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Continuity of Composition Operators in Sobolev Spaces

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Continuity of Composition Operators in Sobolev Spaces Continuity of composition operators in Sobolev spaces G´erard Bourdaud & Madani Moussai February 12, 2019 Abstract We prove that all the composition operators Tf (g) := f ◦ g, which take the Adams- m ˙ 1 Rn m ˙ 1 Rn Frazier space Wp ∩Wmp( ) to itself, are continuous mappings from Wp ∩Wmp( ) to itself, for every integer m ≥ 2 and every real number 1 ≤ p< +∞. The same automatic m Rn continuity property holds for Sobolev spaces Wp ( ) for m ≥ 2 and 1 ≤ p< +∞. 2000 Mathematics Subject Classification: 46E35, 47H30. Keywords: Composition Operators. Sobolev spaces. 1 Introduction We want to establish the so-called automatic continuity property for composition operators in classical Sobolev spaces, i.e. the following statement: Theorem 1 Let us consider an integer m> 0, and 1 ≤ p< +∞. If f : R → R is a function m Rn s.t. the composition operator Tf (g) := f ◦ g takes Wp ( ) to itself, then Tf is a continuous m Rn mapping from Wp ( ) to itself. This theorem has been proved • for m = 1, see [5] in case p = 2, and [22] in the general case, • for m > n/p, m> 1 and p> 1 [14]. arXiv:1812.00212v2 [math.FA] 11 Feb 2019 It holds also trivially in the case of Dahlberg degeneracy, i.e. 1 + (1/p) <m<n/p, see [19]. It does not hold in case m = 0, see Section 2 below. Thus it remains to be proved in the following cases: • m = 2, p = 1, and n ≥ 3. • m = n/p > 1 and p> 1. • m ≥ max(n, 2) and p = 1. 2 Rn If we except the space W1 ( ), all the Sobolev spaces under consideration are particular cases of the Adams-Frazier spaces, or of the Sobolev algebras. We will prove the automatic continuity for those spaces, and for their homogeneous counterparts, conveniently realized. Contrarily to the case m = 1, where the proof of continuity of Tf is much more difficult for p = 1, see [22, p. 219], our proof in case m ≥ 2 will cover all values of p ≥ 1. 1 Plan - Notation 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 set 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 polynomials on R , of n degree less or equal to k. If f is a function on R , we denote by [f]k its equivalence class n modulo Pk. We consider a classical mollifiers sequence 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 composition operator, 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 continuous function 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 norm 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 pointwise 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 apply (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.
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