The boundedness of classical operators on variable L-p spaces
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Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 31, 2006, 239–264 THE BOUNDEDNESS OF CLASSICAL OPERATORS ON VARIABLE LpSPACES D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez Trinity College, Department of Mathematics Hartford, CT 06106-3100, U.S.A.; david.cruzurib[email protected] Universit´a di Napoli, Dipartimento di Costruzioni e Metodi Matematici in Architettura Via Monteoliveto, 3, IT-80134 Napoli, Italy and Consiglio Nazionale delle Ricerche, Istituto per le Applicazioni del Calcolo “Mauro Picone” Sezione di Napoli, via Pietro Castellino, 111, IT-80131 Napoli, Italy; [email protected] Universidad Aut´onoma de Madrid, Departamento de Matem´aticas ES-28049 Madrid, Spain; [email protected] Universidad de Sevilla, Departamento de An´alisis Matem´atico Facultad de Matem´aticas, ES-41080 Sevilla, Spain; carlosp[email protected] Abstract. We show that many classical operators in harmonic analysis—such as maximal operators, singular integrals, commutators and fractional integrals—are bounded on the variable Lebesgue space Lp(·)whenever the Hardy–Littlewood maximal operator is bounded on Lp(·). Further, we show that such operators satisfy vector-valued inequalities. We do so by applying the theory of weighted norm inequalities and extrapolation. As applications we prove the Calder´on–Zygmund inequality for solutions of 4u=fin variable Lebesgue spaces, and prove the Calder´on extension theorem for variable Sobolev spaces. 1. Introduction Given an open set Ω ⊂Rn, we consider a measurable function p: Ω −→ [1,∞), Lp(·)(Ω) denotes the set of measurable functions fon Ω such that for some λ > 0, ZΩ|f(x)| λp(x) dx < ∞. This set becomes a Banach function space when equipped with the norm kfkp(·),Ω= infλ > 0 : ZΩ|f(x)| λp(x) dx ≤1. These spaces are referred to as variable Lebesgue spaces or, more simply, as variable Lpspaces, since they generalize the standard Lpspaces: if p(x) = p0is 2000 Mathematics Subject Classification: Primary 42B25, 42B20, 42B15, 35J05. The third author is partially supported by MEC Grant MTM2004-00678, and the fourth author is partially supported by DGICYT Grant PB980106.
240 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez constant, then Lp(·)(Ω) equals Lp0(Ω). (Here and below we write p(·) instead of pto emphasize that the exponent is a function and not a constant.) They have many properties in common with the standard Lpspaces. These spaces, and the corresponding variable Sobolev spaces Wk,p(·)(Ω), are of interest in their own right, and also have applications to partial differential equations and the calculus of variations. (See, for example, [1], [12], [15], [19], [30], [39], [46] and their references.) In many applications, a crucial step has been to show that one of the classical operators of harmonic analysis—e.g., maximal operators, singular integrals, fractional integrals—is bounded on a variable Lpspace. Many authors have considered the question of sufficient conditions on the exponent function p(·) for given operators to be bounded: see, for example, [13], [15], [27], [28], [29], [40]. Our approach is different. Rather than consider estimates for individual operators, we apply techniques from the theory of weighted norm inequalities and extrapolation to show that the boundedness of a wide variety of operators follows from the boundedness of the maximal operator on variable Lpspaces, and from known estimates on weighted Lebesgue spaces. In order to provide the foundation for stating our results, we discuss each of these ideas in turn. The maximal operator. In harmonic analysis, a fundamental operator is the Hardy–Littlewood maximal operator. Given a function f, we define the maximal function, Mf , by Mf(x) = sup Q3x 1 |Q|ZQ |f(y)|dy, where the supremum is taken over all cubes containing x. It is well known that Mis bounded on Lp, 1 <p<∞, and it is natural to ask for which exponent functions p(·) the maximal operator is bounded on Lp(·)(Ω). For conciseness, define P(Ω) to be the set of measurable functions p: Ω −→ [1,∞) such that p−= ess inf{p(x) : x∈Ω}>1, p+= ess sup{p(x) : x∈Ω}<∞. Let B(Ω) be the set of p(·)∈P(Ω) such that Mis bounded on Lp(·)(Ω). Theorem 1.1. Given an open set Ω⊂Rn, and p(·)∈P(Ω), suppose that p(·)satisfies (1.1) |p(x)−p(y)| ≤ C −log(|x−y|), x, y ∈Ω,|x−y| ≤ 1/2, (1.2) |p(x)−p(y)| ≤ C log(e+|x|), x, y ∈Ω,|y| ≥ |x|. Then p(·)∈B(Ω), that is, the Hardy–Littlewood maximal operator is bounded on Lp(·)(Ω).
The boundedness of classical operators on variable Lpspaces 241 Theorem 1.1 is independently due to Cruz-Uribe, Fiorenza and Neugebauer [10] and to Nekvinda [35]. (In fact, Nekvinda replaced (1.2) with a slightly more general condition.) Earlier, Diening [12] showed that (1.1) alone is sufficient if Ω is bounded. Examples show that the continuity conditions (1.1) and (1.2) are in some sense close to necessary: see Pick and R˚uˇziˇcka [37] and [10]. See also the examples in [33]. The condition p−>1 is necessary for Mto be bounded; see [10]. Very recently, Diening [14], working in the more general setting of Musielak– Orlicz spaces, has given a necessary and sufficient condition on p(·) for Mto be bounded on Lp(·)(Rn). His exact condition is somewhat technical and we refer the reader to [14] for details. Because our proofs rely on duality arguments, we will not need that the maximal operator is bounded on Lp(·)(Ω) but on its associate space Lp0(·)(Ω), where p0(·) is the conjugate exponent function defined by 1 p(x)+1 p0(x)= 1, x ∈Ω. Since |p0(x)−p0(y)| ≤ |p(x)−p(y)| (p−−1)2, it follows at once that if p(·) satisfies (1.1) and (1.2), then so does p0(·)—i.e., if these two conditions hold, then Mis bounded on Lp(·)(Ω) and Lp0(·)(Ω). Furthermore, Diening’s characterization of variable Lpspaces on which the maximal operator is bounded has the following important consequence (see [14, Theorem 8.1]). Theorem 1.2. Let p(·)∈P(Rn). Then the following conditions are equivalent: (a) p(·)∈B(Rn). (b) p0(·)∈B(Rn) (c) p(·)/q ∈B(Rn)for some 1< q < p−. (d) p(·)/q0∈B(Rn)for some 1< q < p−. Weights and extrapolation. By a weight we mean a non-negative, locally integrable function w. There is a vast literature on weights and weighted norm inequalities; here we will summarize the most important aspects, and we refer the reader to [17], [21] and their references for complete information. Central to the study of weights are the so-called Apweights, 1 ≤p≤ ∞. When 1 <p<∞, we say w∈Apif for every cube Q, 1 |Q|ZQ w(x)dx 1 |Q|ZQ w(x)1−p0dxp−1 ≤C < ∞.
242 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez We say that w∈A1if Mw(x)≤Cw(x) for a.e. x. If 1 ≤p < q < ∞, then Ap⊂Aq. We let A∞denote the union of all the Apclasses, 1 ≤p < ∞. Weighted norm inequalities are generally of two types. The first is (1.3) ZRn |Tf(x)|p0w(x)dx ≤CZRn |f(x)|p0w(x)dx, where Tis some operator and w∈Ap0, 1 < p0<∞. (In other words, Tis defined and bounded on Lp0(w).) The constant is assumed to depend only on the Ap0constant of w. The second type is (1.4) ZRn |Tf(x)|p0w(x)dx ≤CZRn |Sf(x)|p0w(x)dx, where Sand Tare operators, 0 < p0<∞,w∈A∞, and fis such that the lefthand side is finite. The constant is assumed to depend only on the A∞constant of w. Such inequalities are known for a wide variety of operators and pairs of operators. (See [17], [21].) Corresponding to these types of inequalities are two extrapolation theorems. Associated with (1.3) is the classical extrapolation theorem of Rubio de Francia [38] (also see [17], [21]). He proved that if (1.3) holds for some operator T, a fixed value p0, 1 < p0<∞, and every weight w∈Ap0, then (1.3) holds with p0replaced by any p, 1 < p < ∞, whenever w∈Ap. Recently, the analogous extrapolation result for inequalities of the form (1.4) was proved in [11]: if (1.4) holds for some p0, 0 < p0<∞and every w∈A∞, then it holds for every p, 0 <p<∞. (More general versions of these results will be stated in Section 6 below.) 1.1. Main results. The proofs of the above extrapolation theorems depend not on the properties of the operators, but rather on duality, the structure of Apweights, and norm inequalities for the Hardy–Littlewood maximal operator. These ideas can be extended to the setting of variable Lpspaces to yield our main result, which can be summarized as follows: If an operator T, or a pair of operators (T, S), satisfies weighted norm inequalities on the classical Lebesgue spaces, then it satisfies the corresponding inequality in a variable Lpspace on which the maximal operator is bounded. To state and prove our main result, we will adopt the approach taken in [11]. There it was observed that since nothing is assumed about the operators involved (e.g., linearity or sublinearity), it is better to replace inequalities (1.3) and (1.4) with (1.5) ZRn f(x)p0w(x)dx ≤CZRn g(x)p0w(x)dx, where the pairs (f, g) are such that the left-hand side of the inequality is finite. One important consequence of adopting this approach is that vector-valued inequalities follow immediately from extrapolation.
The boundedness of classical operators on variable Lpspaces 243 Hereafter Fwill denote a family of ordered pairs of non-negative, measurable functions (f, g). Whenever we say that an inequality such as (1.5) holds for any (f, g)∈Fand w∈Aq(for some q, 1 ≤q≤ ∞), we mean that it holds for any pair in Fsuch that the left-hand side is finite, and the constant Cdepends only on p0and the Aqconstant of w. Finally, note that in the classical Lebesgue spaces we can work with Lpwhere 0< p < 1. (Thus, in (1.4) or (1.5) we can take p0<1.) We would like to consider analogous spaces with variable exponents. Define P0(Ω) to be the set of measurable functions p: Ω −→ (0,∞) such that p−= ess inf{p(x) : x∈Ω}>0, p+= ess sup{p(x) : x∈Ω}<∞. Given p(·)∈P0(Ω), we can define the space Lp(·)(Ω) as above. This is equivalent to defining it to be the set of all functions fsuch that |f|p0∈Lq(·)(Ω), where 0 < p0< p−and q(x) = p(x)/p0∈P(Ω). We can define a quasi-norm on this space by kfkp(·),Ω= |f|p0 1/p0 q(·),Ω. We will not need any other properties of these spaces, so this definition will suffice for our purposes. Theorem 1.3. Given a family Fand an open set Ω⊂Rn, suppose that for some p0,0< p0<∞, and for every weight w∈A1, (1.6) ZΩ f(x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, (f, g)∈F, where C0depends only on p0and the A1constant of w. Let p(·)∈P0(Ω) be such that p0< p−, and p(·)/p00∈B(Ω). Then for all (f, g)∈Fsuch that f∈Lp(·)(Ω), (1.7) kfkp(·),Ω≤Ckgkp(·),Ω, where the constant Cis independent of the pair (f, g). We want to call attention to two features of Theorem 1.3. First, the conclusion (1.7) is an a priori estimate: that is, it holds for all (f, g)∈Fsuch that f∈ Lp(·)(Ω). In practice, when applying this theorem in conjunction with inequalities of the form (1.3) to show that an operator is bounded on variable Lpwe will usually need to work with a collection of functions fwhich satisfy the given weighted Lebesgue space inequality and are dense in Lp(·)(Ω). When working with inequalities of the form (1.3) the final estimate will hold for a suitable family of “nice” functions. Second, the family Fin the hypothesis of and conclusion of Theorem 1.7 is the same, so the goal is to find a large, reasonable family Fsuch that (1.6) holds with a constant depending only on p0and the A1constant of w.
244 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez Remark 1.4. In Theorem 1.3, (1.7) holds if p(·) satisfies (1.1) and (1.2). By Theorem 1.1, setting q(x) = p(x)/p0we have that q(·)∈P(Ω) and |q0(x)−q0(y)| ≤ |p(x)−p(y)| p0(p−/p0−1)2. Remark 1.5. When Ω = Rn, if 1 ≤p0< p−, then by Theorem 1.2 the hypothesis that p(·)/p00∈B(Rn) is equivalent to assuming that p(·)∈B(Rn). As we will see below, this will allow us to conclude that a variety of operators are bounded on Lp(·)(Rn) whenever the Hardy–Littlewood maximal operator is. Remark 1.6. Our approach using pairs of functions leads to an equivalent formulation of Theorem 1.3 in which the exponent p0does not play a role. This can be done by defining a new family Fp0consisting of the pairs (fp0, gp0) with (f, g)∈F. Notice that in this case (1.6) is satisfied by Fp0with p0= 1. Thus, the case p0= 1 will imply that if 1 < p−and p(·)0∈B(Ω) then (1.7) holds. Therefore, if we define r(x) = p(x)p0, we have that r(·)∈P0(Ω), p0< r−, (r(·)/p0)0∈B(Ω) and (1.7) holds with r(·) in place of p(·). But this is exactly the conclusion of Theorem 1.3. Remark 1.7. We believe that a more general version of Theorem 1.3 is true, one which holds for larger classes of weights and yields inequalities in weighted variable Lpspaces. However, proving such a result will require a weighted version of Theorem 1.1, and even the statement of such a result has eluded us. For such a weighted extrapolation result the appropriate class of weights is no longer A1, but Ap(as in [38]) or A∞(as in [11]). We emphasize, though, that the class A1, which is the smallest among the Apclasses, is the natural one to consider when attempting to prove unweighted estimates. Theorem 1.3 can be generalized to give “off-diagonal” results. In the classical setting, the extrapolation theorem of Rubio de Francia was extended in this manner by Harboure, Mac´ıas and Segovia [24]. Theorem 1.8. Given a family Fand an open set Ω⊂Rn, assume that for some p0and q0,0< p0≤q0<∞, and every weight w∈A1, (1.8) ZΩ f(x)q0w(x)dx1/q0 ≤C0ZΩ g(x)p0w(x)p0/q0dx1/p0 ,(f, g)∈F. Given p(·)∈P0(Ω) such that p0< p−≤p+< p0q0/(q0−p0), define the function q(·)by (1.9) 1 p(x)−1 q(x)=1 p0 −1 q0 , x ∈Ω. If q(x)/q00∈B(Ω), then for all (f, g)∈Fsuch that f∈Lq(·)(Ω), (1.10) kfkq(·),Ω≤Ckgkp(·),Ω. Remark 1.9. As before, (1.10) holds if p(·) satisfies (1.1) and (1.2).
The boundedness of classical operators on variable Lpspaces 245 We can generalize Theorem 1.3 by combining it with the two extrapolation theorems discussed above. This is possible since A1⊂Ap, 1 < p ≤ ∞. This has two advantages. First, it makes clear that the hypotheses which must be satisfied correspond to those of the known weighted norm inequalities; see, in particular, the applications discussed in Section 2 below. Second, as in [11], we are able to prove vector-valued inequalities in variable Lpspaces with essentially no additional work. All such inequalities are new. Corollary 1.10. Given a family Fand an open set Ω⊂Rn, assume that for some p0,0< p0<∞, and for every w∈A∞, (1.11) ZΩ f(x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, (f, g)∈F. Let p(·)∈P0(Ω) be such that there exists 0< p1< p−with p(·)/p10∈B(Ω). Then for all (f, g)∈Fsuch that f∈Lp(·)(Ω), (1.12) kfkp(·),Ω≤Ckgkp(·),Ω. Furthermore, for every 0<q<∞and sequence {(fj, gj)}j⊂F, (1.13) X j (fj)q1/q p(·),Ω ≤C X j (gj)q1/q p(·),Ω . Corollary 1.11. Given a family Fand an open set Ω⊂Rn, assume that (1.11) holds for some 1< p0<∞, for every w∈Ap0and for all (f, g)∈F. Let p(·)∈P(Ω) be such that there exists 1< p1< p−with p(·)/p10∈B(Ω). Then (1.12) holds for all (f, g)∈Fsuch that f∈Lp(·)(Ω). Furthermore, for every 1<q<∞and {(fj, gj)}j∈F, the vector-valued inequality (1.13) holds. The rest of this paper is organized as follows. To illustrate the power of our results, we first consider some applications. In Section 2 we give a number of examples of operators which are bounded on Lp(·). These results are immediate consequences of the above results and the theory of weighted norm inequalities. Some of these have been proved by others, but most are new. We also prove vectorvalued inequalities for these operators, all of which are new results. In Section 3 we present an application to partial differential equations: we extend the Calder´on– Zygmund inequality (see [5], [22]) to solutions of 4u=fwith f∈Lp(·)(Ω). In Section 4 we give an application to the theory of Sobolev spaces: we show that the Calder´on extension theorem (see [2], [4]) holds in variable Sobolev spaces. In Section 5 we prove Theorems 1.3 and 1.8. Our proof is adapted from the arguments given in [11]. Finally, in Section 6 we prove Corollaries 1.10 and 1.11. Throughout this paper, we will make use of the basic properties of variable Lpspaces, and will state some results as needed. For a detailed discussion of these spaces, see Kov´aˇcik and R´akosn´ık [30]. As we noted above, in order to emphasize that we are dealing with variable exponents, we will always write p(·) instead of pto denote an exponent function. Throughout, Cwill denote a positive constant whose exact value may change at each appearance.
246 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez 2. Applications: Estimates for classical operators on Lp(·) In this section we give a number of applications of Theorems 1.3 and 1.8, and Corollaries 1.10 and 1.11, to show that a wide variety of classical operators are bounded on the variable Lpspaces. In the following applications we will impose different conditions on the exponents p(·) to guarantee the corresponding estimates. In most of the cases, it will suffice to assume that p(·)∈B(Rn), or in particular that p(·) satisfies (1.1) and (1.2). As we noted in the remarks following Theorem 1.3, to prove these applications we will need to use density arguments. In doing so we will use the following facts: (1) L∞ c, bounded functions of compact support, and C∞ c, smooth functions of compact support, are dense in Lp(·)(Ω). See Kov´aˇcik and R´akosn´ık [30]. (2) If p+<∞and f∈Lp+(Ω) ∩Lp−(Ω), then f∈Lp(·)(Ω). This follows from the fact that |f(x)|p(x)≤ |f(x)|p+χ{|f(x)|≥1}+|f(x)|p−χ{|f(x)|<1}. 2.2. The Hardy–Littlewood maximal function. It is well known that for 1 <p<∞and for w∈Ap, ZRn Mf(x)pw(x)dx ≤CZRn f(x)pw(x)dx. From Corollary 1.11 with the pairs (Mf, |f|), we get vector-valued inequalities for Mon Lp(·), provided there exists 1 < p1< p−with p(·)/p10∈B(Rn); by Theorem 1.2, this is equivalent to p(·)∈B(Rn). To apply Corollary 1.11 we need to restrict the pairs to functions f∈L∞ c, but since these form a dense subset we get the desired estimate for all f∈Lp(·)(Rn). Corollary 2.1. If p(·)∈B(Rn), then for all 1< q < ∞, X j (Mfj)q1/q p(·),Rn ≤C X j |fj|q1/q p(·),Rn . Remark 2.2. From Corollary 1.11 we also get one of the implications of Theorem 1.2: if p(·)/p10∈B(Rn) then p(·)∈B(Rn). It is very tempting to speculate that all of Theorem 1.2 can be proved via extrapolation, but we have been unable to do so. 2.2. The sharp maximal operator. Given a measurable function fand a cube Q, define fQ=1 |Q|ZQ f(y)dy, and the sharp maximal operator by M#f(x) = sup x3Q 1 |Q|ZQ |f(y)−fQ|dy.
The boundedness of classical operators on variable Lpspaces 247 The sharp maximal operator was introduced by Fefferman and Stein [20], who showed that for all p, 0 <p<∞, and w∈A∞, ZRn Mf(x)pw(x)dx ≤CZRn M#f(x)pw(x)dx. (Also see Journ´e [26].) Therefore, by Corollary 1.10 with the pairs (Mf, M#f), f∈L∞ c(Rn), and by Theorem 1.2 we have the following result. Corollary 2.3. Let p(·)∈P0(Rn)be such that there exists 0< p1< p− with p(·)/p1∈B(Rn). Then, (2.1) kMfkp(·),Rn≤CkM#fkp(·),Rn, and for all 0<q<∞, (2.2) X j (Mfj)q1/q p(·),Rn ≤C X j (M#fj)q1/q p(·),Rn . Remark 2.4. Corollary 2.3 generalizes results due to Diening and R˚uˇziˇcka [15, Theorem 3.6] and Diening [14, Theorem 8.10], who proved (2.1) with Mf replaced by fon the left-hand side and under the assumptions that p(·) and p0(·)∈B(Rn) with 1 < p−≤p+<∞in the first paper and p(·)∈B(Rn) in the second. Notice that our result is more general since we allow p(·) to go below 1 and we only need p(·)/p10∈B(Rn) for some small value 0 < p1< p−. Furthermore, we automatically obtain the vector-valued inequalities given in (2.2). 2.3. Singular integral operators. Given a locally integrable function K defined on Rn\ {0}, suppose that the Fourier transform of Kis bounded, and K satisfies (2.3) |K(x)| ≤ C |x|n,|∇K(x)| ≤ C |x|n+1 , x 6= 0. Then the singular integral operator T, defined by Tf(x) = K∗f(x), is a bounded operator on weighted Lp. More precisely, given 1 <p<∞, if w∈Ap, then (2.4) ZRn |Tf(x)|pw(x)dx ≤CZRn |f(x)|pw(x)dx. (For details, see [17], [21].) From Corollary 1.11, we get that Tis bounded on variable Lpprovided there exists 1 < p1< p−with p(·)/p10∈B(Rn); by Theorem 1.2 this is equivalent to p(·)∈B(Rn). Again, to apply the corollary we need to restrict ourselves to a suitable dense family of functions. We use the fact that C∞ cis dense in Lp(·)(Rn), and the fact that if f∈C∞ c, then Tf ∈T1<p<∞Lp⊂Lp(·)(Rn).
254 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez We begin with a few definitions and a lemma. Given p(·)∈P(Ω) and a natural number k, define the variable Sobolev space Wk,p(·)(Ω) to be the set of all functions f∈Lp(·)(Ω) such that X |α|≤k kDαfkp(·),Ω<+∞, where the derivatives are understood in the sense of distributions. Given a function fwhich is twice differentiable (in the weak sense), we define for i= 1,2, Dif=X |α|=i (Dαf)21/2 . We need the following auxiliary result whose proof can be found in [30]. Lemma 3.1. If Ω⊂Rnis a bounded domain, and if p(·), q(·)∈P(Ω) are such that p(x)≤q(x),x∈Ω, then kfkp(·),Ω≤(1 + |Ω|)kfkq(·),Ω. Theorem 3.2. Given an open set Ω⊂Rn,n≥3, suppose p(·)∈P(Ω) with p+< n/2satisfies (1.1) and (1.2). If f∈Lp(·)(Ω), then there exists a function u∈Lq(·)(Ω), where (3.1) 1 p(x)−1 q(x)=2 n, such that (3.2) 4u(x) = f(x),a.e. x∈Ω. Furthermore, kD2ukp(·),Ω≤Ckfkp(·),Ω,(3.3) kD1ukr(·),Ω≤Ckfkp(·),Ω,(3.4) kukq(·),Ω≤Ckfkp(·),Ω,(3.5) where 1 p(x)−1 r(x)=1 n. In particular, if Ωis bounded, then u∈W2,p(·)(Ω). Proof. Our proof roughly follows the proof in the setting of Lebesgue spaces given by Gilbarg and Trudinger [22], but also uses this result in key steps. Fix f∈Lp(·)(Ω); without loss of generality we may assume that kfkp(·),Ω= 1. Decompose fas f=f1+f2=f χ{x:|f(x)|>1}+f χ{x:|f(x)|≤1}.
The boundedness of classical operators on variable Lpspaces 255 Note that |fi(x)| ≤ |f(x)|and so kfikp(·),Ω≤1. Further, we have that f1∈ Lp−(Ω) and f2∈Lp+(Ω) since, by the definition of the norm in Lp(·)(Ω) and since kfkp(·),Ω= 1, ZΩ f1(x)p−dx =Z{x∈Ω:|f(x)|>1} |f(x)|p−dx ≤ZΩ |f(x)|p(x)dx ≤1, ZΩ f2(x)p+dx =Z{x∈Ω:|f(x)|≤1} |f(x)|p+dx ≤ZΩ |f(x)|p(x)dx ≤1. Thus, we can solve Poisson’s equation with f1and f2(see [22]): more precisely, define u1(x) = (Γ ∗f1)(x), u2= (Γ ∗f2)(x), where Γ is the Newtonian potential, Γ(x) = 1 n(2 −n)ωn |x|2−n, and ωnis the volume of the unit ball in Rn. Since p−and q−also satisfy (3.1), by the Calder´on–Zygmund inequality on classical Lebesgue spaces, u1∈Lq−(Ω). Similarly, since p+and q+satisfy (3.1), u2∈Lq+(Ω). Let u=u1+u2; then u∈Lq−(Ω) + Lq+(Ω). Since u1and u2are solutions of Poisson’s equation, 4u(x) = 4u1(x) + 4u2(x) = f1(x) + f2(x) = f(x),a.e. x∈Ω. We show that u∈Lq(·)(Ω) and that (3.5) holds: by inequality (2.12), kukq(·),Ω≤ ku1kq(·),Ω+ku2kq(·),Ω =1 n(2 −n)ωnkI2f1kq(·),Ω+kI2f2kq(·),Ω ≤Ckf1kp(·),Ω+kf2kp(·),Ω ≤C=Ckfkp(·),Ω; the last equality holds since kfkp(·),Ω= 1. Similarly, a direct computation shows that for any multi-index α,|α|= 1, |DαΓ(x)| ≤ 1 n ωn |x|1−n. Therefore, |Dαu(x)| ≤ |Dα(Γ ∗f1)(x)|+|Dα(Γ ∗f2)(x)| =|(DαΓ∗f1)(x)|+|(DαΓ∗f2)(x)| ≤1 n ωnI1(|f1|)(x) + I1(|f2|)(x).
256 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez So again by inequality (2.12) we get kDαukr(·),Ω≤Ckf1kp(·),Ω+kf2kp(·),Ω≤C, which yields inequality (3.4). Given a multi-index α,|α|= 2, another computation shows that DαΓ is a singular convolution kernel which satisfies (2.3). Therefore, the operator Tαg(x) = (DαΓ∗g)(x) = Dα(Γ ∗g)(x) is singular integral operator, and as before (3.3) follows from inequality (2.5) and Remark 2.6 applied to f1and f2. Finally, if Ω is bounded, since p(x)≤q(x) and p(x)≤r(x), x∈Ω, by Lemma 3.1 we have that u∈W2,p(·)(Ω). Remark 3.3. In the previous estimates we could have worked directly with f. Had we done so, however, we would have had to check that all the integrals appearing were absolutely convergent. The advantage of decomposing fas f1+f2 is that we did not need to pay attention to this since f1∈Lp−(Ω), f2∈Lp+(Ω). We also want to stress that u1and u2, as solutions of Poisson’s equation with f1∈Lp−(Ω) and f2∈Lp+(Ω), satisfy Lebesgue space estimates. For instance, as noted above, u∈Lq−(Ω) + Lq+(Ω). However, we have actually proved more, since Lq(·)(Ω) is a smaller space. Similar remarks hold for the first and second derivatives of u. 4. The Calder´on extension theorem In this section we state and prove the Calder´on extension theorem for variable Sobolev spaces. Our proof follows closely the proof of the result in the classical setting; see, for example, R. Adams [2] or Calder´on [4]. First, we give two definitions and a lemma. Definition 4.1. Given a point x∈Rn, a finite cone with vertex at x,Cx, is a set of the form Cx=B1∩ {x+λ(y−x) : y∈B2, λ > 0}, where B1is an open ball centered at x, and B2is an open ball which does not contain x. Definition 4.2. An open set Ω ⊂Rnhas the uniform cone property if there exists a finite collection of open sets {Uj}(not necessarily bounded) and an associated collection {Cj}of finite cones such that the following hold: (1) there exists δ > 0 such that Ωδ={x∈Ω : dist (x, ∂Ω) < δ} ⊂ S j Uj; (2) for every index jand every x∈Ω∩Uj,x+Cj⊂Ω.
The boundedness of classical operators on variable Lpspaces 257 An example of a set Ω with the uniform cone property is any bounded set whose boundary is locally Lipschitz. (See Adams [2].) Finally, in giving extension theorems for variable Lpspaces, we must worry about extending the exponent function p(·). The following result shows that this is always possible, provided that p(·) satisfies (1.1) and (1.2). Lemma 4.3. Given an open set Ω⊂Rnand p(·)∈P(Ω) such that (1.1) and (1.2) hold, there exists a function ˜p(·)∈P(Rn)such that: (1) ˜psatisfies (1.1) and (1.2); (2) ˜p(x) = p(x),x∈Ω; (3) ˜p−=p−and ˜p+=p+. Remark 4.4. Diening [13] proved an extension theorem for exponents p(·) which satisfy (1.1), provided that Ω is bounded and has Lipschitz boundary. It would be interesting to determine if every exponent p(·)∈B(Ω) can be extended to an exponent function in B(Rn). Proof. Since p(·) is bounded and uniformly continuous, by a well-known result it extends to a continuous function on Ω. Straightforward limiting arguments show that this extension satisfies (1), (2) and (3). The extension of p(·) on Ω to ˜p(·) defined on all of Rnfollows from a construction due to Whitney [45] and described in detail in Stein [42, Chapter 6]. For ease of reference, we will follow Stein’s notation. We first consider the case when Ω is unbounded; the case when Ω is bounded is simpler and will be sketched below. When Ω is unbounded, (1.2) is equivalent to the existence of a constant p∞, p−≤p∞≤p+, such that for all x∈Ω, |p(x)−p∞| ≤ C log(e+|x|). Define a new function r(·) by r(x) = p(x)−p∞. Then r(·) is still bounded (though no longer necessarily positive), still satisfies (1.1) on Ω and satisfies (4.1) |r(x)| ≤ C log(e+|x|). We will extend rto all of Rn. If we define ω(t) = 1/log(e/2t), 0 < t ≤1/2, and ω(t) = 1 for t≥1/2, then a straightforward calculation shows that ω(t)/t is a decreasing function and ω(2t)≤C ω(t). Further, since log(e/2t)≈log(1/t), 0< t < 1/2, and since ris bounded, |r(x)−r(y)| ≤ Cω(|x−y|) for all x, y ∈Ω. Therefore, by Corollary 2.2.3 in Stein [42, p. 175], there exists a function ˜r(·) on Rnsuch that ˜r(x) = r(x), x∈Ω, and such that ˜r(·) satisfies (1.1). For x∈Rn\Ω, ˜r(x) is defined by the sum ˜r(x) = X k r(pk)ϕ∗ k(x),
258 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez where {Qk}are the cubes of the Whitney decomposition of Rn\Ω, {ϕ∗ k}is the partition of unity subordinate to this decomposition, and each point pk∈Ω is such that dist(pk, Qk) = dist(Ω, Qk). It follows immediately from this definition that for all x∈Rn,r−≤˜r(x)≤ r+. However, ˜r(·) need not satisfy (4.1) so we must modify it slightly. To do so we need the following observation: if f1,f2are functions such that |fi(x)−fi(y)| ≤ Cω(|x−y|), x, y ∈Rn,i= 1,2, then min(f1, f2) and max(f1, f2) satisfy the same inequality. The proof of this observation consists of a number of very similar cases. For instance, suppose minf1(x), f2(x)=f1(x) and minf1(y), f2(y)=f2(y). Then f1(x)−f2(y)≤f2(x)−f2(y)≤Cω(|x−y|), f2(y)−f1(x)≤f1(y)−f1(x)≤Cω(|x−y|). Hence, minf1(x), f2(x)−minf1(y), f2(y)=|f1(x)−f2(y)| ≤ Cω(|x−y|). It follows immediately from this observation that s(x) = maxmin(˜r(x), C/ log(e+|x|),−C/ log(e+|x|) satisfies (1.1) and (4.1). Therefore, if we define ˜p(x) = s(x) + p∞, then (1), (2) and (3) hold. Finally, if Ω is bounded, we define r(x) = p(x)−p+and repeat the above argument essentially without change. Theorem 4.5. Given an open set Ω⊂Rnwhich has the uniform cone property, and given p(·)∈P(Ω) such that (1.1) and (1.2) hold, then for any natural number kthere exists an extension operator Ek:Wk,p(·)(Ω) →Wk,p(·)(Rn), such that Eku(x) = u(x), a.e. x∈Ω, and kEkukp(·),Rn≤C(p(·),k,Ω)kukp(·),Ω. The proof of Theorem 4.5 in variable Sobolev spaces is nearly identical to that in the classical setting. (See Adams [2].) The proof, beyond calculations, requires the following facts which our hypotheses insure are true.
The boundedness of classical operators on variable Lpspaces 259 – By Lemma 4.3, p(·) immediately extends to an exponent function on Rn. – Functions in C∞(Ω) are dense in Wk,p(·)(Ω). By our hypotheses, the maximal operator is bounded on Lp(·)(Ω), and the density of C∞(Ω) follows from this by the standard argument (cf. Ziemer [47]). For more details, see Diening [12] or Cruz-Uribe and Fiorenza [9]. – If ϕis a smooth function on Rn\ {0}with compact support, and if there exists ε > 0 such that on Bε(0), ϕis a homogeneous function of degree k, k > −n, then kϕ∗fkp(·),Ω≤Cp(·), ϕkfkp(·),Ω. This again follows from the fact that the maximal operator is bounded on Lp(·)(Ω), and from the wellknown inequality |ϕ∗f(x)| ≤ CMf(x). For more details, see Cruz-Uribe and Fiorenza [9]. – Singular integral operators with kernels of the form K(x) = G(x) |x|n, where Gis bounded on Rn\ {0}, has compact support, is homogeneous of degree zero on BR(0) \ {0}for some R > 0, and has RSRG dx = 0, are bounded on Lp(·)(Ω). Such kernels are essentially the same as those given by (2.9), and as discussed above, our hypotheses imply that they are bounded. Remark 4.6. If p(·) satisfies (1.1), then C∞ c(Rn) is dense in Wk,p(·)(Rn). (See [9], [41].) Hence, if the hypotheses of Theorem 4.5 hold, then it follows immediately that the set {uχΩ:u∈C∞ c(Rn)}is dense in Wk,p(·)(Ω). However this result is true under much weaker hypotheses; see [9], [18], [19], [25], [46] for details. 5. Proof of Theorems 1.3 and 1.8 Since Theorem 1.3 is a particular case of Theorem 1.8 with p0=q0, it suffices to prove the second result. We need two facts about variable Lpspaces. First, if p(·), q(·)∈P0(Ω) and p(x)/q(x) = r, then it follows from the definition of the norm that (5.1) kfkr p(·),Ω= |f|r q(·),Ω. Second, given p(·)∈P(Ω), we have the generalized H¨older’s inequality (5.2) ZΩ |f(x)g(x)|dx ≤1 + 1 p− −1 p+kfkp(·),Ωkgkp0(·),Ω, and the “duality” relationship (5.3) kfkp(·),Ω≤sup gZΩ f(x)g(x)dx≤1 + 1 p− −1 p+kfkp(·),Ω,
260 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez where the supremum is taken over all g∈Lp0(·)(Ω) such that kgkp0(·),Ω= 1. For proofs of these results, see Kov´aˇcik and R´akosn´ık [30]. The proof of Theorem 1.8 begins with a version of a construction due to Rubio de Francia [38] (also see [11], [21]). Fix p(·)∈P0(Ω) such that p−> p0, and let ¯p(x) = p(x)/p0. Define q(·) as in (1.9), and let ¯q(x) = q(x)/q0. By assumption, the maximal operator is bounded on L¯q0(·)(Ω), so there exists a positive constant Bsuch that kMfk¯q0(·),Ω≤Bkfk¯q0(·),Ω. Define a new operator Ron L¯q0(·)(Ω) by Rh(x) = ∞ X k=0 Mkh(x) 2kBk, where, for k≥1, Mk=M◦M◦· · ·◦Mdenotes kiterations of the maximal operator, and M0is the identity operator. It follows immediately from this definition that: (a) if his non-negative, h(x)≤Rh(x); (b) kRhk¯q0(·),Ω≤2khk¯q0(·),Ω; (c) for every x∈Ω, M(Rh)(x)≤2BRh(x), so Rh∈A1with an A1constant that does not depend on h. We can now argue as follows: by (5.1) and (5.3), kfkq0 q(·),Ω=kfq0k¯q(·),Ω≤sup ZΩ f(x)q0h(x)dx, where the supremum is taken over all non-negative h∈L¯q0(·)(Ω) with khk¯q0(·),Ω= 1. Fix any such function h; it will suffice to show that ZΩ f(x)q0h(x)dx ≤Ckgkq0 p(·),Ω with the constant Cindependent of h. First note that by (a) above we have that (5.4) ZΩ f(x)q0h(x)dx ≤ZΩ f(x)q0Rh(x)dx. By (5.2), (b), and since f∈Lq(·)(Ω), ZΩ f(x)q0Rh(x)dx ≤Ckfq0k¯q(·),ΩkRhk¯q0(·),Ω ≤Ckfkq0 q(·),Ωkhk¯q0(·),Ω ≤Ckfkq0 q(·),Ω<∞.
The boundedness of classical operators on variable Lpspaces 261 Therefore, we can apply (1.8) to the right-hand side of (5.4) and again apply (5.2), this time with exponent ¯p(·): ZΩ f(x)q0Rh(x)dx ≤CZΩ g(x)p0Rh(x)p0/q0dxq0/p0 ≤Ckgp0kq0/p0 ¯p(·),Ωk(Rh)p0/q0kq0/p0 ¯p0(·),Ω =Ckgkq0 p(·),Ωk(Rh)p0/q0kq0/p0 ¯p0(·),Ω. To complete the proof, we need to show that k(Rh)p0/q0kq0/p0 ¯p0(·),Ωis bounded by a constant independent of h. But it follows from (1.9) that for all x∈Ω, ¯p0(x) = p(x) p(x)−p0 =q0 p0 q(x) q(x)−q0 =q0 p0 ¯q0(x). Therefore, k(Rh)p0/q0kq0/p0 ¯p0(·),Ω=kRhk¯q0(·),Ω≤Ckhk¯q0(·),Ω=C. This completes our proof. 6. Proof of Corollaries 1.10 and 1.11 The proofs of Corollaries 1.10 and 1.11 require the more general versions of the extrapolation theorems discussed in the introduction. For the convenience of the reader we state them both here. Theorem 6.1. Given a family Fand an open set Ω⊂Rn, assume that for some p0,0< p0<∞, and for every w∈A∞, (6.1) ZΩ f(x)p0w(x)dx ≤C0ZΩ g(x)p0w(x)dx, (f, g)∈F. Then for all 0<p<∞and w∈A∞, (6.2) ZΩ f(x)pw(x)dx ≤C0ZΩ g(x)pw(x)dx, (f, g)∈F. Furthermore, for every 0< p, q < ∞,w∈A∞, and sequence {(fj, gj)}j⊂F, (6.3) X j (fj)q1/q Lp(w,Ω) ≤C X j (gj)q1/q Lp(w,Ω) . Theorem 6.2. Given a family Fand an open set Ω⊂Rn, assume that for some p0,1< p0<∞, and for every w∈Ap0,(6.1) holds. Then for every 1< p < ∞and w∈Ap,(6.2) holds. Furthermore, for every 1< p, q < ∞, w∈Ap, and sequence {(fj, gj)}j⊂F,(6.3) holds.
262 D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. P´erez Theorem 6.1 is proved in [11]. The original statement of Theorem 6.2 is only for pairs of the form (|Tf|, f), and does not include the vector-valued estimate (6.3). (See [17], [21], [38].) However, an examination of the proofs shows that they hold without change when applied to pairs (f, g)∈F. Furthermore, as we noted before, this approach immediately yields the vector-valued inequalities: given a family Fand 1 <q<∞, define the new family Fqto consist of the pairs (Fq, Gq), where Fq(x) = X j (fj)q1/q , Gq(x) = X j (gj)q1/q ,{(fj, gj)}j⊂F. Clearly, inequality (6.1) holds for Fqwhen p0=q, so by extrapolation we get (6.3). Corollary 1.10 follows immediately from Theorems 1.3 and 6.1. Since (1.11) holds for some p0, by Theorem 6.1 it holds for all 0 <p<∞and for all w∈A∞. Therefore, we can apply Theorem 1.3 with p1in place of p0to obtain (1.12). To prove the vector-valued inequality (1.13), note that by (6.3) we can apply Theorem 1.3 to the family Fqdefined above, again with p1in place of p0. In exactly the same way, Corollary 1.11 follows from Theorems 1.3 and 6.2. References [1] Acerbi, E., and G. Mingione: Regularity results for stationary electrorheological fluids. - Arch. Ration. Mech. Anal. 164, 2002, 213–259. [2] Adams, R.: Sobolev Spaces. - Academic Press, New York, 1975. [3] Alvarez, J., and C. P´ erez: Estimates with A∞weights for various singular integral operators. - Boll. Unione Mat. Ital. Sez. A 7:8, 1994, 123–133. [4] Calder´ on, A.: Lebesgue spaces of differentiable functions and distributions. - In: Partial Differential Equations, Proc. Symp. Pure Math. 4, edited by C. Morrey, Amer. Math. Soc., Providence, 1961, 33–49. [5] Calder´ on, A., and A. Zygmund: On the existence of certain singular integrals. - Acta Math. 88 1952, 85–139. [6] Capone, C., D. Cruz-Uribe, SFO, and A. Fiorenza: The fractional maximal operator on variable Lpspaces. - Preprint, 2004; available at www.na.iac.cnr.it. [7] Coifman, R., and C. Fefferman: Weighted norm inequalities for maximal functions and singular integrals. - Studia Math. 51, 1974, 241–250. [8] Coifman, R., R. Rochberg, and G. Weiss: Factorization theorems for Hardy spaces in several variables. - Ann. of Math. 103, 1976, 611–635. [9] Cruz-Uribe, D., SFO, and A. Fiorenza: Approximate identities in variable Lpspaces. - Math. Nachr. (to appear). [10] Cruz-Uribe, D., SFO, A. Fiorenza, and C. J. Neugebauer: The maximal function on variable Lpspaces. - Ann. Acad. Sci. Fenn. Math. 28, 2003, 223–238, and 29, 2004, 247–249. [11] Cruz-Uribe, D., SFO, J. M. Martell, and C. P´ erez: Extrapolation from A∞weights and applications. - J. Funct. Anal. 213, 2004, 412–439.
The boundedness of classical operators on variable Lpspaces 263 [12] Diening, L.: Maximal functions on generalized Lp(x)spaces. - Math. Inequal. Appl. 7, 2004, 245–253. [13] Diening, L.: Riesz potentials and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(x)and Wk,p(x). - Math. Nachr. 268, 2004, 31–43. [14] Diening, L.: Maximal Function on Musie lak–Orlicz Spaces and Generalized Lebesgue Spaces. - Preprint, 2004. [15] Diening, L., and M. R˚ uˇ ziˇ cka: Calder´on–Zygmund operators on generalized Lebesgue spaces Lp(·)and problems related to fluid dynamics. - J. Reine Angew. Math. 563, 2003, 197–220. [16] Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. - Trans. Amer. Math. Soc. 336 1993, 869–880. [17] Duoandikoetxea, J.: Fourier Analysis. - Grad. Stud. Math. 29, Amer. Math. Soc., Providence, 2000. [18] Edmunds, D., and J. R´ akosn´ ık: Density of smooth functions in Wk,p(x)(Ω). - Proc. Roy. Soc. London Ser. A 437, 1992, 229–236. [19] Edmunds, D., and J. R´ akosn´ ık: Sobolev embeddings with variable exponent. - Studia Math. 143, 2000, 267–293. [20] Fefferman, C., and E. M. Stein: Hpspaces in several variables. - Acta Math. 129, 1972, 137–193. [21] Garc´ ıa-Cuerva, J., and J. L. Rubio de Francia: Weighted Norm Inequalities and Related Topics. - North-Holland Math. Stud. 116, North Holland, Amsterdam, 1985. [22] Gilbarg, D., and N. Trudinger: Elliptic partial differential equations of second order. - 2nd Edition, Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1983. [23] Gundy, R. F, and R. L. Wheeden: Weighted integral inequalities for the nontangential maximal function, Lusin area integral, and Walsh–Paley series. - Studia Math. 49, 1973, 101–118. [24] Harboure, E., R. Macias, and C. Segovia: Extrapolation results for classes of weights. - Amer. J. Math. 110, 1988, 383–397. [25] H¨ ast¨ o, P.: On the density of continuous functions in variable exponent Sobolev spaces. - Preprint, 2004; available at mathstat.helsinki.fi/analysis/varsobgroup/english/publications.htm [26] Journ´ e, J.-L.: Zygmund Operators, Pseudo-Differential Operators and the Cauchy Integral of Calder´on. - Lecture Notes in Math. 994, Springer-Verlag, New York, 1983. [27] Karlovich, A., and A. Lerner: Commutators of singular integrals on generalized Lp spaces with variable exponent. - Publ. Mat. 49, 2005, 111–125. [28] Kokilashvili, V., and S. Samko: Singular integrals in weighted Lebesgue spaces with variable exponent. - Georgian Math. J. 10, 2003, 145–156. [29] Kokilashvili, V., and S. Samko: On Sobolev theorem for Riesz type potentials in the Lebesgue spaces with variable exponent. - Z. Anal. Anwendungen 22, 2003, 899–910. [30] Kov´ aˇ cik, O., and J. R´ akosn´ ık: On spaces Lp(x)and Wk,p(x). - Czechoslovak Math. J. 41(116), 1991, 4, 592–618. [31] Kurtz, D.: Littlewood–Paley and multiplier theorems on weighted Lpspaces. - Trans. Amer. Math. Soc. 259, 1980, 235–254. [32] Kurtz, D., and R. Wheeden: Results on weighted norm inequalities for multipliers. - Trans. Amer. Math. Soc. 255, 1979, 343–362. [33] Lerner, A.: Some remarks on the Hardy–Littlewood maximal function on variable Lp spaces. - Math. Z. (to appear).