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171 Composition Operator on the Space of Functions Acta Math. Univ. Comenianae 171 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 function f : R ! R satisfying f(0) = 0 and Z sup t−1 sup jf 0(x + h) − f 0(x)jp dx < +1; (0 < p < +1); t>0 R jh|≤t s n we study the composition operator Tf (g) := f◦g on Triebel-Lizorkin spaces Fp;q(R ) in the case 0 < s < 1 + (1=p). 1. Introduction and the main result The study of the composition operator Tf : g ! f ◦ g associated to a Borel- s n measurable function f : R ! R on Triebel-Lizorkin spaces Fp;q(R ), consists in finding a characterization of the functions f such that s n s n (1.1) Tf (Fp;q(R )) ⊆ Fp;q(R ): 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 on f; from which we recall the following results. For s > 0, 1 < p < +1 and 1 ≤ q ≤ +1 n s n s n • if Tf takes L1(R ) \ Fp;q(R ) to Fp;q(R ), then f is locally Lipschitz con- tinuous. n s n • if Tf takes the Schwartz space S(R ) to Fp;q(R ), then f belongs locally to s Fp;q(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]. 1 Bourdaud and Kateb [4] introduced the functions class Up (R), the set of Lipschitz continuous functions f such that their derivatives, in the sense of distri- butions, satisfy Z 1=p 0 −1 0 0 p (1.2) Ap(f ) := sup t sup jf (x + h) − f (x)j dx < +1; t>0 R jh|≤t Received August 8, 2011; revised February 17, 2012. 2010 Mathematics Subject Classification. Primary 46E35, 47H30. Key words and phrases. Triebel-Lizorkin spaces; Besov spaces; functions of bounded variation; composition operators. 172 M. MOUSSAI and are endowed with the seminorm kfk 1 := inf(kgk + A (g)); Up (R) 1 p where the infimum is taken over all functions g such that f is a primitive of g. In s n [4] the authors, proved the acting of the operator Tf on Besov space Bp;q(R ) for 1 1 ≤ p < +1, 1 < s < 1 + (1=p) and f 2 Up (R) with f(0) = 0. In [5] the same result holds for 0 < s < 1 + (1=p). s n In this work we will study the composition operator Tf on Fp;q(R ) for a func- 1 tion f which belongs to Up (R), then we will obtain a result of type (1.1). To do n n this, we introduce the set Vp(R ) of the functions g : R ! R such that n Z 1=p X p 0 kgk n := kg 0 k 1 dx < +1 Vp(R ) xj BV ( ) j n−1 p R j=1 R 1 where BVp (R) is the Wiener space of the primitives of functions of bounded p-variation (see Subsection 2.2 below for the definition) and n (1.3) g 0 (y) := g(x ; : : : ; x ; y; x ; : : : ; x ); y 2 ; x 2 : xj 1 j−1 j+1 n R R We will prove the following statement. Theorem 1.1. Let 0 < p; q < +1 and 0 < s < 1 + (1=p). Then there exists a constant c > 0 such that the inequality (1.4) kf ◦ gk s n ≤ c kfk 1 kgk + kgk n Fp;q (R ) Up (R) p Vp(R ) n n 1 holds for all functions g 2 Lp(R )\Vp(R ) and all f 2 Up (R) satisfying f(0) = 0. n n s n Moreover, for all such f, the operator Tf takes Lp(R ) \Vp(R ) to Fp;q(R ). s n n s n n Remark. (i) Since Fp;q(R ) ,! Lp(R ), then Tf maps from Fp;q(R ) \Vp(R ) s n to Fp;q(R ) under the assumptions of Theorem 1.1. s n s n (ii) Since the Bessel potential spaces Hp (R ) = Fp;2(R ), 1 < p < 1, Theorem 1.1 s n s n covers the results of composition operators in case Hp (R ) instead of Fp;q(R ). The paper is organized as follows. In Section2 we collect some properties of the s n 1 needed function spaces Fp;q(R ) and BVp (R). Section3 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 Section4 we give some corollaries and prove the sharp estimate of (1.4). Notation. We work with functions defined on the Euclidean space Rn. All n spaces and functions are assumed to be real-valued. We denote by Cb(R ) the Banach space of bounded continuous functions on Rn endowed with the supremum. Let D(Rn) (resp. S(Rn) and S0(Rn)) denotes the C1-functions with compact support (resp. the Schwartz space of all C1 rapidly decreasing functions and its topological dual). With k · kp we denote the Lp-norm. We define the differences n by ∆hf := f(· + h) − f for all h 2 R . If E is a Banach function space on Rn, we denote by E`oc the collection of all functions f such that 'f 2 E for all COMPOSITION OPERATOR ON THE SPACE OF FUNCTIONS 173 n ' 2 D(R ). As usual, constants c; c1;::: are strictly positive and depend only on the fixed parameters n; s; p; q; their values may vary from line to line. 2. Function spaces 2.1. Triebel-Lizorkin spaces Let 0 < a ≤ 1. For all measurable functions f on Rn, we set Z Z a Z q=u p=q 1=p s;u;a −sq 1 u dt Mp;q (f) := t n j∆hf(x)j dh dx : n t t R 0 jh|≤t Definition 2.1. Let 0 < p < +1 and 0 < q ≤ +1. Let s be a real satisfying 1 1 1 < s < 2 and s > n max − 1; − 1 : p q n s n Then, a function f 2 Lp(R ) belongs to Fp;q(R ) if n X s−1;1;1 kfk s n := kfk + M (@ f) < +1: Fp;q (R ) p p;q j j=1 s n The set Fp;q(R ) 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 < +1 and 0 < q; u ≤ +1. Let s be a real satisfying 1 1 1 1 1 < s < 2 and s > n max − ; − : p u q u n s n Then, a function f 2 Lp(R ) belongs to Fp;q(R ) if and only if s;u;1 (2.1) kfkp + Mp;q (f) < +1 s n and the expression (2.1) is an equivalent quasi-norm in Fp;q(R ). Moreover, this s;u;1 s;u;a assertion remains true if one replaces Mp;q by Mp;q for any fixed a > 0. The argument of the equivalence of above quasi-norms that we can replace the integration for t 2]0; +1[ by t ≤ a for a fixed positive number a is the part of the integral for which t > a can be easily estimated by the Lp-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: s n (i) The spaces Fp;q(R ) are monotone with respect to s and q, more exactly s n t n t n Fp;1(R ) ,! Fp;q(R ) ,! Fp;1(R ) if t < s and 0 < q ≤ 1. s n s n s n (ii) With Besov spaces, we have Bp;1(R ) ,! Fp;q(R ) ,! Bp;1(R ). s n n (iii) If either s > n=p or s = n=p and 0 < p ≤ 1, then Fp;q(R ) ,! Cb(R ). For various further embeddings we refer to [14, 2.3.2, 2.7.1] or [12, 2.2.2, 2.2.3]. 174 M. MOUSSAI The Fatou property. Well-known the Triebel-Lizorkin space has the Fatou prop- s n erty, cf. [8]. We will briefly recall it. Any f 2 Fp;q(R ) can be approximated (in 0 n the weak sense in S (R )) by a sequence (fj)j≥0 such that any fj is an entire function of exponential type s n f 2 F ( ) and lim sup k f k s n ≤ c k fk s n j p;q R j Fp;q (R ) Fp;q (R ) j!+1 with a positive constant c independent of f. Vice versa, if for a tempered distri- 0 n bution f 2 S (R ), there exists a sequence (fj)j≥0 such that s n f 2 F ( ) and A := lim sup k f k s n < +1 ; j p;q R j Fp;q (R ) j!+1 s n and limj!+1 fj = f in the sense of distributions, then f belongs to Fp;q(R ) and there exists a constant c > 0 independent of f such that kfk s n ≤ cA.
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