Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then we give the weighted strong-type and weak-type estimates for fractional integral operators $I_\gamma$ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators $[b,I_{\gamma}]$ generated by $b$ and $I_{\gamma}$ are established as well. In addition, we are going to study related problems about two-weight, weak type inequalities for $I_{\gamma}$ and $[b,I_{\gamma}]$ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator $M_{\gamma}$ and generalized fractional integrals $\mathcal L^{-\gamma/2}$ in the context of weighted amalgam spaces, where $0<\gamma<n$ and $\mathcal L$ is the infinitesimal generator of an analytic semigroup on $L^2(\mathbb R^n)$ with Gaussian kernel bounds.


Introduction
One of the most significant operators in harmonic analysis is the fractional integral operator. Let n be a positive integer. The n-dimensional Euclidean space R n is endowed with the Lebesgue measure dx and the Euclidean norm j·j. For given γ, 0 < γ < n, the fractional integral operator (or Riesz potential) I γ of order γ is defined by The boundedness properties of I γ between various function spaces have been studied extensively. It is well-known that the Hardy-Littlewood-Sobolev theorem states that the fractional integral operator I γ is bounded from L p ðR n Þ to L q ðR n Þ for 0 < γ < n, 1 < p < n/γ, and 1/q = 1/p − γ/n. Also, we know that I γ is bounded from L 1 ðR n Þ to WL q ðR n Þ for 0 < γ < n and q = n/ðn − γÞ (see [1]). In 1974, Mucken-houpt and Wheeden [2] studied the weighted boundedness of I γ and obtained the following two results (for sharp weighted norm inequalities, see [3]).
For 0 < γ < n, the linear commutator ½b, I γ generated by a suitable function b and I γ is defined by This commutator was first introduced by Chanillo in [4]. In 1991, Segovia and Torrea [5] showed that ½b, I γ is bounded from L p ðw p Þ (1 < p < n/γ) to L q ðw q Þ whenever b ∈ BMOðR n Þ (see [6] for sharp weighted bounds; see also [4] for the unweighted case). This corresponds to the norm inequalities satisfied by I γ . Let us recall the definition of the space of BMOðR n Þ (see [7]). BMOðR n Þ is the Banach function space modulo constants with the norm k⋅k * defined by b k k * ≔ sup where the supremum is taken over all balls B in R n and b B stands for the mean value of b over B; that is, b B ≔ ð1/jBjÞ Ð B bðyÞdy.
In the endpoint case p = 1 and q = n/ðn − γÞ, since linear commutator ½b, I γ has a greater degree of singularity than I γ itself, a straightforward computation shows that ½b, I γ fails to be of weak type (1, n/ðn − γÞ) when b ∈ BMOðR n Þ (see [8] for some counterexamples). However, if we restrict ourselves to a bounded domain Ω in R n , then the following weighted endpoint estimate for commutator ½b, I γ of the fractional integral operator is valid, which was established by Cruz-Uribe and Fiorenza [9] in 2007 (see also [8] for the unweighted case).
Let 1 ≤ p, s ≤ ∞, a function f ∈ L p loc ðR n Þ is said to be in the Wiener amalgam space ðL p , L s ÞðR n Þ of L p ðR n Þ and L s ðR n Þ, if the function y ↦ k f ð⋅Þ ⋅ χ Bðy,1Þ ð⋅Þk L P ðR n Þ belongs to L s ðR n Þ, where Bðy, rÞ = fx ∈ R n : jx − yj < rg is the open ball centered at y and with radius r, χ Bðy,rÞ is the characteristic function of the ball Bðy, rÞ, and k⋅k L p is the usual Lebesgue norm in L p ðR n Þ. Define Then, we know that ðL p , L s ÞðR n Þ becomes a Banach function space with respect to the norm k⋅k ðL p ,L s ÞðR n Þ . This amalgam space was first introduced by Wiener in the 1920's, but its systematic study goes back to the works of Holland [10] and Fournier and Stewart [11]. Let 1 ≤ p, s, α ≤ ∞. We define the amalgam space ðL p , L s Þ α ðR n Þ of L p ðR n Þ and L s ðR n Þ as the set of all measurable functions f satisfying f ∈ L p loc ðR n Þ and k f k ðL p ,L s Þ α ðR n Þ < ∞, where with the usual modification when p = ∞ or s = ∞ and jBðy, rÞj is the Lebesgue measure of the ball Bðy, rÞ. This generalization of amalgam space was originally introduced by Fofana in [12]. As proved in [12], the space ðL p , L s Þ α ðR n Þ is nontrivial if and only if p ≤ α ≤ s; thus, in the remaining of the paper, we will always assume that this condition p ≤ α ≤ s is fulfilled. Note that (i) for 1 ≤ p ≤ α ≤ s ≤ ∞, one can easily see that ðL p , L s Þ α ðR n Þ ⊆ ðL p , L s ÞðR n Þ, where ðL p , L s ÞðR n Þ is the Wiener amalgam space defined by (5) (ii)if 1 ≤ p < α and s = ∞, then ðL p , L s Þ α ðR n Þ is just the classical Morrey space ℒ p,κ ðR n Þ defined by (with κ = 1 − p/α, see [13]) (iii) if p = α and s = ∞, then ðL p , L s Þ α ðR n Þ reduces to the usual Lebesgue space L p ðR n Þ In [14] (see also [15,16]), Feuto considered a weighted version of the amalgam space ðL p , L s Þ α ðwÞ. A nonnegative measurable function w defined on R n is called a weight if it is locally integrable. Let 1 ≤ p ≤ α ≤ s ≤ ∞ and w be a weight on R n . We denote by ðL p , L s Þ α ðwÞ the weighted amalgam space, the space of all locally integrable functions f satisfying with the usual modification when s = ∞ and wðBðy, rÞÞ ≔ Ð Bðy,rÞ wðxÞdx is the weighted measure of Bðy, rÞ. Similarly, for 1 ≤ p ≤ α ≤ s ≤ ∞, we can see that ðL p , L s Þ α ðwÞ becomes a Banach function space with respect to the norm k⋅k ðL p ,L s Þ α ðwÞ . Furthermore, we denote by ðWL p , L s Þ α ðwÞ the weighted weak amalgam space consisting of all measurable functions f such that (see [14]) 2 Journal of Function Spaces Notice that (i) if 1 ≤ p < α and s = ∞, then ðL p , L s Þ α ðwÞ is just the weighted Morrey space ℒ p,κ ðwÞ defined by (with κ = 1 − p/α, see [17]) and ðWL p , L s Þ α ðwÞ is just the weighted weak Morrey space Wℒ p,κ ðwÞ defined by (with κ = 1 − p/α, see [18]) (ii) if p = α and s = ∞, then ðL p , L s Þ α ðwÞ reduces to the weighted Lebesgue space L p ðwÞ and ðWL p , L s Þ α ðwÞ reduces to the weighted weak Lebesgue space WL p ðwÞ Recently, many works in classical harmonic analysis have been devoted to norm inequalities involving several integral operators in the setting of weighted amalgam spaces (see [14][15][16]19] and [20]). These results obtained are extensions of well-known analogues in the weighted Lebesgue spaces.
Let I γ be the fractional integral operator, and let ½b, I γ be its linear commutator. The aim of this paper is twofold. We first define some new classes of weighted amalgam spaces. As the weighted amalgam space may be considered as an extension of the weighted Lebesgue space, it is natural and important to study the weighted boundedness of I γ and ½b, I γ in these new spaces. We will prove that I γ as well as its commutator ½b, I γ , which are known to be bounded on weighted Lebesgue spaces, are bounded on weighted amalgam spaces under appropriate conditions. In addition, we will discuss two-weight, weak-type norm inequalities for I γ and ½b, I γ in the context of weighted amalgam spaces and give some partial results. Using these results and pointwise domination, we will establish the corresponding strong-type and weak-type estimates for fractional maximal operator M γ and generalized fractional integrals ℒ −γ/2 , where 0 < γ < n and L is the infinitesimal generator of an analytic semigroup on L 2 ðR n Þ with Gaussian kernel bounds.
The present paper is organized as follows. In Section 2, we first state some preliminary definitions and results about A p weights, Orlicz spaces, and weighted amalgam spaces, and the main results of the present paper are also given in Section 2. The following Sections 3, 4, and 5 are devoted to their proofs. Finally, in Section 6, we discuss some related two-weight problems.

Notations and Preliminaries.
Let us first recall the definitions of two weight classes: A p and A p,q .
Definition 5 (A p weights [21]). A weight w is said to belong to the class A p for 1 < p < ∞, if there exists a positive constant C such that for any ball B in R n , where we denote the conjugate exponent of p > 1 by p′ = p/ðp − 1Þ. The class A 1 is defined replacing the above inequality by for any ball B in R n . We also define A ∞ = ∪ 1≤p<∞ A p .
Definition 6 (A p,q weights [2]). A weight w is said to belong to the class A p,q for 1 < p, q < ∞, if there exists a positive constant C such that for any ball B in R n , The class A 1,q (1 < q < ∞) is defined replacing the above inequality by for any ball B in R n . There is a close connection between A p weights and A p,q weights (see [22]).
Bðy, rÞ in R n ; that is, Bðy, rÞ c ≔ R n \ Bðy, rÞ. Given a weight w, we say that w satisfies the doubling condition if there exists a universal constant C > 0 such that for any ball B in R n , we have When w satisfies this doubling condition (16), we denote w ∈ Δ 2 for brevity. An important fact here is that if w is in A ∞ , then w ∈ Δ 2 (see [23]). Moreover, if w ∈ A ∞ , then there exists a number δ > 0 such that (see [23]) holds for any measurable subset E of a ball B. Given a weight w on R n , for 1 ≤ p < ∞, the weighted Lebesgue space L p ðwÞ is defined as the set of all functions f such that We also denote by WL p ðwÞ(1 ≤ p < ∞) the weighted weak Lebesgue space consisting of all measurable functions f such that We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of our main results. For further information on this subject, we refer to [24]. A function A : ½0, +∞Þ ⟶ ½0, +∞Þ is said to be a Young function if it is continuous, convex, and strictly increasing satisfying Að0Þ = 0 and AðtÞ ⟶ +∞ as t ⟶ +∞. An important example of Young function is AðtÞ = t p · ð1 + log + tÞ p with some 1 ≤ p < ∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the following Luxemburg norm: In particular, when AðtÞ = t p , 1 ≤ p < ∞, it is easy to see that A is a Young function and that is, the Luxemburg norm coincides with the normalized L p norm. Recall that the following generalization of Hölder's inequality holds where A is the complementary Young function associated with A, which is given by AðsÞ ≔ sup 0≤t<∞ ½st − AðtÞ, 0 ≤ s < ∞. Obviously, ΦðtÞ = t · ð1 + log + tÞ is a Young function, and its complementary Young function is ΦðtÞ ≈ e t − 1. In the present situation, we denote k f k Φ,B and kgk Φ,B by k f k L log L,B and kgk exp L,B , respectively. Now, the above generalized Hölder's inequality reads There is a further generalization of Hölder's inequality that turns out to be useful for our purpose (see [25]): Let A, B, and C be Young functions such that for all t > 0, where A −1 ðtÞ is the inverse function of AðtÞ. Then, for all functions f and g and for all balls B in R n , If ν = w, then we denote ðL p , L s Þ α ðw ; μÞ for brevity, i.e., ðL p , L s Þ α ðw, w ; μÞ ≔ ðL p , L s Þ α ðw ; μÞ. Furthermore, we denote by ðWL p , L s Þ α ðw ; μÞ the weighted weak amalgam space consisting of all measurable functions f for which with the usual modification when s = ∞. The aim of this paper is to extend Theorems 1-4 to the corresponding weighted amalgam spaces. We are going to prove that the fractional integral operator I γ , which is bounded on weighted Lebesgue spaces, is also bounded on our new weighted spaces under appropriate conditions. Our first two results in this paper are stated as follows.
To obtain endpoint estimate for the linear commutator ½b, I γ , we first need to define the weighted A-average of a function f over a ball B by means of the weighted Luxemburg norm; that is, given a Young function A and w ∈ A ∞ , we define (see [24], for instance) When AðtÞ = t, this norm is denoted by k·k LðwÞ,B , and when ΦðtÞ = t · ð1 + log + tÞ, this norm is also denoted by k·k L log LðwÞ,B . The complementary Young function is given by ΦðtÞ ≈ e t − 1 with corresponding mean Luxemburg norm denoted by k·k exp LðwÞ,B . For w ∈ A ∞ and for every ball B in R n , we can also show the weighted version of (23). Namely, the following generalized Hölder's inequality in the weighted setting is true (see [26], for instance). Now, we introduce the new amalgam spaces of L log L type as follows.
Definition 12. Let p = 1, 1 ≤ α ≤ s ≤ ∞, and let ν, w, μ be three weights on R n . We denote by ðL log L, L s Þ α ðν, w ; μÞ the weighted amalgam space of L log L type, the space of all locally integrable functions f defined on R n with finite norm k f k ðL log L,L s Þ α ðν,w;μÞ : Note that t ≤ t · ð1 + log + tÞ for all t > 0. Then, for any ball B in R n and ν ∈ A ∞ , it is immediate that k f k LðvÞ,B ≤ k f k L log LðvÞ,B by definition, i.e., the inequality holds for any ball B in R n . From this, we can further see the following inclusion: when 1 ≤ α ≤ s ≤ ∞ and w, μ are some other weights.
In the endpoint case p = 1, we will prove the following weak-type L log L estimate of linear commutator ½b, I γ in the setting of weighted amalgam spaces.
Moreover, we will discuss the extreme case β = s of Theorem 9. In order to do so, we need to introduce a new BMO-type space given below. Definition 14. Let 1 ≤ s ≤ ∞ and μ ∈ Δ 2 . We define the space ðBMO, L s ÞðμÞ as the set of all locally integrable functions f Here, the L s ðμÞ-norm is taken with respect to the variable y. We also use the notation f Bðy,rÞ to denote the mean value of f over Bðy, rÞ.

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Observe that if s = ∞, then ðBMO, L s ÞðμÞ is the classical BMO space. Now, we can show that I γ is bounded from ðL p , L s Þ α ðw p , w q ; μÞ to our new BMO-type space defined above. This result can be regarded as a supplement of Theorem 9.
Throughout this paper, the letter C always denotes a positive constant that is independent of the essential variables but whose value may vary at each occurrence. We also use A ≈ B to denote the equivalence of A and B; that is, there exist two positive constants C 1 and C 2 independent of quantities A and B such that Equivalently, we could define the above notions of this section with cubes in place of balls and we will use whichever is more appropriate, depending on the circumstances.

Proofs of Theorems 9 and 10
In this section, we will prove the conclusions of Theorems 9 and 10.
Proof of Theorem 9. The proof is inspired by [14,15] For an arbitrary point y ∈ R n , set B = Bðy, rÞ for the ball centered at y and of radius r, 2B = Bðy, 2rÞ. We represent f as where χ 2B is the characteristic function of 2B. By the linearity of the fractional integral operator I γ , one can write Here and in what follows, for any positive number τ > 0, we use the convention f τ ðxÞ ≔ ½ f ðxÞ τ . Below, we will give the estimates of I 1 ðy, rÞ and I 2 ðy, rÞ, respectively. By the weighted ðL p , L q Þ-boundedness of I γ (see Theorem 1), we have Observe that 1/β − 1/q − 1/s = 1/α − 1/p − 1/s when 1/β = 1/α − γ/n. This fact implies that Since w ∈ A p,q , we get w q ∈ A q ⊂ A ∞ by Lemma 7(i). Moreover, since 1/α − 1/p − 1/s < 0, then by doubling inequality (16), we obtain Substituting the above inequality (40) into (39), we can see that Let us now turn to the estimate of I 2 ðy, rÞ. First, it is clear that when x ∈ Bðy, rÞ and z ∈ Bðy, 2rÞ c , we get jx − zj≈ jy −zj. We then decompose R n into a geometrically increasing sequence of concentric balls and obtain the following pointwise estimate: From this estimate (42), it then follows that Journal of Function Spaces By using Hölder's inequality and A p,q condition on w, we get Hence, where in the last equality we have used the relation then, by using the inequality (17) with exponent δ > 0 and our assumption β < s, we find that where the last series is convergent since δð1/β − 1/sÞ > 0. Therefore, by taking the L s ðμÞ-norm of both sides of (37)(with respect to the variable y), and then using Minkowski's inequality, (41), (45), and (46), we have Thus, by taking the supremum over all r > 0, we complete the proof of Theorem 9.
Proof of Theorem 10.
For an arbitrary ball B = Bðy, rÞ in R n , we represent f as then, by the linearity of the fractional integral operator I γ , one can write We first consider the term I ′ 1 ðy, rÞ. By the weighted weak ð1, qÞ-boundedness of I γ (see Theorem 2), we have

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Observe that 1/β − 1/q − 1/s = 1/α − 1 − 1/s when 1/β = 1/α − γ/n and q = n/ðn − γÞ. Then, we have Since w is in the class A 1,q , we get w q ∈ A 1 ⊂ A ∞ by Lemma 7(ii). Moreover, since 1/α − 1 − 1/s < 0, then we apply inequality (16) to obtain that Substituting the above inequality (52) into (51), we thus obtain As for the second term I ′ 2 ðy, rÞ, it follows directly from Chebyshev's inequality and the pointwise estimate (42) that Moreover, by applying Hölder's inequality and then the reverse Hölder's inequality in succession, we can show that w q ∈ A 1 if and only if w ∈ A 1 ∩ RH q (see [27]), where RH q denotes the reverse Hölder class (see [28] for further details). Another application of A 1 condition on w gives that In addition, note that w ∈ RH q . We are able to verify that for any positive integer j ∈ ℤ + , which is equivalent to Consequently, where in the last equality we have used the relation 1/β − 1/q = 1/α − 1. Recall that w q ∈ A 1 ⊂ A ∞ , then by using the inequality (17) with exponent δ * > 0 and the assumption β < s, we find that where the last series is convergent since δ * ð1/β − 1/sÞ > 0. Therefore, by taking the L s ðμÞ-norm of both sides of where the last inequality follows from (59). Thus, by taking the supremum over all r > 0, we finish the proof of Theorem 10.
Suppose that L is a linear operator which generates an analytic semigroup fe −tL g t>0 on L 2 ðR n Þ with a kernel p t ðx, yÞ satisfying Gaussian upper bound; that is, there exist two positive constants C and A such that for all x, y ∈ R n and all t > 0, we have For any 0 < γ < n, the generalized fractional integrals L −γ/2 associated with the operator L is defined by Note that if L = −Δ is the Laplacian on R n , then L −γ/2 is the classical fractional integral operator I γ , which is given by (1). Since the semigroup e −tL has a kernel p t ðx, yÞ which satisfies the Gaussian upper bound (64), it is easy to check that for all x ∈ R n , In fact, if we denote the kernel of L −γ/2 by K γ ðx, yÞ, then it follows immediately from (65) that (see [29,30]) where p t ðx, yÞ is the kernel of e −tL . Thus, by using the Gaussian upper bound (64) and the expression (67), we can deduce that (see [29,30]) which implies (66). Taking into account this pointwise inequality, as a consequence of Theorems 9 and 10, we have the following results.

Proofs of Theorems 11 and 13
To prove our main theorems in this section, we need the following lemma about BMOðR n Þ functions.
Lemma 20. Let b be a function in BMOðR n Þ.
(i)For any ball B in R n and for any positive integer j ∈ ℤ + , then (ii)Let 1 < q < ∞. For any ball B in R n and for any weight ν ∈ A ∞ , then Proof. For the proof of (i), we refer the reader to [31]. For the proof of (ii), we refer the reader to [32].
Proof of Theorem 11. Let 1 < p ≤ α < s ≤ ∞ and f ∈ ðL p , L s Þ α ðw p , w q ; μÞ with w ∈ A p,q and μ ∈ Δ 2 . For each fixed ball B = Bðy, rÞ in R n , as before, we represent f as f = f 1 + f 2 , where f 1 = f · χ 2B , 2B = Bðy, 2rÞ ⊂ R n . By the linearity of the commutator operator ½b, I γ , we write Since w is in the class A p,q , we get w q ∈ A q ⊂ A ∞ by Lemma 7(i). Also, observe that 1/β − 1/q = 1/α − 1/p by our assumption. By using Theorem 3, we obtain where the last inequality is due to (16) and the fact that 1/α − 1/p − 1/s < 0. Let us now turn to the estimate of J 2 ðy, rÞ. By definition, for any x ∈ Bðy, rÞ, we have In the proof of Theorem 9, we have already shown that (see (42)) By the same manner as in the proof of (42), we can also show that Hence, from the above two pointwise estimates (74) and (75), it follows that 10 Journal of Function Spaces Below, we will give the estimates of J 3 ðy, rÞ, J 4 ðy, rÞ, and J 5 ðy, rÞ, respectively. To estimate J 3 ðy, rÞ, note that w q ∈ A q ⊂ A ∞ with 1 < q < ∞. Using the second part of Lemma 20, Hölder's inequality, and the A p,q condition on w, we compute To estimate J 4 ðy, rÞ, applying the first part of Lemma 20, Hölder's inequality, and the A p,q condition on w, we can deduce that It remains to estimate the last term J 5 ðy, rÞ. An application of Hölder's inequality gives us that If we set νðzÞ = wðzÞ −p ′ , then we have ν ∈ A p ′ ⊂ A ∞ because w ∈ A p,q by Lemma 7(i). Thus, it follows from the second part of Lemma 20 and the A p,q condition on w that

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Therefore, in view of the estimate (80), we get Summarizing the estimates derived above, we conclude that where in the last equality we have used the relation 1/β − 1/q = 1/α − 1/p again. Since w q ∈ A q with 1 < q < ∞, then by using the inequality (17) with exponent δ > 0 together with the fact that β < s, we obtain where the last series is convergent since the exponent δð1/β − 1/sÞ is positive. Therefore, by taking the L s ðμÞ-norm of both sides of (71) (with respect to the variable y) and then using Minkowski's inequality, (72) and (82), we can get where the last inequality follows from (83). Thus, by taking the supremum over all r > 0, we complete the proof of Theorem 11.
Proof of Theorem 13. For any fixed ball B = Bðy, rÞ in R n , as before, we represent f as Then, for any given λ > 0, by the linearity of the commutator operator ½b, I γ , one can write We first consider the term J′ 1 ðy, rÞ. By using Theorem 4, we get where in the last equality we have used our assumption 1/β = 1/α − γ/n. Since w is a weight in the class A 1,q , one has w q ∈ A 1 ⊂ A ∞ by Lemma 7(ii). This fact, together with the inequalities (52) and (32), gives us that 12 Journal of Function Spaces We now turn to deal with the term J′ 2 ðy, rÞ. Recall that the following inequality is valid. Thus, we can further decompose J′ 2 ðy, rÞ as Applying the previous pointwise estimate (42), Chebyshev's inequality together with Lemma 20(ii), we deduce that Furthermore, note that t ≤ ΦðtÞ = t · ð1 + log + tÞ for any t > 0. As we pointed out in Theorem 10 that w q ∈ A 1 if and only if w ∈ A 1 ∩ RH q , it then follows from the A 1 condition that where in the last inequality we have used the estimate (32). In view of (57) and our assumption 1/β = 1/α − γ/n, we have On the other hand, applying the pointwise estimate (75) and Chebyshev's inequality, we get

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For the term J ′ 5 ðy, rÞ, since w ∈ A 1 , it follows directly from the A 1 condition and the inequality t ≤ ΦðtÞ that Furthermore, we use the generalized Hölder's inequality (29) to obtain In the last inequality, we have used the well-known fact that (see [26], for instance) It is equivalent to the inequality which is just a corollary of the well-known John-Nirenberg's inequality (see [7]) and the comparison property of A 1 weights. In addition, by the estimate (57) For the last term J ′ 6 ðy, rÞ we proceed as follows. Using the first part of Lemma 20 together with the facts that w ∈ A 1 and t ≤ ΦðtÞ = t · ð1 + log + tÞ, we deduce that Making use of the inequalities (32) and (57), we further obtain Summarizing the above discussions, we conclude that Recall that w q ∈ A 1 ⊂ A ∞ with 1 < q < ∞. We can now argue exactly as we did in the estimation of (83) to get (now, choose δ * in (17)) 14 Journal of Function Spaces Notice that the exponent δ * ð1/β − 1/sÞ is positive by our assumption, which guarantees that the last series is convergent. Therefore, by taking the L s ðμÞ-norm of both sides of (85) (with respect to the variable y) and then using Minkowski's inequality, (87) and (101), we have where the last inequality follows from (102). This completes the proof of Theorem 13.
Let bðxÞ be a BMO function on R n and 0 < γ < n. The related commutator formed by fractional maximal operator M γ and b is given by where the supremum is taken over all balls B containing x.
Obviously, ½b, M γ is a sublinear operator. It should be pointed out that ½b, M γ ð f Þ can be controlled pointwise by the expression given below. For any 0 < γ < n, x ∈ R n , and r > 0, we have ð Taking the supremum for all r > 0 on both sides of the above inequality, we get ð which is our desired result. Moreover, on the commutator ½b, M γ of the fractional maximal operator M γ , we also have the following result.
Taking into account (106) and Theorem 21 and then using the same arguments as in the proof of Theorem 11, we know that the conclusion of Theorem 11 still holds for the sublinear operator ½b, M γ .
Let L −γ/2 be the generalized fractional integrals of L for 0 < γ < n, and let b be a locally integrable function on R n . The generalized commutator generated by b and L −γ/2 is defined as follows: By the kernel estimate (68),
Hence, as a direct consequence of the above results, we can also obtain the following.

Proof of Theorem 15
This section is concerned with the proof of Theorem 15 and the corresponding result for generalized fractional integrals L −γ/2 .
Proof of Theorem 15. Let 1 < p ≤ α < s ≤ ∞ and f ∈ ðL p , L s Þ α ðw p , w q ; μÞ with w ∈ A p,q and μ ∈ Δ 2 . For any fixed ball B = Bðy, rÞ in R n , we are going to estimate the following expression: Decompose c , and 4B = Bðy, 4rÞ. By the linearity of the fractional integral operator I γ , the above expression (109) can be divided into two parts. That is, Let us first consider the term Iðy, rÞ. Applying the weighted ðL p , L q Þ-boundedness of I γ (see Theorem 1) and Hölder's inequality, we obtain Since w is a weight in the class A p,q , one has w q ∈ A q ⊂ A ∞ by Lemma 7(i). By definition, it reads which implies Since w q ∈ A q ⊂ A ∞ , then w q ∈ Δ 2 . Using the inequalities (113) and (16), we have where in the last equality we have used the hypothesis 1/s = 1/α − γ/n and 1/q = 1/p − γ/n. We now turn to estimate the second term IIðy, rÞ. For any x ∈ Bðy, rÞ, 16 Journal of Function Spaces Since both x and z are in Bðy, rÞ, ζ ∈ Bðy, 4rÞ c , by a purely geometric observation, we must have jx − ζj ≥ 2jx − zj. This fact along with the mean value theorem yields Furthermore, by using Hölder's inequality and A p,q condition on w, the last expression in (116) can be estimated as follows: where in the last equality we have used the fact that 1/α − 1/p − 1/s = −1/q again. From this, it readily follows that for any x ∈ Bðy, rÞ, Consequently, Therefore, by taking the L s ðμÞ-norm of (109) (with respect to the variable y) and then using Minkowski's inequality, (114) and (119), we get By taking the supremum over all r > 0, we are done.
For any f ∈ L p ðR n Þ, 1 ≤ p < ∞, Martell [35] defined a kind of sharp maximal function M # L f associated with the semigroup fe −tL g t>0 by the following expression: where t B = r 2 B and r B is the radius of the ball B. We say that f ∈ BMO L if the sharp maximal function M # L f ∈ L ∞ ðR n Þ, and we define k f k BMO L = kM # L f k L ∞ . Inspired by this notion and Theorem 15, a natural question for the generalized fractional integrals L −γ/2 is the following: can we get any result corresponding to Theorem 15 for the limiting case β = s? For this purpose, we need to introduce the following BMO -type space associated with the semigroup fe −tL g t>0 . Journal of Function Spaces Definition 25. Let 1 ≤ s ≤ ∞ and μ ∈ Δ 2 . We define the space ðBMO L , L s ÞðμÞ as the set of all locally integrable functions f satisfying k f k * * * < ∞, where and the L s ðμÞ-norm is taken with respect to the variable y. Based on the above notion, we can prove the following result.
Proof. Let 1 < p ≤ α < s ≤ ∞ and f ∈ ðL p , L s Þ α ðw p , w q ; μÞ with w ∈ A p,q and μ ∈ Δ 2 . In this situation, for any given ball B = Bðy, rÞ in R n , we need to consider the following expression: Decompose c , and 4B = Bðy, 4rÞ. Similarly, the above expression (123) can be divided into three parts. That is, First, let us consider the term I′ðy, rÞ. By (66) and Theorem 1, we know that the generalized fractional integral L −γ/2 is also bounded from L p ðw p Þ to L q ðw q Þ whenever w ∈ A p,q . This fact along with Hölder's inequality implies We now proceed exactly as we did in the proof of Theorem 15 For any x ∈ Bðy, rÞ and z ∈ Bðy, 4rÞ, by (64), we have jp r 2 ðx, zÞj ≤ C · ðr 2 Þ −n/2 .

Lemma 27.
For 0 < γ < n, the difference operator ðI − e −tL Þ L −γ/2 has an associated kernelK γ,t ðx, zÞ which satisfies the following estimate: Let us return to the proof of III ′ ðy, rÞ. By the above kernel estimate (133) Therefore, by taking the L s ðμÞ-norm of (123) (with respect to the variable y) and then using Minkowski's inequality, (126), (132), and (136), we get We end the proof by taking the supremum over all r > 0.

Some Results on Two-Weight Problems
In the last section, we consider related problems about twoweight, weak-type norm inequalities for I γ and ½b, I γ on weighted amalgam spaces. In [37], Cruz-Uribe and Perez considered the problem of finding sufficient conditions on a pair of weights ðw, νÞ which ensure the boundedness of the operator I γ from L p ðνÞ to WL p ðwÞ, where 1 < p < ∞. They gave a sufficient A p -type condition (see (138) below) and proved a two-weight, weak-type ðp, pÞ inequality for I γ (see also [38] for another, more simpler proof), which solved a problem posed by Sawyer and Wheeden in [39].