In order to better study the solutions of the Poisson equation Δ u = f in partial differential equations, Sobolev [1] showed that the fractional integral operator is bounded from the classical Lebesgue space L p ( ℝ n ) to L q ( ℝ n ) . After that, Harboure and Salinas [2] provided necessary and sufficient conditions for the boundedness of fractional integral operators and their commutators on L p ( ω ) . Capone, Cruz-Uribe and Fiorenza [3] have proved the boundedness on L p ( ⋅ ) , provided that the exponents satisfy the log-Hölder continuous conditions. Bernardis, Dalmasso and Pradolini [4] proved the boundedness of fractional integral operators and their commutators on L p ( ⋅ ) ( ω ) . Cruz-Uribe and Wang [5] have also obtained the boundedness of fractional integrals on weighted Lebesgue spaces with variable exponent by applying the extrapolation theorem. We recall the so-called fractional integral operators.

Let S n − 1 , with n ≥ 2 , be the unit sphere in ℝ n with the normalized Lebesgue measure d σ ( x ′ ) . Assuming that Ω ∈ L s ( S n − 1 ) , with s > n / ( n − β ) , the fractional integral operator with homogeneous kernel T Ω , β is defined by

If β = 1 , the integral returns the well-known Cauchy principal value.

If Ω = 1 , T Ω , β is the fractional integral operator (Riesz potential operator)

If β = 0 and Ω satisfy the vanishing condition on S n − 1 :

∫ S n − 1 Ω ( x ′ ) d σ ( x ′ ) = 0 s

then T Ω , β becomes a Calderon-Zygmund operator.

If b ∈ L loc 1 ( ℝ n ) , set

where b B = ( 1 / | B | ) ∫ B b ( y ) d y and the supremum is taken over all B ⊂ ℝ n , and what follows | B | is the Lebesgue measure of measurable set B in ℝ n . A function b is called bounded mean oscillation if ‖ b ‖ ∗ < ∞ . Denote by BMO ( ℝ n ) the set of all bounded mean oscillation functions on ℝ n .

Let b ∈ BMO ( ℝ n ) , the commutator of fractional integral operator [ b , T Ω , β ] is defined by

Variable exponent function spaces play a vital role in fluid dynamics, elasticity dynamics, and differential equations with nonstandard growth. Since Kováčik and Rákosník developed the theory of variable exponent function spaces in [6], the variable exponent Lebesgue spaces L p ( ⋅ ) ( ℝ n ) have been extensively investigated, see [7]-[9]. A class of function spaces called Herz spaces has played an important role in real analysis because the interesting norm includes explicitly both local and global information of the function. Izuki introduced the variable exponent Herz spaces in [10][11] and considered the boundedness of commutators of fractional integrals [12]. In [13], Izuki has proved the boundedness of commutators with fractional integrals on Herz spaces with variable exponent. In 2016, Izuki and Noi [14] defined weighted Herz spaces with variable exponent and proved the boundedness of fractional integrals on those spaces under proper assumptions on weights and exponents.

Recently, Izuki and Noi introduced two weight Herz spaces with variable exponents in [15]. Motivated by the mentioned works, we will consider the boundedness of the fractional operator with homogeneous kernels on two weight Herz spaces with variable exponents. Compared with reference [15], this paper introduces the two weight settings in the variable exponent Herz space for the first time. Although [15] established the theoretical framework of the two weight variable exponent Herz space, it did not involve the fractional integral operator. Compared with reference [16], this paper extends its single weight results to a more general and challenging two weight case, while retaining the homogeneity assumption of the kernel function. In terms of the proof method, it combines the fine characterization of variable exponent Muckenhoupt weight classes with the ring decomposition technique.

In this paper we use the following symbols and notations:

1) For any measurable set E , | E | denotes the Lebesgue measure and χ E means the characteristic function.

2) A locally integrable and positive function defined on ℝ n is said to be a weight. We write w ( E ) : = ∫ E w ( x ) d x for a weight w and a measurable set E .

3) Given k ∈ ℤ , we write B k : = B ( 0 , 2 k ) ¯ = { x ∈ ℝ n : | x | ≤ 2 k } .

4) Give k ∈ ℤ , we define B k : = { x ∈ ℝ n : | x | ≤ 2 k } , D k : = B k \ B k − 1 = { x ∈ ℝ n : 2 k − 1 < | x | ≤ 2 k } , χ k : = χ D k .

5) For any quantities A and B, if there exists a constant C > 0 such that A ≤ C B , we write A ≲ B . If A ≲ B and B ≲ A , we write A ≈ B .

We introduce Lebesgue spaces with variable exponent. Let p ( ⋅ ) be a measurable function on ℝ n taking values in [ 1 , ∞ ) , the Lebesgue space with variable exponent L p ( ⋅ ) ( ℝ n ) is defined by

Then L p ( ⋅ ) ( ℝ n ) is a Banach function space equipped with the norm

Denote by P ( ℝ n ) the set of all measurable functions p ( ⋅ ) : ℝ n → ( 1 , ∞ ) such that

and P 0 ( ℝ n ) consists of all p ( ⋅ ) satisfying p − > 0 and p + < ∞ .

Definition 2.1 ([9]) Let α ( ⋅ ) be a real-valued function on ℝ n .

(i) For any x , y ∈ ℝ n , | x − y | < 1 / 2 , if

then α ( ⋅ ) is said local log-Hölder continuous on ℝ n .

(ii) For all x ∈ ℝ n , if

then α ( ⋅ ) is said log-Hölder continuous functions at origin and denote by P 0 log ( ℝ n ) .

(iii) For some real number α ∞ ∈ ℝ , for x ∈ ℝ n , if

then α ( ⋅ ) is said log-Hölder continuous at infinity and denote by P ∞ log ( ℝ n ) .

(iv) The function α ( ⋅ ) satisfying (ii) and (iii) is denoted by P log ( ℝ n ) . It is also well known that the Hardy-Littlewood maximal operator M , defined by

is bounded on L p ( ⋅ ) ( ℝ n ) whenever p ( ⋅ ) ∈ P log ( ℝ n ) ∩ P ( ℝ n ) [17].

Now we define the Muckenhoupt classes. We begin with the classical Muckenhoupt A 1 weight.

Definition 2.2 ([18]) Let ω be a weighted function on ℝ n , that is, ω is real-valued, non-negative and locally integrable. ω is said to be a Muckenhoupt A 1 weight if

For 1 < p < ∞ , we say that ω is an A p weight if

Definition 2.3 ([19]) Let

. For some constant C , a weight ω is said to be an A p ( ⋅ ) weight, if for all balls B in ℝ n such that

Diening and Hästä [20], have proved the next monotone property.

Lemma 2.1 ([20]) If p ( ⋅ ) , q ( ⋅ ) ∈ P log ( ℝ n ) ∩ P ( ℝ n ) and p ( ⋅ ) ≤ q ( ⋅ ) , then we have

In order to state the boundedness of fractional integrals on weighted function spaces we shall define the class A ( p 1 ( ⋅ ) , p 2 ( ⋅ ) ) as follows.

Definition 2.4 Let 0 < β < n and p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P ( ℝ n ) such that 1 / p 2 ( x ) ≡ 1 / p 1 ( x ) − β / n . A weight w is said to be an A ( p 1 ( ⋅ ) , p 2 ( ⋅ ) ) weight if

holds for all balls B ⊂ ℝ n .

Lemma 2.2 ([5]) Let 0 < β < n and p 1 ( ⋅ ) , p 2 ( ⋅ ) ∈ P ( ℝ n ) such that 1 / p 2 ( x ) ≡ 1 / p 1 ( x ) − β / n . Then w ∈ A ( p 1 ( ⋅ ) , p 2 ( ⋅ ) ) if and only if w p 2 ( ⋅ ) ∈ A 1 + p 2 ( ⋅ ) / p ′ 1 ( ⋅ ) .

Definition 2.5 Let p ( ⋅ ) ∈ P ( ℝ n ) and w ∈ A p ( ⋅ ) . The weight variable exponent Lebesgue space L p ( ⋅ ) ( w ) denotes the set of all complex-valued measurable functions f satisfying

This is a Banach space equipped with the norm:

Lemma 2.3 ([16]) Let p 1 ( ⋅ ) ∈ P log ( ℝ n ) ∩ P ( ℝ n ) and 0 < β < n / p 1 + . Define p 2 ( ⋅ ) by 1 / p 2 ( ⋅ ) ≡ 1 / p 1 ( ⋅ ) − β / n . If w ∈ A ( p 1 ( ⋅ ) , p 2 ( ⋅ ) ) , then T Ω , β is bounded from L p 1 ( ⋅ ) ( w p 1 ( ⋅ ) ) to L p 2 ( ⋅ ) ( w p 2 ( ⋅ ) ) .

Let w be a non-negative measurable function and let { f j } j ∈ ℤ be a sequence of functions. The modulus is defined by

where λ 1 ∞ = 1 . If q + < ∞ or q ( ⋅ ) ≤ p ( ⋅ ) , the above can be written as

The norm is

Definition 3.1 ([15]) Let w 1 ∈ A p ,   w 2 ∈ A p ( ⋅ ) ,   p ( ⋅ ) , q ( ⋅ ) ∈ P 0 ( ℝ n ) and α ( ⋅ ) ∈ L ∞ ( ℝ n ) .

The homogeneous two weight Herz space K ˙ p ( ⋅ ) α ( ⋅ ) , q ( ⋅ ) ( w 1 , w 2 ) and non-homogeneous two weight Herz space K p ( ⋅ ) α ( ⋅ ) , q ( ⋅ ) ( w 1 , w 2 ) are defined respectively by

and

where

and

Regarding function spaces, we have the following Lemma:

Lemma 3.1 ([15]) If α ( ⋅ ) ∈ L ∞ ( ℝ n ) and α ( ⋅ ) ∈ P log ( ℝ n ) , ω ∈ A p for some p ∈ [ 1 , ∞ ) then for all k ∈ ℤ and x ∈ D k ,

Lemma 3.2 ([15]) Let α ( ⋅ ) ∈ L ∞ ( ℝ n ) ,   p ( ⋅ ) , q ( ⋅ ) ∈ P 0 ( ℝ n ) ,   w 1 ∈ A p for some p ∈ [ 1 , ∞ ) and w 2 ∈ A p ( ⋅ ) . If q ( ⋅ ) , α ( ⋅ ) ∈ P ∞ log ( ℝ n ) , then

Additionally, if q ( ⋅ ) , α ( ⋅ ) ∈ P log ( ℝ n ) , then

Lemma 3.3 ([15]) Let k , l ∈ ℤ , w ∈ A p , where p ∈ [ 1 , ∞ ) and δ ∈ ( 0 , 1 ) . w − : = { δ α − ≥ 0 q α − < 0 , w + : = { q α + ≥ 0 δ α + < 0 If α ( ⋅ ) ∈ L ∞ ( ℝ n ) and α ( ⋅ ) ∈ P log ( ℝ n ) , then for any x ∈ C k and y ∈ C l , we have

where the implicit constants are independent of x , y , k and l .

Lemma 3.4 ([21]) Let X be a Banach function space. Suppose that the Hardy-Littlewood maximal operator M is weakly bounded on X , that is,

is true for all f ∈ X and all λ > 0 . Then we have

The weighted Banach function space X ( ℝ n , W ) is a Banach function space equipped

The associate space of X ( ℝ n , W ) is a Banach function space and equals X ′ ( ℝ n , W − 1 ) . The properties above naturally arise from those of the usual Banach function spaces and the proof is found in [22].

If we take X = L p ( ⋅ ) ( ℝ n ) and W = w , then we have L p ( ⋅ ) ( ℝ n , w ) = L p ( ⋅ ) ( w p ( ⋅ ) ) . If we take X = L p ′ ( ⋅ ) ( ℝ n ) and W = w − 1 , then we have L p ′ ( ⋅ ) ( ℝ n , w − 1 ) = L p ′ ( ⋅ ) ( w − p ′ ( ⋅ ) ) .

Thus we have:

Lemma 3.5 ([15]) If p ( ⋅ ) ∈ P log ( ℝ n ) ∩ P ( ℝ n ) and ω ∈ A p ( ⋅ ) , then there exist constants δ 1 , δ 2 ∈ ( 0 , 1 ) , such that for all balls B in ℝ n and all measurable subsets S ⊂ B ,

Lemma 3.6 ([23]) Let 0 < p < ∞ ,   δ > 0 . Then there is a positive constant C such that

for non-negative sequences { a j } j = − ∞ ∞ .

Theorem 4.1Let 0 < q 1 ( ⋅ ) < q 2 ( ⋅ ) < ∞ , p ( ⋅ ) ∈ P log ( ℝ n ) ∩ P ( ℝ n ) , q ( ⋅ ) ∈ P log ( ℝ n ) ∩ P 0 ( ℝ n ) , α ( ⋅ ) ∈ L ∞ ( ℝ n ) and α ( ⋅ ) ∈ P log ( ℝ n ) , v ∈ A p for some p ∈ [ 1 , ∞ ) , w p 2 ( ⋅ ) ∈ A ( p 1 ( ⋅ ) , p 2 ( ⋅ ) ) and δ 1 , δ 2 ∈ ( 0 , 1 ) . 0 < β < n ( δ 1 + δ 2 ) , − n δ 1 < w − α − − n / r and β + w + α + + n / r < n δ 2 .Define p 1 ( ⋅ ) by 1 / p 2 ( ⋅ ) ≡ 1 / p 1 ( ⋅ ) − β / n .Then the fractional integral operator T Ω , β is a bounded operator from K ˙ p 2 ( ⋅ ) α ( ⋅ ) , q 2 ( ⋅ ) ( v , w p 2 ( ⋅ ) ) to K ˙ p 1 ( ⋅ ) α ( ⋅ ) , q 1 ( ⋅ ) ( v , w p 1 ( ⋅ ) ) for all f ∈ K ˙ p 2 ( ⋅ ) α ( ⋅ ) , q 2 ( ⋅ ) ( v , w p 2 ( ⋅ ) ) .

Proof of Theorem 4.1 Let f ∈ K ˙ p 2 ( ⋅ ) α ( ⋅ ) , q 2 ( ⋅ ) ( v , w p 2 ( ⋅ ) ) . Then, by the Jensen inequality, we have

Let f j : = f χ j for any j ∈ ℤ . Then f = ∑ j = − ∞ ∞ f j . So we have By Lemma 3.3, we decompose f i into the following three parts as

It is easy to see that

where

We estimate E 1 . Note that if x ∈ B k , y ∈ B j , and j ≤ k − 2 , then | x − y | ≈ | x | ≈ 2 k . By generalized Hölder inequality, we obtain

By virtue of the generalized Hölder’s inequality, we have

Then, for any Ω ∈ L s ( S n − 1 ) , we have the following inequality

By taking the L p 2 ( ⋅ ) ( w p 2 ( ⋅ ) ) -norm and using Lemma 3.4, we have