THE COMPOSITION IN LIZORKIN-TRIEBEL SPACES VIA PARA-DIFFERENTIAL OPERATORS

VIA PARA-DIFFERENTIAL OPERATORS

MADANI MOUSSAI

We will use the para-differential operators for the study of the composition opera- torTf : u → f◦u on Lizorkin-Triebel space Fp , q^s (Rⁿ), in the following sense:

Letf:R→Rbe a function which belongs locally to Besov spaceB^s+1∞, q(R) such that f(0) = 0, then Tf takes either B∞^s , q(Rⁿ)∩Fp , q^s (Rⁿ) (in case s > 1) or W∞¹(Rⁿ)∩Fp , q^s (Rⁿ) (in case 0< s <1) toFp , q^s (Rⁿ).

AMS 2010 Subject Classification: 46E35, 47H30.

Key words: Besov spaces, Littlewood-Paley decomposition, Lizorkin-Triebel spa- ces, pseudo-differential operators.

1. INTRODUCTION AND MAIN RESULT

LetE_p,q^s (Rⁿ) denote the real space of either the Besov spaceB_{p , q}^s (Rⁿ) or the Lizorkin-Triebel space F_{p , q}^s (Rⁿ), when there is no need to distinguish the B_{p , q}^s (Rⁿ) space and theF_{p , q}^s (Rⁿ) one (abbreviated toB-space andF-space in the following). For a Borel-measurable functionf :R→Rwe are interested in composition by the study of the associate nonlinear operatorTf :u→f◦uon E^s_p,q(Rⁿ). The problem of the composition consists in the full characterization of functions f such that T_f takes E_p,q^s (Rⁿ) to itself (such function f is also said to act on E_p,q^s (Rⁿ) by composition). In this sense, the following results are well-known:

– For 1≤p <∞and 0< s <1, a functionf :R→Racts onE_p,q^s (Rⁿ) by composition if and only iff(0) = 0, and eitherfis locally Lipschitz continuous, i.e.,f ∈W∞^{1,`oc</sup>(R) (in the caseE<sub>p,q</sub><sup>s</sup> (R<sup>n</sup>)⊂L∞(R)), orf is uniformly Lipschitz continuous i.e. f ∈ W<sub>∞</sub><sup>1</sup>(R) (in the case E<sub>p,q</sub><sup>s</sup> (R) 6⊂ L∞(R)). For p = ∞, a function f :R → Racts on B<sub>∞</sub><sup>s</sup> <sub>, q</sub>(R) by composition if and only if f(0) = 0 and f ∈W∞<sup>1,`oc}(R), cf. [2, 17].

If we consider an investigation such thatT_f takes E_p,q^s (Rⁿ)∩L∞(Rⁿ) to E^s_p,q(Rⁿ), the problem of the composition is not trivial in the sense that f is not linear. We recall the following results:

MATH. REPORTS13(63),2 (2011), 151–170

– Dahlberg [9] proved: For 1≤ p ≤ +∞ and 1 + (1/p) < m < n/p an integer, if T_f maps the Sobolev spacesW_p^m(Rⁿ) into itself, then f is a linear function.

– The result of Dahlberg was extended toE_p,q^s (Rⁿ). More precisely, for either 1 + (1/p) < s < n/p or 1 + (1/p) = s < n/p (with 1< q ≤ ∞ in the case ofB-space and 1< p <∞in the case of F-space), the operator T_f takes E^s_p,q(Rⁿ) into itself, if and only if f(t) =c tfor some real c, see e.g. [17].

Many authors studied analytically the boundedness of the composition operator Tf on E_p,q^s (Rⁿ), as in the different works of Bourdaud and his col- laborators [2, 3, 4, 6, 7], as well as in [14] and Runst-Sickel [17], for example.

Using the para-differential operators theory, Meyer [12] proved the bounded- ness of Tf, associated to smooth functions f ∈C^∞(R) withf(0) = 0, on the Bessel potential spaces H_p^s(Rⁿ) in the algebra case, i.e.,s > n/p .

In this work, we will be interested in the composition problem using the para-differential operators. Then we will essentially prove the following results, in which our principal contribution concerns the part of F-space. But before formulating this, we introduce the following notation, which depends on the choice of the parameter s

(1.1) K =

( W¹_∞(Rⁿ) in the case 0< s <1, B_∞^s _{, q}(Rⁿ) in the case s >1.

Theorem 1.1. Let 0 < s 6= 1 and 1 ≤ p, q ≤ ∞, with 1 ≤ p < ∞ in the case of F-space. Let f : R → R be a Borel measurable function such that f(0) = 0 and f ∈ B∞^{s+1, `oc}, q (R). Then the composition operator T_f maps K ∩E_p,q^s (Rⁿ) into E_p,q^s (Rⁿ).

As an immediate consequence, we have the following statement:

Corollary 1.2. Let p, q and s be as in Theorem 1.1. Let f : R → R be a Borel measurable function such that f(0) = 0. Then the following assertions hold:

(i) If f^(j) ∈ L^`oc_∞(R) for j = 1, . . . ,[s] + 2, then Tf maps K ∩E_p,q^s (Rⁿ) into E_p,q^s (Rⁿ).

(ii)Assume in addition s−[s]>1/p. If f ∈Ep , q^{s+2, `oc}(R), then T_f maps K ∩E_p,q^s (Rⁿ) into E_p,q^s (Rⁿ).

Remark 1.3. We have the identities (equivalent norms):

–W_p^m(Rⁿ) =F_{p ,}^m₂(Rⁿ), 1< p <∞,m= 1,2, . . ..

–H_p^s(Rⁿ) =F_{p ,}^s₂(Rⁿ), 1< p <∞.

– The H¨older spaces C^s(Rⁿ) = B_∞^s _,∞(Rⁿ), 0 < s 6= integer, see [18, 2.2.2, 2.5.7].

– Let W_p^s(Rⁿ) denote the Slobodeckij spaces, see [18, 2.2.2] for a defi- nition. Then W_p^s(Rⁿ) = F_{p , p}^s (Rⁿ), 1 < p < ∞, 0 < s 6= integer, see [18, 2.2.2(18), 2.5.7(5)].

Hence by Theorem 1.1 it holds that, fors >0 and f ∈B^{s+1, `oc}∞, q (R), the operator T_f maps C^s(Rⁿ) into itself. If in addition f(0) = 0, then T_f maps K ∩E intoE, whereE =W_p^m(Rⁿ) orH_p^s(Rⁿ) orW_p^s(Rⁿ).

Remark 1.4. As mentioned before the main result, the part ofB-space in Theorem 1.1 can be compared with the previous works of Peetre, which proved in [15] that for s > n/p and a function f : R→Rsuch that f(0) = 0 and f ∈C^∞(R), the operatorTf maps B^s_{p , q}(Rⁿ) to itself.

Again Meyer in [12] studied the boundedness of the composition opera- tor modulo the para-product on algebra spaces H_p^s(Rⁿ), in which the “para- linearization” was used to transform a partial differential nonlinear equation to an almost linear one. We will extend the result of the H_p^s-boundedness to E^s_p,q-spaces, in the same way of Theorem 1.1.

We turn to the main result. The claimed boundedness in Theorem 1.1 will be reflected by certain refined inequalities. For this reason we found it convenient to recall the Littlewood–Paley decomposition, see Subsection 2.1 below. This leads us to give the outline of the paper. In Section 2, we first collect the needed material about Besov and Lizorkin-Triebel spaces and some classical inequalities. Second, as B_∞^s _{, q}(Rⁿ) plays an important role in this work, we recall, in Subsections 2.2.1–2.2.2, some properties and characteriza- tion by differences of this space. Section 3 is devoted to the proof of the main result, in which we give in Subsections 3.1–3.2 a study of the composition op- erator on B_∞^s _{, q}(Rⁿ), the basic tool and the complete proof, respectively; here we extend our investigation to the case s= 1. Finally, in Section 4 we prove a result for the composition operator modulo the para-product.

Notation. All functions and spaces are assumed to be real valued. For brevity, we employ some reductions. The parameters p, q are fixed as follows:

p ∈ [1,+∞] in the case of B-space, and p ∈ [1,+∞[ in the case of F-space, and q ∈[1,+∞]. We put

E_p,q=

`q(Lp) in the case B-space, Lp(`q) in the case F-space.

We will use acut-offfunction denoted byρ: we fixρ∈C₀^∞(R), a function such that suppρ⊂[−2,2] andρ(x) = 1 if x∈[−1,1]. We put

(1.2) ρt(x) =ρ(x/t), t >0, x∈R.

For a space E of functions defined on Rⁿ, the associated local space is defined as

E^`oc={f ∈ S⁰:ϕf ∈E, ∀ϕ∈C₀^∞(Rⁿ)}.

We denote byW_∞¹(Rⁿ) the space of bounded functions such that the first order weak derivatives are bounded, equipped with the norm

kfk_W1

∞(Rⁿ)=kfk_∞+

n

X

j=1

k∂_jfk_∞.

Also, we will use several times the following elementary convolution es- timate: For any real b ∈]0,1[ and any sequence {ε_j}j∈N of positive numbers, such that k{ε_j}_j∈Nk_`q =C <+∞, it holds

(1.3)

∞

X

k=0 k

X

j=0

b^k−jεj

!q!1/q

+

∞

X

k=0

∞

X

j=k

b^j−kεj

!q!1/q

≤ 2C 1−b. Now, the standard notations: Smeans the Schwartz space of all infinitely differentiable and rapidly decreasing functions, andS⁰its topological dual, the space of tempered distributions. Iff ∈ S⁰, letFf =fbbe the Fourier transform and F⁻¹f denotes its inverse. Ifϕ∈ S then we set

ϕ(ξ) =b Z

Rⁿ

e^−ix·ξϕ(x) dx and F⁻¹ϕ(ξ) = (2π)⁻ⁿϕ(−ξ),b ∀ξ ∈Rⁿ. For s∈R, [s] denotes the greatest integer less than or equal to s. Withk · k_p we denote the L_p-norm. For 1 ≤ p ≤ ∞ we denote by p⁰ = p/(p−1) the conjugate exponent. Constants c, c1, . . . are strictly positive and depend only on the fixed parameters n, s, p, q, and probably on auxiliary functions; their values may vary from line to line.

2. BESOV AND LIZORKIN-TRIEBEL SPACES

We begin this section by noting that, for basic properties of B-space and F-space, such equivalent norms, embeddings, approximation by smooth functions, . . .etc, we refer to Bergh and L¨ofstr¨om [1], Franke [10], Peetre [16], Runst and Sickel [17] and Triebel [18, 19].

2.1. Definition and first properties

Letγ be a C^∞(Rⁿ) even and positive function, such that 0≤γ(ξ) ≤1, γ(ξ) = 1 for |ξ| ≤1 andγ(ξ) = 0 for |ξ| ≥3/2. We put

(2.1) φ(ξ) =γ(ξ)−γ(2ξ), ∀ξ ∈Rⁿ,

whose support is contained in the compact annulus (1/2)≤ |ξ| ≤(3/2), and the identity PM

j=1 φ(2^−jξ) = γ(2^−Mξ)−γ(ξ) for all integer M ≥ 1 yields (when M →+∞), the following resolution of unity

γ(ξ) +

∞

X

j=1

φ(2^−jξ) = 1, ∀ξ∈Rⁿ.

We introduce the pseudo-differential operators S_j and Q_j defined on S⁰(Rⁿ) by

S0 =Q0=γ(D), Sj =γ(2^−jD), Qj =φ(2^−jD) forj= 1,2, . . . . Since γ is even,S_jf and Q_jf are real valued as soon asf is real valued function. Hence for every f ∈ S⁰(Rⁿ), the Littlewood–Paley decomposition is described by the following series

(2.2) f =

∞

X

j=0

Qjf =S_kf +

∞

X

j=k+1

Qjf, ∀k∈N,convergence in S⁰(Rⁿ).

Remark 2.1. According to Young inequality in L_p(Rⁿ), the families of operators {S_j}j∈N and {Q_j}j∈N constitute bounded subsets of the normed space L(L_p(Rⁿ)), for anyp∈[1,+∞].

Definition 2.2. Lets∈R. The space E_p,q^s (Rⁿ) is the collection off ∈ S⁰ such that

(2.3) kfk_Es

p,q(Rⁿ)=k{2^skQkf}_k∈Nk_Ep,q <+∞.

By means of (2.2), we have introduced the Banach space E_p,q^s (Rⁿ); its norm is independent of the chosen functions (2.1), see e.g. [18].

Lemma 2.3. The families of operators {S_j}_j∈N and {Q_j}_j∈N are uni- formly bounded in E_p,q^s (Rⁿ).

Proof. The proof of the boundedness in the case ofB-space is obvious, and will be omitted. Now the case ofF-space. We introduce the maximal function (2.4) Q^∗,a_k f(x) = sup

y∈Rⁿ

|Q_kf(x−y)|

1 + (2^k|y|)^a , with a >0.

Then in (2.3), by replacingQ_kf byQ^∗,a_k f witha > _min(p,q)ⁿ , we obtain an equi- valent norm in F-space, see e.g. [19, Theorem 2.3.2]. We will use this norm.

Clearly,QjQkf(x) = 0 if|k−j| ≥4, which implies the following estimate

|Q_jQ_kf(x)| ≤ Z

Rⁿ

|F⁻¹φ(y)| |Q_kf(x−2^−jy)|dy

≤ Z

Rⁿ

(1 + 2^3a|y|^a)|F⁻¹φ(y)||Q_kf(x−2^−jy)|

1 + (2^k−j|y|)^a dy

≤c|Q^∗,a_k f(x)|, and gives the desired result.

Now, we recall some estimates of the Yamazaki type for the conver- gent series.

Proposition2.4. Letb >1. (i)Let s∈R. Then there exists a constant c >0, such that for all sequence{g_j}_j∈NofS⁰(Rⁿ)’s functions, with thesuppbg_j is contained in the annulus b⁻¹2^j ≤ |ξ| ≤ b2^j and k{2^sjg_j}j∈NkE_p,q = A <

+∞, the series P∞

j=0 g_j converges in S⁰(Rⁿ) to a limit in E_p,q^s (Rⁿ), and

∞

X

j=0

g_{j Es}

p,q(Rⁿ)

≤c A.

(ii) If s > 0 the assertion (i) remains true if one replaces the annuls b⁻¹2^j ≤ |ξ| ≤b2^j by the ball |ξ| ≤b2^j.

Proof. The proof of (i) can be found in [20, Theorem 3.6] or [17, Propo- sition 2.3.1/1, p. 59], while the proof of (ii) is given in [20, Theorem 3.7] or [17, Proposition 2.3.1/2, p. 60].

Proposition 2.5. Let b > 1 and s ∈ R. Then there exists a constant c >0, such that for all g∈E_p,q^s (Rⁿ), and all function θ∈C₀^∞(Rⁿ) supported by the annulus b⁻¹ ≤ |ξ| ≤b, the sequence {g_j}_j∈N, which is defined by bg_j = θ(2^−j·)bg, satisfies

k{2^sjgj}_j∈Nk_Ep,q ≤ckgk_Es p,q(Rⁿ). Proof. Step1.The case of F-space.

Substep 1.1. The case q = 2. Let L_j be a pseudo-differential operator with a symbol θ(2^−jξ). We introduce the operator T : g → {L_j(g)}_j∈N, defined from S(Rⁿ) toS⁰(Rⁿ, `2(N)) the space of tempered distributions with values in the Hilbert space `₂(N). We haveT(g)(ξ) =[ M(ξ)bg(ξ), withM(ξ) = {θ(2^−jξ)}_j∈N. We claim that

(2.5) kM^(α)(ξ)k_`2(N) ≤c(1 +|ξ|)^−|α|, ∀α∈Nⁿ. Indeed, observe that

–|ξ| ∼2^j on support ofθ(2^−jξ), then

|2^−j|α|θ^(α)(2^−jξ)| ≤c(1 +|ξ|)^−|α|, c=c(b, α, n)>0;

– there exists a positive integerr =r(b), such that for allξ there existr annulus b⁻¹2^j ≤ |ξ| ≤b2^j for whichξ belongs, hence (2.5) is obtained with a constantc r^1/2kθ^(α)k∞.

We now apply the vectorial version of Mihlin–Marcinkiewicz multipliers theorem, cf. [1, Theorem 6.1.6], then we obtain

kT(g)k_Lp(Rn,`2(N))≤ckgk_p.

Substep1.2. The general case. By considering the supports we have (2.6) θ(2^−jξ)Qd_kg(ξ) = 0, for |k−j| ≥1 +ν, ν = 1 +hlog 2b

log 2 i

. Using (2.4), then we automatically replaceQkgbyQ^∗,a_k gin (2.3), and we have

|g_j(x)| ≤ Z

Rⁿ

X

|k−j|≤ν

|F⁻¹θ(y)| |Q_kg(x−2^−jy)|dy

≤ Z

Rⁿ

(1 + 2^νa|y|^a)|F⁻¹θ(y)| X

|k−j|≤ν

|Q_kg(x−2^−jy)|

1 + (2^k−j|y|)^a dy

≤(2ν+ 1) Z

Rⁿ

(1 + 2^νa|y|^a)|F⁻¹θ(y)|dy

X

|k−j|≤ν

|Q^∗,a_k g(x)|

and we obtain

|g_j(x)| ≤





















 c

∞

P

k=j−ν

2^−sk(2^sk|Q^∗,a_k g(x)|) if s >0, c

j+ν

P

k=j−ν

|Q^∗,a_k g(x)| if s= 0, c

j+ν

P

k=0

2^−sk(2^sk|Q^∗,a_k g(x)|) if s <0.

Consequently, ifs >0 ors <0, the estimate of P∞

j=0 2^sjq|g_j(x)|^q1/q

follows using (1.3). However, the cases= 0 can be obtained by H¨older inequality, i.e.,

∞

X

j=0

|g_j(x)|^q ≤c₁

∞

X

j=0

^j+ν X

k=j−ν

1

q/q⁰ j+ν

X

k=j−ν

|Q^∗,a_k g(x)|^q

≤c₂(2ν+ 1)^q/q0

∞

X

j=0 j+ν

X

k=j−ν

|Q^∗,a_k g(x)|^q

≤c3(2ν+ 1)^1+(q/q0)

∞

X

k=0

|Q^∗,a_k g(x)|^q.

Now, we choose the number a > _min(p,q)ⁿ , then the result follows.

Step2.The case of B-space. We have

k2^jnF⁻¹θ(2^j·)∗Q_kgk_p ≤ kF⁻¹θk₁kQ_kgk_p. Again, from (2.6) we arrive at

kg_jk_p ≤





















 c

∞

P

k=j−ν

2^−sk(2^skkQ_kgk_p) if s >0, c

j+ν

P

k=j−ν

kQ_kgk_p if s= 0, c

j+ν

P

k=0

2^−sk(2^skkQ_kgk_p) if s <0.

Then we proceed as in Step 1. We omit details.

2.2. The special caseB∞^s , q(Rⁿ) 2.2.1. Some properties of B∞^s , q(Rⁿ)

We need the following statements, which the first and the second can be found in [17, Theorem 2.2.4/1] and [19, Theorem 2.3.8, pp. 58–59], respectively.

Proposition 2.6. Let s > 0. Then the following continuous embed- ding holds:

(2.7) B_∞^s _{, q}(Rⁿ),→L∞(Rⁿ).

Proposition 2.7. Let s∈R. Let m= 1,2,3, . . .. Then the expression X

|α|≤m

kf^(α)k_Bs−m

∞, q(Rⁿ)

defines an equivalent norm in B^s_{∞, q}(Rⁿ).

Proposition 2.8. Let 0 < s ≤ 1 < t. Then the following chain, of continuous embeddings, holds:

B_∞^t _{, q}(Rⁿ),→W¹_∞(Rⁿ),→B_∞^s _{, q}(Rⁿ).

Proof. SinceB_∞^t−1_{, q}(Rⁿ),→L∞(Rⁿ), then the first one is satisfied. Now, by Proposition 2.7, and the embedding L∞(Rⁿ),→B_∞⁰ _,∞(Rⁿ) see [17, Theo- rem 2.2.2/(8)], it holds

kfk_B1

∞,∞(Rⁿ)≤c1

kfk_∞+

n

X

j=1

k∂_jfk_B0

∞,∞(Rⁿ)

≤c2kfk_W1

∞(Rⁿ).

Now by the embedding B_∞¹ _,∞(Rⁿ) ,→ B_∞^s _{, q}(Rⁿ), for 0 < s < 1, we obtain the desired result.

Proposition 2.9. Let s > 0. Then the Banach space B_∞^s _{, q}(Rⁿ) is an algebra for the pointwise multiplication of functions.

Proof. Letf, g∈B_∞^s _{, q}(Rⁿ). Form (2.2), we can write (2.8) f g=

∞

X

j=0

Q_jf

S_jg+

∞

X

k=j+1

Q_kg

=

∞

X

j=0

(Q_jf) (S_jg) +

∞

X

k=1

(Sk−1f) (Q_kg).

Since F(Q_jf S_jg) is supported by the ball |ξ| ≤ 3(2^j), then by Propositions 2.4(ii) and 2.6, and Remark 2.1, it holds

∞

X

j=0

(Q_jf) (S_jg) _Bs

∞, q(Rⁿ)

≤c₁kfk_Bs

∞, q(Rⁿ)kgk_∞≤c₂kfk_Bs

∞, q(Rⁿ)kgk_Bs

∞, q(Rⁿ). Similarly for the last term in (2.8), and details will be omitted.

Remark 2.10. We recall the definition of the pointwise multiplication of functions. Let f, g∈ S⁰. The productf g is defined by the formula

(2.9) f g= lim

j→+∞(Sjf)(Sjg)

if the limit on the right hand side of (2.9) exists in S⁰ (see [17, 4.2]).

2.2.2. Characterization of B∞^s , q(Rⁿ)by differences

The Littlewood-Paley approach is the simplest to Besov space, as far as we are concerned with estimate of families of functions, duality and approxi- mation . . . , see e.g. [18]. Alternative descriptions by using differences turn out to be more convenient for the study of composition operators. Further, we need the following characterization of the B-space; we give it forB_∞^s _{, q}(Rⁿ).

Proposition 2.11. (i) Let 0< s <1. Then the expression kfk_∞+

Z

|h|≤1

|h|^−sqkf(·+h)−fk^q_∞ dh

|h|ⁿ 1/q

defines an equivalent norm in B^s_{∞, q}(Rⁿ).

(ii) Let 0< s < 1. Then a function f ∈L∞(Rⁿ) belongs to B_∞^s _{, q}(Rⁿ), if and only if

(2.10) kfk_∞+ Z 1

0

τ^−sq sup

|h|≤τ

|f(·+h)−f|

q

∞

dτ τ

1/q

<+∞.

(iii) Let 0< s <2. Then a function f ∈L∞(Rⁿ) belongs to B_∞^s _{, q}(Rⁿ), if and only if

(2.11) kfk_∞+ Z 1

0

τ^−sq sup

|h|≤τ

|f(·+h) +f(· −h)−2f|

q

∞

dτ τ

1/q

<+∞.

Moreover, the expressions (2.10) and (2.11) generate equivalent norms in the Besov space B_∞^s _{, q}(Rⁿ). Also, the above assertions (i)–(iii) remain true if one replaces R

|h|≤1. . .dh and R1

0 . . . dτ by R

Rⁿ. . .dh and R∞

0 . . .dτ, respectively.

Proof. See e.g. [18, Theorem 2.5.12] and [19, Theorem 3.5.3, p. 194].

3. PROOF OF THE MAIN RESULT

3.1. Preparation, the composition operator on B∞^s , q(Rⁿ)

The boundedness of the composition operator onB^s_{∞, q}(Rⁿ) is one of the main tools when we prove Theorem 1.1. That is the following statement.

Proposition 3.1. Let 0< s6= 1. Letf :R→Rbe a function such that f ∈B^{s, `oc}∞, q(R). If u∈ K thenf ◦u∈B_∞^s _{, q}(Rⁿ).

Proof.Let t≥max(1,kuk∞). For the brevity we putfet=f ρt (see (1.2) for the definition of ρ_t), and we will use the equality f ◦u = fe_t◦u. The inequality

(3.1) |f(u(x))| ≤ kfk_∞, ∀x∈Rⁿ,

implies that f ◦u belongs to L∞(Rⁿ). Now we divide our proof into some steps with respect to s:

Step1. The case0< s <1. Clearly, it holds Z

Rⁿ

|h|^−sq

sup

|v|≤k∇uk∞|h|

|fet(·+v)−fet|

q

∞

dh

|h|ⁿ 1/q

≤

≤c1k∇uk^s_∞ Z ∞

0

τ^−sq sup

|v|≤τ

|fet(·+v)−fet|

q

∞

dτ τ

1/q

≤

≤c2kfetk_Bs

∞, q(R)kuk^s_W1

∞(Rⁿ), which, together with (3.1), gives us

kf◦gk_Bs

∞, q(Rⁿ) ≤ ckfetk_Bs

∞, q(R)

1 +kuk^s_W1

∞(Rⁿ)

.

Step2. The case 1< s6= integer. Let|α| ≤[s], then ∂^αfet(u) is the sum of terms of type

(fe_t^{(`)</sup>◦u)∂<sup>β1</sup>u . . . ∂<sup>β`}u, |α| ≥`≥0, |β<sub>1</sub>|+· · ·+|β<sub>`</sub>|=|α|.

Then it suffices to estimate

(3.2) U =

Z

Rⁿ

|h|^{−(s−[s])q}kA(·, h)k^q_∞ dh

|h|ⁿ 1/q

, where

A(x, h) =fe_t^{(`)</sup>(u(x+h))·∂<sup>β1</sup>u . . . ∂<sup>β`}u(x+h)−fe_t^{(`)</sup>(u(x))·∂<sup>β1</sup>u . . . ∂<sup>β`}u(x).

The Lipschitz continuous of uyields

|fe_t^{(`)</sup>(u(x+h))−fe<sub>t</sub><sup>(`)}(u(x))| ≤

sup

|v|≤k∇uk∞|h|

|fe_t^{(`)</sup>(·+v)−fe<sub>t</sub><sup>(`)}| ∞, and it holds

|A(x, h)| ≤ |fe_t^{(`)</sup>(u(x+h))−fe<sub>t</sub><sup>(`)}(u(x))| |∂^β1u . . . ∂^β`u(x+h)|+

+|fe_t^{(`)</sup>(u(x))| |∂<sup>β1</sup>u . . . ∂<sup>β`}u(x+h)−∂^β1u . . . ∂^β`u(x)| ≤

≤c₁

sup

|v|≤k∇uk∞|h|

|fe_t^{(`)</sup>(. . .+v)−fe<sub>t</sub><sup>(`)}|

_∞k∂^β1uk∞. . .k∂^{β`</sup>uk∞+ +kfe<sub>t</sub><sup>(`)}k_∞

|∂^β1u(x+h)−∂^β1u(x)| k∂^β2uk_∞ . . .k∂^{β`</sup>uk<sub>∞</sub>+ +|∂<sup>β2</sup>u(x+h)−∂<sup>β2</sup>u(x)| k∂<sup>β1</sup>uk∞k∂<sup>β3</sup>uk∞. . .k∂<sup>β`}uk∞ +. . .

≤

≤c₂ max

kfe_t^(`)k_∞,

sup

|v|≤k∇uk∞|h|

|fe_t^{(`)</sup>(·+v)−fe<sub>t</sub><sup>(`)}| _∞

×

×

k∂^β1uk

B∞^s−[s], q(Rⁿ). . .k∂^β`uk

B^s−[s]∞, q(Rⁿ)+ +|∂^β1u(x+h)−∂^β1u(x)| k∂^β2uk

B^s−[s]∞, q(Rⁿ). . .k∂^β`uk

B∞^s−[s], q(Rⁿ)+. . .

. Then we obtain the following inequality

|A(x, h)| ≤ c3 max

kfe_t^(`)k_∞,

sup

|v|≤k∇uk∞|h|

|fe_t^{(`)</sup>(·+v)−fe<sub>t</sub><sup>(`)}| ∞

(3.3) ×

×

kuk^`_Bs

∞, q(Rⁿ) +

`

X

j=1

|∂^βju(x+h)−∂^βju(x)|kuk^`−1_Bs

∞, q(Rⁿ)

.

Here, we have employed the Sobolev embedding B∞^s−[s], q(Rⁿ) ,→ L∞(Rⁿ), (s−[s]>0). Now by the embeddingB_∞^s _{, q}(Rⁿ),→B∞^s−[s], q(Rⁿ) and the follow- ing inequalities

(3.4) Z

Rⁿ

|h|^{−(s−[s])q}k∂^βju(·+h)−∂^βjuk^q_∞ dh

|h|ⁿ 1/q

≤ kuk_Bs

∞, q(Rⁿ), the estimate of the last term in (3.3), which is defined byP`

j=1. . ., is obvious;

it is bounded byckuk^s_Bs

∞, q(Rⁿ). Also, by the embedding (2.7) and Proposition 2.11(ii) we obtain

Z

Rⁿ

|h|^{−(s−[s])q}

sup

|v|≤k∇uk∞|h|

|fe_t^{(`)</sup>(·+v)−fe<sub>t</sub><sup>(`)}|

q

∞

dh

|h|ⁿ 1/q

≤

≤c₁k∇uk^s−[s]_∞ Z ∞

0

τ^{−(s−[s])q} sup

|v|≤τ

|fe_t^{(`)</sup>(·+v)−fe<sub>t</sub><sup>(`)}|

q

∞

dτ τ

1/q

≤

≤c2kuk^s−[s]_Bs

∞, q(Rⁿ)kfe_t^(`)k

B^s−[s]∞, q(R)≤c3kuk^s−[s]_Bs

∞, q(Rⁿ)kfetk_Bs

∞, q(R). All together with the inequality kfe_t^(`)k_∞≤ckfetk_Bs

∞, q(R), yield (3.5) U ≤ckfe_tk_Bs

∞, q(R)kuk^s_Bs

∞, q(Rⁿ).

Step3. The cases= 2,3, . . .. We fix a numberσ such that 1< σ <2.

Substep3.1. The case s= 2. Assume first

fe_t∈B_∞¹ _{, q}(R) and u∈B²_{∞, q}(Rⁿ).

Let us prove the following inequality (3.6) kf ◦uk_B1

∞, q(Rⁿ)≤ckfetk_B1

∞, q(R)

1 +kuk_B2

∞, q(Rⁿ)+kuk^1/σ_B2

∞, q(Rⁿ)

. We start with the following identity

fe_t(u(x+h)) +fe_t(u(x−h))−2fe_t(u(x)) = 1 2

2

X

j=1

A_j(x, h) +A_j(x,−h) , where the Aj’s are defined by

A1(x, h) =fet(u(x+h))−fet(2u(x)−u(x−h)), A2(x, h) =fet(u(x+h)) +fet(2u(x)−u(x+h))−2fet(u(x)).

Using the norms defined in (2.10)–(2.11), then it suffices to estimate (3.7) U_j =

Z ∞ 0

τ⁻¹

sup

|h|≤τ

|A_j(·, h)|

_∞

q

dτ τ

1/q

, forj= 1,2.

Estimate of U1. Since u∈B_∞^σ _,∞(R), it holds

|u(x+h) +u(x−h)−2u(x)| ≤ckuk_B2

∞, q(Rⁿ)|h|^σ. Hence for all |h| ≤τ

|A₁(x, h)| ≤c

sup

|v|≤cτ^σkuk_B2

∞, q(Rn)

|fet(·+v)−fet| _∞. Consequently, in (3.7) the change of variable w = c τ^σkuk_B2

∞, q(Rⁿ) in the integral w.r.t. τ, and the continuous embeddingB_∞¹ _{, q}(R),→B∞^1/σ, q(R) yield (3.8) U₁ ≤c₁kfe_tk

B^1/σ∞, q(R)kuk^1/σ_B2

∞, q(Rⁿ)≤c₂kfe_tk_B1

∞, q(R)kuk^1/σ_B2

∞, q(Rⁿ). Estimate of U2. For |h| ≤τ and allx∈Rⁿ, it holds

|A₂(x, h)| ≤ sup

|v|≤k∇uk∞|h|

|fe_t(u(x) +v) +fe_t(u(x)−v)−2fe_t(u(x))|

≤

sup

|v|≤k∇uk∞τ

|fe_t(·+v) +fe_t(· −v)−2fe_t| _∞.

In (3.7), using the change of variable w = k∇uk_∞τ in the integral w.r.t. τ, then by the embedding B_∞² _{, q}(Rⁿ),→W¹_∞(Rⁿ) (see Proposition 2.8) it holds (3.9) U₂ ≤c₁kfe_tk_B1

∞, q(R)k∇uk_∞≤c₂kfe_tk_B1

∞, q(R)kuk_B2

∞, q(Rⁿ). Now (3.6) follows from (3.1), (3.8) and (3.9).

Substep3.2. The case s= 2. Letfet∈B_∞² _{, q}(R) and u∈B²_{∞, q}(Rⁿ). The function fe_t⁰ satisfies (3.6), because fe_t⁰ ∈ B_∞¹ _{, q}(R), then by Propositions 2.7 and 2.9, it holds

kfe_t◦uk_B2

∞, q(Rⁿ)≤c₁

n

X

j=1

k(fe_t⁰◦u)∂_juk_B1

∞, q(Rⁿ)

≤c₂

n

X

j=1

k(fe_t⁰◦u)k_B1

∞, q(Rⁿ)k∂_juk_B1

∞, q(Rⁿ), which is bounded by

(3.10) c3kfetk_B2

∞, q(R)kuk_B2

∞, q(Rⁿ)

1 +kuk_B2

∞, q(Rⁿ)+kuk^1/σ_B2

∞, q(Rⁿ)

. Substep3.3. Now we argue by induction on the integers, then we prove (3.10) with B_∞,q^s instead of B_∞,q² . Let f ∈ B^{s+1, `oc}∞, q (R) and u ∈ B_∞^s+1_{, q}(Rⁿ).

We apply the inductive assumption to fe_t⁰. Then by Proposition 2.7 and the algebra property of B_∞^t _{, q}(Rⁿ),t >0, see Proposition 2.9, we obtain

kfet◦uk_Bs+1

∞, q(Rⁿ)≤c1 n

X

j=1

k(fe_t⁰◦u)∂juk_Bs

∞, q(Rⁿ)≤

≤c₂

n

X

j=1

k(fe_t⁰◦u)k_Bs

∞, q(Rⁿ)k∂_juk_Bs

∞, q(Rⁿ) ≤

≤c3kfe_t⁰k_Bs

∞, q(R)kuk^s_Bs

∞, q(Rⁿ)kuk_Bs+1

∞, q(Rⁿ)

1 +kuk_Bs

∞, q(Rⁿ)+kuk^1/σ_Bs

∞, q(Rⁿ)

≤

≤c4kfetk_Bs+1

∞, q(R)kuk^s+1

B∞^s+1, q(Rⁿ)

1 +kuk_Bs+1

∞, q(Rⁿ)+kuk^1/σ

B∞^s+1, q(Rⁿ)

. This completes the proof.

Remark 3.2. In the case s > 1, Proposition 3.1 has been proved by Bourdaud and Lanza de Cristoforis, see [3, Theorem 4].

If s = 1, Proposition 3.1 can be studied, but by replacing K (see (1.1) for a definition) by B_∞^1+ε_{, q}(Rⁿ) for ε >0 arbitrary small.

Proposition 3.3. Let f :R→Rbe a function such that f ∈B∞^{1, `oc}, q(R).

Let ε >0. If u∈B_∞^1+ε_{, q}(Rⁿ) thenf ◦u∈B_∞¹ _{, q}(Rⁿ).

Proof. Let f ∈ B^{1, `oc}∞, q(R) and u ∈ B_∞^1+ε_{, q}(Rⁿ), then we will prove the following inequality

kf ◦uk_B1

∞, q(Rⁿ)≤ckfetk_B1

∞, q(R)

1 +kuk_B1+ε

∞, q(Rⁿ)+kuk^1/(1+ε)

B∞^1+ε, q(Rⁿ)

; with fet = f ρt and t ≥ max(1,kuk∞). Using the splitting and notation of Substep 3.1 of the proof of Proposition 3.1.

Assume that 0< ε <1. Since fe_t∈B_∞¹ _{, q}(R),→B∞^1/(1+ε),∞ (R), it holds

|A₁(x, h)| ≤ckfetk_B1

∞, q(R)|u(x+h) +u(x−h)−2u(x)|^1/(1+ε). Since 0<1 +ε <2, then we have

Z ∞ 0

τ⁻¹

sup

|h|≤τ

|u(·+h)+u(·−h)−2u|^1/(1+ε) ∞

q

dτ τ

1/q

≤ kuk^1/(1+ε)

B_∞^1+ε_{, q/(1+ε)}(Rⁿ), Consequently, the continuous embedding B_∞^1+ε_{, q/(1+ε)}(Rⁿ),→B_∞^1+ε_{, q}(Rⁿ) yields

U₁≤ckfe_tk_B1

∞, q(R)kuk^1/(1+ε)

B∞^1+ε, q(Rⁿ).

The estimate of U2 is unchanged, we just use in (3.9) the embedding B_∞^1+ε_{, q}(Rⁿ),→W¹_∞(Rⁿ),

i.e., k∇uk∞≤ckuk_B1+ε

∞, q(Rⁿ).

Proposition 3.4. Let 0 < s 6= integer. Let f : R → R be a function such thatf^(j)∈L^`oc_∞(R) forj= 1, . . . ,[s] + 1. Ifu∈ Kthenf◦u∈B_∞^s _{, q}(Rⁿ).

Proof. As in the Step 2 of the proof of Proposition 3.1 with s > 0, we arrive to estimate (3.2). The locally Lipschitz continuous of f^(j) yields

|fe_t^{(`)</sup>(u(x+h))−fe<sub>t</sub><sup>(`)}(u(x))| ≤ckfe_t<sup>(`+1)</sup>k<sub>∞</sub>|u(x+h)−u(x)|, 0≤`≤[s], and we have (see (3.3))

|A(x, h)| ≤cmax

kfe_t^{(`)</sup>k<sub>∞</sub>,kfe<sub>t</sub><sup>(`+1)}k_∞

|u(x+h)−u(x)|kuk^`_Bs

∞, q(Rⁿ)+ +

`

X

j=1

|∂^βju(x+h)−∂^βju(x)|kuk^`−1_Bs

∞, q(Rⁿ)

.

Hence, using (3.4) we obtain an estimate similar to (3.5). Notice that, in the case 0< s <1, we also employ the embeddingW_∞¹(Rⁿ),→B^s_{∞, q}(Rⁿ).

3.2. Basic estimates

It is well-known that fors >0 all function ofB_∞^s _{, q}(Rⁿ) acts by pointwise multiplication on E_p,q^s (Rⁿ), cf. [16, Theorem 9, p. 144] and [17]. We use this idea in the present section. Also, the proof of Theorem 1.1 relies upon the fol- lowing statement which is an almost orthogonality type, and has been proved previously for L2(Rⁿ),H_p^s(Rⁿ) and B_{p , q}^s (Rⁿ) in [8], [13] and [5], respectively.

Proposition 3.5. Let b > 1 and s > 0. Then there exists a constant c >0, such that the inequality

∞

X

j=0

χjfj

E^s_p,q(Rⁿ)

≤c

sup

k∈N

kχ_kk_Bs

∞, q(Rⁿ)

k{2^sjfj}_j∈Nk_Ep,q

holds, for every sequence {χ_j}_j∈N of functions in B_∞^s _{, q}(Rⁿ), and every se- quence {f_j}_j∈N such that suppfb_j ⊆ {ξ∈Rⁿ:|ξ| ≤b2^j}.

Proof. We begin by a splitting of the area of summing with respect toj, i.e., we write

χj =Sjχj+

∞

X

k=j+1

Qkχj, j∈N,

which implies P∞

j=0χjfj =g1+g2, where g1 =

∞

X

j=0

fjSjχj and g2=

∞

X

k=1 k−1

X

j=0

fjQkχj.

Estimate of g₁. The function fb_j ∗(γ(2^−j·)χb_j) is supported by the ball

|ξ| ≤2^j(b+ 2), then Proposition 2.4(ii) and the inequality kS_jχ_jk∞≤ckF⁻¹γk₁ sup

k∈N

kχ_kk∞, ∀j∈N give the desired result.

Estimate of g2. Also, since the function Pk−1

j=0fbj ∗(φ(2^−k·)χbj) is sup- ported by the ball |ξ| ≤2^k(2 + (b/2)), then Proposition 2.4(ii) yields

kg₂k_Es

p,q(Rⁿ)≤c

2^sk

k−1

X

j=0

f_jQ_kχ_j

k∈N

_E

p,q

.

Now using in the order Minkowski (w.r.t. `_q, q ≥ 1) and H¨older inequalities, we obtain the desired estimate, i.e., for theB-space

kg₂k_Bs

p , q(Rⁿ)≤ ∞

X

k=1

2^skq k−1

X

j=0

kf_jQkχjk_p q1/q

≤ ^∞

X

k=1

2^{skq k−1}

X

j=0

kf_jk_pkQ_kχjk_∞ q1/q

≤

∞

X

j=0

∞

X

k=j+1

2^skqkQ_kχjk^q_∞ 1/q

2^−sj(2^sjkf_jk_p)

≤csup

j∈N

kχ_jk_Bs

∞, q(Rⁿ)

^∞ X

`=0

2<sup>s`q</sup>kf<sub>`</sub>k^q_p 1/q

, and for the F-space

kg₂k_Fs

p , q(Rⁿ)≤

^∞ X

k=1

2^skq

k−1

X

j=0

f_j(·)Q_kχ_j(·)

q1/q p

≤

∞

X

j=0

^∞ X

k=j+1

2^skq|f_j(·)Q_kχ_j(·)|^q 1/q

p

≤

∞

X

j=0

∞

X

k=j+1

2^skqkQ_kχjk^q_∞ 1/q

2^−sj(2^sj|f_j(·)|) p