Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces

Journal of Inequalities and Applications, Jan 2017

In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators I α in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators [ b , I α ] formed by b and I α are established. Also we study related problems about two-weight, weak type inequalities for I α and [ b , I α ] in the Morrey-type spaces and give partial results. MSC: 42B20, 42B25, 42B35.

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Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces

Wang Journal of Inequalities and Applications Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces Hua Wang 0 0 Furthermore, by using Hölder's inequality and the A In this paper, we first introduce some new Morrey-type spaces containing generalized Morrey space and weighted Morrey space with two weights as special cases. Then we give the weighted strong type and weak type estimates for fractional integral operators Iα in these new Morrey-type spaces. Furthermore, the weighted strong type estimate and endpoint estimate of linear commutators [b, Iα] formed by b and Iα are established. Also we study related problems about two-weight, weak type inequalities for Iα and [b, Iα] in the Morrey-type spaces and give partial results. MSC: Primary 42B20; secondary 42B25; 42B35 fractional integral operators; commutators; Morrey-type spaces; BMO(Rn); weights; Orlicz spaces 1 Introduction  f (y) Iαf (x) := γ (α) Rn |x – y|n–α dy, and It is well known that the Hardy-Littlewood-Sobolev theorem states that the fractional © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. In , Segovia and Torrea [] proved that [b, Iα] is also bounded from Lp(wp) ( < p < n/α) to Lq(wq) whenever b ∈ BMO(Rn) (see also [] for the unweighted case). Theorem . ([]) Let  < α < n,  < p < n/α, /q = /p – α/n and w ∈ Ap,q. Suppose that b ∈ BMO(Rn), then the linear commutator [b, Iα] is bounded from Lp(wp) to Lq(wq). In , Cruz-Uribe and Fiorenza [] discussed the weighted endpoint inequalities for commutator of fractional integral operator and proved the following result (see also [] for the unweighted case). Theorem . ([]) Let  < α < n, p = , q = n/(n – α) and wq ∈ A. Suppose that b ∈ BMO(Rn), then, for any given σ >  and any bounded domain ⊂ Rn, there is a constant C > , which does not depend on f , and σ > , such that wq x ∈ /q ≤ C On the other hand, the classical Morrey space was originally introduced by Morrey in [] to study the local behavior of solutions to second order elliptic partial differential equations. This classical space and various generalizations on the Euclidean space Rn have been extensively studied by many authors. In [], Mizuhara introduced the generalized Morrey space Lp, (Rn) which was later extended and studied in []. In [], Komori and Shirai defined a version of the weighted Morrey space Lp,κ (v, u) which is a natural generalization of the weighted Lebesgue space. Let Iα be the fractional integral operator, and let [b, Iα] be its linear commutator. The main purpose of this paper is twofold. We first define a new kind of Morrey-type spaces Mp,θ (v, u) containing generalized Morrey space Lp, (Rn) and weighted Morrey space Lp,κ (v, u) as special cases. As the Morrey-type spaces may be considered as an extension of the weighted Lebesgue space, it is natural and important to study the weighted boundedness of Iα and [b, Iα] in these new spaces. Then we will establish the weighted strong type and endpoint estimates for Iα and [b, Iα] in these Morrey-type spaces Mp,θ (v, u) for all  ≤ p < ∞. In addition, we will discuss two-weight, weak type norm inequalities for Iα and [b, Iα] in Mp,θ (v, u) and give some partial results. 2 Statements of the main results 2.1 Notations and preliminaries centered at x of radius r, B(x, r)c denote its complement and |B(x, r)| be the Lebesgue measure of the ball B(x, r). A non-negative function w defined on Rn is called a weight if it is locally integrable. We first recall the definitions of two weight classes; Ap and Ap,q. Definition . (Ap weights []) A weight w is said to belong to the class Ap for  < p < ∞, if there exists a positive constant C such that, for any ball B in Rn, w(x)–p /p dx ≤ C < ∞, where p is the dual of p such that /p + /p = . The class A is defined replacing the above inequality by w(x) dx ≤ C · esxs∈iBnf w(x), for any ball B in Rn. We also define A∞ = ≤p<∞ Ap. Definition . (Ap,q weights []) A weight w is said to belong to the class Ap,q ( < p, q < ∞), if there exists a positive constant C such that, for any ball B in Rn, w(x)–p dx ≤ C < ∞. ≤ C < ∞. w(B) ≤ C · w(B). Given a ball B and λ > , λB denotes the ball with the same center as B whose radius is λ times that of B. For a given weight function w and a Lebesgue measurable set E, we denote the characteristic function of E by χE, the Lebesgue measure of E by |E| and the weighted measure of E by w(E), where w(E) := E w(x) dx. Given a weight w, we say that w satisfies the doubling condition if there exists a universal constant C >  such that, for any ball B in Rn, we have When w satisfies this doubling condition (.), we denote w ∈  for brevity. We know that if w is in A∞, then w ∈  (see []). Moreover, if w ∈ A∞, then, for any ball B and any measurable subset E of B, there exists a number δ >  independent of E and B such that (see []) f Lp(w) := < ∞. We also denote by WLp(w) ( ≤ p < ∞) the weighted weak Lebesgue space consisting of all measurable functions f such that f WLp(w) := sup λ · w x ∈ Rn : f (x) > λ λ> /p < ∞. We next recall some definitions and basic facts about Orlicz spaces needed for the proofs of the main results. For further information on this subject, we refer to []. A function A : [, +∞) → [, +∞) is said to be a Young function if it is continuous, convex and strictly increasing satisfying A() =  and A(t) → +∞ as t → +∞. An important example of Young function is A(t) = tp( + log+ t)p with some  ≤ p < ∞. Given a Young function A, we define the A-average of a function f over a ball B by means of the Luxemburg norm: dx ≤  . f A,B = f (x) · g(x) dx ≤  f A,B g A¯,B, f (x) · g(x) dx ≤  f L log L,B g exp L,B. A–(t) · B–(t) ≤ C–(t), f · g C,B ≤  f A,B g B,B. that is, the Luxemburg norm coincides with the normalized Lp norm. Recall that the following generalization of Hölder’s inequality holds: where A¯ is the complementary Young function associated to A, which is given by A¯(s) := sup≤t<∞[st – A(t)],  ≤ s < ∞. Obviously, (t) = t · ( + log+ t) is a Young function and its complementary Young function is ¯ (t) ≈ et – . In the present situation, we denote f ,B and g ¯ ,B by f L log L,B and g exp L,B, respectively. So we have There is a further generalization of Hölder’s inequality that turns out to be useful for our purpose (see []): Let A, B, and C be Young functions such that, for all t > , Let us now recall the definition of the space of BMO(Rn) (see []). BMO(Rn) is the Banach function space modulo constants with the norm · ∗ defined by b(x) – bB dx < ∞, bB := 2.2 Morrey-type spaces Let us begin with the definitions of the weighted Morrey space with two weights and generalized Morrey space. Lp,κ (v, u) := f ∈ Llpoc(v) : f Lp,κ (v,u) < ∞ , Definition . Let  ≤ p < ∞,  < κ <  and w be a weight on Rn. We denote by W Lp,κ (w) the weighted weak Morrey space of all measurable functions f for which /p < ∞. Let = (r), r > , be a growth function; that is, a positive increasing function on (, +∞) and satisfy the following doubling condition: (r) ≤ D · (r), for all r > , Definition . ([]) Let  ≤ p < ∞ and be a growth function on (, +∞). Then the generalized Morrey space Lp, (Rn) is defined by f Lp, (Rn) := Definition . Let  ≤ p < ∞ and be a growth function on (, +∞). We denote by W Lp, (Rn) the generalized weak Morrey space of all measurable functions f for which f WLp, (Rn) := sup sup B(x,r) λ> In order to unify the definitions given above, we now introduce Morrey-type spaces associated to θ as follows. Let  ≤ κ < . Assume that θ (·) is a positive increasing function defined in (, +∞) and satisfies the following Dκ condition: θ (ξ ) θ (ξ ) ξ κ ≤ C · (ξ )κ , for any  < ξ < ξ < +∞, Definition . Let  ≤ p < ∞,  ≤ κ <  and θ satisfy the Dκ condition (.). For two weights u and v on Rn, we denote by Mp,θ (v, u) the generalized weighted Morrey space, the space of all locally integrable functions f with finite norm. Mp,θ (v, u) := f ∈ Llpoc(v) : f Mp,θ (v,u) < ∞ , where the norm is given by Here the supremum is taken over all balls B in Rn. If v = u, then we denote Mp,θ (v), for short. Furthermore, we denote by W Mp,θ (v) the generalized weighted weak Morrey space of all measurable functions f for which /p < ∞. Lp,κ (v, u) = Mp,θ (v, u)|θ(x)=xκ , W Lp,κ (v) = W Mp,θ (v)|θ(x)=xκ . Also, note that if θ (x) ≡ , then Mp,θ (v) = Lp(v) and W Mp,θ (v) = WLp(v), the classical weighted Lebesgue and weak Lebesgue spaces. The aim of this paper is to extend Theorems .-. to the corresponding Morrey-type spaces. Our main results on the boundedness of Iα in the Morrey-type spaces associated to θ can be formulated as follows. Theorem . Let  < α < n,  < p < n/α, /q = /p – α/n and w ∈ Ap,q. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < p/q, then the fractional integral operator Iα is bounded from Mp,θ (wp, wq) into Mq,θq/p (wq). Theorem . Let  < α < n, p = , q = n/(n – α) and w ∈ A,q. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < /q, then the fractional integral operator Iα is bounded from M,θ (w, wq) into W Mq,θq (wq). Theorem . Let  < α < n,  < p < n/α, /q = /p – α/n and w ∈ Ap,q. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < p/q and b ∈ BMO(Rn), then the commutator operator [b, Iα] is bounded from Mp,θ (wp, wq) into Mq,θq/p (wq). To obtain endpoint estimate for the linear commutator [b, Iα], 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 [, ] for instance) |f (σx)| · w(x) dx ≤  . When A(t) = t, this norm is denoted by · L(w),B, and when (t) = t · ( + log+ t), this norm is also denoted by · L log L(w),B. The complementary Young function of (t) is ¯ (t) ≈ et –  with mean Luxemburg norm denoted by · exp L(w),B. For w ∈ A∞ and for every ball B in Rn, we can also show the weighted version of (.). Namely, the following generalized Hölder inequality in the weighted setting  w(B) B f (x) · g(x) w(x) dx ≤ C f L log L(w),B g exp L(w),B Here the supremum is taken over all balls B in Rn. If v = u, then we denote ML log L(v) for ,θ brevity. holds for any ball B ⊂ Rn. From this, we can further see that, when θ satisfies the Dκ condition (.) with  ≤ κ < , and u is another weight function, f (x) · v(x) dx ≤ f L log L(v),B θ (u(B)) B f (x) · v(x) dx = θ (vu((BB))) · v(B) B f (x) · v(x) dx = θ (vu((BB))) · f L(v),B v(B) ≤ θ (u(B)) · f L log L(v),B. Definition . Let p =  and  < κ < . For two weights u and v on Rn, we denote by ,κ LL log L(v, u) the weighted Morrey space of L log L type, the space of all locally integrable functions f defined on Rn with finite norm f LL,κlogL(v,u). We have In this situation, we have LL,κlog L(v, u) ⊂ L,κ (v, u). In the endpoint case p = , we will prove the following weak type L log L estimate of the linear commutator [b, Iα] in the Morrey-type space associated to θ . Theorem . Let  < α < n, p = , q = n/(n – α) and w ∈ A,q. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < /q and b ∈ BMO(Rn), then, for any given σ >  and any ball B ⊂ Rn, there exists a constant C >  independent of f , B and σ >  such that θ (wq(B)) wq x ∈ B : [b, Iα](f )(x) > σ /q ≤ C · where (t) = t · ( + log+ t). From the definitions, we can roughly say that the commutator operator [b, Iα] is bounded from ML log L(w, wq) into W Mq,θq (wq). ,θ In particular, if we take θ (x) = xκ with  < κ < , then we immediately get the following strong type estimate and endpoint estimate of Iα and [b, Iα] in the weighted Morrey spaces. Corollary . Let  < α < n,  < p < n/α, /q = /p – α/n and w ∈ Ap,q. If  < κ < p/q, then the fractional integral operator Iα is bounded from Lp,κ (wp, wq) into Lq,κq/p(wq). Corollary . Let  < α < n, p = , q = n/(n – α) and w ∈ A,q. If  < κ < /q, then the fractional integral operator Iα is bounded from L,κ (w, wq) into W Lq,κq(wq). Corollary . Let  < α < n, p = , q = n/(n – α) and w ∈ A,q. If  < κ < /q and b ∈ BMO(Rn), then, for any given σ >  and any ball B ⊂ Rn, there exists a constant C >  independent of f , B and σ >  such that wq(B)κ wq x ∈ B : [b, Iα](f )(x) > σ /q ≤ C · where (t) = t · ( + log+ t). Theorem . Let  < α < n,  < p < n/α, /q = /p – α/n and w ∈ Ap,q. If κ = p/q, then the fractional integral operator Iα is bounded from Lp,κ (wp, wq) into BMO(Rn). It should be pointed out that Corollaries . through . were given by Komori and Shirai in []. Corollary . and Theorem . are new results. Definition . In the unweighted case (when u = v ≡ ), we denote the corresponding unweighted Morrey-type spaces associated to θ by Mp,θ (Rn), W Mp,θ (Rn) and ML log L(Rn), respectively. That is, let  ≤ p < ∞ and θ satisfy the Dκ condition (.) with ,θ  ≤ κ < , we define ML,θlog L Rn := f ∈ Lloc Rn : f θ (||BB||) · f L log L,B < ∞ . Corollary . Let  < α < n,  < p < n/α and /q = /p – α/n. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < p/q, then the fractional integral operator Iα is bounded from Mp,θ (Rn) into Mq,θq/p (Rn). Corollary . Let  < α < n, p = , and q = n/(n – α). Assume that θ satisfies the Dκ condition (.) with  ≤ κ < /q, then the fractional integral operator Iα is bounded from M,θ (Rn) into W Mq,θq (Rn). Corollary . Let  < α < n,  < p < n/α, and /q = /p – α/n. Assume that θ satisfies the Dκ condition (.) with  ≤ κ < p/q and b ∈ BMO(Rn), then the commutator operator [b, Iα] is bounded from Mp,θ (Rn) into Mq,θq/p (Rn). Corollary . Let  < α < n, p = , and q = n/(n – α). Assume that θ satisfies the Dκ condition (.) with  ≤ κ < /q and b ∈ BMO(Rn), then, for any given σ >  and any ball B ⊂ Rn, there exists a constant C >  independent of f , B and σ >  such that x ∈ B : [b, Iα](f )(x) > σ /q ≤ C · where (t) = t · ( + log+ t). We also introduce the generalized Morrey space of L log L type. Definition . Let p =  and be a growth function on (, +∞). We denote by LL log L(Rn) the generalized Morrey space of L log L type, which is given by , f LL,logL(Rn) := r>s;Bu(px,r) Below we are going to show that our new Morrey-type spaces can be reduced to generalized Morrey spaces. In fact, assume that θ (·) is a positive increasing function defined in (, +∞) and satisfies the Dκ condition (.) with some  ≤ κ < . For any fixed x ∈ Rn and r > , we set (r) := θ (|B(x, r)|). Observe that (r) = θ B(x, r) = θ n B(x, r) . Then it is easy to verify that (r), r > , is a growth function with doubling constant D( ) :  ≤ D( ) < n. Hence, by the choice of mentioned above, we get Mp,θ (Rn) = Lp, (Rn) and W Mp,θ (Rn) = W Lp, (Rn) for p ∈ [, +∞), and ML,θlog L(Rn) = LL,log L(Rn). Therefore, by the above unweighted results (Corollaries .-.), we can also obtain strong type estimate and endpoint estimate of Iα and [b, Iα] in the generalized Morrey spaces. Corollary . Let  < α < n,  < p < n/α and /q = /p – α/n. Suppose that satisfies the doubling condition (.) and  ≤ D( ) < np/q, then the fractional integral operator Iα is bounded from Lp, (Rn) into Lq, q/p (Rn). Corollary . Let  < α < n, p =  and q = n/(n – α). Suppose that satisfies the doubling condition (.) and  ≤ D( ) < n/q, then the fractional integral operator Iα is bounded from L, (Rn) into W Lq, q (Rn). Corollary . Let  < α < n,  < p < n/α and /q = /p – α/n. Suppose that satisfies the doubling condition (.) with  ≤ D( ) < np/q and b ∈ BMO(Rn), then the commutator operator [b, Iα] is bounded from Lp, (Rn) into Lq, q/p (Rn). Corollary . Let  < α < n, p =  and q = n/(n – α). Suppose that satisfies the doubling condition (.) with  ≤ D( ) < n/q and b ∈ BMO(Rn), then, for any given σ >  and any ball B(x, r) ⊂ Rn, there exists a constant C >  independent of f , B(x, r) and σ >  such that x ∈ B(x, r) : [b, Iα](f )(x) > σ /q ≤ C · where (t) = t · ( + log+ t). Theorem . Let  < α < n,  < p < n/α and /q = /p – α/n. Suppose that following condition: (r) ≤ C · rnp/q, for all r > , where C = C( ) >  is a universal constant independent of r. Then the fractional integral operator Iα is bounded from Lp, (Rn) into BMO(Rn). It is worth pointing out that Corollaries . through . were obtained by Nakai in []. Corollary . and Theorem . seem to be new, as far as we know. 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, C independent of quantities A and B such that CA ≤ B ≤ CA. 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. 3 Proofs of Theorems 2.1 and 2.2 Proof of Theorem . Here and in the following, for any positive number γ > , we denote f γ (x) := [f (x)]γ by convention. For example, when  < p < q < ∞, we have [f q/p(x)]/q = [f (x)]/p. Let f ∈ Mp,θ (wp, wq) with  < p, q < ∞ and w ∈ Ap,q. For an arbitrary point x ∈ Rn, set B = B(x, rB) for the ball centered at x and of radius rB, B = B(x, rB). We represent f as := I + I. Below we will give the estimates of I and I, respectively. By the weighted (Lp, Lq)boundedness of Iα (see Theorem .), we have Since w ∈ Ap,q, we get wq ∈ Aq ⊂ A∞ by Lemma .(i). Moreover, since  < wq(B) < wq(B) < +∞ when wq ∈ Aq with  < q < ∞, then by the Dκ condition (.) of θ and inequality (.), we obtain As for the term I, it is clear that, when x ∈ B and y ∈ (B)c, we get |x – y| ≈ |x – y|. We then decompose Rn into a geometrically increasing sequence of concentric balls, and we obtain the following pointwise estimate: ≤ C (B)c |x|f–(yy)||n–α dy From this, it follows that wq(B)/q ∞  I ≤ C · θ (wq(B))/p j= |j+B|–α/n j+B By using Hölder’s inequality and the Ap,q condition on w, we get w(y)–p dy I ≤ C f Mp,θ (wp,wq) × ∞ θ (wq(j+B))/p wq(B)/q θ (wq(B))/p · wq(j+B)/q . Notice that wq ∈ Aq ⊂ A∞ for  < q < ∞, then by using the Dκ condition (.) of θ again, the inequality (.) with exponent δ >  and the fact that  ≤ κ < p/q, we find that ≤ C ≤ C ≤ C, which gives our desired estimate I ≤ C f Mp,θ (wp,wq). Combining the above estimates for I and I, and then taking the supremum over all balls B ⊂ Rn, we complete the proof of Theorem .. f = f · χB + f · χ(B)c := f + f; then, for any given σ > , by the linearity of the fractional integral operator Iα , one can write  ≤ θ (wq(B)) σ · wq x ∈ B : Iα(f)(x) > σ / + θ (wq(B)) σ · wq x ∈ B : Iα(f)(x) > σ / := I + I. We first consider the term I. By the weighted weak (, q)-boundedness of Iα (see Theorem .), we have f (x) w(x) dx wq(B)/q ∞  ≤ C · θ (wq(B)) j= |j+B|–α/n j+B Since w is in the class A,q, we get wq ∈ A ⊂ A∞ by Lemma .(ii). Moreover, since  < wq(B) < wq(B) < +∞ when wq ∈ A, then we apply the Dκ condition (.) of θ and inequality (.) to obtain As for the term I, it follows directly from Chebyshev’s inequality and the pointwise estimate (.) that Moreover, by applying Hölder’s inequality and then the reverse Hölder inequality in succession, we can show that wq ∈ A if and only if w ∈ A ∩ RHq (see []), where RHq denotes the reverse Hölder class. Another application of A condition on w shows that j+B ≤ C f M,θ (w,wq) · |wj(+jB+|αB/)n · θ wq j+B . f (y) w(y) dy wq j+B /q = ≤ C · j+B /q– · w j+B , which is equivalent to Recall that wq ∈ A ⊂ A∞, therefore, by using the Dκ condition (.) of θ again, the inequality (.) with exponent δ∗ >  and the fact that  ≤ κ < /q, we get wq(B)/q ∞ wq(B)/q–κ · wq(j+B)/q ≤ C j= wq(j+B)/q–κ ≤ C ≤ C ≤ C, which implies our desired estimate I ≤ C f M,θ (w,wq). Summing up the above estimates for I and I, and then taking the supremum over all balls B ⊂ Rn and all σ > , we finish the proof of Theorem .. 4 Proofs of Theorems 2.3 and 2.4 To prove our main theorems in this section, we need the following lemma about BMO functions. Lemma . Let b be a function in BMO(Rn). (i) For every ball B in Rn and for all j ∈ Z+, then |bj+B – bB| ≤ C · (j + ) b ∗. Proof of Theorem . Let f ∈ Mp,θ (wp, wq) with  < p, q < ∞ and w ∈ Ap,q. For each fixed ball B = B(x, rB) ⊂ Rn, as before, we represent f as f = f + f, where f = f · χB, B = B(x, rB) ⊂ Rn. By the linearity of the commutator operator [b, Iα], we write Since w is in the class Ap,q, we get wq ∈ Aq ⊂ A∞ by Lemma .(i). By using Theorem ., the Dκ condition (.) of θ and inequality (.), we obtain := J + J. In the proof of Theorem ., we have already shown that (see (.)) Following the same argument as in (.), we can also prove that ≤ C ≤ C b(y) – bB · f (y) dy. Hence, from the above two pointwise estimates for |Iα(f)(x)| and |Iα([bB – b]f)(x)|, it follows that wq(B)/q ∞  + C · θ (wq(B))/p j= |j+B|–α/n j+B wq(B)/q ∞  + C · θ (wq(B))/p j= |j+B|–α/n j+B |bj+B – bB| · f (y) dy b(y) – bj+B · f (y) dy := J + J + J. Below we will give the estimates of J, J and J, respectively. To estimate J, note that wq ∈ Aq ⊂ A∞ with  < q < ∞. Using the second part of Lemma ., Hölder’s inequality, and the Ap,q condition on w, we obtain wq(B)/q ∞  ≤ C b ∗ · θ (wq(B))/p j= |j+B|–α/n w(y)–p dy ∞ θ (wq(j+B))/p wq(B)/q θ (wq(B))/p · wq(j+B)/q wq(B)/q ∞ (j + ) J ≤ C b ∗ · θ (wq(B))/p × j= |j+B|–α/n j+B wq(B)/q ∞ (j + ) ≤ C b ∗ · θ (wq(B))/p j= |j+B|–α/n w(y)–p dy θ (wq(j+B))/p wq(B)/q θ (wq(B))/p · wq(j+B)/q . where in the last inequality we have used the estimate (.). To estimate J, applying the first part of Lemma ., Hölder’s inequality, and the Ap,q condition on w, we can deduce that For any j ∈ Z+, since  < wq(B) < wq(j+B) < +∞ when wq ∈ Aq with  < q < ∞, by using the Dκ condition (.) of θ and the inequality (.) with exponent δ > , we thus obtain θ (wq(j+B))/p wq(B)/q θ (wq(B))/p · wq(j+B)/q ≤ C ≤ C ≤ C ≤ C, where the last series is convergent since the exponent δ(/q – κ/p) is positive. This implies our desired estimate J ≤ C f Mp,θ (wp,wq). It remains to estimate the last term J. An application of Hölder’s inequality shows that wq(B)/q ∞  J ≤ C · θ (wq(B))/p j= |j+B|–α/n If we set μ(y) = w(y)–p , then we have μ ∈ Ap ⊂ A∞ because w ∈ Ap,q by Lemma .(i). Thus, it follows from the second part of Lemma . and the Ap,q condition that ≤ C b ∗ · μ j+B /p = C b ∗ · w(y)–p dy Therefore, in view of the estimates (.) and (.), we conclude that wq(B)/q ∞  J ≤ C b ∗ · θ (wq(B))/p j= wq(j+B)/q ∞ θ (wq(j+B))/p wq(B)/q θ (wq(B))/p · wq(j+B)/q Summarizing the estimates derived above and then taking the supremum over all balls B ⊂ Rn, we complete the proof of Theorem .. Proof of Theorem . For any fixed ball B = B(x, rB) in Rn, as before, we represent f as f = f + f, where f = f · χB, B = B(x, rB) ⊂ Rn. Then, for any given σ > , by the linearity of the commutator operator [b, Iα], we write θ (wq(B)) · wq x ∈ B : [b, Iα](f )(x) > σ ≤ θ (wq(B)) · wq x ∈ B : [b, Iα](f)(x) > σ / + θ (wq(B)) · wq x ∈ B : [b, Iα](f)(x) > σ / := J + J. We first consider the term J. By using Theorem . and the previous estimate (.), we get θ (wq(B))  = C · θ (wq(B)) · θ (wq(B)) B θ (wq(B)) w(B) ≤ C · θ (wq(B)) · θ (wq(B)) · wq(B)κ w(B) J ≤ C · wq(B)κ · θ (wq(B)) · ≤ C · L log L(w),B L log L(w),B Since w is a weight in the class A,q, one has wq ∈ A ⊂ A∞ by Lemma .(ii). Moreover, since  < wq(B) < wq(B) < +∞ when wq ∈ A, by the Dκ condition (.) of θ and inequality (.), we have which is our desired estimate. We now turn to dealing with the term J. Recall that the inequality is valid. So we can further decompose J as := J + J. By using the previous pointwise estimate (.), Chebyshev’s inequality together with Lemma .(ii), we deduce that wq(B)/q |f σ(y)| dy × θ (wq(B)) . Furthermore, note that t ≤ (t) = t · ( + log+ t) for any t > . As we pointed out in Theorem . that wq ∈ A if and only if w ∈ A ∩ RHq, it then follows from the A condition and the previous estimate (.) that In view of (.) and (.), we have J ≤ C ≤ C ≤ C = C J ≤ C · ≤ C · ≤ C · j= w(j+B) j+B LlogL(w),j+B |j+B|α/n θ (wq(j+B)) × w(j+B) · θ (wq(B)) LlogL(w),j+B ∞ |j+B|α/n θ (wq(j+B)) × j= w(j+B) · θ (wq(B)) ∞ θ (wq(j+B)) wq(B)/q θ (wq(B)) · wq(j+B)/q On the other hand, applying the pointwise estimate (.) and Chebyshev’s inequality, we get wq(B)/q C ∞  ≤ θ (wq(B)) · σ j= |j+B|–α/n j+B wq(B)/q C ∞  ≤ θ (wq(B)) · σ j= |j+B|–α/n j+B b(y) – bB · f (y) dy b(y) – bj+B · f (y) dy wq(B)/q C ∞  + θ (wq(B)) · σ j= |j+B|–α/n j+B |bj+B – bB| · f (y) dy := J + J. C J ≤ σ · θ (wq(B)) j= w(j+B) j+B b(y) – bj+B · f (y) w(y) dy ≤ C · θ (wq(B)) j= w(j+B) j+B b(y) – bj+B · Furthermore, we use the generalized Hölder inequality with weight (.) to obtain wq(B)/q ∞ J ≤ C · θ (wq(B)) j= wq(B)/q ∞ ≤ C b ∗ · θ (wq(B)) j= L log L(w),j+B L log L(w),j+B In the last inequality, we have used the well-known fact that (see []) It is equivalent to the inequality which is just a corollary of the well-known John-Nirenberg inequality (see []) and the comparison property of A weights. Hence, by the estimates (.) and (.), J ≤ C b ∗ ≤ C · ≤ C · L log L(w),j+B For the last term J we proceed as follows. Using the first part of Lemma . together with the facts w ∈ A and t ≤ (t) = t · ( + log+ t), we deduce that wq(B)/q ∞  J ≤ C · θ (wq(B)) j= (j + ) b ∗ · |j+B|–α/n j+B wq(B)/q ∞ |j+B|α/n ≤ C · θ (wq(B)) j= (j + ) b ∗ · w(j+B) j+B wq(B)/q ∞ (j + )|j+B|α/n ≤ C b ∗ · θ (wq(B)) j= w(j+B) Making use of the inequalities (.) and (.), we further obtain wq(B)/q ∞ J ≤ C · θ (wq(B)) j= = C · L log L(w),j+B L log L(w),j+B Recall that wq ∈ A ⊂ A∞ with  < q < ∞. We can now argue exactly as we did in the estimation of J to get (now choose δ∗ in (.)) ≤ C ≤ C ≤ C. Notice that the exponent δ∗(/q – κ) is positive by the choice of κ , which guarantees that the last series is convergent. If we substitute this estimate (.) into the term J, then we get the desired inequality J ≤ C · This completes the proof of Theorem .. 5 Proofs of Theorems 2.5 and 2.6 := I + II. Proof of Theorem . Let f ∈ Mp,θ (wp, wq) with  < p, q < ∞ and w ∈ Ap,q. For any given ball B = B(x, rB) in Rn, it suffices to prove that the following inequality holds. Decompose f as f = f + f, where f = f · χB, f = f · χ(B)c , B = B(x, rB). By the linearity of the fractional integral operator Iα , the left-hand side of (.) can be divided into two parts. That is, Iαf (x) – (Iαf )B dx ≤ C f Lp,κ (wp,wq) First let us consider the term I. Applying the weighted (Lp, Lq)-boundedness of Iα (see Theorem .) and Hölder’s inequality, we obtain  I ≤ |B| B w(x)–q dx w(x)–q dx w(x)–q dx . Using the inequalities (.) and (.) and noting the fact ≤ C, which implies Since wq ∈ Aq ⊂ A∞, wq ∈ that κ = p/q, we have w(x)–q dx Now we estimate II. For any x ∈ B,   (B)c |x – z|n–α – |y – z|n–α · f (z) dz dy. Since both x and y are in B, z ∈ (B)c, by a purely geometric observation, we must have |x – z| ≥ |x – y|. This fact along with the mean value theorem yields ≤ C (B)c |z – x|n–α+ · f (z) dz ≤ C w(y)–p dy From the pointwise estimate (.), it readily follows that  II = |B| B Iαf(x) – (Iαf)B dx ≤ C f Lp,κ (wp,wq). By combining the above estimates for I and II, we are done. Proof of Theorem . Let f ∈ Lp, (Rn) with  < p < ∞. For any given ball B = B(x, rB) in Rn, it is sufficient to prove that the following inequality Iαf (x) – (Iαf )B dx ≤ C f Lp, (Rn) holds. Decompose f as f = f + f, where f = f · χB, f = f · χ(B)c , B = B(x, rB). As in the proof of Theorem ., we can also divide the left-hand side of (.) into two parts. That is, |B(x, rB)| B(x,rB) |B(x, rB)| B(x,rB) |B(x, rB)| B(x,rB) := I + II . |B(x, rB)| B(x,rB) I ≤ |B(x, rB)| B(x,rB) ≤ C ≤ C f Lp, (Rn) × ≤ C f Lp, (Rn) × ≤ C f Lp, (Rn). Applying our assumption (.) on , we further have  j= j · |B(x, j+rB)|–α/n B(x,j+rB) Moreover, by using Hölder’s inequality and the assumption (.) on , we can deduce that Iαf(x) – (Iαf)B dx ≤ C f Lp, (Rn). By combining the above estimates for I and II , we are done. 6 Partial results on two-weight problems In the last section, we consider related problems about two-weight, weak type norm inequalities for Iα and [b, Iα]. In [], Cruz-Uribe and Pérez considered the problem of finding sufficient conditions on a pair of weights (u, v) which ensure the boundedness of the operator Iα from Lp(v) to WLp(u), where  < p < ∞. They gave a sufficient Ap-type condition (see (.) below), and proved a two-weight, weak type (p, p) inequality for Iα (see also [] for another, simpler proof ), which solved a problem posed by Sawyer and Wheeden in []. Theorem . ([, ]) Let  < α < n and  < p < ∞. Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, v(x)–p /p dx ≤ C < ∞. u x ∈ Rn : Iαf (x) > σ ≤ σCp Rn f (x) pv(x) dx, for any σ > , Moreover, in [], Li improved this result by replacing the ‘power bump’ in (.) by a smaller ‘Orlicz bump’. On the other hand, in [], Liu and Lu obtained a sufficient Ap-type condition for the commutator [b, Iα] to satisfy the two-weight weak type (p, p) inequality, where  < p < ∞. That condition is an Ap-type condition in the scale of Orlicz spaces (see (.) below). Theorem . ([]) Let  < α < n,  < p < ∞ and b ∈ BMO(Rn). Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, v–/p A,Q ≤ C < ∞, u x ∈ Rn : [b, Iα](f )(x) > σ ≤ σCp Rn f (x) pv(x) dx, for any σ > , Here and in the following, all cubes are assumed to have their sides parallel to the coordinate axes, Q(x, ) will denote the cube centered at x and has side length . For any cube Q(x, ) and any λ > , we denote by λQ the cube with the same center as Q whose side length is λ times that of Q, i.e., λQ := Q(x, λ ). We now extend the results mentioned above to the Morrey-type spaces associated to θ . Theorem . Let  < α < n and  < p < ∞. Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, (.) holds. If θ satisfies the Dκ condition (.) with  ≤ κ <  and u ∈ , then the fractional integral operator Iα is bounded from Mp,θ (v, u) into W Mp,θ (u). Theorem . Let  < α < n,  < p < ∞ and b ∈ BMO(Rn). Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, (.) holds. If θ satisfies the Dκ condition (.) with  ≤ κ <  and u ∈ A∞, then the linear commutator [b, Iα] is bounded from Mp,θ (v, u) into W Mp,θ (u). Proof of Theorem . Let f ∈ Mp,θ (v, u) with  < p < ∞. For arbitrary x ∈ Rn, set Q = Q(x, ) for the cube centered at x and with the side length . Let f = f · χQ + f · χ(Q)c := f + f,  ≤ θ (u(Q))/p σ · u x ∈ Q : Iα(f)(x) > σ / + θ (u( Q))/p σ · u x ∈ Q : Iα(f)(x) > σ / := K + K. Using Theorem ., the Dκ condition (.) of θ and inequality (.) (consider cube Q instead of ball B), we get As for the term K, using the same methods and steps as in dealing with I in Theorem ., we can also obtain, for any x ∈ Q, This pointwise estimate (.) together with Chebyshev’s inequality implies u(Q)/p  ≤ C · θ (u(Q))/p j= |j+Q|–α/n j+Q Moreover, an application of Hölder’s inequality shows that u(Q)/p ∞  K ≤ C · θ (u(Q))/p j= |j+Q|–α/n v(y)–p /p dy v(y)–p /p dy u(Q) ≥ D · u(Q), for any cube Q ⊂ Rn, ≤ C, The last inequality is obtained by the Ap-type condition (.) on (u, v). Furthermore, since u ∈ , we can easily check that there exists a reverse doubling constant D = D(u) >  independent of Q such that (see Lemma . in []) For any j ∈ Z+, since  < u(Q) < u(j+Q) < +∞ when u is a weight function, by the Dκ condition (.) of θ with  ≤ κ < , we can see that In addition, we apply Hölder’s inequality with exponent r to get u j+Q = u(y) dy ≤ j+Q /r Hence, in view of (.) and (.) derived above, we have v(y)–p /p dy v(y)–p /p dy where the last series is convergent since the reverse doubling constant D >  and  ≤ κ < . This yields our desired estimate K ≤ C f Mp,θ (v,u). Summing up the above estimates for K and K, and then taking the supremum over all cubes Q ⊂ Rn and all σ > , we finish the proof of Theorem .. f = f + f,  θ (u(Q))/p σ · u x ∈ Q : [b, Iα](f )(x) > σ  ≤ θ (u(Q))/p σ · u x ∈ Q : [b, Iα](f)(x) > σ / + θ (u( Q))/p σ · u x ∈ Q : [b, Iα](f)(x) > σ / := K + K. Applying Theorem ., the Dκ condition (.) of θ and inequality (.) (consider cube Q instead of ball B), we get θ (u(Q))/p ≤ C f Mp,θ (v,u) · θ (u(Q))/p Consequently, we can further divide K into two parts,  K ≤ θ (u(Q))/p σ · u x ∈ Q : ξ (x) > σ / + θ (u( Q))/p σ · u x ∈ Q : η(x) > σ / := K + K. For the term K, it follows from the pointwise estimate (.) mentioned above and Chebyshev’s inequality that u(Q)/p ∞  ≤ C · θ (u(Q))/p j= |j+Q|–α/n j+Q where in the last inequality we have used the fact that Lemma .(ii) still holds when B replaced by Q and u is an A∞ weight. Repeating the arguments in the proof of Theorem ., we can show that K ≤ C f Mp,θ (v,u). As for the term K, we can show the following pointwise estimate in the same manner as in the proof of Theorem .: ≤ C  j= |j+Q|–α/n j+Q b(y) – bQ · f (y) dy. This, together with Chebyshev’s inequality yields u(Q)/p ∞  ≤ C · θ (u(Q))/p · j= |j+Q|–α/n j+Q b(y) – bQ · f (y) dy u(Q)/p ∞  ≤ C · θ (u(Q))/p · j= |j+Q|–α/n j+Q b(y) – bj+Q · f (y) dy u(Q)/p ∞  + C · θ (u(Q))/p · j= |j+Q|–α/n j+Q |bj+Q – bQ| · f (y) dy := K + K. An application of Hölder’s inequality leads to u(Q)/p ∞  K ≤ C · θ (u(Q))/p · j= |j+Q|–α/n × j+Q /p (b – bj+Q) · v–/p C,j+Q, A(t) ≈ tp  + log+ t p B(t) ≈ et – . ≤ C b ∗ · v–/p A,j+Q. . Moreover, in view of (.) and (.), we can K ≤ C b ∗ f Mp,θ (v,u) ∞ u(j+Q)κ/p u(Q)/p u(Q)κ/p · |j+Q|/p–α/n · v–/p A,j+Q The last inequality is obtained by the Ap-type condition (.) on (u, v) and the estimate (.). It remains to estimate the last term K. Applying Lemma .(i) (use Q instead of B) and Hölder’s inequality, we get u(Q)/p ∞ (j + ) b ∗ K ≤ C · θ (u(Q))/p j= |j+Q|–α/n j+Q f (y) dy u(Q)/p ∞ (j + ) b ∗ ≤ C · θ (u(Q))/p j= |j+Q|–α/n v(y)–p /p dy v(y)–p /p dy Let C(t), A(t) be the same as before. Obviously, C(t) ≤ A(t) for all t > , then it is not difficult to see that, for any given cube Q ⊂ Rn, we have f C,Q ≤ f A,Q by definition, which implies that condition (.) is stronger than condition (.). This fact together with (.) and (.) yields u(Q)(–κ)/p u(j+Q)/p (j + ) · u(j+Q)(–κ)/p · |j+Q|–α/n v(y)–p /p dy u(Q)(–κ)/p |j+Q|/(r p) (j + ) · u(j+Q)(–κ)/p · |j+Q|–α/n v(y)–p /p dy u(Q)(–κ)/p (j + ) · u(j+Q)(–κ)/p . Moreover, by our additional hypothesis on u : u ∈ A∞ and inequality (.) with exponent δ >  (use Q instead of B), we finally obtain u(Q)(–κ)/p (j + ) · u(j+Q)(–κ)/p ≤ C ≤ C ≤ C, which in turn shows that K ≤ C f Mp,θ (v,u). Summing up all the above estimates, and then taking the supremum over all cubes Q ⊂ Rn and all σ > , we therefore conclude the proof of Theorem .. In particular, if we take θ (x) = xκ with  < κ < , then we immediately get the following two-weight, weak type (p, p) inequalities for Iα and [b, Iα] in the weighted Morrey spaces. Corollary . Let  < p < ∞,  < κ <  and  < α < n. Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, (.) holds. If u ∈ , then the fractional integral operator Iα is bounded from Lp,κ (v, u) into W Lp,κ (u). Corollary . Let  < p < ∞,  < κ < , b ∈ BMO(Rn) and  < α < n. Given a pair of weights (u, v), suppose that, for some r >  and for all cubes Q, (.) holds. If u ∈ A∞, then the linear commutator [b, Iα] is bounded from Lp,κ (v, u) into W Lp,κ (u). Competing interests The author declares that he has no competing interests. 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Hua Wang. Weighted inequalities for fractional integral operators and linear commutators in the Morrey-type spaces, Journal of Inequalities and Applications, 2017, 6, DOI: 10.1186/s13660-016-1279-6