Singular Integrals and Potentials in Some Banach Function Spaces with Variable Exponent

We introduce a new Banach function space a Lorentz type space with variable exponent. In this space the boundedness of singular integral and potential type operators is established, including the weighted case. The variable exponent p(t) is assumed to satisfy the logarithmic Dini condition and the exponent β of the power weight ω(t) = |t|β is related only to the value p(0). The mapping properties of Cauchy singular integrals defined on Lyapunov curves and on curves of bounded rotation are also investigated within the framework of the introduced spaces. AMS Classification 2000: 42B20, 47B38

The theory of the spaces L p(•) (Ω) nowadays is quickly developed.After the first disappointment caused by 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) a rapid progress followed for continuous exponents p(x) satisfying the logarithmic Dini condition.We mention in particular the result on denseness of C ∞ 0 -functions in the Sobolev space W m,p(x) (R n ), see [25], and the breakthrough connected with the study of maximal operators, see [5], [6].
Because of applications, a reconsideration of the main theorems of harmonic analysis is actual, with the aim to find new proofs of those theorems which remain valid for variable exponents, or to find their substituting analogs.Among the challenging problems there were: the Sobolev type theorem on boundedness of the Riesz potential operator − α n and the boundedness in L p(•) of singular integral operators.Boundedness of I α (Sobolev type theorem) for bounded domains was proved in [23] conditionally, under the assumption that the maximal operator is bounded in the spaces L p (•) , which turns to be unconditional after the result of [5] - [6] on maximal operators (we refer also to [3] for maximal operators on unbounded domains).
Singular operators within the framework of the spaces with variable exponents were treated in [18], [17] and [7].
We introduce a new form of spaces with variable exponents for which the problem of boundedness of singular type integral operators may be resolved positively in a natural way, including the case of weighted spaces with variable exponents.We consider the Calderon -Zygmund operators, singular operators with the Cauchy kernel along Lyapunov curves or curves of bounded variation in the complex plane, the Riesz potential operator and the Poisson integral and its conjugates.The main statements are given in Theorems 3.1-3.5,4.1-4.4.

On Some Banach Function Spaces
Let (Ω, µ) be a measure space and M (Ω, µ) a space of measurable functions on Ω.
Every Banach function space is a Banach space.For definition and fundamental properties of Banach function space we refer to [2].
We shall deal with some special Banach function space.
Let Ω be a bounded open subset of R n and p(x) a measurable function on Ω such that 1 This is a Banach function space with respect to the norm (see e.g.[9]).We denote .
The following integral transforms will be treated: a) the potential operator c) the Hardy-type operator where 0 < < ∞.
In [16] (see also [15], [14] ) the following theorems were proved: Theorem III .Under assumptions (1.1), (1.2) and the condition inf Observe that Theorem IV provides norm estimates for Hardy operators in spaces with variable exponent.In [28] there was proved a natural fact that the modular inequality for the Hardy operator (and more generally for some integral operators) is impossible in the case of variable exponents, see [28], Theorem 2.2.
On the base of L p(

Definition 2
The subset of all functions of M (Ω, m) for which According to Theorem IV we conclude that there exists a constant c > 0 such that (1.9) Note that f * * L p(•) is a norm.The triangle inequality follows from the inequality (f + g) * * (t) ≤ f * * (t) + g * * (t).
Proof.Most of requirements of Definition 1 follow directly from properties of non-increasing rearrangements of functions and properties of the space L p(•) .For example, iv) is valid since for 0 ≤ f n f we have f * n f * (see e.g.[29], Lemma 3.5, Chapter 5).Then by the property of L p (•) .
Applying the Hölder inequality for L p(•) , we get Let w(t) be a nonnegative function defined on [0, mΩ] such that

Definition 3
The subset of all functions in M (Ω, m) for which

The space Λ p(•)
w is a Banach space.The proof is similar as above.In the sequel for w(t) = t β we put

Integral Transforms in R n
We begin with the mapping properties of singular operators ) , where k is an odd function on R n homogeneous of degree 0 and satisfying the Dini condition on the unit sphere As particular cases one may mention the Hilbert transform (n = 1, k(x) = x |x| ) and the Riesz transforms Proof.As it is known (see [1]) we have the following corollary.In the sequel we discuss the boundedness in Λ p(•) of Riesz potentials and give an application to imbedding of certain spaces of differentiable functions.
The next theorem deals with the Riesz potential operator

Theorem 3.3 Let p(t) satisfy the assumptions of Theorem 2.2 and s(x) be a measurable function on [0, mΩ] such that 1 ≤ s(x) < S < ∞ for all
x ∈ [0, mΩ] and then the inequality holds with the constant c not depending on f .
Proof.We make use of the estimate for the decreasing rearrangements of I α (see [21]): Applying Theorem IV, we obtain Similarly, according to the same theorem we have For a multi-index of nonnegative integers for all real-valued functions u on Ω where the continuation by zero beyond Ω has weak derivatives up to the order k over R n .(ii) If Ω is convex, then a positive constant c exists such that for all real valued functions u on Ω which have weak derivatives up to the order k in Ω.
In the case k = 1, inequality (2.5) holds with Q equal to the mean value of u over Ω, Q = 1 mΩ Ω u(x)dx.
Proof.The part i).It is clear that D K u ∈ L 1 (Ω).Then by Theorem 1.1.10/2 of [20] we have the estimate Applying Theorem 2.1, we arrive at the desired result.The proof of part ii) is similar, since by Theorem 1.1.10/1 of [9], there exists a constant c depending only on n, k and Ω, and a polynomial Now we pass to the mapping properties of Poisson integral and conjugate Poisson integrals in Λ p(•) β spaces.We consider the Poisson integral (see [29], Chapters 6 and 2).From (2.7), by the known estimate (see [1]) we have (sup By means of Theorem IV we derive the following result.By inequality (2.9) we have Remark 1 Applying the results of [5] and using the idea which was developed above, we can deduce that the theorems of this section are also valid in R n if a function p(t) is assumed to satisfy the local logarithmic Dini condition and in addition is constant outside some large interval (0, t 0 ), i.e. p(t) = p, t > t 0 .For the power weight we assume that max − 1 p , − 1 p(0) < β < min 1 q , 1 q(0) , where q = p p−1 .

Cauchy Singular Integrals on Lyapunov Curves and Curves of Bounded Rotation
In this section we deal with the Cauchy singular integral where Γ is a finite rectifiable Jordan curve on which the arc-length is chosen as a parameter, starting from any fixed point.Γ is called Lyapunov curve if t (s) ∈ Lip α, 0 < α ≤ 1.When t (s) is a function of bounded variation, Γ is called a curve of bounded rotation.
Our goal is to study the mapping properties of S Γ when Γ is a Lyapunov curve or a curve of bounded rotation without cusps.
We assume the function p(s) to be defined on [0, l].The function f (t(s)) will be denoted by f 0 (s).
is satisfied in a neighbourhood of the origin.
Theorem 4.2 Let Γ be a curve of bounded rotation without cusps.Let p(s) satisfy the condition of Theorem 2.1 with m denoting the arc-length measure on Γ.Then the operator S Γ is bounded in Λ p(s) .
Proof.We have As t (s) is a function of bounded variation, we have Since Γ is a curve of bounded rotation without cusps, it satisfies the arc-chord condition, i.e.
Therefore, we can derive the estimate Note that for the p(s) a constant function p(s) = p the boundedness of S Γ on Lyapunov curve and on curve of bounded rotation without cusps was proved in [13] and [4], respectively.Then the Cauchy singular operator S Γ is bounded in Λ p w .
Proof.As mentioned in the proof of Theorem 3.2, Γ satisfies the arc-chord condition.Thus w(s) ∼ s β .
Therefore, we may follow the scheme of the proof of Theorems 3.1 and 3.2 and apply Theorems 1.2 and 2.1 to obtain the boundedness of S Γ in Λ p w .
Basing on the recent results on the singular integrals from [7] and on the proofs of Theorems 3.1 and 3.2 we conclude the validity of the following theorem.

Theorem 3 . 4
Let n ≥ 2 and let k = |K| be any positive integer smaller than n.Suppose that p(x) and s(x) satisfy the assumptions of Theorem 2.1.Then i) there exists a positive constant c such that

Theorem 3 . 5
Let p(t) and β satisfy the conditions of Theorem I. Then T is bounded in Λ p(•) β .Now consider the operator T j f (x) = sup y |v j f (x, y)|.

. 10 )Theorem 3 . 6
Basing on(2.10)  and Corollary 1 we obtain the following result.Let a function p(t) and a number β satisfy the assumptions of Theorem 2.1.Then the operators T j are bounded in Λ p(•) β .

Theorem 4 . 4
Let Γ be a Lyapunov curve or a curve of bounded rotation without cusps.If the function p(s) satisfies the conditions (1.1) and (1.2) on Ω = [0, l], then S Γ is bounded in L p(s) .