Georgian Mathematical Journal Volume 10 (2003), Number 1, 145–156

SINGULAR INTEGRALS IN WEIGHTED LEBESGUE SPACES WITH VARIABLE EXPONENT

V. KOKILASHVILI AND S. SAMKO

Abstract. In the weighted Lebesgue space with variable exponent the boundedness of the Calderón–Zygmund operator is established. The variable exponent p(x) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ρ(x) = |x − x_{0}|^{β} is related only to the value p(x_{0}). The mapping properties of Cauchy singular integrals defined on the Lyapunov curve and on curves of bounded rotation are also investigated within the framework of the above-mentioned weighted space.

2000 Mathematics Subject Classification: 42B20, 47B38.

Key words and phrases: Variable exponent, singular integral operators, Lyapunov curve, curve of bounded rotation.

1. Introduction

The generalized Lebesgue spaces L^{p(·)}(Ω) and the related Sobolev type spaces W^{m,p(x)}(Ω) with variable exponent have proved to be an appropriate tool in studying models with non-standard local growth (in elasticity theory, fluid mechanics, differential equations, see for example Ružička [18], [6] and the references therein).

These applications stimulate a quick progress of the theory of the spaces L^{p(·)}(Ω) and W^{m,p(x)}(Ω). We mention the papers by Sharapudinov [23] (1979),

[24] (1996), Kováčik and Rákosník [17] (1991), Edmunds and Rákosník [11] (1992), Samko [19]–[20] (1998), [21] (1999), Edmunds, Lang, and Nekvinda [8] (1999), Cruz-Uribe, Fiorenza, and Neugebauer [3] (2002), Diening [4]–[5] (2002), Diening and Ružička [6] (2002), Edmunds abd Nekvinda [10] (2002), Edmunds and Meskhi [9], Fiorenza [12](2002), Kokilashvili abd Samko [14]–[16] (2002), see also the references therein.

Although the spaces L^{p(·)}(Ω) possess some undesirable properties (functions from these spaces are not p(x)-mean continuous, the space L^{p(·)}(Ω) is not translation invariant, convolution operators in general do not behave well and so on), there is an evident progress in their study, stimulated by applications, first of all for continuous exponents p(x) satisfying the logarithmic Dini condition. We mention in particular the result on the denseness of C_{0}^{∞}-functions in the Sobolev space W^{m,p(x)}(Ω), see [21], and the breakthrough connected with the study of maximal operators in [4], [5].

Because of applications, the reconsideration of the main theorems of harmonic analysis is topical in order to find out what theorems remain valid for variable exponents or to find their substituting analogs. Among the challenging

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

problems there were: a Sobolev type theorem for the Riesz potential operator I^{α} and the boundedness of singular integral operators. A Sobolev type theorem for bounded domains is proved in [19] under the condition that the maximal operator is bounded in the spaces L^{p(·)}. This condition is fulfilled due to the result on maximal operators obtained in [4]–[5] (we refer also to [3] for maximal operators on unbounded domains).

Singular operators were treated in [6] and [16] within the framework of the spaces with variable exponents.

The main goal of the present paper is to establish the boundedness of Calderón–Zygmund singular operators in weighted spaces L_{ρ}^{p(·)}. In particular, we obtain a weighted mapping theorem for finite Hilbert transform and apply this result to the boundedness of Cauchy singular operators on curves in the complex plane.

2. Preliminaries

Let Ω be a bounded open subset of ℝ^{n} and p(x) a measurable function on Ω such that

1 < \underline{p} ≤ p(x) ≤ \overline{p} < ∞, x ∈ Ω  (2.1)

and

|p(x) − p(y)| ≤ \frac{A}{ln \frac{1}{|x−y|}} , |x − y| ≤ \frac{1}{2}, x, y ∈ Ω.  (2.2)

We denote by P = P(Ω) the set of functions p(x) satisfying conditions (2.1)– (2.2). We refer to Appendix A for examples of non-Hölderian functions satisfying condition (2.2). By L^{p(·)} we denote the space of functions f(x) on Ω such that

A_{p}(f) = ∫_{Ω} |f(x)|^{p(x)}dx < ∞.

This is a Banach function space with respect to the norm

∥f∥_{L^{p(·)}} = inf \{λ > 0 : A_{p} (\frac{f}{λ}) ≤ 1\}  (2.3)

(see, e.g., [5]). We denote

\frac{1}{q(x)} = 1 − \frac{1}{p(x)}.

Under condition (2.1) the space L^{p(·)} coincides with the space

{f(x) : |∫_{Ω} f(x)φ(x) dx| < ∞ for all φ(x) ∈ L^{q(·)}(Ω)}  (2.4)

up to the equivalence of the norms

∥f∥_{L^{p(·)}} ∼ sup_{∥φ∥_{L^{q(·)}}≤1} |∫_{Ω} f(x)φ(x) dx| ∼ sup_{A_{q}(φ)≤1} |∫_{Ω} f(x)φ(x) dx| ,  (2.5)

see [17], Theorem 2.3 or [20], Theorem 3.5.

Let ρ be a measurable almost everywhere positive integrable function. Such functions are usually called weights. The weighted Lebesgue space L_{ρ}^{p(·)} is defined as the set of all measurable functions for which

∥f∥_{L_{ρ}^{p(·)}} = ∥ρf∥_{L^{p(·)}} < ∞.

The space L_{ρ}^{p(·)} is a Banach space.

We deal with the following integral operators: the Calderón–Zygmund singular operator

T f(x) = ∫_{Ω} K(x, y)f(y) dy,  (2.6)

(as treated in [6]), the maximal operator

Mf(x) = sup_{r>0} \frac{1}{|B(x, r)|} ∫_{B(x,r)} |f(y)| dy,  (2.7)

the Riesz potential operator

I_{α}f(x) = ∫_{Ω} \frac{f(y)}{|x − y|^{n−α}} dy, α > 0,

and the Cauchy singular operator

S_{Γ}f(t) = ∫_{Γ} \frac{f(τ)dτ}{τ − t} , t = t(s), 0 ≤ s ≤ ℓ,  (2.8)

along a finite rectifiable Jordan curve Γ of the complex plane on which the arc-length is chosen as a parameter starting from any fixed point.

In the definition of the maximal function we assume that f(x) = 0 when x ∉ Ω.

In [5] the boundedness of the maximal operator in the space L^{p(·)} was proved. Later in [6] an analogous result for Calderón–Zygmund operator (2.6) was obtained.

The boundedness of the maximal operator M in the weighted Lebesgue space L_{ρ}^{p(·)} with the power weight ρ(x) = |x − x_{0}|^{β} was established by the authors in [15], see also [14]. The main point of the result in [14], [15] is that the exponent β is related to the value of p(x) at the point x_{0}. Recently, we have also established the boundedness of various integral operators, in particular Calderón–Zygmund operators, in weighted Lorentz type spaces with variable exponent [15], see also [16]. However the result of [15], [16] does not imply the boundedness of singular operators in the Lebesgue spaces with variable exponent.

3. Statements of the Main Results

Let

T^{∗}f(x) = sup_{ε>0} |T_{ε}f(x)|

be a maximal singular operator, where T_{ε}f(x) is the usual truncation

T_{ε}f(x) = ∫_{|x−y|≥ε} K(x, y)f(y) dy

and we assume that f(x) = 0 outside Ω. In what follows,

ρ(x) = ∏_{k=1}^{m} |x − a_{k}|^{β_{k}},  (3.1)

where a_{k} ∈ \overline{Ω}, k = 1, ⋯, m.

Theorem 1. Let p(x) ∈ P(Ω) and ρ(x) be weight function (3.1). Then the operators T and T^{∗} are bounded in the space L_{ρ}^{p(·)}(Ω) if

−\frac{n}{p(a_{k})} < β_{k} < \frac{n}{q(a_{k})},  k = 1, …, m.  (3.2)

Besides operator (2.8), we also consider the corresponding maximal singular operator

S^{∗}f(t) = sup_{ε>0} | ∫_{|s−σ|>ε} \frac{f[τ(σ)]τ′(σ)}{τ(σ) − τ(s)} dσ|  (3.3)

where it is supposed that f[t(σ)] = 0 when s ∉ [0, ℓ].

We remind that Γ is called the Lyapunov curve if t′(s) ∈ Lip γ, 0 < γ ≤ 1 and that in this case

\frac{t′(s)}{t(σ) − t(s)} = \frac{1}{σ − s} + h(s, σ), with |h(s, σ)| ≤ \frac{c}{|σ − s|^{1−γ}}  (3.4)

see [13]; observe that

h(s, σ) = \frac{1}{t(σ) − t(s)} ∫_{0}^{1}[t′(s) − t′(s + ξ(σ − s))] dξ  (3.5)

from which the estimate for h(s, σ) follows.

If t′(s) is a function of bounded variation, Γ is called a curve of bounded rotation. When Γ is a curve of bounded rotation without cusps, Γ satisfies the chord-arc condition

|\frac{t(s) − t(σ)}{s − σ}| ≥ m > 0.  (3.6)

Thus we have |t(s) − t(s_{0})| ≈ |s − s_{0}|.

When dealing with the operators S and S^{∗}, we assume the functions p(s) and ρ(s) ≥ 0 to be defined on [0, ℓ] and put

L_{ρ}^{p(·)} = {f : ∥f[t(s)]ρ(s)∥_{L^{p(s)}} < ∞} .

In the next theorem we take

ρ(s) = ∏_{k=1}^{m} |t(s) − t(c_{k})|^{β_{k}} ≈ ∏_{k=1}^{m} |s − c_{k}|^{β_{k}},  (3.7)

where c_{k} ∈ [0, ℓ], k = 1, 2, …, m.

Theorem 2. Let Γ be a Lyapunov curve or a curve of bounded rotation without cusps and let p(s) ∈ P. The operators S_{Γ} and S^{∗} are bounded in the space L_{ρ}^{p(·)}(Γ) with the weight function (3.7) if and only if

−\frac{1}{p(c_{k})} < β_{k} < \frac{1}{q(c_{k})},  k = 1, 2, …, m.  (3.8)

4. Auxiliary Results

In this section we present some basic results which we need to prove our main statements. Let

M^{β}f(x) = sup_{r>0} \frac{|x − x_{0}|^{β}}{|B(x, r)|} ∫_{B(x,r)} \frac{|f(y)| dy}{|y − x_{0}|^{β}} ,  (4.1)

where x_{0} ∈ \overline{Ω}.

Theorem A ([15]). Let p(x) ∈ P. The operator M^{β} with x_{0} ∈ Ω is bounded in the space L^{p(·)}(Ω) if and only if

−\frac{n}{p(x_{0})} < β < \frac{n}{q(x_{0})}.  (4.2)

When x_{0} ∈ ∂Ω, condition (4.2) is sufficient in the case of any point x_{0} and necessary if the point x_{0} satisfies the condition |Ω_{r}(x_{0})| ∼ r^{n}, where Ω_{r}(x_{0}) = {y ∈ Ω : r < |y − x_{0}| < 2r}.

Theorem B ([15]). Let p(x) ∈ P. The Riesz potential operator I_{α} acts boundedly from the space L_{ρ}^{p(·)}(Ω) with the weight ρ(x) = |x − x_{0}|^{β}, x_{0} ∈ Ω, into itself if condition (4.2) is satisfied.

Let F ∈ L_{loc}(ℝ^{1}) and

F^{#}(x) = sup_{r>0} \frac{1}{|B(x, r)|} ∫_{B(x,r)} |F (y) − F_{B(x,r)}| dy,  (4.3)

where

F_{B(x,r)} = \frac{1}{|B(x, r)|} ∫_{B(x,r)} f(z) dz.

Proposition A ([1]). Let T be a Calderón–Zygmund operator. Then for arbitrary s, 0 < s < 1, there exists a constant c_{s} > 0 such that

[(|T f|^{s})^{#}(x)]^{\frac{1}{s}} ≤ c_{s}Mf(x)

for all f ∈ C_{0}^{∞}(ℝ^{n}), x ∈ ℝ^{n}.

The following statement holds (see [4], Lemma 3.5).

Proposition B. Let p(x) ∈ P . Then for all f ∈ L^{p(·)}(Ω) and g ∈ L^{q(·)}(Ω) there holds

|∫_{Ω} f(x)g(x) dx| ≤ c ∫_{Ω} f^{#}(x)Mg(x) dx

with a constant c > 0 not depending on f.

150 V. KOKILASHVILI AND S. SAMKO

Lemma 4.1. Let p(x) ∈ P, w(x) = |x − x_{0}|^{γ}, x_{0} ∈ \overline{Ω}, −\frac{n}{p(x_{0})} < γ < \frac{n}{q(x_{0})}. Then

∥fw∥_{L^{p(·)}} ≤ c∥f^{#}w∥_{L^{p(·)}}

with a constant c > 0 not depending on f. Proof. By (2.5) we have

∥fw∥_{L^{p(·)}} ≤ c sup_{∥g∥_{L^{q(·)}}≤1} |∫_{Ω} f(x)g(x)w(x) dx| .

According to Proposition B,

∥fw∥_{L^{p(·)}} ≤ c sup_{∥g∥_{L^{q(·)}}≤1} |∫_{Ω} f^{#}(x)w(x)[w(x)]^{−1}M(gw) dx| .

Making use of the Hölder inequality for L^{p(·)}, we derive

∥fw∥_{L^{p(·)}} ≤ c sup_{∥g∥_{L^{q(·)}}≤1} ∥f^{#}w∥_{L^{p(·)}}∥w^{−1}M(gw)∥_{L^{q(·)}}.

We observe that −\frac{1}{q(x_{0})} < −γ < \frac{1}{p(x_{0})}. Therefore we may apply Theorem A for the space L^{q(·)} with β = γ and conclude that

∥fw∥_{L^{p(·)}} ≤ c sup_{∥g∥_{L^{q(·)}}≤1} ∥f^{#}w∥_{L^{p(·)}}∥g∥_{L^{q(·)}} ≤ ∥f^{#}w∥_{L^{p(·)}}.

□

Lemma 4.1 in the case of the constant p(·) and w ≡ 1 is well known [25]. For variable exponent and w ≡ 1 it was proved in [6].

Theorem 4.1. Let p(x) be a measurable function on ℝ^{n} such that 1 ≤ \underline{p} ≤ p(x) < \overline{p} < ∞, ρ(x) ≥ 0 and |\{x ∈ ℝ^{n} : ρ(x) = 0\}| = 0 and

w(x) = [ρ(x)]^{p(x)} ∈ L_{loc}^{1}(ℝ^{n}).  (4.4)

Then C_{0}^{∞}(ℝ^{n}) is dense in the space L_{ρ}^{p(·)}(ℝ^{n}). Proof. I. First we prove that the class C_{0}(ℝ^{n}) of continuous functions with a compact support is dense in the space L_{ρ}^{p(·)}(ℝ^{n}).

Let f ∈ L_{ρ}^{p(·)}(ℝ^{n}). Since |{x ∈ ℝ^{n} : ρ(x) = 0}| = 0, the function f(x) is a.e. finite.

1st step. The functions f_{N}(x) = \{ \begin{matrix}f(x), & |x| < N \\ 0, & |x| > N \end{matrix} approximate f in L_{ρ}^{p(·)} since A_{p}(ρ|f − f_{N}|) → 0 as N → ∞. Therefore there exists a function g ∈ L_{ρ}^{p(·)} with a compact support such that ∥f − g∥_{L_{ρ}^{p(·)}} < ε.

2nd step. The function g can be approximated in L_{ρ}^{p(·)} by the bounded functions with a compact support g̃_{N}(x) = \{ \begin{matrix}g(x), & |g(x)| < N \\ 0, & |g(x)| > N \end{matrix} . Indeed, the passage to the limit

A_{p}(ρ|g − g̃_{N}|) = ∫_{ℝ^{n}} w(x)|g(x) − g̃_{N}(x)|^{p(x)} dx → 0 as n → ∞ (4.5)

is justified by the Lebesgue dominated convergence theorem since w(x)|g(x) − g_{N}(x)|^{p(x)} ≤ 2w(x)|g(x)|^{p(x)} and the integrand tends to zero a.e. (at any point x at which f(x) is finite). So we choose g̃_{N} such that ∥g − g̃_{N}∥_{L_{ρ}^{p(·)}} < ε.

3rd step. To approximate the function g̃_{N} by continuous bounded functions, we choose δ > 0 so that

∫_{E} w(x) dx < ε_{1} = \frac{ε}{(2N)\overline{p}}  (4.6)

for any measurable set E ⊂ supp g with |E| < δ, which is possible by (4.4). Then we choose a function φ(x) ∈ C(ℝ^{n}) with ∥φ∥_{C} ≤ N which coincides with g̃_{N}(x) everywhere except possibly for the set A ⊂ supp g with |A| < δ, by the Luzin theorem. Then

A_{p}(ρ|g̃_{N} − φ|) = ∫_{A} w(x)|g̃_{N}(x) − φ(x)|^{p(x)} dx ≤ (2N)\overline{p} ∫_{A} w(x) dx < ε .

4th step. It remains to approximate in L_{ρ}^{p(·)} the function φ by a continuous function with a compact support, which is done in the standard way by means of smooth truncation.

II. Approximation in L_{ρ}^{p(·)} of a continuous function with a compact support by C_{0}^{∞}-functions can already be realized via the identity approximation

K_{t}φ = \frac{1}{t^{n}} ∫_{ℝ^{n}} a (\frac{y}{t} ) φ(x − y) dy, t > 0,

with a(x) ∈ C_{0}^{∞} and ∫_{ℝ^{n}} a(x) dx = 1. Obviously, K_{t}φ ∈ C_{0}^{∞} for φ ∈ C_{0}(ℝ^{n}) and |K_{t}φ − φ| < ε as t → 0 uniformly on any given compact set. Therefore,

A_{p}(ρ|K_{t}φ − φ|) = ∫_{B} w(x)|K_{t}φ − φ(x)|^{p(x)} dx ≤ ε\underline{p} ∫_{B} w(x) dx

for t small enough, B being a sufficiently large ball. □

Corollary. Let p(x) and ρ(x) satisfy the assumptions of Theorem 4.1 in Ω. Then the set C^{∞}(\overline{Ω}) is dense in the space L_{ρ}^{p(·)}(Ω).

Indeed, it suffices to observe that functions from C^{∞}(\overline{Ω}) can be continued outside Ω as C_{0}^{∞}(ℝ^{n})-functions; likewise, p(x) and ρ(x) can be continued with preservation of their properties.

The statement of Theorem 4.1 is well known in the case of constant exponent, while in the case of variable exponent and ρ(x) ≡ 1 it was proved in [17]. The denseness of C_{0}^{∞}(ℝ^{n}) in non-weighted Sobolev spaces W^{m,p(·)} was proved in [21]–[22].

5. Proofs of the Main Results

It suffices to deal with a weight of the form ρ(x) = |x − x_{0}|^{β} because the general case (3.1) is easily reduced to this special case by separation of the points a_{k} by means of the partition of unity, which provides the representation

\frac{∏_{k=1}^{m} |x − a_{k}|^{β_{k}}}{∏_{k=1}^{m} |y − a_{k}|^{β_{k}}} = ∑_{k=1}^{m} c_{k}(x, y)\frac{|x − a_{k}|^{β_{k}}}{|y − a_{k}|^{β_{k}}}

with bounded “coefficients” c_{k}(x, y).

Proof of Theorem 1. Let us consider the operator T. Let f ∈ C^{∞}(\overline{Ω}) and 0 < s < 1. Obviously,

∥ρT f∥_{L^{p(·)}} = ∥ρ^{s}|T f|^{s}∥L^{\frac{p(·)}{s}}^{\frac{1}{s}} .  (5.1)

Applying Lemma 4.1 with w(x) = [ρ(x)]^{s} and p(·) replaced by \frac{p(·)}{s}, we obtain

∥ρT f∥_{L^{p(·)}} ≤ c ∥ρ^{s}(|T f|^{s})^{#}∥L^{\frac{p(·)}{s}}^{\frac{1}{s}} ,

which is possible since the condition

− \frac{n}{\frac{p(x_{0})}{s}} < sβ < \frac{n}{(\frac{p(x_{0})}{s})′}

is satisfied. Thus we have

∥ρT f∥_{L^{p(·)}} ≤ c ∥ρ [(|T f|^{s})^{#}]^{\frac{1}{s}} ∥_{L^{p(·)}}  (5.2)

because ∥f∥L^{\frac{p(·)}{s}}^{\frac{1}{s}} = ∥|f|^{\frac{1}{s}} ∥_{L^{p(·)}}. Since a function f ∈ C^{∞}(\overline{Ω}) can be continued

outside Ω as a C_{0}^{∞}(ℝ^{n})-function, Proposition A is applicable. Therefore, by Proposition A, from estimate (5.2) we get

∥ρT f∥_{L^{p(·)}} ≤ c ∥ρ(Mf)∥_{L^{p(·)}} .

Now we apply Theorem A and conclude that

∥ρT f∥_{L^{p(·)}} ≤ c ∥ρf∥_{L^{p(·)}}

for all f ∈ C_{0}^{∞}(ℝ^{n}). Since C_{0}^{∞}(ℝ^{n}) is dense in L_{ρ}^{p(·)} by Theorem 4.1, we complete the proof of Theorem 1. □

The boundedness of the operator T^{∗} follows from the known estimate

T^{∗}f(x) ≤ c[M(T f)(x) + Mf(x)],

from Theorem A and Theorem 1.

Corollary 1. Let Ω = [a, b], ρ(x) = ∏_{k=1}^{m} |x−a_{k}|^{β_{k}}, a_{k} ∈ [a, b], k = 1, …, m, and p(x) ∈ P. Then the finite Hilbert transform and its maximal version

H_{[a,b]}f = ∫_{a}^{b} \frac{f(y) dy}{y − x} and H_{[a,b]}^{∗} = sup_{ε>0} | ∫_{|y−x|>ε} \frac{f(y) dy}{y − x} |  (5.3)

are bounded in the space L_{ρ}^{p(·)}(a, b), if −\frac{1}{p(a_{k})} < β_{k} < \frac{1}{q(a_{k})}, k = 1, 2, …, m.

Proof of Theorem 2. We assume the function p(s) to be defined on [0, l]. The function f(t(σ)) will be denoted by f_{0}(σ). In the case of a Lyapunov curve we use equality (3.4) and apply Corollary 1 and Theorem B, which immediately gives the statement of Theorem 2.

Let Γ be a curve of bounded rotation without cusps and let V be the total variation of t′(s) on [0, l]. In this case the function h(s, σ) can be estimated as

|h(s, σ)| ≤ c\frac{V (s) − V (σ)}{s − σ}  (5.4)

according to (3.5) and (3.6) (see [13], Chapter II, Subsection 2.3). Then we have

|∫_{|s−σ|>ε} \frac{f_{0}(σ) dσ}{t(σ) − t(s)}|

≤ c |∫_{|s−σ|>ε} \frac{f_{0}(σ) dσ}{σ − s} | + c ∫_{|s−σ|>ε} \frac{|f_{0}(σ)| (V (σ) − V (s))}{σ − s}dσ

≤ c |∫_{|s−σ|>ε} \frac{f_{0}(σ) dσ}{σ − s} | + cV (s) |∫_{|s−σ|>ε} \frac{|f_{0}(σ)| dσ}{σ − s} |

+ c |∫_{|s−σ|>ε} \frac{|f_{0}(σ)| V (σ)}{σ − s} dσ| .

From here by Corollary 1 and the boundedness of the function V (s) we conclude that the operator S_{Γ}^{∗} is bounded in L_{ρ}^{p(·)}.

Let us prove the necessity part. From the boundedness of S_{Γ} in L_{ρ}^{p(s)} it follows that S_{Γ}f(t) exists almost everywhere for arbitrary f ∈ L_{ρ}^{p(s)}. Thus ρ should be such that f ∈ L^{1}(Γ) for arbitrary f ∈ L_{ρ}^{p(s)}. The function f = fρρ^{−1} belongs to L^{1}(Γ) for arbitrary f ∈ L_{ρ}^{p(s)} if and only if ρ^{−1} ∈ L^{q(s)}, which follows from equivalence (2.5). Then function ρ^{−1}(s) = |s − s_{0}|^{−β}, s_{0} ∈ [0, l], belongs to L^{q(s)}[0, l] if and only if β < \frac{1}{q(s_{0})}. Indeed, we have

|s − s_{0}|^{−βq(s)} = m(s)|s − s_{0}|^{−βq(s_{0})},

where the function m(s) = |s − s_{0}|^{−β(q(s)−q(s_{0}))} satisfies the condition

0 < c ≤ m(s) ≤ C < ∞

in view of (2.2). On the other hand, from |s−s_{0}|^{−βq(s_{0})} ∈ L^{q(s)} we have β < \frac{1}{q(s_{0})}.

The necessity of the condition −\frac{1}{p(s_{0})} < β follows from the duality argument.

□

6. Appendix

The following is an example of the function which satisfies condition (2.2) but is not a Hölder function:

p(x) = a(x) + \frac{b(x)}{(ln \frac{A}{|x|})^{γ}} , x ∈ Ω,  (6.1)

where a(x) and b(x) are Hölder functions, a(x) ≥ 1, b(x) ≥ 0, A > sup_{x∈Ω} |x| and

γ ≥ 1. One may write a little bit more complicated example:

p(x) = a(x) + \frac{b(x)}{(ln \frac{A}{|x|})^{γ}} (ln ln ln ⋯ ln \frac{C}{|x|})^{μ}  (6.2)

with arbitrary sufficiently large C > 1 and μ > 0, and the same assumptions on a(x), b(x), and A, but γ > 1. It is also possible to take different powers of different logarithms as factors or superpositions in (6.2).

To prove condition (2.2) for functions (6.1) or (6.2) or functions of a similar type, we do not need to check condition (2.2) directly. For this purpose we can use properties of continuity moduli. It suffices to deal with the case where a(x) ≡ 0 and b(x) = 1 since we consider differences p(x) − p(y).

We remind that a non-negative function f(t) on [0, ℓ] is called a continuity modulus if

ω(f, h) ∼ f(h),

where ω(f, h) = sup_{{|t_{1}−t_{2}|≤h \atop t_{1},t_{2}∈[0,ℓ]}} |f(t_{1}) − f(t_{2})|. The sufficient conditions for a function

f(x) to be a continuity modulus are known, see, for example [2] or [7]: 1) f(x) is continuous on [0, ℓ],

2) f(0) = 0 and f(x) > 0 for x > 0,

3) f(x) is non-decreasing and \frac{f(x)}{x} is non-increasing in a neighborhood of the point x = 0. It is easy to check that functions (6.1)–(6.2) satisfy the above conditions 1)-3).

Acknowledgement

This work was supported by Centro de Matemática Aplicada of Instituto Superior Técnico in Lisbon during the first author’s visit to IST in February-March of 2002.

References

1. T. Alvarez and C. Pérez, Estimates with A_{∞} weights for various singular integral operators. Boll. Un. Mat. Ital., A (7) 8 (1994), No. 1, 123–133.

2. N. K. Bari and S. B. Stechkin, Best approximations and differential properties of two conjugate functions. (Russian) Trudy Moskov. Mat. Obshch. 5(1956), 483–522.

3. D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, The maximal function on variable L^{p} spaces. Preprint. Intituto per le Applicaziopni del Calculo “Mauro Picone”, Sezione di Napoli, 249, 2002.

4. L. Deining, Maximal functions on generalized Lebesgue spaces L^{p(x)}. Preprint. Mathematische Fakultät, Albert-Ludwigs-Universitẗ Freiburg, No. 2, 2002.

5. L. Diening, Maximal functions on generalized Lebesgue spaces L^{p(x)}. Math. Inequal. Appl. (to appear).

6. L. Diening and M. Ružička, Calderón–Zygmund operators on generalized Lebesgue spaces L^{p(x)} and problems related to fluid dynamics. Preprint Mathematische Fakultät, Albert-Ludwigs-Universitẗ Freiburg, 21/2002, 04.07.2002, 1-20, 2002.

7. V. K. Dzjadyk, Introduction to the theory of uniform approximation of functions by polynomials. (Russian) Nauka, Moscow, 1977.

8. D. E. Edmunds, J. Lang, and A. Nekvinda, On L^{p(x)} norms. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455(1999), No. 1981, 219–225.

9. D. E. Edmunds and A. Meskhi, Potential-type operators in L^{p(x)} spaces. Z. Anal. Anwendungen 21(2002), No. 3, 681–690.

10. D. E. Edmunds and A. Nekvinda, Averaging operators on l^{{p_{n}}} and L^{p(x)}. Math. Inequal. Appl. 5(2002), No. 2, 235–246.

11. D. E. Edmunds and J. Rákosník, Density of smooth functions in W^{k,p(x)}(Ω). Proc. Roy. Soc. London Ser. A 437(1992), No. 1899, 229–236.

12. A. Fiorenza, A mean continuity type result for certain Sobolev spaces with variable exponent. Commun. Contemp. Math. 4(2002), No. 3, 587–605.

13. B. V. Khvedelidze, The method of Cauchy type integrals for discontinuous boundary value problems of the theory of holomorphic functions of one variable. (Russian) Itogi Nauki. i Tekhniki, Sovrem. Probl. Mat. 7(1977), 5–162.

14. V. Kokilashvili and S. Samko Maximal and fractional operators in weighted L^{p(x)}- spaces. Proc. A. Razmadze Math. Inst. 129(2002), 145–149.

15. V. Kokilashvili and S. Samko, Maximal and fractional operators in weighted L^{p(x)} spaces. Rev. Mat. Iberoamericana (accepted for publication).

16. V. Kokilashvili and S. Samko, Singular integrals and potentials in some Banach spaces with variable exponent. J. Funct. Spaces Appl. (accepted for publication).

17. O. Kováčik and J. Rákosník,. On spaces L^{p(x)} and W^{k,p(x)}. Czechoslovak Math. J. 41(116)(1991), No. 4, 592–618.

18. M. Ružička, Electrorheological fluids: modeling and mathematical theory. Lecture Notes in Mathematics, 1748. Springer, Berlin, 2000.

19. S. G. Samko, Convolution and potential type operators in L^{p(x)}(R^{n}). Integral Transform. Spec. Funct. 7(1998), No. 3-4, 261–284.

20. S. G. Samko, Differentiation and integration of variable order and the spaces L^{p(x)}. Operator theory for complex and hypercomplex analysis (Mexico City, 1994), 203–219, Contemp. Math., 212, Amer. Math. Soc., Providence, RI, 1998.

21. S. G. Samko, Density C_{0}^{∞}(R^{n}) in the generalized Sobolev spaces W^{m,p(x)}(R^{n}). (Russian) Dokl. Ross. Akad. Nauk 369(1999), No. 4, 451–454.

22. S. Samko, Denseness of C_{0}^{∞}(R^{N}) in the generalized Sobolev spaces W^{M,P (X)}(R^{N}). Direct and inverse problems of mathematical physics (Newark, DE, 1997), 333–342, Int. Soc. Anal. Appl. Comput., 5, Kluwer Acad. Publ., Dordrecht, 2000.

23. I. I. Sharapudinov, The topology of the space L^{p(t)}([0, 1]). (Russian) Mat. Zametki 26(1979), No. 4, 613–632.

24. I. I. Sharapudinov, On the uniform boundedness in L^{p}(p = p(x)) of some families of convolution operators. (Russian) Mat. Zametki 59(1996), No. 2, 291–302; translation in Math. Notes 59(1996), No. 1-2, 205–212

25. E. M. Stein, Harmonic Analysis: real-variable methods, orthogonality and oscillatory integrals. Princeton Univ. Press, Princeton, NJ, 1993.

(Received 25.11.2002)

Authors’ addresses:

V. Kokilashvili A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: kokil@rmi.acnet.ge

S. Samko University of Algarve Unidade de Ciencias Exactas e Humanas Campus de Cambelas Faro, 800 Portugal E-mail: ssamko@ualg.pt