JOURNAL OF c 2006, Scientific Horizon FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net Volume 4, Number 2 (2006), 113-144
Characterization of Riesz and Bessel potentials on variable Lebesgue spaces
Alexandre Almeida and Stefan Samko
(Communicated by Vakhtang Kokilashvili)
2000 Mathematics Subject Classification. 46E35, 26A33, 42B20, 47B38. Keywords and phrases. Lebesgue space with variable exponent, hypersingular integral, Riesz potential, Bessel potential.
Abstract. Riesz and Bessel potential spaces are studied within the framework of the Lebesgue spaces with variable exponent. It is shown that the spaces of these potentials can be characterized in terms of convergence of hypersingular integrals, if one assumes that the exponent satisfies natural regularity conditions. As a consequence of this characterization, we describe a relation between the spaces of Riesz or Bessel potentials and the variable Sobolev spaces.
1. Introduction
The Lebesgue spaces Lp(·) with variable exponent and the corresponding m Sobolev spaces Wp(·) have been intensively investigated during the last years. We refer to the papers [16], [27], where the basics of such spaces were developed, to the papers [9], [23], where the denseness of nice functions
The first author is partially supported by Unidade de Investiga¸c˜ao “Matem´atica e Aplica¸c˜oes” of Universidade de Aveiro, through Programa Operacional “Ciˆencia, Tecnologia e Inova¸c˜ao” (POCTI) of the Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT), cofinanced by the European Community Fund FEDER. 114 Characterization of Riesz and Bessel potentials in Sobolev variable spaces was considered, and to the papers [3], [5], [7], [8], [13], [15],[18], [21], [22] and the recent preprints [2], [4] and references therein, where various results on maximal, potential and singular operators in variable Lebesgue spaces were obtained (see also the surveys [12], [25]). The interest to the Lebesgue spaces with variable exponent during the last decade was in particular roused by applications in problems of fluid dynamics, elasticity theory and differential equations with non-standard growth conditions (see [8], [19]). We deal with the spaces of Riesz and Bessel potentials with densities n in the spaces Lp(·)(R ). For the constant p, it is known that the left inverse operator to the Riesz potential operator Iα within the frameworks n of the spaces Lp(R ) may be constructed in terms of the hypersingular α integrals, and the range I [Lp] is described in terms of convergence of those hypersingular integrals, see [20], [24]. The extension of the statement on the inversion to the case of variable exponents was recently given in [1]. In this paper we give a description of the range of the Riesz and α α Bessel potential operators, I [Lp(·)]andB [Lp(·)], respectively, in terms of convergence of hypersingular integrals. As a consequence, we also establish a connection of the spaces of Riesz and Bessel potentials with the Sobolev m spaces Wp(·) . This partially extends the known results for constant p (see [24]) to the variable exponent setting. The paper is organized as follows. In Section 2 we provide notation and necessary preliminaries and auxiliary results which will be often used throughout the text. The first main result, Theorem 3.2, given in Section 3 contains a characterization of the space of Riesz potentials in terms of fractional derivatives. In Section 4 we consider some spaces of fractional smoothness defined in terms of hypersingular integrals and study their connection with the space of Riesz potentials, the main results being given in Theorems 4.1 and 4.4. The study of the space of Bessel potentials on Lp(·) and its description are made in Section 5, see Theorem 5.7. In the last section the connection between the spaces of potentials and the Sobolev spaces with variable exponent is studied. Throughout the paper, we shall consider standard notation or it will be properly introduced whenever needed.
2. Preliminaries
∞ n ∞ n As usual, C0 (R ) stands for the class of all C functions on R with compact support. By S(Rn) we denote the Schwartz class of all rapidly decreasing C∞ -functions on Rn ,andbyS(Rn) its dual. For ϕ ∈S(Rn), A. Almeida and S. Samko 115 by Fϕ (or ϕ ) we denote the Fourier transform of ϕ, (2.1) (Fϕ)(ξ)= eix·ξ ϕ(x) dx, ξ ∈ Rn. Rn
n By W0(R ) we denote the class of Fourier transforms of integrable functions. By Φ(Rn) we denote the topological dual of the Lizorkin space Φ(Rn) consisting of all functions ϕ ∈S(Rn) such that (Dβ ϕ )(0) = 0, for all n β n β ∈ N0 ,whereD is the usual partial derivative. Two elements of S (R ) differing by a polynomial are indistinguishable as elements of Φ(Rn)(see [24], Section 2.2). By C (or c) we denote a general positive constant whose value is irrelevant and may change at different occurrences.
2.1 On Lebesgue spaces with variable exponent
A detailed discussion of properties of the variable Lebesgue spaces may be found in the papers [9], [10], [16], [27]. We recall here some important tools and definitions which will be used throughout this paper. Let p : Rn → [1, ∞) be a (Lebesgue) measurable function. Put
p := ess sup p(x)andp := ess inf p(x). x∈Rn x∈Rn
n n By Lp(·)(R ) we denote the space of all measurable functions f on R such that the modular p(x) Ip(·)(f):= |f(x)| dx Rn is finite. Under this definition, this is a linear space if and only if p<∞, and we only consider bounded exponents. Lp(·)(Rn)isaBanachspace endowed with the norm f n (2.2) fp(·) := inf λ>0: Ip(·) ≤ 1 ,f∈ Lp(·)(R ). λ
This space inherits some properties from the classical Lebesgue spaces with constant exponent. In fact, under the additional assumption p > 1, n Lp(·)(R ) is uniformly convex, reflexive and its dual space is (isomorphic n to) Lp (·)(R ), where p (·) is the natural conjugate exponent given by 1 1 p(x) + p (x) ≡ 1. An important property of this space is that the convergence in norm is equivalent to the modular convergence: given {fk}k∈N0 ⊂ n Lp(·)(R ), then fkp(·) → 0 if and only if Ip(·)(fk) → 0, as k →∞. Nevertheless, this generalized space has some undesirable properties. As 116 Characterization of Riesz and Bessel potentials
n mentioned before, Lp(·)(R ) is not translation invariant contrarily to its classical counterpart. Similarly to the classical case, one defines the Sobolev space of variable m n exponent Wp(·)(R ), m ∈ N0 , as the space of all measurable functions f β n such that its (weak) derivatives D f up to order m are in Lp(·)(R ). This is a Banach space equipped with the norm β m n fm,p(·) := D fp(·),f∈ Wp(·)(R ). |β|≤m
In order to emphasize that we are dealing with variable exponents, we always write p(·) instead of p to denote an exponent function. In general, we will consider function spaces defined on the whole Euclidean space Rn . So, in what follows, we shall omit the “Rn ” from their notation.
2.2 The maximal operator in Lp(·)
For a locally integrable function g on Rn , the Hardy-Littlewood maximal operator M is defined by 1 Mg(x)=sup |g(y)| dy, r>0 |B(x, r)| B(x,r) where B(x, r) denotes the open ball centered at x ∈ Rn and of radius r>0. The boundedness of the maximal operator M was first proved by L. Diening [5] over bounded domains, under the assumption that p(·) is locally log-H¨older continuous, that is, C (2.3) |p(x) − p(y)|≤ ,x,y∈ Rn, |x − y|≤1/2. − ln |x − y|
He later extended the result to unbounded domains by supposing, in addition, that the exponent p(·) is constant outside some large fixed ball. The general case of the exponent p(·) non-constant at infinity, was considered in [18], where some integral condition was imposed and in [3] where it was assumed that the exponent is log-H¨older continuous at infinity: C (2.4) |p(x) − p(y)|≤ ,x,y∈ Rn, |y|≥|x|. ln(e + |x|)
Condition (2.4) is equivalent to the logarithmic decay condition
C (2.5) |p(x) − p(∞)|≤ ,x∈ Rn, ln(e + |x|) A. Almeida and S. Samko 117 where p(∞) = lim p(x). |x|→∞ Let P(Rn) be the class of all exponents p(·), 1
(ii) f ∗ ϕεp(·) ≤ c fp(·) (with c independent of ε and f ) and, if in addition Rn ϕ(x) dx =1,then (iii) f ∗ ϕε → f as ε → 0 in Lp(·) and almost everywhere. Theorem 2.1 is an important tool which allows us to obtain boundedness of various concrete convolution operators, even in the case where they are defined by the Fourier transform of their kernel.
2.3 The Riesz potential operator on variable Lebesgue spaces
We recall that the Riesz potential operator Iα is defined by Iα g(y) g(x):= n−α dy, Rn |x − y| where g is a locally integrable function with an appropriate behavior at infinity and 0 <α Diening [7] proved the Sobolev theorem on Rn for p(·) satisfying the local logarithmic condition (2.3) and constant at infinity. Some weighted version of the Sobolev theorem for Rn with the power weight fixed to infinity, was obtained in [14]. Recently, C. Capone, D. Cruz- Uribe and A. Fiorenza [2] proved the Sobolev theorem on arbitrary domains for p(·) non-constant at infinity. Their statement for the case of the whole space Rn runs as follows. Theorem 2.2. Let 0 <α α (2.6) I fq(·) ≤ c fp(·),f∈ Lp(·), 1 1 α n where q(·) is the Sobolev exponent given by q(x) = p(x) − n , x ∈ R . 2.4 Hypersingular integrals on Lp(·) spaces We refer to [24] and [26] for the theory of hypersingular integrals. A typical hypersingular integral has the form 1 Δyf (x) (2.7) n+α dy, α > 0, dn,(α) Rn |y| where Δyf denotes the finite difference of order ∈ N of the function f and dn,(α) is a certain normalizing constant, which is chosen so that the construction in (2.7) does not depend on (see [24], Chapter 3, for details). It is well-known that the integral (2.7) exists (for each x ∈ Rn )iff ∈S(Rn) and >α, for instance. Following [24], we shall consider both a centered difference and a non- centered one in the construction of the hypersingular integral. However, m when we write Δh without any specification we mean a non-centered difference. The important fact here is that the order should be chosen according to the following rule (as stated in [24], page 65), which will be always assumed in the sequel: α (1) in the case of a non-centered difference we take >2 with the 2 obligatory choice = α for α odd; (2) in the case of a centered difference we take even and >α>0. In general, the integral in (2.7) may be divergent, and hence it needs to α α be properly defined. We interpret hypersingular operators as D := lim Dε , ε→0 α where Dε denotes the truncated hypersingular operator A. Almeida and S. Samko 119 Δ f (x) Dα 1 y (2.8) ε f(x)= n+α dy , ε > 0. dn,(α) |y|>ε |y| Sometimes Dα is also called the Riesz fractional derivative since it can be α interpreted as a positive fractional power (−Δ) 2 of the minus Laplacian. In what follows, the limit above is always taken in the sense of convergence in the Lp(·) norm. This makes sense in view of the Proposition 2.3 below. Proposition 2.3. If p(·) ∈P(Rn), then the truncated hypersingular α integral operator Dε is bounded in Lp(·) , for every ε>0. Another important property of the truncated hypersingular integrals is the following uniform estimate: n n Proposition 2.4. Let 0 <α α α lim Dε I ϕ(x)=ϕ(x), ε→0 for almost all x ∈ Rn . 120 Characterization of Riesz and Bessel potentials 3. Characterization of the Riesz potentials on Lp(·) spaces We define the space of Riesz potentials on Lp(·) in a natural way as α α I [Lp(·)]={f : f = I ϕ, ϕ ∈ Lp(·)}, 0 <α α Following approaches in [24], we will show below that the space I [Lp(·)] can be described in terms of convergence of hypersingular integrals. n n Lemma 3.1. Let 1 Proof. The proof is direct. n Theorem 3.2. Let 0 <α α Proof. First, assume that f ∈I [Lp(·)]. The fact that f ∈ Lq(·) follows α from Theorem 2.2. On the other hand, as f = I ϕ for some ϕ ∈ Lp(·) , then we have α α α (3.1) D f = lim Dε I ϕ = ϕ ε→0 (convergence in Lp(·) ) according to Theorem 2.5. α Conversely, let f ∈ Lq(·) and suppose that its Riesz derivative D f exists. α α α Our aim is to prove that f = I D f and hence that f ∈I [Lp(·)]. Both f and IαDαf canberegardedaselementsofΦ .Letusshowthat they coincide in this sense. For all φ ∈ Φ, we have: IαDα Dα φ(x) f(x) φ(x) dx = f(y) | − |n−α dx dy Rn Rn Rn x y (Δzf)(y) Iα = lim n+α dz φ(y) dy ε→0 Rn |z|>ε dn,(α) |z| ⎛ ⎞ k α ⎜ (−1) k f(u)I φ(u+kz) ⎟ ⎜ k=0 ⎟ = lim ⎜ n+α du⎟dz ε→0 |z|>ε⎝ Rn dn,(α) |z| ⎠ A. Almeida and S. Samko 121 α (Δ−zI φ)(u) = lim f(u) n+α dz du ε→0 n dn,(α) |z| R |z|>ε α α = lim f(u) Dε I φ(u) du ε →0 Rn = f(u) φ(u) du. Rn The first equality follows from the Fubini theorem since the double integral c converges absolutely. In fact, since |φ(y)|≤ (1+|y|)N with an arbitrary large N , then by Lemma 1.38 in [24], we have that Iα(|φ|)(x) is bounded and α c I (|φ|)(x) ≤ (1+|x|)n−α as |x|→∞.Thus α dx I | | ≤ ∞ Ip (·)( ( φ )) c1 + c2 (n−α)p (x) < |x|>1 (1 + |x|) because inf (n − α)p(x) >n. Hence, using H¨older inequality we arrive at x∈Rn α α α α |D f(y)|I (|φ|)(y) dy ≤ c D fp(·) I (|φ|)p (·) < ∞. Rn In the second equality, we notice that the convergence with respect to the α Lp(·) norm implies weak convergence in Φ (note that I φ ∈ Φ). The third equality follows from similar arguments to those used in the first one, and then by the change of variables y → u + kz,withfixedz . We only observe α now that Dε f ∈ Lq(·) (cf. Proposition 2.3 and Lemma 3.1). Finally, the last passage is obtained by making use of the Lebesgue theorem. Indeed, α α since φ ∈ Lq (·) ,thenDε I φ ∈ Lq (·) (cf. Proposition 2.3 and Lemma 3.1), α α so that f(·)(Dε I φ)(·) ∈ L1 . The result follows now from the inversion Theorem 2.5. To finish the proof, we observe that since both f and IαDαf are tempered distributions, then IαDαf = f + P ,whereP is a polynomial. Therefore f + P ∈ Lq(·) , which implies P ∈ Lq(·) . Thus we should have P ≡ 0, which means IαDαf(x)=f(x) almost everywhere. We generalize another characterization which is contained in Theorem 7.11 in [24]. Theorem 3.3. In Theorem 3.2 above one can replace the assertion on the existence of the Riesz derivative of f by the following uniform boundedness condition: there exists C>0 such that α (3.2) Dε fp(·) ≤ C for all ε>0. 122 Characterization of Riesz and Bessel potentials α Proof. If f = I ϕ, ϕ ∈ Lp(·) , then (3.2) is immediate by Proposition 2.4. α Conversely, if sup Dε fp(·) < ∞ then there exists a subsequence of ε>0 {Dα } {Dα } ε f ε>0 ,say εk f k∈N , which converges weakly in Lp(·) (note that Lp(·) is a reflexive Banach space under conditions 1 The second equality follows directly from the weak convergence in Lp(·) by α noticing that I φ ∈ Lp (·) , while the last one is justified by the convergence α α D I of εk φ to φ in Lq (·) (taking into account the inversion theorem and the Lemma 3.1 once again) and from the fact that f ∈ (Lq (·)) = Lq(·) . α α Hence, as previously, one arrives at f = I g ,sothatf ∈I [Lp(·)]. 4. Function spaces on Lp(·) defined by fractional derivatives Hypersingular integrals can also be used to construct function spaces of fractional smoothness. Similarly to the classical case let us consider the space α α Lp(·) = {f ∈ Lp(·) : D f ∈ Lp(·)},α>0, where the fractional derivative Dα is treated in the usual way as convergent in the Lp(·) norm. We remark that this space does not depend on the order of the finite differences, and it is a Banach space with respect to the norm α α f | Lp(·) := fp(·) + D fp(·). These spaces will be shown the same as the spaces of Bessel potentials. They are connected with the space of Riesz potentials in the following way. Theorem 4.1. Assume that the exponent p(·) satisfies the usual logarithmic conditions (2.3) and (2.4).Letalso0 <α Then α α Lp(·) = Lp(·) ∩I [Lp(·)]. α Proof. By Theorem 3.2 we only need to prove the embedding Lp(·) ⊂ α α Lp(·) ∩I [Lp(·)]. So, let f ∈ Lp(·) . As in the proof of Theorem 3.2 (but here under the assumption that f ∈ Lp(·) instead of f ∈ Lq(·) as there), we α α α have f(x)=I D f(x) almost everywhere, so that f ∈I [Lp(·)]. Remark 4.2. Theorem 4.1 also holds if one takes centered differences (everything in the proof of Theorem 3.2 works in a similar way). Hence, α from this theorem we conclude that the space Lp(·) does not depend on the type of finite differences used to construct the derivative Dα , at least when n p< α . ∞ α 4.1 Denseness of C0 in the space Lp(·) α ∞ Before to prove that functions from Lp(·) can be approximated by C0 ∞ functions, let us show a preliminary denseness result. By Wp(·) we denote the Sobolev space of all functions of Lp(·) for which all their (weak) derivatives are also in Lp(·) . ∞ ∞ α Proposition 4.3. The set C ∩ Wp(·) is dense in Lp(·) for all p(·) ∈ P(Rn). ∞ ∞ α Proof. Step 1: Let us show that C ∩ Wp(·) ⊂ Lp(·) , which is not obvious ∞ ∞ in the case of variable exponents. If f ∈ C ∩ Wp(·) then we already know that (Δ f)(x) y ∈ (4.1) n+α dy Lp(·) |y|>ε |y| for any ε>0 (see Proposition 2.3). On the other hand, we have (Δ f)(x) y → → (4.2) n+α dy 0asδ 0. |y|≤δ |y| p(·) To prove (4.2), we use the representation j y r−k r (Δyf)(x)=r (−1) k j! k (4.3) |j|=r k=1 1 × (1 − t)r−1(Dj f)(x − kty) dt, 0 124 Characterization of Riesz and Bessel potentials (see [24], formula (3.31)) with the choice ≥ r>α. Hence (Δyf)(x) n+α dy |y|≤δ |y| 1 j r−1 y j = cr,j,k (1 − t) (D f)(x − kty) dy dt. |y|n+α |j|=r k=1 0 |y|≤δ The change of variables y → δz yields (4.4) (Δyf)(x) n+α dy |y|≤δ |y| 1 · r−α − r−1 1 ∗ j = δ cr,j,k (1 t) nKj D f (x) dt δk t δk t |j|=r k=1 0 ( ) ( ) where Kj is given by zj Kj(z)= when |z|≤1andKj(z)=0 otherwise, |z|n+α and δk(t)=kδt.Since|j| = r>α,thekernelKj has a decreasing radial integrable dominant, so that Theorem 2.1 is applicable and we have (4.5) 1 (Δyf)(x) r−α r−1 j dy ≤ δ |cr,j,k | (1 − t) c M(D f)(x) dt, |y|n+α |y|≤δ |j|=r k=1 0 where c>0 is independent of δk(t). Hence, (4.6) (Δyf)(x) (r−α)p j Ip(·) dy ≤ cδ Ip(·)(M(D f)) → 0asδ → 0. |y|n+α |y|≤δ |j|=r We remind that the convergence in norm is equivalent to the modular convergence since the exponent is bounded. Further, under the present assumptions on p(·), the maximal operator M maps Lp(·) into itself. (Δyf)(x) From (4.1) and (4.5) we see that the integral Rn |y|n+α dy converges absolutely for all x and defines a function belonging to Lp(·) .Moreover,by (4.6), it coincides with the Riesz derivative: A. Almeida and S. Samko 125 (Δ f)(x) (Δ f)(x) y − y n+α dy n+α dy Rn |y| |y|>ε |y| p(·) (Δ f)(x) y → → = n+α dy 0asε 0, |y|≤ε |y| p(·) α so that D f ∈ Lp(·) . We remark that some modification is needed if α is odd. In this case, we should consider centered differences (and hence >α; recall the previous rule) and then proceed in a similar way from (4.3). Step 2: We use the standard approximation by using mollifiers (as in ∈ ∞ ≥ [24], Lemma 7.14). Let ϕ C0 such that ϕ 0, Rn ϕ(x) dx =1with n ∞ supp ϕ ⊂ B(0, 1). Put ϕm(x):=m ϕ(mx),m∈ N.Thenϕm ∈ C0 and α supp ϕm ⊂ B(0, 1/m). Given f ∈ Lp(·) let us define y fm(x):=ϕm ∗ f(x)= ϕ(y) f x − dy. Rn m ∞ Hence fm ∈ C .Wealsohavefm ∈ Lp(·) by Theorem 2.1 and j j D fm = D (ϕm) ∗ f ∈ Lp(·) . In the case of fractional derivatives we have, for each ε>0andm ∈ N, α α Dε fm =(Dε f)m, that is, α α Dε (ϕm ∗ f)=ϕm ∗ Dε f, which can be easily proved by Fubini theorem. Hence α α α α D fm = lim (ϕm ∗ Dε f)=ϕm ∗ D f =(D f) , ε→0 m where the second equality follows from the continuity of the convolution α operator. In particular one concludes that D fm ∈ Lp(·) . It remains to show that the functions fm approximate the function f in α the Lp(·) norm. Of course, f − fmp(·) → 0asm →∞ by Theorem 2.1 once again. On the other hand, using similar arguments we have α α α α α D (f − fm)p(·) = D f − D fmp(·) = D f − (D f)mp(·) → 0 α as m →∞,sinceD f ∈ Lp(·) . n Theorem 4.4. If p(·) is as in Proposition 4.3 with p< α , then the class ∞ α C0 is dense in Lp(·) . 126 Characterization of Riesz and Bessel potentials Proof. By Proposition 4.3, it is sufficient to show that every function ∞ ∞ ∞ f ∈ C ∩ Wp(·) can be approximated by functions in C0 in the norm α · |Lp(·). As in the case of constant p, we will use the “smooth truncation” of functions. ∞ Let μ ∈ C0 with μ(x)=1if |x|≤ 1, supp μ ⊂ B(0, 2) and 0 ≤ μ(x) ≤ 1 x n for every x. Define μm(x):=μ m , x ∈ R , m ∈ N. We are to show that α the sequence of truncations {μmf}m∈N converges to f in Lp(·) . The passage to the limit lim f − μmfp(·) =0⇐⇒ lim Ip(·)(f − m→∞ m→∞ μmf) = 0 is directly checked by means of the Lebesgue dominated α convergence theorem. It remains to show that Ip(·)(D (f − μmf)) also tends to zero as m →∞. Taking Remark 4.2 into account, we may consider centered differences in the fractional derivative (under the choice >α with even). For brevity we denote νm =1− μm .Thenwehave k −k (Δ νm)(x+ y)(Δ f)(x+( −k)y) Dα 1 y 2 y 2 (νmf)(x)= n+α dy n, n | | d (α) k=0 k R y 1 =: Am,kf(x). n, d (α) k=0 k So we need to show that Ip(·)(Am,kf) → 0asm →∞,fork =0, 1,...,. We separately treat the cases k =0,k= and 1 ≤ k ≤ − 1. The case k =0:wehave α (4.7) Am,0f(x)=dn,(α) νm(x) D f(x)+Bmf(x) where [νm(x + 2 y) − νm(x)](Δy f)(x + 2 y) Bmf(x)= n+α dy Rn |y| [μm(x) − μm(x + 2 y)](Δyf)(x + 2 y) = n+α dy Rn |y| The convergence of the first term in (4.7) is clear, so that it remains to prove that Ip(·)(Bmf) → 0asm →∞.Put 0 1 Bmf(x)= (···) dy + (···) dy := Bmf(x)+Bmf(x). |y|≤1 |y|>1 0 To estimate the term Bmf , we make use of the Taylor formula (of order 1) with the remainder in the integral form and obtain A. Almeida and S. Samko 127 n 1 t ∂μ x + 2 y μm x + y − μm(x)= yj dt. 2 2m j=1 0 ∂xj m Hence c (4.8) μm x + y − μm(x) ≤ |y|, 2 m where c>0 does not depend on x, y and m. 0 As in the proof of Proposition 4.3, we can estimate Bmf in terms of the convolution of the derivatives of f with a “good kernel” in the sense of Theorem 2.1. In fact, taking (4.8) and (3.31) in [24] into account, we get |B0 f(x)| m 2ν · ≤ c − r−1 1 ∗| j | (1 t) n K D f (x) dt m (−θν (t)) −θν(t) |j|=r 0 ν= 2 1 · − r−1 1 ∗| j | + (1 t) n K D f (x) dt θν (t) θν (t) 2ν −1 2 1 · c − r−1 1 ∗| j | (4.9) + (1 t) n K D f (x) dt m −θν t −θν t |j|=r ν=1 0 ( ( )) ( ) where K is given by K(z)=|z|r+1−n−α if |z| < 1andK(z)=0 otherwise, with θν (t)=νt − 2 and under the choice r>α− 1. So 0 c j Ip(·)(B f) ≤ Ip(·) M(|D f|) → 0asm →∞. m m |j|=r 1 For the term Bmf we may proceed as follows. Since μ is infinitely differentiable and compactly supported, then it satisfies the H¨older continuity condition. Hence, for an arbitrary ε ∈ (0, 1], there exists c = cε > 0 not depending on x, y , such that c ε μm x + y − μm(x) ≤ |y| . 2 mε When α>1, we may proceed as before by considering r<α<. 1 Putting all these things together, one estimates Bmf(x) as in (4.9) with 128 Characterization of Riesz and Bessel potentials the correspondent kernel K given by |y|r K(y)= when |y| > 1andK(y)=0 otherwise. |y|n+α−ε Under the choice 0 <ε The case 0 <α≤ 1 can be treated without passing to the derivatives of f . In fact, in this case, we may take = 2, and hence | | | | | − | | 1 |≤ c f(x + y) f(x) f(x y) Bmf(x) ε n+α−ε dy+ n+α−ε dy+ n+α−ε dy . m |y|>1 |y| |y|>1 |y| |y|>1 |y| Each term can be managed by using similar arguments as above but now with the choice 0 <ε<α. The case k = :let (Δyνm)(x + 2 y) f(x − 2 y) Am,f(x)= | |n+α dy Rn y = (···) dy + (···) dy |y|≤1 |y|>1 0 1 =: Bm,f(x)+Bm,f(x) z Notice that (Δyνm)(z)=−(Δyμm)(z)=−(Δ y μ) . So, according m m to (4.3), one gets the estimate x + 2 y (Δyνm) x + y = (Δ y μ) 2 m m r |y| j ≤ c D μ∞ m |j|=r c ≤ |y|r mr (with ≥ r>α). Hence, c |B0 f(x)|≤ (K ∗|f|)(x) m, mr A. Almeida and S. Samko 129 where K is now given by 1 K(y)= if |y|≤ and K(y)=0 otherwise. |y|n+α−r 2 Since r>α,thekernelK is under the assumptions of Theorem 2.1. As 0 before, we get Bm,fp(·) → 0asm →∞. 1 As far as the term Bm,f is concerned, when α>1wemaychoose >α>r and proceed in a similar way as in the case k = 0 above. When 0 <α≤ 1wemaytake =2 andget 2 x+y Δ y μ m |f(x − y)| | 1 |≤ m Bm,f(x) n+α dy |y|>1 |y| x+y x x−y μ m − 2 μ m + μ m |f(x − y)| = n+α dy |y|>1 |y| x+y μ − μ x |f(x − y)| ≤ m m dy |y|n+α |y|>1 x−y μ − μ x |f(x − y)| + m m dy |y|n+α |y |>1 | |ε | − | ≤ c y f(x y) ε n+α dy m |y|>1 |y| for any ε ∈ (0, 1] (and c>0 independent of m). Thus we arrive at the desired conclusion by taking ε<α. The case k ∈{1, 2,...,− 1}: as in the previous case, we have k −k (Δyνm)(x + 2 y)(Δy f)(x +(2 − k)y) Am,kf(x)= | |n+α dy Rn y = (···) dy + (···) dy |y|≤1 |y|>1 0 1 =: Bm,kf(x)+Bm,kf(x). 0 We may estimate the term Bm,kf by noticing that k k |y| (Δyνm)(x + y) ≤ c 2 m and then by proceeding as above with an appropriate choice of r. 130 Characterization of Riesz and Bessel potentials 1 k k For the term Bm,k we first consider the case α>1. Since l = k−l , for l =0, 1,...,k,wemaywrite k−1 2 k x + 2 y k x + 2 y y x + 2 y y (Δ y μ) = μ − l −μ − (k − l) m m l=0 l m m m m if k is odd. When k is even, we can also represent our finite difference as the sum of the first order differences of two appropriate terms since k l k (−1) l = 0. In both situations we may again make use of the H¨older l=0 continuity (of order ε) of the function μ. Finally, we shall arrive at the desired estimate by using arguments as above, but under the assumption 0 <ε 5. Bessel potentials on Lp(·) spaces and their characterization The main aim of this section is to describe the range of the Bessel potential operator on Lp(·) in terms of convergence of hypersingular integrals. This isknowninthecaseofconstantp, see [24], Section 7.2, or [26], Section n 27.3, and references therein. Here we consider the case 0 <α 5.1 Basic properties. The Bessel kernel Gα can be introduced in terms of Fourier transform by 2 −α/2 n Gα(x)=(1+|x| ) ,x∈ R ,α>0. It is known that ∞ 2 − π|x| − t α−n dt n Gα(x)=c(α) e t 4π t 2 ,x∈ R , 0 t where c(α) is a certain constant (see, for example, [29], Section V.3.1), so that Gα is a non-negative, radially decreasing function. Moreover, Gα A. Almeida and S. Samko 131 is integrable with Gα1 = Gα(0) = 1 and it can also be represented by means of the McDonald function: α−n Gα(x)=c(α, n) |x| 2 K n−α (|x|). 2 The Bessel potential of order α>0ofthedensity ϕ is defined by α (5.1) B ϕ(x)= Gα(x − y) ϕ(y) dy. Rn For convenience, we also denote B0ϕ = ϕ. Theorem 5.1. If p(·) ∈P(Rn) then the Bessel potential operator Bα is bounded in Lp(·) . Proof. The boundedness of the operator Bα follows from the properties of the kernel Gα described above. Taking into account Theorem 2.1, there exists a constant c>0 such that α B ϕp(·) = Gα ∗ ϕp(·) ≤ c ϕp(·), for all ϕ ∈ Lp(·). We are interested in Bessel potentials with densities in Lp(·) . One defines the space of Bessel potentials as the range of the Bessel potential operator α α B [Lp(·)]={f : f = B ϕ, ϕ ∈ Lp(·)},α≥ 0. α According to Theorem 5.1, the space B [Lp(·)], also called sometimes Liouville space of fractional smoothness, is well defined, being a subspace of Lp(·) if the maximal operator is bounded in Lp(·) , in particular, if the exponent p(·)islog-H¨older continuous both locally and at infinity. α B [Lp(·)] is a Banach space endowed with the norm α (5.2) f |B [Lp(·)] := ϕp(·), where ϕ is the density from (5.1). The symbol“→” below denotes continuous embedding. n α Proposition 5.2. If p(·) ∈P(R ) and α>γ≥ 0,thenB [Lp(·)] → γ B [Lp(·)]. Proof. The proof follows immediately from the properties of the Bessel kernel and from the boundedness of the Bessel potential operator. Indeed, α γ α−γ if f = B ϕ for some ϕ ∈ Lp(·) then one can write f = B (B ϕ). Thus γ f ∈B [Lp(·)] by Theorem 5.1. Furthermore, γ α−γ α f |B [Lp(·)] := B ϕp(·) ≤ c ϕp(·) =: c f |B [Lp(·)]. 132 Characterization of Riesz and Bessel potentials α 5.2 Characterization of the space B [Lp(·)] via hypersingular integrals. The comparison of the ranges of the Bessel and Riesz potential operators is naturally made via the convolution type operator whose symbol is the ratio of the Fourier transforms of the Riesz and Bessel kernels. This operator is the sum of the identity operator and the convolution operator with a radial integrable kernel. Keeping in mind the application of Theorem 2.1, we have to show more, namely that this kernel has an integrable decreasing dominant. We have to show the existence of integrable decreasing dominants for two important kernels gα and hα , one defined in (5.3), another in (5.4). This will require substantial efforts. Let gα and hα be the functions defined via the following Fourier transforms | |α x ∈ Rn (5.3) 2 α =1+gα(x),α>0,x , (1 + |x| ) 2 2 α (1 + |x| ) 2 n (5.4) =1+hα(x),α>0,x∈ R . 1+|x|α Observe that | |α 1+ x (5.5) 2 α = Gα(x)+gα(x)+1. (1 + |x| ) 2 It is known that gα and hα are integrable (see, for example, Lemma 1.25 in [24]). The following two lemmas are crucial for our further goals. Lemma 5.3. The function gα defined in (5.3) has an integrable and radially decreasing dominant. Lemma 5.4. The kernel hα given by (5.4) admits the bounds c (5.6) |hα(x)|≤ as |x| < 1,a=min{1,α}, |x|n−a and c (5.7) |hα(x)|≤ as |x|≥1, |x|n+α where c>0 is a constant not depending on x. The proof of these lemmas being somewhat technical is postponed till the next subsection. A. Almeida and S. Samko 133 Before to formulate the main result of this section, we prove the following two statements. Proposition 5.5. Let 0 <α α α (5.8) ϕ = B (I + Uα)(ϕ + D ϕ), where I denotes the identity operator and Uα is the convolution operator with the kernel hα . ∞ Proof. Identity (5.8) holds for functions ϕ ∈ C0 . This follows immediately ∞ α from equality (5.4) above (cf. (7.39) in [24]). The denseness of C0 in Lp(·) α (stated in Theorem 4.4) allows us to write (5.8) for all functions in Lp(·) . α To this end, we observe that both operators B and Uα are continuous in α Lp(·) . In fact, the boundedness of B was proved in Theorem 5.1. On the other hand, the convolution operator Uα is bounded since its kernel has a radially decreasing and integrable dominant by Lemma 5.4. Proposition 5.6. Let 0 <α α α (5.9) B ψ = I (I + Kα) ψ, for all ψ ∈ Lp + Lp ,whereI is the identity operator and Kα is the convolution operator with the kernel gα . Proof. Representation (5.9) holds for densities belonging to classical Lebesgue spaces (see, for instance, (7.38) in [24]), where the kernel of Kα is precisely the function gα from (5.3). By the Sobolev theorem one concludes α that either B or I (I + Kα) are linear operators from Lp into Lq(p) ,with 1 1 α 1 1 α q(p) = p − n ,andfromLp into Lq(p) ,with q(p) = p − n .So,wecan define these operators on the sum Lp + Lp in the usual way. Hence, if ψ = ψ0 + ψ1 ,withψ0 ∈ Lp and ψ1 ∈ Lp , then we may make use the already known representation for each term and then arrive at equality (5.9). Finally, we are able to characterize the Bessel potentials in terms of convergence of hypersingular integrals. The following theorem in the case of constant p, 1 Theorem 5.7. Let 0 <α α α α α c1 f | Lp(·)≤f |B [Lp(·)]≤c2 f | Lp(·), ∀f ∈B [Lp(·)]. α Proof. Assume first that f ∈B [Lp(·)]. Then f ∈ Lp(·) by Theorem 5.1. It remains to show that its Riesz derivative also belongs to Lp(·) .Since α f = B ϕ for some ϕ ∈ Lp(·) and Lp(·) ⊂ Lp + Lp , then by Proposition 5.6 one gets the representation α α B ϕ = I (I + Kα) ϕ. Lemma 5.3 combined with Theorem 2.1, allow us to conclude that Kα α is bounded in Lp(·) , and hence f ∈I[Lp(·)]. So, according to the characterization given in Theorem 3.2, the Riesz derivative Dαf exists in α the sense of convergence in Lp(·) . Therefore, f ∈ Lp(·) .Moreover, α α α α f | Lp(·) = B ϕp(·) + D B ϕp(·) α α α = B ϕp(·) + D I (I + Kα) ϕp(·) α = B ϕp(·) + (I + Kα) ϕp(·) α α ≤ c ϕp(·) = c f |B [Lp(·)]. The third equality follows from the inversion Theorem 2.5, while the inequality is obtained from Theorem 5.1 and from the boundedness of Kα . α Conversely, suppose that f ∈ Lp(·) . Proposition 5.5 yields the representation α α f = B (I + Uα)(f + D f). Taking into account Lemma 5.4 and Theorem 2.1, we arrive at the α conclusion that f ∈B [Lp(·)]and α α f |B [Lp(·)] = (I + Uα)(f + D f)p(·) α α ≤ c (fp(·) + D fp(·))=c f | Lp(·). Corollary 5.8. If the exponent p(·) is under the conditions of ∞ α Theorem 5.7,thenC0 is dense in B [Lp(·)]. 5.3 Proof of Lemmas 5.3 and 5.4. We start by proving Lemma 5.3. 2 1/2 |x|α −2 α/2 Let us denote ρ =(1+|x| ) .Then α − 1=(1− ρ ) − 1. (1+|x|2) 2 Taking the expansion into the binomial series we get ∞ ∞ −2 α α/2 −2 k k α/2 −2k (1 − ρ ) 2 − 1= −ρ − 1= (−1) ρ ,ρ>1. k=0 k k=1 k A. Almeida and S. Samko 135 Hence, for each x =0, α ∞ ∞ |x| k α/2 α − 1= (−1) G2k(x):= c(α, k) G2k(x), | |2 2 (1 + x ) k=1 k k=1 where G2k is the Bessel kernel of order 2k. Hence ∞ n (5.10) gα(x)= c(α, k) G2k(x),x∈ R . k=1 Now, we stress that ∞ mα(x):= |c(α, k)| G2k(x) k=1 defines a radial decreasing dominant of gα .Furthermore,mα is integrable, ∞ α/2 mα1 ≤ < ∞, k=1 k α/2 ≤ c →∞ since k k1+α/2 as k (cf. [26], p. 14). As regards Lemma 5.4, we split its proof into two parts. Step 1 (Proof of (5.6)): Let us start by representing the function hα as a finite sum of Fourier transforms of Bessel kernels plus an integrable 1 1 − function. To this end, we denote t = 1+|x|2 .Thenhα(x)= tβ +(1−t)β 1, α with β = 2 .But 1 − 1 · 1 − β β 1= β β 1 t +(1− t) (1 − t) t 1+ 1−t ∞ kβ 1 − k t − = − β ( 1) − 1 (1 t) k=0 1 t t 1 where the series converges if 1−t < 1, that is, if t< 2 or |x| > 1. Since 1 1+|x|2 t 1 1−t = |x|2 and 1−t = |x|2 ,weget 2 α ∞ k | | 2 − (1 + x ) ( 1) − | | hα(x)= | |α | |αk 1, x > 1. x k=0 x 136 Characterization of Riesz and Bessel potentials For each natural number N ,wecanwrite N k α − 2 2 ( 1) (5.11) hα(x)= 1+|x| − 1+AN (x), |x| > 1, | |α(k+1) k=0 x | |≤ c 1 → → ∞ where AN (x) |x|αN . Indeed, since |x|αk 0ask + (recall that |x| > 1), we have (5.12) 2 α ∞ k 2 α α (1 + |x| ) 2 (−1) (1 + |x| ) 2 1 2 | | ≤ ≤ AN (x) = | |α | |αk | |2α | |αN | |αN . x k=N+1 x x x x 1 Now it remains to represent the powers α(k+1) in terms of the powers |x| √ 1 .Weobservethatforanyγ>0, taking ρ = 1+|x|2 ,wehave 1+|x|2 (5.13) ⎛ ⎞ −γ/2 M − 1 −γ − 1 −γ ⎝ − j γ/2 −2j ⎠ | |γ = ρ 1 2 = ρ ( 1) ρ + φM (ρ) , x ρ j=0 j where M ∈ N and ∞ j −γ/2 −2j φM (ρ)= (−1) ρ j=M+1 j converges absolutely for ρ>1, that is, for x =0. Obviously ∞ ∞ φM (ρ) ≤ 1 1 ≤ c 1 1 (5.14) γ c γ 2j+γ M+1 γ j+γ 1− 2 1− 2 ρ j=M+1 j ρ ρ j=M+1 j 2 2 √ where we took into account that |x| > 1 ⇐⇒ ρ ≥ 2. Hence φM (ρ) c1 c2 (5.15) ≤ ≤ . ργ ρM+1 ρM Then from (5.13) and (5.15) M − j −γ/2 1 ( 1) j γ B x , (5.16) | |γ = | |2 j+γ/2 + M ( ) x j=0 (1 + x ) A. Almeida and S. Samko 137 where γ C (5.17) |B (x)|≤ as |x| > 1. M |x|2M Substituting (5.16) into (5.11) (with γ = α(k + 1)), and taking M = N , we arrive at N − k+j −α(k+1)/2 ( 1) j (5.18) hα(x)= | |2 j+αk + rN (x), k,j=0 (1 + x ) k+j =0 where the function N α 2 2 α(k+1) (5.19) rN (x)=AN (x)+ 1+|x| BN (x) k=0 satisfies the estimate c |rN (x)|≤ ,μ= N min(2,α), |x|μ for all |x| > 1 according to (5.12) and (5.17). Hence, we only have to choose n N>min(2,α) in order to get the integrability of rN at infinity. The estimate at infinity was given for |x| > 1, but the equality (5.18) n itself may be written for all x ∈ R , just by defining rN as N n n rN (x):=hα(x) − c(k, j) G2j+αk(x),N> ,x∈ R , k,j=0 min(2,α) k+j =0 k+j −α(k+1)/2 where G2j+αk are Bessel kernels and c(k, j):=(−1) j . So, we have rN ∈W0 .Inparticular,rN is a bounded continuous function. Also, rN is integrable at infinity in view of the estimate above −1 and hence, rN ∈W0 ∩ L1 . On the other hand, F rN ∈W0 ∩ L1 .Thus, −1 F rN is a bounded continuous function too. So N −1 |hα(x)|≤ |c(k, j)||G2j+αk(x)| + |F rN (x)| k,j=0 k+j =0 N ≤ |c(k, j)||G2j+αk(x)| + C. k,j=0 k+j =0 138 Characterization of Riesz and Bessel potentials We know that 1 c G2j+αk(x) ∼ ≤ as |x| < 1, |x|n−2j−αk |x|n−min(1,α) when 2j + αk < n. Thus, we arrive at (5.6) with a =min(1,α). In the case 2j + αk > n we arrive at the same estimate since G2j+αk(x) ∼ C(2j + αk), |x| < 1. For the case 2j + αk = n we have the following logarithmic behavior: 1 G2j+αk(x) ∼ ln , |x| < 1. |x| 1 1 But ln |x| ≤ |x|n−a for any a ∈ (0,n). The proof of (5.6) is completed. Step 2 (Proof of (5.7)): To obtain (5.7), we transform the Bochner formula for the Fourier transform of radial functions via integration by parts and arrive at the formula ∞ c n −1 (m) 2 α n n +m−1 | | (5.20) F h (x)= +m−1 ψα (t) t J 2 (t x ) dt, x =0, |x| 2 0 α 2 (1+t ) 2 n where ψα(t)= 1+tα and m is arbitrary such that m>1+ 2 (the latter condition on m) guarantees the convergence of the integral at infinity. To justify formula (5.20), we make use of the standard regularization of the integral (cf. [30]): −1 −n −ε|y| −ix·y F hα(x)=(2π) lim e e hα(|y|) dy ε→0 Rn n/2 ∞ −n (2π) −εt n/2 | | =(2π) lim n/2−1 e hα(t) t Jn/2−1(t x ) dt ε→0 |x| 0 −ν ∞ (2π) ν | | = ν−1 lim fε(t) t Jν−1(t x ) dt |x| ε→0 0 −ν − m ∞ (2π) ( 1) (m) ν | | (5.21) = ν−1 lim m fε (t) t Jν+m−1(t x ) dt |x| ε→0 |x| 0 n where Jν−1(t) denotes the Bessel function of the first kind, ν = 2 , m ∈ N and −εt fε(t):=e hα(t),ε>0. The second equality follows from the Bochner formula for Fourier transforms of radial functions, while the last is obtained via integration by parts and the A. Almeida and S. Samko 139 d ν ν relation du [u Jν (u)] = u Jν−1(u) (cf. (8.133) in [24]). Here we assumed that some quantities vanish, namely ∞ (k) ν (5.22) fε (t) t Jν+k(t|x|) =0,k=0, 1,...,m− 1. 0 2 α To check this for ψα(t)=ϕα(t) · φα(t)withϕα(t)=(1+t ) 2 and 1 φα(t)= 1+tα ,weobservethat [ k/2] k−2j (k) t ϕα (t)=ϕα(t) cj(α) 2 k−j ,k=0, 1,... j=0 (1 + t ) and k jα (k) −k t φα (t)=φα(t) t dj(α) α j ,k=1, 2,... j=1 (1 + t ) where the constants cj(α)anddj(α) may vanish (but not all simultaneously), which may be directly proved. For k ≥ 1, we have k (k) k (r) (k−r) ψα (t)= ϕα (t) φα (t) r=0 r ⎛ ⎞ ⎛ ⎞ k −1 [ r/2] r−2j k −r jα k ⎝ cj(α) t ⎠ ⎝ −(k−r) dj(α) t ⎠ = ψα(t) 2 r−j t α j r=0 r j=0 (1 + t ) j=1 (1 + t ) ⎛ ⎞ [ k/2] k−2j ⎝ cj(α) t ⎠ + ψα(t) 2 k−j . j=0 (1 + t ) −εt Since fε(t)=e (ψα(t) − 1), then k (k) k k−j −εt (j) k −εt (5.23) fε (t)= (−ε) e ψα (t) − (−ε) e . j=0 j Let k ∈{0, 1, 2,...,m− 1}. Taking into account that Jν+k(u) behaves like uν+k for small values of u,weobtain (k) ν fε (t) t Jν+k(t|x|) −→ 0ast → 0. √1 On the other hand, Jν+k(u) behaves like u for large values of u.Since −εt ν−1/2 (j) e t goes to zero as t →∞ and |ψα (t)| behaves like a constant (0 140 Characterization of Riesz and Bessel potentials or 1, if j>0orj = 0, respectively) when t →∞,then (k) ν fε (t) t Jν+k(t|x|) −→ 0ast →∞, which completes the verification of (5.22). (m) ν To derive (5.20) from (5.21), we notice that the functions fε (t) t × Jν+m−1(t|x|), ε>0,areintegrablein(0, ∞). In fact, the integrability at the origin follows from the asymptotic behavior of the Bessel function, while its integrability at infinity follows from the definition of the Gamma function. (m) (m) It suffices to note that fε (t) −→ ψα (t)asε → 0, by (5.23), and the passage to the limit in (5.21) is easily justified, which yields (5.20). To obtain (5.7) from (5.20), we observe that the following estimates hold: c (5.24) |ψ(m)(t)|≤ as t ≥ 1 α tm and (m) α−m m−2[ m ] (5.25) |ψα (t)|≤c (t + t 2 )ast<1. So, we have 1 c | (m) | ν | | | | ν+m−1 ψα (t) t Jν+m−1(t x ) dt |x| 0 1 ≤ c α−m+ν | | | | ν+m−1 t Jν+m−1(t x ) dt |x| 0 |x| ≤ c α−m+ν | | n+α t Jν+m−1(t) dt |x| 0 ∞ ≤ c α−m+ν | | c1 n+α t Jν+m−1(t) dt = n+α |x| 0 |x| if m>1+ν + α, which guarantees the convergence of the last integral at infinity. The proof is completed. 6. Connection of the Riesz and Bessel potentials with the Sobolev spaces of variable exponent The identification of the spaces of Bessel potentials of integer smoothness with Sobolev spaces is a well-known result within the framework of the classical Lebesgue spaces. The result is due to A. Calder´on and states that m m B [Lp]=Wp ,ifm ∈ N0 and 1 The key point is the following characterization: n α Theorem 6.1. Let p(·) ∈P(R ) and let α ≥ 1.Thenf ∈B [Lp(·)],if α−1 ∂f α−1 ∈B p(·) ∈B p(·) and only if f [L ] and ∂xj [L ] for every j =1,...,n. Furthermore, there exist positive constants c1 and c2 such that n α α−1 ∂f α−1 c1 f |B [Lp(·)]≤f |B [Lp(·)] + |B [Lp(·)] (6.1) j=1 ∂xj α ≤ c2 f |B [Lp(·)]. α Proof. Suppose first that f = B ϕ for some ϕ ∈ Lp(·) .Thenforeach j =1, 2,...,n,wehave ∂f α−1 (6.2) = B [−Rj(I + K1)ϕ], ∂xj where I is the identity operator and K1 is the convolution operator whose kernel is g1 , given by (5.3) with α = 1. This identity, obvious in Fourier transforms, is known to be valid for ϕ ∈ Lp when p is constant, see [29], p. 136. Then it is also valid for variable p(·), since Lp(·) ⊂ Lp + Lp . The right-hand side inequality in (6.1) follows from (6.2) and from the mapping properties of the Bessel potential operator on spaces Lp(·) . The proof of the left-hand side inequality follows the known scheme for constant p. However, we need to refine the connection with the Riesz transforms and the derivatives, in order to overcome the difficulties associated to the convolution operators in the variable exponent setting. Wewriteherethemainstepsoftheproof for the completeness of the presentation. ∂f α−1 α−1 B p(·) B Assume that both f and ∂xj belong to [L ]. If f = ϕ, with ϕ ∈ Lp(·) , then the first order derivatives of ϕ exist in the weak ∂f α−1 ∂ϕ 1 p(·) B ∈ sense and belong to L .Moreover,∂xj = ∂xj .Sinceϕ Wp(·) there exists a sequence of infinitely differentiable and compactly supported ∂ϕk ∂ϕ functions {ϕk}k∈N such that lim ϕk = ϕ and lim = in Lp(·) , k→∞ k→∞ ∂xj ∂xj ∞ j =1, 2,...,n. This follows from the denseness of C0 in the Sobolev space 1 Wp(·) (see [23]), which holds under the assumptions on the exponent. This completes the proof. 142 Characterization of Riesz and Bessel potentials Corollary 6.2. Let p(·) be as in Theorem 6.1 and let m ∈ N0 .Then m m B [Lp(·)]=Wp(·), up to the equivalence of the norms. The theorem below provides a connection of the spaces of Riesz potentials with the Sobolev spaces. It partially extends the facts known for constant p (see, for instance, [24], p. 181) to the variable exponent setting. Theorem 6.3. Let p(·) be log-H¨older continuous both locally and at infinity, with 1 m α (6.3) Wp(·) ⊂ Lp(·) ∩I [Lp(·)] if 0 <α m m (6.4) Wp(·) = Lp(·) ∩I [Lp(·)] when 0 References [1] A. Almeida, Inversion of the Riesz potential operator on Lebesgue spaces with variable exponent, Fract. Calc. Appl. Anal., 6 (3) (2003), 311–327. [2] C. Capone, D. Cruz-Uribe and A. Fiorenza, The fractional maximal function on variable Lp spaces, Preprint, (2004); available at www.na.iac.cnr.it. [3] D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math., 28 (2003), 223–238. [4] D. Cruz-Uribe, A. Fiorenza, J. M. 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Soc. Anal. Appl. Comput., 5, 333– 342, Kluwer Acad. Publ. Dordrecht, 2000. [24] S. G. Samko, Hypersingular integrals and their applications,Ser. Analytical Methods and Special Functions, Taylor & Francis, London 2002. [25] S. G. Samko, On a progress in the theory of Lebesgue spaces with variable exponent: maximal and singular operators,Integral Transforms Spec. Funct. 16 (5-6) (2005), 461-482. [26] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York 1993. [27] I. I. Sharapudinov, On a topology of the space Lp(t)([0, 1]), Math. Notes, 26 (1979), 796–806. [28] E. Stein, The characterization of functions arising as potentials, I, Bull. Amer. Math. Soc., 67 (1) (1961), 102–104. [29] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton 1970. [30] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton 1971. Departamento de Matem´atica Universidade de Aveiro 3810-193 Aveiro Portugal (E-mail : [email protected]) Faculdade de Ciˆencias e Tecnologia Universidade do Algarve Campus de Gambelas 8005-139 Faro Portugal (E-mail : [email protected]) (Received : March 2005 ) Advances in Advances in Journal of Journal of Operations Research Decision Sciences Applied Mathematics Algebra Probability and Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 The Scientific International Journal of World Journal Differential Equations Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Mathematical Physics Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 Journal of Journal of Mathematical Problems Abstract and Discrete Dynamics in Complex Analysis Mathematics in Engineering Applied Analysis Nature and Society Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 International Journal of Journal of Mathematics and Mathematical Discrete Mathematics Sciences Journal of International Journal of Journal of Function Spaces Stochastic Analysis Optimization Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2014 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 http://www.hindawi.com http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014 http://www.hindawi.com Volume 2014