171–183 COMPOSITION OPERATOR ON THE SPACE OF FUNCTIONS TRIEBEL-LIZORKIN AND BOUNDED VARIATION TYPE M

Vol. LXXXI, 2 (2012), pp. 171–183

COMPOSITION OPERATOR ON THE SPACE OF FUNCTIONS TRIEBEL-LIZORKIN AND BOUNDED VARIATION TYPE

M. MOUSSAI

Abstract. For a Borel-measurable functionf:R→Rsatisfyingf(0) = 0 and sup

t>0

t⁻¹ Z

R

sup

|h|≤t

|f⁰(x+h)−f⁰(x)|^pdx <+∞, (0< p <+∞), we study the composition operatorTf(g) :=f◦gon Triebel-Lizorkin spacesF_p,q^s (Rⁿ) in the case 0< s <1 + (1/p).

1. Introduction and the main result

The study of the composition operator T_f: g → f ◦g associated to a Borel- measurable function f:R → R on Triebel-Lizorkin spaces F_p,q^s (Rⁿ), consists in finding a characterization of the functionsf such that

T_f(F_p,q^s (Rⁿ))⊆F_p,q^s (Rⁿ).

(1.1)

The investigation to establish (1.1) was improved by several works, for example the papers of Adams and Frazier [1, 2], Brezis and Mironescu [6], Maz’ya and Shaposnikova [9], Runst and Sickel [12] and [10]. There were obtained some necessary conditions onf; from which we recall the following results. Fors >0, 1< p <+∞and1≤q≤+∞

• if Tf takes L_∞(Rⁿ)∩F_p,q^s (Rⁿ)toF_p,q^s (Rⁿ), thenf is locally Lipschitz con- tinuous.

• if Tf takes the Schwartz space S(Rⁿ)toF_p,q^s (Rⁿ), thenf belongs locally to F_p,q^s (R).

The first assertion is proved in [3, Theorem 3.1]. The proof of the second assertion can be found in [12, Theorem 2, 5.3.1].

Bourdaud and Kateb [4] introduced the functions class U_p¹(R), the set of Lipschitz continuous functionsf such that their derivatives, in the sense of distri- butions, satisfy

Ap(f⁰) :=

sup

t>0

t⁻¹ Z

R

sup

|h|≤t

|f⁰(x+h)−f⁰(x)|^pdx^1/p

<+∞, (1.2)

Received August 8, 2011; revised February 17, 2012.

2010Mathematics Subject Classification. Primary 46E35, 47H30.

Key words and phrases. Triebel-Lizorkin spaces; Besov spaces; functions of bounded variation;

composition operators.

and are endowed with the seminorm kfk_U1

p(R):= inf(kgk_∞+A_p(g)),

where the infimum is taken over all functionsg such thatf is a primitive ofg. In [4] the authors, proved the acting of the operatorT_f on Besov spaceB^s_p,q(Rⁿ) for 1≤p < +∞, 1 < s <1 + (1/p) andf ∈U_p¹(R) with f(0) = 0. In [5] the same result holds for 0< s <1 + (1/p).

In this work we will study the composition operatorT_f onF_p,q^s (Rⁿ) for a func- tionf which belongs to U_p¹(R), then we will obtain a result of type (1.1). To do this, we introduce the setVp(Rⁿ) of the functionsg:Rⁿ→Rsuch that

kgk_Vp(Rn):=

n

X

j=1

Z

Rⁿ⁻¹

kgx⁰_jk^p_BV1

p(R)dx⁰_j^1/p

<+∞

where BV_p¹(R) is the Wiener space of the primitives of functions of bounded p-variation (see Subsection 2.2 below for the definition) and

g_x⁰

j(y) :=g(x₁, . . . , x_j−1, y, x_j+1, . . . , x_n), y∈R, x∈Rⁿ. (1.3)

We will prove the following statement.

Theorem 1.1. Let 0< p, q <+∞and 0< s <1 + (1/p). Then there exists a constantc >0such that the inequality

kf◦gk_Fs

p,q(Rⁿ)≤ckfk_U1 p(R)

kgkp+kgk_Vp(Rn)

(1.4)

holds for all functionsg∈L_p(Rⁿ)∩ Vp(Rⁿ)and allf ∈U_p¹(R)satisfyingf(0) = 0.

Moreover, for all suchf, the operator Tf takesLp(Rⁿ)∩ Vp(Rⁿ) toF_p,q^s (Rⁿ).

Remark. (i) SinceF_p,q^s (Rⁿ),→L_p(Rⁿ), thenT_f maps fromF_p,q^s (Rⁿ)∩ Vp(Rⁿ) toF_p,q^s (Rⁿ) under the assumptions of Theorem 1.1.

(ii) Since the Bessel potential spacesH_p^s(Rⁿ) =F_p,2^s (Rⁿ), 1< p <∞, Theorem 1.1 covers the results of composition operators in caseH_p^s(Rⁿ) instead ofF_p,q^s (Rⁿ).

The paper is organized as follows. In Section 2 we collect some properties of the needed function spacesF_p,q^s (Rⁿ) andBV_p¹(R). Section 3 is devoted to the proof of the main result where in a first step we study the case of 1-dimensional which is the main tool when we prove Theorem 1.1. Also, our proof uses various Sobolev and Peetre embeddings, Fubini and Fatou properties, etc. In Section 4 we give some corollaries and prove the sharp estimate of (1.4).

Notation. We work with functions defined on the Euclidean space Rⁿ. All spaces and functions are assumed to be real-valued. We denote by C_b(Rⁿ) the Banach space of bounded continuous functions onRⁿendowed with the supremum.

Let D(Rⁿ) (resp. S(Rⁿ) and S⁰(Rⁿ)) denotes the C^∞-functions with compact support (resp. the Schwartz space of allC^∞ rapidly decreasing functions and its topological dual). Withk · kp we denote theLp-norm. We define the differences by ∆hf := f(·+h)−f for all h ∈ Rⁿ. If E is a Banach function space on Rⁿ, we denote byE^`oc the collection of all functionsf such thatϕf ∈E for all

ϕ∈ D(Rⁿ). As usual, constantsc, c1, . . .are strictly positive and depend only on the fixed parametersn, s, p, q; their values may vary from line to line.

2. Function spaces 2.1. Triebel-Lizorkin spaces

Let 0< a≤ ∞. For all measurable functionsf onRⁿ, we set M_p,q^s,u,a(f) :=Z

Rⁿ

Z a

0

t^−sq1 tⁿ

Z

|h|≤t

|∆_hf(x)|^udhq/udt t

p/q

dx1/p

.

Definition 2.1. Let 0< p <+∞and 0< q≤+∞. Letsbe a real satisfying 1< s <2 and s > nmax1

p−1,1 q−1

.

Then, a functionf ∈L_p(Rⁿ) belongs toF_p,q^s (Rⁿ) if kfk_Fs

p,q(Rⁿ):=kfkp+

n

X

j=1

M_p,q^s−1,1,∞(∂jf)<+∞.

The set F_p,q^s (Rⁿ) is a quasi Banach space for the quasi-norm defined above.

For the equivalence of the above definition with other characterizations we refer to [15, Theorem 3.5.3] from which we recall the following statement.

Proposition 2.2. Let 0 < p < +∞ and 0 < q, u ≤ +∞. Let s be a real satisfying

1< s <2 and s > nmax1 p−1

u,1 q− 1

u

. Then, a functionf ∈L_p(Rⁿ)belongs toF_p,q^s (Rⁿ)if and only if

kfkp+M_p,q^s,u,∞(f)<+∞

(2.1)

and the expression (2.1) is an equivalent quasi-norm in F_p,q^s (Rⁿ). Moreover, this assertion remains true if one replacesM_p,q^s,u,∞ byM_p,q^s,u,a for any fixeda >0.

The argument of the equivalence of above quasi-norms that we can replace the integration fort∈]0,+∞[ byt≤afor a fixed positive numberais the part of the integral for whicht > acan be easily estimated by theLp-norm.

Embeddings. Triebel-Lizorkin spaces are spaces of equivalence classes w.r.t. al- most everywhere equality. However, if such an equivalence class contains a contin- uous representative, then usually we work with this representative and call also the equivalence class a continuous function. Later on we need the following continuous embeddings:

(i) The spacesF_p,q^s (Rⁿ) are monotone with respect to s and q, more exactly F_p,∞^s (Rⁿ),→F_p,q^t (Rⁿ),→F_p,∞^t (Rⁿ) ift < s and 0< q≤ ∞.

(ii) With Besov spaces, we haveB^s_p,1(Rⁿ),→F_p,q^s (Rⁿ),→B_p,∞^s (Rⁿ).

(iii) If eithers > n/por s=n/pand 0< p≤1, thenF_p,q^s (Rⁿ),→Cb(Rⁿ).

For various further embeddings we refer to [14, 2.3.2, 2.7.1] or [12, 2.2.2, 2.2.3].

The Fatou property. Well-known the Triebel-Lizorkin space has the Fatou prop- erty, cf. [8]. We will briefly recall it. Any f ∈F_p,q^s (Rⁿ) can be approximated (in the weak sense in S⁰(Rⁿ)) by a sequence (f_j)_j≥0 such that any f_j is an entire function of exponential type

fj ∈F_p,q^s (Rⁿ) and lim sup

j→+∞

kfjkF_p,q^s (Rⁿ)≤ckfkF_p,q^s (Rⁿ)

with a positive constantc independent of f. Vice versa, if for a tempered distri- butionf ∈ S⁰(Rⁿ), there exists a sequence (f_j)_j≥0 such that

fj∈F_p,q^s (Rⁿ) and A:= lim sup

j→+∞

kfjkF_p,q^s (Rⁿ)<+∞,

and lim_j→+∞f_j=f in the sense of distributions, thenf belongs toF_p,q^s (Rⁿ) and there exists a constantc >0 independent off such thatkfk_Fs

p,q(Rⁿ)≤cA.

2.2. Functions of bounded variation

For a functiong:R→R, we set ν_p(g) := supX^N

k=1

|g(bk)−g(a_k)|^p1/p , (2.2)

taken over all finite sets{]ak, bk[ ;k= 1, . . . , N}of pairwise disjoint open intervals.

A function g is said to be of bounded p-variation if νp(g) < +∞. Clearly, by considering a finite sequence with only two terms, we obtain|g(x)−g(y)| ≤νp(g), for allx, y∈R, henceg is a bounded function. The set of (generalized) primitives of functions of boundedp-variation is denoted by BV_p¹(R) and endowed with the seminorm

kfkBV_p¹(R):= inf ν_p(g),

where the infimum is taken over all functionsgwhosef is the primitive. For more details about this space we refer to [11] or [5]. However, we need to recall some embeddings

BV_p¹(R),→U_p¹(R) (2.3)

(equality in case p = 1), see [5, Theorem 5] for the proof which is given for 1< p <+∞and can be easily extended to 0< p≤1, see also [7, Theorem 9.3].

The Peetre embedding theorem

B˙^1+(1/p)_p,1 (R),→BV_p¹(R),→B˙^1+(1/p)_p,∞ (R), (1≤p <+∞), (2.4)

where the dotted space is thehomogeneousBesov space.

Example. Letα∈R. We putuα(x) :=|x+α| − |α|for allx∈R, and fα(x, y) :=uα(x)χ[0,1](y) +uα(y)χ[0,1](x), ∀x, y∈R,

whereχ_[0,1] denotes the indicatrix function of [0,1]. Clearly thatνp(u⁰_α) = 2 and kχ_[0,1]k_BV1

p(R)= 0. Then it holdsfα∈ Vp(R²) with kfαk_Vp(R2) = 4. TheVp(Rⁿ) space is defined in Section 1.

3. Proof of the result Theorem 1.1 can be obtained from the following statement.

Proposition 3.1. Let 0 < p, q < +∞, 0 < u < min(p, q) and 0 < s < 1/p.

Then there exists a constantc >0 such that the inequality M_p,q^s,u,∞((f◦g)⁰)≤ckfk_U1

p(R)kgk_BV1 p(R)

(3.1)

holds for allf ∈U_p¹(R)∩C¹(R)and all real analytic functionsg in BV_p¹(R).

Proof. For a better readability we split our proof in two steps.

Step 1. Let us prove

M_p,q^s,u,a((f◦g)⁰)≤c a^(1/p)−skfkU_p¹(R)kgkBV_p¹(R)

(3.2)

for alla >0 and allf ∈U_p¹(R)∩C¹(R) and all real analytic functionsginBV_p¹(R).

Assume firsta= 1. By the assumptions onf andgit holds (f◦g)⁰= (f⁰◦g)g⁰. We havek(f◦g)⁰k_∞ ≤ kf⁰k_∞kg⁰k_∞ and

|∆h((f⁰◦g)g⁰)(x)| ≤ kf⁰k_∞|∆hg⁰(x)| + |g⁰(x)| |∆h(f⁰◦g)(x)|.

Hence

M_p,q^s,u,1((f◦g)⁰)≤ kf⁰k_∞M_p,q^s,u,1(g⁰) +V(f;g), where

V(f;g) :=Z

R

Z 1

0

t^−sq1 t

Z t

−t

|∆h(f⁰◦g)(x)|^u|g⁰(x)|^udh^q/udt t

^p/q dx^1/p (3.3) .

Estimate ofM_p,q^s,u,1(g⁰).By writingR1

0 · · ·=P∞ j=0

R2^−j

2^−j−1· · · and by an elementary computation, we have

Z 1

0

t^−sq1 t

Z t

−t

|∆hg⁰(x)|^udh^q/udt t ≤c₁

∞

X

j=0

Z 2^−j

2^−j−1

t^−sq sup

|h|≤t

|∆hg⁰(x)|^q dt t

≤c₂

∞

X

j=0

2^jsq sup

|h|≤2^−j

|∆_hg⁰(x)|^q.

Letα:= min(1, p/q). By using the monotonicity of the`<sub>r</sub>-norms (i.e. `₁,→`_1/α) and by the Minkowski inequality w.r.tL_p/(αq), sinceq <+∞, we obtain

M_p,q^s,u,1(g⁰)≤c1

Z

R

X^∞

j=0

2^jsαq sup

|h|≤2^−j

|∆hg⁰(x)|^αqp/(αq)

dx1/p

≤c2

X^∞

j=0

2^jsαqZ

R

sup

|h|≤2^−j

|∆hg⁰(x)|^pdx(αq)/p1/(αq)

≤c3

X^∞

j=0

2j(s−(1/p))αq^1/(αq) kgk_U1

p(R).

From the embedding (2.3) and the assumption ons, the desired estimate holds.

Estimate of V(f;g). In (3.3) the integral with respect to hcan be limited to the interval [0, t] denoting the corresponding expression by V+(f;g). Let us notice that the estimate with respect to [−t,0] will be completely similar.

Again, by applying the Minkowski inequality twice, it holds V₊(f;g)

≤ Z

R

Z 1

0

Z 1

h

t−(s+(1/u))q|∆_h(f⁰◦g)(x)|^q|g⁰(x)|^q dt t

^u/q dh

^p/u dx

^1/p

≤ Z 1

0

Z

R

|∆_h(f⁰◦g)(x)|^p|g⁰(x)|^pdx

^u/p Z ∞

h

t−(s+(1/u))q dt t

^u/q dh

^1/u

≤c Z 1

0

h^−(su+1) Z

R

|∆_h(f⁰◦g)(x)|^p|g⁰(x)|^pdx ^u/p

dh ^1/u

.

Case 1: Assume thatg⁰ does not vanish on R. By the Mean Value Theorem and by the change of variabley=g(x), we find

V₊(f;g)

≤c1kg⁰k^1−(1/p)_∞ Z 1 0

h^−(su+1)Z

R

sup

|v|^≤hkg⁰k∞

|f⁰(v+y)−f⁰(y)|^pdyu/p

dh1/u

≤c₂kfk_U1

p(R)kg⁰k_∞Z 1 0

hu((1/p)−s)−1dh1/u

≤c3kfk_U1

p(R)kgk_BV1 p(R).

Case 2: Assume that the set of zeros of g⁰ is nonempty. Then it is a discrete set whose complement in R is the union of a family (Il)l of open disjoint intervals.

For anyh >0, we denote byI_l,h⁰ the set ofx∈Il whose distance to the boundary ofIl is greater thanh. We set

I_l,h⁰⁰ :=Il\I_l,h⁰ and gl:=g_|Il.

Clearly the functiong_lis a diffeomorphism ofI_l ontog(I_l). Let us notice thatI_l,h⁰ is an open interval, possibly empty. In case it is not empty, we have

|g(g_l⁻¹(y) +h)−y| ≤hsup

I_l

|g⁰|, ∀y∈g_l(I_l,h⁰ ).

(3.4)

The setI_l,h⁰⁰ is an interval of length at most 2hor the union of two such intervals, andg⁰ vanishes at one of the endpoints of these or those intervals.

We writeV₊(f;g)≤V₁(f;g) +V₂(f;g), where V₁(f;g) :=

Z 1

0

h^−(su+1)

X

l

Z

I_l,h⁰

|∆h(f⁰◦g)(x)|^p|g⁰(x)|^pdx u/p

dh 1/u

andV2(f;g) is defined in the same way by replacingI_l,h⁰ byI_l,h⁰⁰ .

Estimate ofV1(f;g). By the change of variabley=gl(x) and by (3.4), we deduce V1(f;g)≤

Z 1

0

h^−(su+1)

X

l

sup

Il

|g⁰|^p−1

× Z

g(I_l,h⁰ )

sup

|v|≤hsup_Il|g⁰|

|f⁰(v+y)−f⁰(y)|^pdy ^u/p

dh ^1/u

≤c1kfk_U1 p(R)

X

l

sup

Il

|g⁰|^p

^1/p Z 1

0

hu((1/p)−s)−1dh ^1/u

≤c₂kfkU_p¹(R)

X

l

sup

I_l

|g⁰|^p 1/p

. Hence it suffices to show

X

l

sup

t∈Il

|g⁰(t)|^p1/p

≤ ckgk_BV1 p. (3.5)

Indeed, by the assumption ong, for any Il there existsξl∈Il such that

|g⁰(ξl)| = sup

t∈Il

|g⁰(t)|.

Furthermore, set β_l the right endpoint of I_l. The open intervals {]ξ_l, β_l[}l are pairwise disjoint. Then the assertion (3.5) follows from

X

l

sup

t∈Il

|g⁰(t)|^p=X

l

|g⁰(ξl)−g⁰(βl)|^p≤νp(g⁰)^p. (See (2.2) for the definition ofνp).

Estimate of V₂(f;g). Using both the elementary inequality |∆h(f⁰ ◦ g)(x)| ≤ 2kf⁰k∞ and the properties ofI_l,h⁰⁰ , it holds

V₂(f;g)≤c₁kf⁰k_∞

X

l

sup

I_l

|g⁰|^p

^1/p Z 1

0

hu((1/p)−s)−1dh ^1/u

≤c2kfk_U1

p(R)kgk_BV1 p(R).

Hence we obtain (3.2) witha= 1. We put gλ(x) := g(λx) for all x∈R and all λ >0. Then (3.2) can be obtained for alla >0 since kgak_BV1

p(R) =akgk_BV1 p(R)

and

M_p,q^s,u,a((f ◦g)⁰) =a^{(1/p)−s−1}M_p,q^s,u,1((f◦ga)⁰).

Step 2: Proof of (3.1). Let a > 0. Let f and g be as in Proposition 3.1. By Proposition 2.2 it holds

M_p,q^s,u,∞((f◦g)⁰)≤ k(f ◦g)⁰k_Fs

p,q(R)=k(f◦g)⁰kp+M_p,q^s,u,a((f◦g)⁰).

Applying (3.2), we obtain

M_p,q^s,u,∞((f◦g)⁰)≤ kf⁰k∞kg⁰kp+c1a^(1/p)−skfk_U1

p(R)kgk_BV1 p(R)

(3.6)

with a positive constantc1 depending only ons, p andq (see the end of Step 1).

Now, by replacing g by gλ in (3.6), (gλ is defined in Step 1), and by using the equality

M_p,q^s,u,∞ (f ◦gλ)⁰

=λ^s+1−(1/p)M_p,q^s,u,∞((f◦g)⁰), we deduce

M_p,q^s,u,∞((f◦g)⁰)

≤λ^−skf⁰k_∞kg⁰k_p+c₁a^(1/p)−sλ^(1/p)−skfk_U1

p(R)kgk_BV1 p(R)

(3.7)

for alla, λ > 0. Taking a= 1/λ. Now letting λ→ +∞ in (3.7), we obtain the

desired result.

Remark. Proposition 3.1 is also valid in then-dimensional case. The inequality (3.1) becomes

M_p,q^s−1,u,∞(∂j(f ◦g))≤ckfkU_p¹(R)kgk_Vp(Rⁿ), (j= 1, . . . , n) for allf ∈U_p¹(R)∩C¹(R) and all real analytic functionsg inVp(Rⁿ).

Proof of Theorem1.1. Step 1. Observe that the conditions f(0) = 0 andf⁰ ∈ L_∞(R) imply

kf ◦gkp ≤ kf⁰k_∞kgkp

which is sufficient for the estimateT_f(g) with respect toL_p(Rⁿ)-norm.

Step 2: The case 1 < s < 1 + (1/p) and n = 1. We first consider a function f ∈ U_p¹(R), of class C¹ and a function g real analytic in Lp(R)∩BV_p¹(R). By Proposition 3.1, it holds

kf◦gk_Fs

p,q(R)≤ckfk_U1 p(R)

kgkp+kgk_BV1 p(R)

. (3.8)

Now we prove (3.8) in the general case. Letg∈Lp(R)∩BV_p¹(R) andf ∈U_p¹(R).

We introduce a function ρ ∈ D(R) satisfying ρ(0) = 1, and we set ϕ_j(x) :=

2^jnF⁻¹ρ(2^jx) for allx∈Rand allj ∈N; hereF⁻¹ρdenotes the inverse Fourier transform ofρ. We set also

fj :=ϕj∗f−ϕj∗f(0) and gj :=ϕj∗g.

Then the functiong_j is real analytic andg_j→g inL_p(R). We have also kgjk_BV1

p(R)≤ckgk_BV1

p(R), ∀j∈N. (3.9)

To prove (3.9), let{]a_k, b_k[, k = 1, . . . , N} be a set of pairwise disjoint intervals.

By the Minkowski inequality, it holds X^N

k=1

Z

R

ϕj(y)

g⁰(bk−y)−g⁰(ak−y) dy

p^1/p

≤ Z

R

|ϕj(y)|X^N

k=1

g⁰(b_k−y)−g⁰(a_k−y)

p^1/p dy.

Now, for ally∈R, the intervals ]ak−y, bk−y[ (k= 1, . . . , N) are pairwise disjoint.

Then

X^N

k=1

|g⁰_j(b_k)−g⁰_j(a_k)|^p1/p

≤ kF⁻¹ρk1ν_p(g⁰), ∀j∈N. Hence we obtain (3.9).

The functionsfj areC^∞ such thatfj(0) = 0 and satisfy kfjk_U1

p(R)≤ckfk_U1

p(R), ∀j ∈N. (3.10)

To prove (3.10), for allt >0 and all h∈[−t, t] we trivially have

|ϕj∗f⁰(x+h)−ϕj∗f⁰(x)| ≤ Z

R

|ϕj(y)|sup

|z|≤t

|f⁰(x−y+z)−f⁰(x−y)|dy.

By the Minkowski inequality, we have Z

R

sup

|h|≤t

|ϕj∗f⁰(x+h)−ϕj∗f⁰(x)|^pdx

≤Z

R

|ϕ_j(y)|Z

R

sup

|z|≤t

|f⁰(x−y+z)−f⁰(x−y)|^pdx1/p

dyp

≤tkF⁻¹ρk^p₁A_p(f⁰)^p, (see (1.2) for the definition ofA_p).

Consequently,

Ap(f_j⁰) +kf_j⁰k_∞≤ kF⁻¹ρk1(Ap(f⁰) +kf⁰k_∞) and we obtain the desired result.

On the other hand, we have

j→+∞lim kfj−fk∞= 0.

(3.11)

To prove (3.11), since lim_j→+∞ϕj∗f(0) =f(0) = 0, the Lipschitz continuous of f yields

|fj(x)−f(x)| ≤ kf⁰k∞

Z

R

|x−y||ϕj(x−y)|dy +|ϕj∗f(0)|

≤c2^−jkf⁰k_∞ +|ϕj∗f(0)|.

Then the desired result holds. By the same argument, we obtain kgj−gk∞≤c2^−jkg⁰k∞.

(3.12)

Now we apply (3.8) tof_j andg_j. Then by (3.9) and (3.10), we obtain kf_j◦g_jk_Fs

p,q(R)≤ckfk_U1 p(R)

kgk_p+ kgk_BV1 p(R)

. (3.13)

The elementary inequality

kf◦g−f_j◦g_jk∞≤ kf⁰k∞kg−g_jk∞+kf−f_jk∞

complemented by (3.11)–(3.12) yields the convergence of the sequence{f_j◦g_j}_j∈N tof◦g inL_∞(R). Since

|hfj◦gj−f◦g, ψi| ≤ kfj◦gj−f◦gk_∞kψk1, ∀ψ∈ D(R),

thus we conclude that lim_j→+∞fj◦gj =f◦gin the sense of distributions. Hence, by the Fatou property ofF_p,q^s (R), see Subsection 2.1, we deduce (3.8).

Step 3: The case1< s <1 + (1/p)andn≥2. We use the notation (1.3). Since Triebel-Lizorkin space has the Fubini property (see [12, p. 70]), by (3.1) it holds

kf◦gk_Fs

p,q(Rⁿ)≤c1 n

X

j=1

Z

Rⁿ⁻¹

kf◦g_x⁰

jk^p_Fs

p,q(R)dx⁰_j1/p

≤c2kfk_U1 p(R)

n

X

j=1

Z

Rⁿ⁻¹

kg_x⁰

jk^p_p+kg_x⁰

jk^p_BV1 p(R)

dx⁰_j^1/p

≤c₃kfk_U1 p(R)

kgk_p+kgk_Vp(Rn)

.

Step 4: The case 0 < s ≤1. Due to the monotonicity of the Triebel-Lizorkin scale with respect to the smoothness parameter s, the result holds. Indeed, let 1 < t < 1 + (1/p). From Step 3, we have (1.4) with kf ◦gk_Ft

p,q(Rⁿ) instead of kf ◦gk_Fs

p,q(Rⁿ). Now we apply the continuous embedding F_p,q^t (Rⁿ),→ F_p,q^s (Rⁿ).

This completes the proof.

Remark. In case n= 1 and 1≤p, q <+∞the inequality (1.4) becomes kf◦gk_Fs

p,q(R)≤ckfk_U1 p(R)

kgk_Fs

p,q(R)+kgk_BV1 p(R)

for allg ∈Lp(R)∩BV_p¹(R), since F_p,q^s (R)∩BV_p¹(R) = Lp(R)∩BV_p¹(R) if s <

1 + (1/p). To prove this equality, we have ˙Bp,∞^1+(1/p)(R)∩Lp(R) = B^1+(1/p)p,∞ (R) (see [12, 2.6.2, p. 95]). Then by (2.4) and by bothB^1+(1/p)_p,∞ (R) ,→B_p,1^s (R) and B_p,1^s (Rⁿ),→F_p,q^s (Rⁿ), it holdsLp(R)∩BV_p¹(R),→F_p,q^s (R).

4. Concluding remarks 4.1. Some corollaries

In this section we fix a smooth cut-off functionϕ∈ D(R) such thatϕ(x) = 1 for

|x| ≤ 1. We putϕ_t(x) :=ϕ t⁻¹x

, ∀x∈Rand for all t >0. Also for brevity we introduce the spaceF_p,q^s (Rⁿ) :=F_p,q^s (Rⁿ)∩L_∞(Rⁿ) endowed with the quasi-norm

kfk_Fs

p,q(Rⁿ):=kfk_Fs

p,q(Rⁿ)+kfk_∞.

Theorem 1.1 has a consequence for the case of functionsf which are only locally inU_p¹(R).

Corollary 4.1. Lets, p, q be real numbers as in Theorem1.1. Then there exists a constantc >0such that the inequality

kf◦gk_Fs

p,q(Rⁿ)≤ckf ϕ_kgk∞k_U1 p(R)

kgk_Fs

p,q(Rⁿ)+kgk_Vp(Rn)

(4.1)

holds for all functions g ∈ F_p,q^s (Rⁿ)∩ Vp(Rⁿ) and all f ∈ U_p^1,`oc(R) satisfying f(0) = 0. Moreover, for all such functions f, the composition operator Tf takes F_p,q^s (Rⁿ)∩ Vp(Rⁿ)toF_p,q^s (Rⁿ).

Proof. Since f ◦g = (f ϕ_kgk∞)◦g and (f ϕt)(0) = 0, the result follows from

Theorem 1.1.

There is consequence of Theorem 1.1 that we can obtain the equivalence of acting condition and boundedness.

Corollary 4.2. Lets, p, q be real numbers as in Theorem1.1. Letf be a func- tion inU_p^1,`oc(R)satisfyingf(0) = 0. Then the following assertions are equivalent:

(i) Tf satisfies the acting conditionTf(F_p,q^s (Rⁿ)∩ Vp(Rⁿ))⊆ F_p,q^s (Rⁿ).

(ii) Tf maps bounded sets inF_p,q^s (Rⁿ)∩ Vp(Rⁿ)into bounded sets inF_p,q^s (Rⁿ).

Proof. Lett >0. By (4.1), it holds

kf◦gk_Fp,q^s (Rⁿ)≤c tkf ϕtkU_p¹(R)

(4.2)

for all g ∈ F_p,q^s (Rⁿ)∩ V_p(Rⁿ) such that kgk_Fs

p,q(Rⁿ)+kgk_Vp(Rn) ≤t. Now, from (4.2), we conclude thatTf maps bounded sets inF_p,q^s (Rⁿ)∩ Vp(Rⁿ) into bounded

sets inF_p,q^s (Rⁿ).

Remark. Ifn/p < s <1 + (1/p), then we can replaceF_p,q^s (Rⁿ) byF_p,q^s (Rⁿ) in Corollaries 4.1–4.2, sinceF_p,q^s (Rⁿ),→Cb(Rⁿ).

We show that Theorem 1.1 can be extended to the case of the boundedness between Besov spaces and Triebel-Lizorkin spaces.

Corollary 4.3. Let 1≤p, q <+∞and0< s <1 + (1/p). Then there exists a constantc >0such that the inequality

kf◦gkF_p,q^s (Rⁿ)≤ckfkU_p¹(R)kgk_B1+(1/p) p,1 (Rⁿ)

holds for all functions g ∈ B_p,1^1+(1/p)(Rⁿ) and all f ∈U_p¹(R) satisfying f(0) = 0.

Moreover, for all such functionsf, the operatorT_f takesB_p,1^1+(1/p)(Rⁿ)toF_p,q^s (Rⁿ).

Proof. This is an easy consequence of Theorem 1.1 and the following continuous embedding

B_p,1^1+(1/p)(Rⁿ),→ V_p(Rⁿ).

(4.3)

To prove (4.3), we use the notation (1.3) and the equivalent norm in Besov space given by

kfk_p +

n

X

j=1

Z 1

0

t^−sqk∆²_te

jfk^q_p dt t

^1/q

, (0< s <2), where{e1, . . . , en} denotes the canonical basis ofRⁿ, see [15, p. 96].

Letf ∈B_p,1^1+(1/p)(Rⁿ). Since ˙B_p,1^1+(1/p)(R)∩Lp(R) =B_p,1^1+(1/p)(R) (in the sense of equivalent norms, see, e.g. [15]), then by (2.4), we get

kfk_Vp(Rⁿ)≤c

n

X

j=1

Z

Rⁿ⁻¹

kfx⁰_jk^p

B^1+(1/p)_p,1 (R)dx⁰_j 1/p

.

Using the Minkowski inequality with respect toLp(Rⁿ⁻¹), it follows Z

Rⁿ⁻¹

Z 1

0

t^−(1+(1/p))k∆²_tekf_x⁰

jkp

dt t

p

dx⁰_j≤ Z 1

0

t^−(1+(1/p))k∆²_tekfkp

dt t

p

forj, k∈ {1, . . . , n}. Then we obtain the desired result.

Remark. As in Corollary 4.1 we can see the case when the functionfassociated to the composition operatorT_f belongs locally toU_p¹(R). Indeed, if 1≤p, q <+∞

and 0< s <1 + (1/p), it holds that kf◦gk_Fs

p,q(Rⁿ)≤ckf ϕ_kgk∞k_U1

p(R)kgk_B1+(1/p) p,1 (Rⁿ)

for allf ∈U_p^1,`oc(R) such thatf(0) = 0 and all g∈B_p,1^1+(1/p)(Rⁿ)∩L_∞(Rⁿ).

4.2. Sharpness of estimate For simplicity we define

kgk:=kgk_Fs

p,q(Rⁿ)+kgk_Vp(Rn).

According to Corollary 4.1, there is a substantial class ofnonlinearfunctionsf for which there exist constantscf =c(f)>0 such that

kf ◦gk_Fs

p,q(Rⁿ)≤c_fkgk, ∀g∈F_p,q^s (Rⁿ)∩ V_p(Rⁿ).

In this form the inequality isoptimalif we avoid linearfunctions in the following sense.

Proposition 4.4. LetΩ : [0,+∞)→[0,+∞)be a continuous function satisfy- ing

t→+∞lim t^1/pΩ(t) = 0.

(4.4)

Iff is a function such that the inequality

kf ◦gkF_p,q^s (Rⁿ) ≤ Ω(kgk) (4.5)

holds for all g ∈ F_p,q^s (Rⁿ)∩ Vp(Rⁿ), then f is an affine function (linear, if we assume thatf(0) = 0).

Proof. Let us define a smooth function ϕ∈ D(Rⁿ) such thatϕ(x) = 1 on the cubeQ:= [−1,1]ⁿ andϕ(x) = 0 ifx /∈2Q. We put ∆²_h:= ∆_h◦∆_hand

g_a(x) :=ax₁ϕ(x), (x= (x₁, x⁰)∈R×Rⁿ⁻¹, a >0).

We havekgak ∼aand

∆²_h(f◦ga)(x) = ∆²_ah1f(ax1), (∀x∈ ₂₍^√1_n)Q, ∀h∈ ₄₍^√1_n)Q, ∀a >0).

On the other hand, for allh∈ ₄₍^√1_n)Q(i.e. |h| ≤1/4), we have k∆²_h(f ◦g_a)k_p≥Z

x∈(1/(2√ n))Q

|∆²_h(f◦g_a)(x)|^pdx1/p

≥c a^−1/pZ a/(2√ n)

−a/(2√ n)

|∆²_ah1f(y)|^pdy^1/p .

By the above inequality, the embeddingF_p,q^s (Rⁿ),→B_p,∞^s (Rⁿ) and the assumption (4.5), we obtain

Z a/(2√ n)

−a/(2√ n)

|∆²_ah1f(y)|^pdy^1/p

≤c₁|h|^sa^1/pΩ(kgak)

≤c2a^1/pΩ(kgak), (∀h:|h| ≤1/4).

By settingu:=ah1, we deduce that Z a/(2√

n)

−a/(2√ n)

|∆²_uf(y)|^pdy1/p

≤c1a^1/pΩ(c2a), ∀a >0, ∀u:|u| ≤a.

By applying the assumption (4.4) on Ω and takingato +∞, we obtain Z +∞

−∞

|f(y+ 2u)−2f(y+u) +f(y)|^pdy= 0, ∀u∈R. Hencef(y+ 2u)−2f(y+u) +f(y) = 0 a.e.,∀y, u∈R. Then

f⁰(y+ 2u)−f⁰(y+u) = 0, i.e.,

it impliesf⁰(u) =f⁰(0) (∀u∈R). We deduce thatf⁰ is a constant.

References

1. Adams D. and Frazier M.,BMO and smooth truncation in Sobolev spaces, Studia Math.89 (1988), 241–260.

2. ,Composition operators on potential spaces, Proc. Amer. Math. Soc.114(1992), 155–165.

3. Allaoui S. E.,Remarques sur le calcul symbolique dans certains espaces de Besov `a valeurs vectorielles, Ann. Math. Blaise Pascal16(2)(2009), 399–429.

4. Bourdaud G. and Kateb M. E. D.,Fonctions qui op`erent sur les espaces de Besov, Math.

Ann.303(1995) 653–675.

5. Bourdaud G., Lanza de Cristoforis M. and Sickel W.,Superposition operators and functions of boundedp-variation, Rev. Mat. Iberoamer.22(2006), 455–485.

6. Brezis H. and Mironescu P.,Gagliardo-Nirenberg, compositions and products in fractional Sobolev spaces, J. Evol. Equ.1(2001), 387–404.

7. DeVore R. and Lorentz G. G.,Constructive Approximation, Springer, Berlin, 1993.

8. Franke J.,On the spacesF_p,q^s (R)of Triebel-Lizorkin type: Pointwise multipliers and spaces on domains, Math. Nachr.125(1986), 29–68.

9. Maz’ya V. and Shaposnikova T.,An elementary proof of the Brezis and Mironescu theorem on the composition operator in fractional Sobolev spaces, J. Evol. Equ.2(2002), 113–125.

10. Moussai M., The composition in multidimensional Triebel-Lizorkin spaces, Math. Nachr.

284(2-3)(2011), 317–331.

11. Peetre J.,New thoughts on Besov spaces, Duke Univ. Math. Series I, Durham, N. C., 1976.

12. Runst T. and Sickel W., Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, de Gruyter, Berlin, 1996.

13. Triebel H.,Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978.

14. ,Theory of Function Spaces, Birkh¨auser, Basel, 1983.

15. ,Theory of Function Spaces II, Birkh¨auser, Basel, 1992.

M. Moussai, Department of Mathematics, University of M’Sila, P.O. Box 166, 28000 M’Sila, Algeria,e-mail:[email protected]