Revista Colombiana de Matem´aticas Volumen 49(2015)2, p´aginas293-305

Multiplication operators in variable Lebesgue spaces

Operador multiplicaci´onen los espacios de Lebesgue con exponente variable

Rene´ Erlin Castillo1, Julio C. Ramos Fernandez´ 2,B, Humberto Rafeiro3

1Universidad Nacional de Colombia, Bogot´a,Colombia 2Universidad de Oriente, Cuman´a,Venezuela 3Pontificia Universidad Javeriana, Bogot´a,Colombia

Abstract. In this note we will characterize the boundedness, invertibility, compactness and closedness of the range of multiplication operators on vari- able Lebesgue spaces.

Key words and phrases. Multiplication operator, variable Lebesgue spaces, com- pactness.

2010 Mathematics Subject Classification. Primary 47B38; Secondary 46E30.

Resumen. En esta nota vamos a caracterizar los operadores multiplicaci´onque son continuos, invertibles y que tienen rango cerrados sobre los espacios de Lebesgue con exponente variable.

Palabras y frases clave. Operador multiplicaci´on,espacios de Lebesgue variables, compacidad.

1. Introduction Variable Lebesgue spaces are a generalization of Lebesgue spaces where we allow the exponent to be a measurable function and thus the exponent may vary. It seems that the first occurence in the literature is in the 1931 paper of Orlicz [29]. The seminal work on this field is the 1991 paper of Kov´aˇcikand R´akosn´ık[27] where many basic properties of Lebesgue and Sobolev spaces were shown. To see a more detailed history of such spaces see, e.g., [17, §1.1]. These

293 294 R.E. CASTILLO, J.C. RAMOS FERNANDEZ´ & H. RAFEIRO variable exponent function spaces have a wide variety of applications, e.g., in the modeling of electrorheological fluids [4, 5, 31] as well as thermorheological fluids [6], in the study of image processing [1, 2, 10, 11, 14, 15, 34] and in differential equations with non-standard growth [23, 28]. For details on variable Lebesgue spaces one can refer to [16, 17, 21, 27, 30] and the references therein.

The multiplication operator, defined roughly speaking as the pointwise mul- tiplication by a real-valued measurable function, is a well-studied transfor- mation. This operator received considerable attention over the past several decades, specially on Lebesgue and Bergman spaces and they played an impor- tant role in the study of operators on Hilbert spaces. For more details on these operators we refer to [3, 9, 18, 22, 32]. Studies of the multiplication operator on various spaces can be seen, e.g. [7, 8, 13, 12, 25, 26, 33], in particular on Lp space in [25, 33], on Orlicz space in [26], on Lorentz space in [7] and on Lorentz-Bochner space in [8, 20]. It is natural to extend the study to variable Lebesgue spaces.

The main goal of the present note is to establish boundedness, invertibility, compactness and closedness of multiplication operators in the framework of variable Lebesgue spaces Lp(·)(X, µ).

2. Preliminaries 2.1. On Lebesgue spaces with variable exponent

The basics on variable Lebesgue spaces may be found in the monographs [16, 17] (see also [24, 27]), but we recall here some necessary definitions. Let (X, Σ, µ) + be a σ-finite, complete measure space. For A ⊂ X we put pA := ess supx∈A p(x) − + + − − and pA := ess infx∈A p(x); we use the abbreviations p = pX and p = pX . For a measurable function p : X → [1, ∞), we call it a variable exponent, and define the set of all variable exponents with p+ < ∞ as P(X, µ). In this note, all the variable exponents are tacitly assumed to belong to the class P(X, µ). For a real-valued µ-measurable function ϕ : X → R we define the modular Z p(x) ρp(·)(ϕ) := |ϕ(x)| dµ(x) X and the Luxemburg norm n ϕ o kϕk p(·) := inf λ > 0 : ρ 1 . (1) L (X,µ) p(·) λ 6 Definition 2.1. Let p ∈ P(X, µ). The variable Lebesgue space Lp(·)(X, µ) is introduced as the set of all real-valued µ-measurable functions ϕ : X → R for which ρp(·)(ϕ) < ∞. Equipped with the Luxemburg norm (1) this is a Banach space.

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We gather here some useful properties of variable exponent Lebesgue spaces, see [17, p. 77]

Remark 2.2. We say that a function is simple if it is the linear combination of Pk indicator functions of measurable sets with finite measure, i=1 siχAi (x) with µ(A1) < ∞, . . . , µ(Ak) < ∞ and s1, . . . , sk ∈ R. We denote the set of simple functions by S(X, µ).

Proposition 2.3. Let p ∈ P(X, µ). Then the set of simple functions S(X, µ) is contained in Lp(·)(X, µ) and

min{1, µ(E)} 6 kχEkLp(·)(X,µ) 6 max{1, µ(E)}

for every measurable set E ⊂ X.

Proposition 2.4. If p ∈ P(X, µ), then the set of simple functions is dense in Lp(·)(X, µ).

Remark 2.5. The previous proposition is explicitly stated in [17, Corollary 3.4.10] for a domain Ω ⊂ Rn, but the result can be stated in full general- ity as done above, since the result is a corollary of Theorem 2.5.9 and Theo- rem 3.4.1(c) from [17] which are given for a σ-finite, complete measure spaces (X, Σ, µ).

Proposition 2.6. The variable Lebesgue space Lp(·)(X, µ) is circular, solid, satisfies Fatou’s lemma (for the norm) and has the Fatou property, namely:

p(·) circular kfkLp(·)(X,µ) = k|f|kLp(·)(X,µ) for all f ∈ L (X, µ);

solid If f ∈ Lp(·)(X, µ), g ∈ L0(X, µ) (where L0(X, µ) stands for the space of all real-valued, µ-measurable functions on X) and 0 6 |g| 6 |f| µ-almost p(·) everywhere, then g ∈ L (X, µ) and kgkLp(·)(X,µ) 6 kfkLp(·)(X,µ);

Fatou’s lemma If fk → f µ-almost everywhere, then we have the following: kfkLp(·)(X,µ) 6 lim infk→∞ kfkkLp(·)(X,µ);

p(·) Fatou property If |fk| % |f| µ-almost everywhere with fk ∈ L (X, µ) p(·) and supk kfkkLp(·)(X,µ) < ∞, then f ∈ L (X, µ) and kfkkLp(·)(X,µ) % kfkLp(·)(X,µ).

3. Multiplication operators Definition 3.1. Let F (X) be a function space on a non-empty set X. Let u : X → C be a function such that u · f ∈ F (X) whenever f ∈ F (X). Then the aplication f 7→ u · f on F (X) is denoted by Mu. In case F (X) is a topological space and Mu is continuous we call it a multiplication operator induced by u.

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Multiplication operators generalize the notion of operator given by a diag- onal matrix. More precisely, one of the results of operator theory is a spectral theorem, which states that every self-adjoint operator on a Hilbert space is unitarily equivalent to a multiplication operator on an L2 space. Consider the Hilbert space X = L2[−1, 3] of complex-valued square inte- grable functions on the interval [−1, 3]. Define the operator 2 Mu(x) = u(x)x , for any function u ∈ X. This will be a self-adjoint bounded linear operator with norm 9. Its spectrum will be the interval [0, 9] (the range of the function x → x2 defined on [−1, 3]). Indeed, for any complex number λ, the operator Mu − λ is given by 2 (Mu − λ)(x) = (x − λ)u(x). It is invertible if and only if λ is not in [0, 9], and then its inverse is u(x) (M − λ)−1(x) = . u x2 − λ which is another multiplication operator. For a systematic study of the multiplication operators on different spaces we refer, e.g., to [3, 7, 9, 26]. Remark 3.2. In general, the multiplication operators on measure spaces are not 1−1. Indeed, let (X, Σ, µ) be a measure space and A = X\supp (u) = {x ∈ X : u(x) = 0}.

If µ(A) 6= 0 and f = χA then for any x ∈ X we have f(x)u(x) = 0 which implies that Mu(f) = 0, therefore ker(Mu) 6= {0} and hence Mu is not 1−1.

If, on the contrary, Mu is 1−1, then µ(X\supp(u)) = 0. On the other hand, if µ(X\supp(u)) = 0 and µ is a complete measure, then Mu(f) = 0 implies f(x)u(x) = 0 ∀ x ∈ X, then {x ∈ X : f(x) 6= 0} ⊆ X\supp(u) and so f = 0 µ-a.e. on X.

Hence, if µ(X\supp(u)) = 0 and µ is a complete measure, then Mu is 1−1. p(·) Proposition 3.3. Mu is 1−1 on Y = L (supp (u)).

p(·) p(·) Proof. Let Y = L (supp (u)) = {fχsupp (u) : f ∈ L (X, µ)}. Indeed, if ˜ ˜ Mu(f) = 0 with f = fχsupp (u) ∈ Y , then f(x)χsupp (u)(x)u(x) = 0 for all x ∈ X, and so f(x)u(x) = 0 ∀ x ∈ supp (u), ⇒ f(x) = 0 ∀ x ∈ supp (u),

⇒ f(x)χsupp (u)(x) = 0 ∀ x ∈ X. ˜ Then f = 0 and the proof is complete. X

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In this section we want to characterize the boundedness and compactness of p(·) the multiplication operator Mu in variable Lebesgue space L (X, µ) in terms of the boundedness and invertibility of the real-valued measurable u function.

3.1. Boundedness results

In the next theorems we will obtain necessary and sufficient conditions related with boundedness and invertibility of the multiplication operator in the frame- work of variable Lebesgue spaces.

p(·) p(·) Theorem 3.4. The linear transformation Mu : L (X, µ) → L (X, µ) is bounded if and only if u is essentially bounded. Moreover

kMukLp(·)(X,µ)→Lp(·)(X,µ) = kukL∞(X,µ).

∞ Proof. Letting u ∈ L (X, µ) we then have the pointwise estimate |(u·f)(x)| 6 p(·) kukL∞(X,µ)|f(x)|. Using the fact that L (X, µ) is solid, we have

kMufkLp(·)(X,µ) 6 kukL∞(X,µ)kfkLp(·)(X,µ). (2)

On the other hand, suppose that Mu is a bounded operator. If u is not essentially bounded, then for every n ∈ N, the set Un = {x ∈ X : |u(x)| > n} has positive measure. Since the measure is σ-finite, there exists a measur- able subset of Un with finite positive measure, denote it by Un. Then χ ∈ e Uen Lp(·)(X, µ) and since Lp(·)(X, µ) is solid and (|u|χ )(x) nχ (x) we obtain Uen > Uen kMuχ k p(·) nkχ k p(·) . This contradicts the supposition that Uen L (X,µ) > Uen L (X,µ) Mu is bounded, therefore u is essentially bounded.

To evaluate the norm of the operator Mu we proceed as follows. For a fixed ε > 0, let us define U = {x ∈ X : |u(x)| > kukL∞(X,µ) − ε} which has positive measure. Since the variable Lebesgue space is solid, we have

(kuk ∞ − ε)χ L (X,µ) U 6 1 kMuχU kLp(·)(X,µ) Lp(·)(X,µ) which yields kMukLp(·)(X,µ)→Lp(·)(X,µ) > kukL∞(X,µ)−ε. Since ε > 0 is arbitrary and taking (2) into account, we prove that the norm is equal to kuk∞. X

Theorem 3.5. Let (X, Σ, µ) be a finite measure space. The set of all multiplica- tion operators on Lp(·)(X, µ) is a maximal Abelian subalgebra of B(Lp(·)(X, µ)), the algebra of all bounded operators on Lp(·)(X, µ).

Proof. Let ∞ H = {Mu : u ∈ L (X, µ)} (3)

Revista Colombiana de Matem´aticas 298 R.E. CASTILLO, J.C. RAMOS FERNANDEZ´ & H. RAFEIRO and consider the operator product

Mu · Mv = Muv,

where Mu,Mv ∈ H . Let us check that H is a Banach algebra. Let u, v ∈ ∞ L (X, µ), then by the pointwise estimates |u| 6 kukL∞(X,µ), |v| 6 kvkL∞(X,µ) we have kuvkL∞ 6 kukL∞(X,µ)kvkL∞(X,µ) which implies that product is an inner operation, moreover, the usual function product is associative, commutative and distributive with respect to the sum and scalar product, thus H is a subalgebra of B(Lp(·)(X, µ)). We will now prove that it is a maximal Abelian subalgebra. Consider the unit function e : X → R given by x 7→ 1 for all x ∈ X. Let N ∈ B(Lp(·)(X, µ)) be an operator which commutes with H and let χE be the indicator function of a measurable set E. Then

N(χE) = N[MχE (e)] = MχE [N(e)] = χE · N(e) = N(e) · χE = Mw(χE)

where w := N(e). Similarly

N(s) = Mw(s), (4)

for any simple function. Now, let us check that w ∈ L∞(X, µ). By way of contradiction, we assume that w∈ / L∞(X, µ), then the set

Wn = {x ∈ X : |w(x)| > n},

has positive measure for each n ∈ N. Note that we have the pointwise estimate

Mw(χWn )(x) = (wχWn )(x) > nχWn (x) (5) for all x ∈ X. Using the fact that Lp(·)(X, µ) is solid and the pointwise estimate (5) we obtain

kMw(χWn )kLp(·)(X,µ) = kw(χWn )kLp(·)(X,µ) > nkχWn kLp(·)(X,µ). (6) Using (4) and (6) we obtain

kN(χWn )kLp(·)(X,µ) > nkχWn kLp(·)(X,µ) which contradicts the fact that N is a bounded operator. Therefore w ∈ ∞ L (X, µ) and by Theorem 3.4 Mw is bounded. To obtain the result for all functions in Lp(·)(X, µ) we proceed with a lim- iting argument. Taking, without loss of generality, a non-negative function

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p(·) ∞ u ∈ L (X, µ), there exists a nondecreasing sequence {sn}s=1 of measurable simple functions such that limn→∞ sn = u. Using (4) we have     N(u) = N lim sn = lim N(sn) = lim Mw(sn) = Mw lim sn = Mw(u), n→∞ n→∞ n→∞ n→∞

p(·) which gives that N(u) = Mw(u) for all u ∈ L (X, µ) and thus N ∈ H . X Corollary 3.6. Let (X, Σ, µ) be a finite measure space. The multiplication op- p(·) ∞ erator Mu in L (X, µ) is invertible if and only if u is invertible in L (X, µ).

p(·) Proof. Let Mu be invertible, then there exists N ∈ B(L (X, µ)) such that

Mu · N = N · Mu = 1 (7) where 1 represents the identity operator. Let us check that N commutes with H (where H is defined in (3)). Let Mw ∈ H , then

Mw · Mu = Mu · Mw. (8)

Applying N to both sides of (8) and using (7) we obtain

N ·Mw = N ·Mw ·1 = N ·Mw ·Mu ·N = N ·Mu ·Mw ·N = 1·Mw ·N = Mw ·N, and thus we conclude that N commutes with H . Then N ∈ H by Theorem 3.5 and using again Theorem 3.5 we have that there exists g ∈ L∞(X, µ) such that N = Mg, hence Mu · Mg = Mg · Mu = 1 and this implies ug = gu = 1 µ-almost everywhere, this means that u is invertible in L∞(X, µ). On the other hand, assume that u is invertible on L∞(X, µ), that is, u−1 ∈ L∞(X, µ), then

Mu · Mu−1 = Mu−1 · Mu = Mu−1u = M1 = 1

p(·) which means that Mu is invertible on B(L (X, µ)). X

3.2. Compactness results

In the next theorems we will obtain necessary and sufficient conditions related with compactness of the multiplication operator in the framework of variable Lebesgue spaces. We need some definitions for further results, namely Definition 3.7. For the set X, the real-valued essentially bounded function u and non-negative ε we define the set

X (u, ε) := {x ∈ X : |u(x)| > ε}.

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With this newly defined set X (u, ε) we restrict our space Lp(·)(X, µ), namely Definition 3.8. We define the space Lp(·)(X (u, ε)) as

p(·) p(·) L (X (u, ε)) := {fχX (u,ε) : f ∈ L (X, µ)}.

Lemma 3.9. The space Lp(·)(X (u, ε)) is a closed invariant subspace of the p(·) variable Lebesgue space L (X, µ) under Mu.

p(·) Proof. Let h, s ∈ L (X (u, ε)) and α, β ∈ R. Then h = fχX (u,ε) and s = p(·) gχX (u,ε), where f, g ∈ L (X, µ), thus

p(·) αh + βs = α fχX (u,ε) + β gχX (u,ε) = (αf + βg)χX (u,ε) ∈ L (X (u, ε))

yielding that Lp(·)(X (u, ε)) is a subspace of Lp(·)(X, µ). We have that

Muh = uh = ufχX (u,ε) = (uf)χX (u,ε)

p(·) p(·) where uf ∈ L (X, µ). Therefore, Muh ∈ L (X (u, ε)), which means that p(·) p(·) L (X (u, ε)) is an invariant subspace of L (X, µ) under Mu. Now, let us show that Lp(·)(X (u, ε)) is a closed set. Indeed, let us take a function g ∈ Lp(·)(X (u, ε)) which also belongs to Lp(·)(X, µ) since it is a ∞ p(·) Banach space. Then there exists a sequence {gn}n=1 in L (X (u, ε)) such p(·) that gn → g in L (X (u, ε)). It remains just to show that g belongs to Lp(·)(X (u, ε)). Note that

g = gχX (u,ε) + gχ(X (u,ε)){ .

Next, we want to show that gχ(X (u,ε)){ = 0. In fact, given ε1 > 0 there exists n0 ∈ N such that

kgχ(X (u,ε)){ kLp(·)(X,µ) = k(g − gn0 + gn0 )χ(X (u,ε)){ kLp(·)(X,µ)

= k(g − gn0 )χ(X (u,ε)){ kLp(·)(X,µ)

6 kg − gn0 kLp(·)(X,µ)

< ε1

p(·) implying that gχ(X (u,ε)){ = 0, hence g ∈ L (X (u, ε)). X ∞ Theorem 3.10. Let u ∈ L (X, µ). Then the operator Mu is compact if and only if the space Lp(·)(X (u, ε)) is finite dimensional for each ε > 0.

Proof. For each f ∈ Lp(·)(X, µ), we have the pointwise estimate

|ufχX (u,ε)(x)| > ε|f|χX (u,ε)(x).

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Since Lp(·)(X, µ) is solid, we obtain

kMufχX (u,ε)kLp(·)(X,µ) > εkfχX (u,ε)kLp(·)(X,µ). (9)

From Lemma 3.9 we have that Lp(·)(X (u, ε)) is a closed invariant subspace of p(·) L (X, µ) under Mu which implies that

M u Lp(·)(X (u,ε)) is a compact operator. Then by (9) the operator M has closed u Lp(·)(X (u,ε)) range in Lp(·)(X (u, ε)) and it is invertible, being compact, Lp(·)(X (u, ε)) is finite dimensional. Conversely, suppose that Lp(·)(X (u, ε)) is finite dimensional for each ε > 0. In particular, for n ∈ N, Lp(·)(X (u, 1/n)) is finite dimensional, then for each n, define un : X → R as ( 1 u(x), if |u(x)| > n , un(x) = 1 0, if |u(x)| < n . Then we find that

|Mun f − Muf| = |(un − u) · f| 6 kun − uk∞|f|, yielding, since Lp(·)(X, µ) is solid,

kMun f − MufkLp(·)(X,µ) 6 kun − uk∞kfkLp(·)(X,µ).

Consequently 1 kM f − M fk p(·) kfk p(·) un u L (X,µ) 6 n L (X,µ) which implies that Mun converges to Mu uniformly. Therefore Mu is compact since it is the limit of operators with finite range. X

∞ Theorem 3.11. Let u ∈ L (X, µ). Then Mu has closed range if and only if there exists a δ > 0 such that |u(x)| > δ µ-almost everywhere on supp (u).

Proof. If there exists δ > 0 such that |u(x)| > δ µ-almost everywhere on supp (u), then for all f ∈ Lp(·)(X, µ) we have

kMufχsupp (u)kLp(·)(X,µ) > δkfχsupp (u)kLp(·)(X,µ)

and hence Mu has closed range.

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Conversely, by [19, Lemma VI.6.1], if Mu has closed range, then there exists an ε > 0 such that

kMufkLp(·)(X,µ) > εkfkLp(·)(X,µ) for all f ∈ Lp(·)(supp (u)), where

p(·) n p(·) o L (supp (u)) := fχsupp (u) : f ∈ L (X, µ) .

Let E = {x ∈ supp (u): |u(x)| < ε/2}. If µ(E) > 0, then by the σ-finiteness of p(·) measure we can find a measurable set F ⊆ E such that χF ∈ L (supp (u)), implying that ε kM χ k p(·) kχ k p(·) u F L (X,µ) 6 2 F L (X,µ) which is a contradiction. Therefore µ(E) = 0, and this completes the proof. X

Acknowledgements. H. Rafeiro was partially supported by Pontificia Uni- versidad Javeriana under the research project “Multiplication operators in vari- able Lebesgue spaces”, Id-Pppta: 6613.

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Volumen 49, N´umero2, A˜no2015 MULTIPLICATION OPERATORS IN VARIABLE LEBESGUE SPACES 305

(Recibido en marzo de 2014. Aceptado en octubre de 2015)

Departamento de Matematicas´ Universidad Nacional de Colombia Facultad de Ciencias, Carrera 30, calle 45 Bogota,´ Colombia e-mail: [email protected]

Departamento de Matematicas´ Universidad de Oriente 6101 Cumana,´ Estado Sucre Republica´ Bolivariana de Venezuela e-mail: [email protected]

Departamento de Matematicas´ Pontificia Universidad Javeriana Facultad de Ciencias, Cra 7a No 43-82 Bogota,´ Colombia e-mail: [email protected]

Revista Colombiana de Matem´aticas