θ-type generalized fractional integral and its commutator on some non-homogeneous variable exponent spaces

Abstract: Let (X, d, μ) be a non-homogeneous space satisfying certain growth conditions. In this paper, the authors obtain the boundedness of θ-type generalized fractional integral Tα on variable exponent Lebesgue spaces Lp(·)(X) and variable exponent Morrey spaces M q(·) (X)N . Furthermore, by establishing the sharp maximal function for commutator [b,Tα] generated by b ∈ RBMO(μ) and Tα, the authors prove that the [b,Tα] is bounded from spaces Lp(·)(X) into spaces Lq(·)(X) with 1 q(x) = 1 p(x) −α and α ∈ (0, 1), and bounded from spaces M q(·) (X)N into spaces M s(·) t(·) (X)Nā, where t(·) s(·) = q(·) p(·) , 1 s(·) = 1 p(·)−α and ā > 0 is a constant.


Introduction
In 1931, Orlicz first obtained the definition of Lebesgue space with variable exponent L p(·) (Ω) (see [15]), i.e., for any measurable functions f and sets Ω ⊂ R n , if there exists a positive constant η such that, where p is a function on Ω satisfying 1 < p(x) < ∞. Respectively, the norm of Luxemburg-Nakano is defined by f L p(·) (Ω) = inf η > 0 : Since then, many papers focus on the variable exponent spaces and their applications. For example, Kováčik and Rákosník [9] systematically researched variable exponent Lebesgue spaces L p(·) (R n ) and Sobolev spaces W k,p(·) (R n ). In [16], Radulescu and Repovs studied the Lebesgue and Morrey spaces with variable exponent on R n , and also obtained some applications in partial differential equations. In [17], Ragusa and Tachikawa established the C 1,γ loc (Ω)-regularity result for W 1,1 -local minimizers µ of the double phase functional with x-dependent exponents. In 2021, with the nonstandard growth conditions, Mingione and Rȃdulescu provide an overview of recent results concerning elliptic variational problems (see [12]). The more development and research on the variable exponents, we refer readers to see [3,4,8,11,13,[21][22][23] and reference therein.
On the other hand, fractional integrals, which regard as an important class of operators in harmonic analysis, have played a key role in the fields of harmonic analysis, applied probability and physics communities. For example, Sawano and Tanaka in [18] proved that fractional integral is bounded on Morrey space over non-doubling measures. Based on this work, the boundedness of fractional integral on Morrey space over non-homogeneous metric measure space is obtained by Cao and Zhou in [1]. Shen et.al used the generalization of a parameterized inexact Uzawa method to solve such a kind of saddle point problem for fractional diffusion equations (see [19]). However, in this paper, we will mainly consider the boundedness of θ-type generalized fractional integrals, which are slightly modified in [5], on Lebesgue and Morrey spaces with variable exponents over non-homogeneous spaces. What's more, the results of this paper extend the contents of fractional integral on variable exponent spaces over R n and non-homogeneous spaces.
A measure µ on X is said to satisfy the following growth condition, if there exists a constant C > 0 such that, for all x ∈ X and r > 0, µ(B(x, r)) ≤ Cr. (1.1) Then the space (X, d, µ) with measure µ satisfying (1.1) is called a non-homogeneous space. In this setting, Kokilashvili and Meskhi obtained the boundedness of Maximal function and Riesez potential on variable Morrey spaces(see [7]). In [10], Lu proved that parameter Marcinkiewicz integral and its commutator are bounded on Morrey spaces with variable exponent and so on.
In this paper, we set that p is a µ-measurable function on X, and respectively define where E ⊂ X is a µ-measurable. Moreover, we also denote p − = p − (X) and p + = p + (X). We now recall the following definitions introduced in [7]. Definition 1.1. Let N ≥ 1 be a constant. Suppose that p is a function on X such that 0 < p − < p + < ∞. We say that p ∈ P(N) if there exists a constant C > 0 such that, for all x ∈ X and r > 0.
We say that a function p on X satisfies the Log-Hölder continuity condition p ∈ LH(X) if where constant A > 0 does not depend on x, y ∈ X.
For any ball B, we respectively denote its center and radius by c B and r B (or r(B)). Let η > 1 and β > η, a ball B is said to be an (η, β)-doubling ball if µ(ηB) ≤ βµ(B), where ηB denotes the ball with the same center as B and r(ηB) = ηr(B). Especially, for any given ball B, we denote by B the smallest doubling ball which contains B and has the same center as B. Given two balls B ⊂ S in X, set , (1.4) where N B,S is denoted by the smallest integer k such that r(2 k B) ≥ r(S ).
The following notion of regular bounded mean oscillation (RBMO) space is from [20]. Definition 1.3. Let τ > 1. A function f ∈ L 1 loc (µ) is said to be in the space RBMO(µ) if there exists a constant C > 0 such that for any ball B centered at some point of supp(µ), and for any two doubling balls B ⊂ S , where m B ( f ) represents the mean value of function f over ball B, that is, Moreover, the minimal constant C satisfying (1.5) and (1.6) is defined to be the norm of f in the space RBMO(µ) and denoted by f RBMO(µ) . Now we state the definition of θ-type generalized fractional integral kernel as follows. Definition 1.4. Let α ∈ (0, 1), and θ be a non-negative and non-decreasing function on (0, ∞) satisfying (1.9) Remark 1.1. If we take the function θ(t) ≡ t δ with δ ∈ (0, 1], then the θ-type generalized fractional integral kernel K α is just the fractional kernel of order 1 (see [7]).
Let L ∞ b (µ) be the space of all L ∞ (µ) functions with bounded support. A linear T α is called an θ-type generalized fractional integral with K α satisfying (1.8) and (1.9) if, for all f ∈ L ∞ b (µ) and x supp( f ), The following definition of variable exponent Morrey space M p(·) q(·) (X) N is from [7].
Finally, we make some conventions on notation. Throughout the whole paper, C represents a positive constant being independent of the main parameters. For any subset E of X, we use χ E to denote its characteristic function.

Estimate for T α on variable exponent spaces
In this section, by applying some known results, the boundedness of θ-type generalized fractional integral T α on variable Lebsgue spaces L p(·) (X) and on variable exponent Morrey spaces M p(·) q(·) (X) N is obtained. Now we state the main theorems as follows.
Proof of Theorem 2.2. For any x ∈ X, by (1.9), we can deduce that where I α represents the homogeneous fractional integral operator (see [7]), namely, for any x ∈ X, set

Estimate for
In this section, by establishing the sharp maximal function for commutator [b, T α ], which is generated by T α and b ∈ RBMO(µ), we prove that the [b, T α ] is bounded from space L p(·) (X) into space L q(·) (X). The main theorem of this section is as follows.
To prove the above theorem, we need to recall and establish the following corollary and lemmas, see [6,7], respectively. Corollary 3.1. If f ∈ RBMO(µ), then there exists a constant C > 0 such that, for any balls B, ρ ∈ (1, ∞) and r ∈ [1, ∞), Lemma 3.1. Let µ(X) < ∞, N ≥ 1 be a constant, 1 < p − ≤ p(x) ≤ p + < ∞ and s ∈ (1, p − ). If there exists a positive constant C such that for all x ∈ X and r > 0, the following inequality and the supremum is taken over all balls B x. Remark 3.1. With a way similar to that used in the proof of Theorem 1.3 in [2], it is easy to show that Lemma 3.4 hold on (X, d, µ).
From [6], the sharp maximal function M ,α is defined by, for all x ∈ X, α ∈ [0, 1) and f ∈ L 1 loc (µ), where ∆ x = {x ∈ B ⊂ S and B, S are doubling balls} and coefficient K (α) B,S is defined by Lemma 3.4. Let τ ∈ (0, 1), g ∈ L 1 loc (X) and µ-measurable function f satisfy the following condition Lemma 3.5. Let K α satisfy the conditions (1.8) and (1.9), s ∈ (1, ∞) and p 0 ∈ (1, ∞). If T α is bounded on L 2 (µ), then there exists a positive constant C such that, for all f ∈ L ∞ (µ) ∩ L p 0 (µ), Proof. By applying the definition of sharp maximal function M ,α defined as in (3.4), for any ball B, it suffices to show that, for all x and B with B x, and, for all balls B, S with B ⊂ S and B x, From Hölder inequality, Corollary 3.2 and (3.2), it follows that To estimate D 2 , take t = √ s and 1 r = 1 t − α. By applying Hölder inequality, Corollary 3.2 and Lemma 3.6, we obtain that For all y, z ∈ B, by applying (1.7), (1.9), Corollary 3.2 and Hölder inequality, we have where we have used the following fact that which, together with D 1 and D 2 , implies (3.8). Now let us estimate (3.9). Consider two balls B ⊂ S with x ∈ B and let N := N B,S + 1. Write With arguments similar to that used in the estimate of D 3 and Theorem 1 in [22], it is not difficult to obtain that and For any y ∈ B, by applying Hölder inequality, Corollary 3.2 and (3.2), we obtain that Taking the mean over ball B, we get E 3 Which, combining the estimates E 1 , E 2 and E 3 , implies (3.9).

Conclusions
In this paper, we mainly obtain the boundedness of θ-type generalized fractional integral T θ and its commutator [b, T θ ] generated by b and T θ on variable Lebesgue space L p(·) (X) and Morrey space M p(·) q(·) (X) N .