(D g) |Dg| , T2(g) := |Dg| . In case m = 2 and p = 1, a similar computation starting with

′′ 2 kSD(g)k1 = |f ◦ g| (Dg) dx , Rn Z shows that (17) is also valid.

By Gagliardo-Nirenberg and by the definition of E, the linear mappings g 7→ D2g, g 7→ Dg n n are continuous from E to Lmp/2(R ) and Lmp(R ) respectively. Thus, by using Proposition 2, the mappings g 7→ D2g, g 7→ |Dg|mp−2 , g 7→ |Dg|mp n n n are continuous from E to L mp (R ), L mp (R ) and L (R ) respectively. Since 2 mp−2 1 2 mp − 2 + =1 , mp mp we can use Proposition 12, and conclude that T1 and T2 are nonlinear continuous mappings n from E to L1(R ). By the continuity of Ψℓ, and by (15), it follows that F and G belong to Cb(R). From (17) and Proposition 14, we deduce the following property:

g 7→ kSD(g)kp is a continuous nonlinear functional on E. (18)

(m) Now consider a sequence (gν) which converges to g in E. By the continuity of f , it holds

lim SD(gν)= SD(g) a.e., up to replacement by a subsequence. By (18), it holds

lim kSD(gν)kp = kSD(g)kp .

2 Then we can apply the Theorem of Scheff´e[27] and conclude that lim SD(gν) = SD(g) in n Lp(R ).

2Usually attributed to Scheff´e, but first proved by F. Riesz [24], see the survey of N. Kusolitsch [21].

13 ˙ m 5.2.3 Continuity to Wp : end of the proof

(α) n n We have to prove the continuity of g 7→ (f ◦ g) , from E to Lp(R ), for all α ∈ N s.t. |α| = m. We use the classical formula

(α) (s) (γ1) (γs) (f ◦ g) = cα,s,γ(f ◦ g) g ··· g , (19) where the parameters satisfy the conditionsX

s

s =1, . . . , m , |γr| > 0(r =1,...,s) , γr = α , (20) r=1 X and the cα,s,γ’s are some combinatorial constants. Formula (19) holds true for any smooth function g. We need to prove it for any function g ∈ E.3 We associate with g the sequence (α) (α) gν := θν ∗ g, ν ≥ 1. It is easily seen that (f ◦ gν) → (f ◦ g) in the sense of distributions. Now we prove that the r.h.s. of (19) behaves similarly.

Case 1: terms with s < m. Let us consider

S(g) := (f (s) ◦ g) g(γ1) ··· g(γs) , (21) for a set of parameters satisfying (20), and s < m.

Lemma 3 The nonlinear operator S, given by (21), with s < m, is well defined and contin- n uous from E to Lp(R ).

Proof. According to Gagliardo-Nirenberg (see e.g. [11, (7), p. 6108]), we have the imbed- dings ˙ m Rn ˙ N , Ap ( ) ֒→ Wpm/N for all N =1,...,m − 1. Applying this to N = |γr|, for r =1,...,s, and Proposition 12, we (γ1) (γs) n obtain the continuity of the mapping g 7→ g ··· g from E to Lp(R ). By Proposition (s) 6, we have f ∈ Cb(R). Hence we can apply Proposition 14, and conclude that S is continuous. This ends up the proof of Lemma 3.

As a consequence, it holds

(s) (γ1) (γs) (s) (γ1) (γs) lim (f ◦ gν) gν ··· gν =(f ◦ g) g ··· g , ν→∞

n in Lp(R ).

Case 2 : terms with s = m. Let us consider the operator

(m) S(g) := (f ◦ g) ∂j1 g ··· ∂jm g , with jr ∈{1,...,n} for r =1,...,m. For every subset H of {1,...,m}, let us denote

DH := ∂jk . kX∈H 3 ′ m−1 R In [11, par. 2.4, step 2] we obtained (19) for f ∈ Cb ( ), but here the assumption on f implies only ′ m−2 R f ∈ Cb ( ).

14 According to Proposition 13, it holds (−1)m S = (−1)|H| S , m! DH H X n see (14) for the definition of SD. By Lemma 2, it holds limν→∞ S(gν)= S(g) in Lp(R ).

As a consequence of the above cases 1 and 2, the formula (19) holds for all g ∈ E. This ends up the proof.

5.3 Proofs of Theorems 5 and 6 Proof of Theorem 5. We refer to the proof of Theorem 2. By Theorem 4 and by the first ˙ m Rn ˙ m Rn imbedding given in (3), Tfωr is a continuous mapping from Wp ∩ L∞( ) to Wp ( ). ′ R ˙ m Rn Rn Since (fωr) ∈ L∞( ), Tfωr is also continuous from Wp ∩ L∞( ) to L∞( ). We conclude ˙ m Rn that Tfωr is continuous from Wp ∩ L∞( ) to itself for every r > 0. The continuity of Tf follows by (5).

Proof of Theorem 6. It is similar to that of Theorem 4. We assume n =6 2, since the case n = 2 is covered by Theorem 5. By the estimations (6), (9), (13), and by the density of D(R) ′′ into L1(R), we can restrict ourselves to the case f ∈ D(R). Notice that such a property ′ ∞ R j,k j,k j,k ′ implies f ∈ Cb ( ), hence ∂j∂k(f ◦ g)= U (g)+ S (g), where U (g) := (f ◦ g)∂j∂kg and j,k ′′ S (g) := (f ◦ g)(∂jg)(∂kg), see [11, step 2, p. 6111, and the proof of prop. 18, p. 6128]. j,k n The continuity of U : E → L1(R ) follows by Proposition 14. To prove the continuity of Sj,k, we introduce ′′ 2 SD(g) := (f ◦ g)(Dg) , where D is a first order differential operators with constant coefficients, and we set

+∞ h(x) := |f ′′(t)| dt . Zx Then we must discuss according to n.

˙ 1 Rn Case n > 2. For all g ∈ E, and j = 1,...,n, it holds ∂jg ∈ W1 ∩ Lr( ), with 1 1 r := 1 − n , see [11, prop. 15]. This implies the following property: for almost every n−1 (x1,...,xj−1, xj+1,...,xn) ∈ R , the function

t 7→ ∂jg(x1,...,xj−1, t, xj+1,...,xn)

is absolutely continuous on R, with limit 0 at ±∞. Thus we can make integrations by parts w.r.t. each coordinate, obtaining

2 kSD(g)k1 = (h ◦ g)(D g) dx , (22) Rn Z j,k n for all g ∈ E. The continuity of S : E → L1(R ) follows, exactly as in the proof Theorem 2 Rn 4. By Sobolev’s Theorem, the inhomogeneous space W1 ( ) is imbedded into E. The 2 Rn continuity of Tf on W1 ( ) follows at once if f(0) = 0.

15 Case n =1. Now D = d/dx and the formula (22) becomes

′′ ′ ′ kSD(g)k1 = (h ◦ g) g dx − h (g(+∞)) g (+∞)+ h (g(−∞)) g (−∞) . R Z

If lim gν = g in E, then, by the proof of Lemma 1, lim gν(x) = g(x) for every x ∈ R, and also for x = ±∞. Since h is continuous on R, we conclude that lim kSD(gν)k1 = kSD(g)k1. The rest of the proof is unchanged.

Conclusion

Let us mention possible continuations of the present work: 1- Generalization of theTheorem 1 to Sobolev spaces with fractional order of . The automatic continuity is known to hold in the following cases:

s Rn 1 • Besov spaces Bp,q( ) with 0

s R s R • Besov spaces Bp,q( ) and Lizorkin-Triebel spaces Fp,q( ) with s > 1+(1/p), p, q ∈ [1, +∞[, see [14, cor. 2] and [16, thm. 8] (in those papers, they are some restrictions in case of Besov spaces, which have been removed in [15]).

The extension to the spaces on Rn, for n > 1 and s > 1 noninteger, is completely open: the first difficulty is that we have not even a full characterization of functions which act by composition. 2- Proof of the higher-order chain rule. In the proof of Theorem 4, we have established the formula (19) for all g ∈ E, but only for smooth functions f. Could we generalize it to any f s.t. f ′ ∈ W m−1(R)? In this respect, we Lp,lu can refer to the partial results of F. Isaia [20].

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G´erard Bourdaud Universit´eParis Diderot, I.M.J. - P.R.G (UMR 7586) Case 7012 75205 Paris Cedex 13 [email protected]

Madani Moussai Laboratory of and Geometry of Spaces, M. Boudiaf University of M’Sila, 28000 M’Sila, Algeria [email protected]

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