Two-weight, weak type norm inequalities for a class of sublinear operators on weighted Morrey and amalgam spaces

Let $\mathcal T_\alpha~(0\leq\alpha<n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by $\mathrm{BMO}(\mathbb R^n)$ functions and $\mathcal T_\alpha$. This paper is concerned with two-weight, weak type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criterions for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our main results, we can obtain the weak-type norm inequalities for several integral operators as well as the corresponding commutators in the framework of weighted Morrey and amalgam spaces.


Introduction and Main Results
1.1. Sublinear Operators. Let R n be the n-dimensional Euclidean space equipped with the Euclidean norm j·j and the Lebesgue measure dx. Suppose that T represents a linear or a sublinear operator, which satisfies that for any f ∈ L 1 ðR n Þ with compact support and x ∉ supp f , where c 1 is a universal constant independent of f and x ∈ R n . The size condition (1) was first introduced by Soria and Weiss in [1]. It can be proved that (1) is satisfied by many integral operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, Ricci-Stein's oscillatory singular integrals, and Bochner-Riesz operators at the critical index and so on. Similarly, for any given 0 < γ < n, we assume that T γ represents a linear or a sublinear operator with order γ, which satisfies that for any f ∈ L 1 ðR n Þ with compact support and x ∉ supp f , where c 2 is also a universal constant independent of f and x ∈ R n . It can be easily checked that (2) is satisfied by some important operators such as the fractional maximal operator, Riesz potential operators, and fractional oscillatory singular integrals. Let b be a locally integrable function on R n ; suppose that the commutator operator ½b, T stands for a linear or a sublinear operator, which satisfies that for any f ∈ L 1 ðR n Þ with compact support and x ∉ supp f , where c 3 is an absolute constant independent of f and x ∈ R n . Similarly, for any given 0 < γ < n, we assume that the commutator operator ½b, T γ stands for a linear or a sublinear operator, which satisfies that for any f ∈ L 1 ðR n Þ with compact support and x ∉ supp f , where c 4 is also an absolute constant independent of f and x ∈ R n . Clearly, based on the above assumptions, 1.2. Weighted Morrey Spaces. The classical Morrey space was introduced by Morrey [2] in connection with elliptic partial differential equations. Let 1 ≤ p < ∞ and 0 ≤ λ ≤ n. We recall that a real-valued function f is said to belong to the space L p,λ on the n-dimensional Euclidean space R n , if the following norm is finite: where Bðx, rÞ = fy ∈ R n : jx − yj < rg is the Euclidean ball with center x ∈ R n and radius r ∈ ð0,∞Þ. In particular, one has In [3], Komori and Shirai considered the weighted case and gave the definitions of the weighted Morrey spaces as follows.
Definition 1. Let 1 < p < ∞ and 0 ≤ κ < 1. For two weights w and ν on R n , the weighted Morrey space L p,κ ðν, wÞ is defined by where the norm is given by and the supremum is taken over all cubes Q in R n .
Definition 2. Let 1 < p < ∞, 0 ≤ κ < 1, and w be a weight on R n . We define the weighted weak Morrey space WL p,κ ðwÞ as the set of all measurable functions f satisfying By definition, it is clear that 1.3. Weighted Amalgam Spaces. Let 1 ≤ p, q ≤ ∞; a function f ∈ L p loc ðR n Þ is said to be in the Wiener amalgam space ðL p , L q ÞðR n Þ of L p ðR n Þ and L q ðR n Þ, if the function y ↦ k f ð·Þ · χ Bðy,1Þ k L p belongs to L q ðR n Þ, where Bðy, 1Þ is the open ball in R n centered at y with radius 1, χ Bðy,1Þ is the characteristic function of the ball Bðy, 1Þ, and k·k L p is the usual Lebesgue norm in L p ðR n Þ. In [4], Fofana introduced a new class of function spaces ðL p , L q Þ α ðR n Þ which turn out to be the subspaces of ðL p , L q ÞðR n Þ. More precisely, for 1 ≤ p, q, α ≤ ∞, we define the amalgam space ðL p , L q Þ α ðR n Þ of L p ðR n Þ and L q ðR n Þ as the set of all measurable functions f satisfying with the usual modification when p = ∞ or q = ∞, and jBðy, rÞj is the Lebesgue measure of the ball Bðy, rÞ. It was shown in [4] that the space ðL p , L q Þ α ðR n Þ is nontrivial if and only if p ≤ α ≤ q. Throughout this paper, we will always assume that the condition p ≤ α ≤ q is satisfied. Let us consider the following special cases: (1) If we take p = q, then p = α = q. It is easy to check that Hence, the amalgam space ðL p , L q Þ α ðR n Þ is equal to the Lebesgue space L p ðR n Þ with the same norms provided that p = α = q (2) If q = ∞, then we can see that the amalgam space ðL p , L q Þ α ðR n Þ is equal to the Morrey space L p,λ ðR n Þ with equivalent norms, where λ = ðpnÞ/α In this paper, we will consider the weighted version of ðL p , L q Þ α ðR n Þ. Definition 3. Let 1 ≤ p ≤ α ≤ q ≤ ∞, and let ν, w, and μ be three weights on R n . We denote by ðL p , L q Þ α ðν, w ; μÞ the weighted amalgam space, the space of all locally integrable functions f such that 2 Abstract and Applied Analysis with wðQðy, ℓÞÞ = Ð Qðy,ℓÞ wðxÞ dx and the usual modification when q = ∞. Definition 4. Let 1 ≤ p ≤ α ≤ q ≤ ∞, and let w and μ be two weights on R n . We denote by ðWL p , L q Þ α ðw ; μÞ the weighted weak amalgam space consisting of all measurable functions f such that with wðQðy, ℓÞÞ = Ð Qðy,ℓÞ wðxÞ dx and the usual modification when q = ∞.
Note that when μ ≡ 1, this kind of weighted (weak) amalgam space was introduced by Feuto in [5] (see also [6]). We remark that Feuto considered ball B instead of cube Q in his definition, but these two definitions are apparently equivalent. Also, note that when 1 ≤ p ≤ α and q = ∞, then ðL p , L q Þ α ðν, w ; μÞ is just the weighted Morrey space L p,κ ðν, wÞ with κ = 1 − p/α, and ðWL p , L q Þ α ðw ; μÞ is just the weighted weak Morrey space WL p,κ ðwÞ with κ = 1 − p/α.
The two-weight problem for classical integral operators has been extensively studied. In [7][8][9] and [10], the authors gave some A p -type conditions which are sufficient for the two-weight, weak-type ðp, pÞ inequalities for Calderón-Zygmund operators, fractional integral operators, and their commutators on the weighted Lebesgue spaces. In [11], the authors established the two-weight, weak-type ðp, pÞ estimates for the maximal Bochner-Riesz operators and their commutators. Inspired by the above results, it is natural and interesting to study the weak-type estimates for sublinear operators (1) and (2), as well as the corresponding commutators (3) and (4).
Let p′ be the conjugate index of p whenever p > 1; that is, 1/p + 1/p′ = 1. The main purpose of this paper is to investigate the two-weight, weak-type norm inequalities in the setting of weighted Morrey and amalgam spaces. Our main results can be stated as follows. On the boundedness properties of the sublinear operators and their commutators on weighted Morrey spaces, we will prove the following. Theorem 5. Let 1 < p < ∞, 0 < κ < 1, and T satisfy condition (1). Given a pair of weights ðw, νÞ, suppose that for some r > 1 and for all cubes Q in R n , Furthermore, we suppose that T satisfies the weak-type ðp, pÞ inequality where C does not depend on f and σ > 0. If w ∈ Δ 2 , then the operator T is bounded from L p,κ ðν, wÞ into WL p,κ ðwÞ.
Theorem 6. Let 1 < p < ∞, 0 < κ < 1, and T γ satisfy condition (2) with 0 < γ < n. Given a pair of weights ðw, νÞ, suppose that for some r > 1 and for all cubes Q in R n , Furthermore, we suppose that T γ satisfies the weak-type ðp, pÞ inequality where C does not depend on f and σ > 0. If w ∈ Δ 2 , then the operator T γ is bounded from L p,κ ðν, wÞ into WL p,κ ðwÞ.
Concerning the boundedness properties on weighted amalgam spaces for these operators, we have the following results.
Given a pair of weights ðw, νÞ, assume that for some r > 1 and for all cubes Q in R n , Furthermore, we assume that T satisfies the weak-type ðp, pÞ inequality (17). If w ∈ Δ 2 , then the operator T is bounded from ðL p , L q Þ α ðν, w ; μÞ into ðWL p , L q Þ α ðw ; μÞ.
Given a pair of weights ðw, νÞ, assume that for some r > 1 and for all cubes Q in R n , Furthermore, we assume that T γ satisfies the weak-type ðp, pÞ inequality (19). If w ∈ Δ 2 , then the operator T γ is bounded from ðL p , L q Þ α ðν, w ; μÞ into ðWL p , L q Þ α ðw ; μÞ.
Given a pair of weights ðw, νÞ, assume that for some r > 1 and for all cubes Q in R n , where AðtÞ = t p ′ ½log ðe + tÞ p ′ . Furthermore, we assume that ½b, T γ satisfies the weak-type ðp, pÞ inequality (23). If w ∈ A ∞ , then the commutator operator ½b, T γ is bounded from Remark 13. It should be pointed out that the conclusions of our main theorems are natural generalizations of the corresponding weak-type estimates on the weighted Lebesgue spaces. The operators satisfying the assumptions of the above theorems include Calderón-Zygmund operators, Bochner-Riesz operators, and fractional integral operators. Hence, we are able to apply our main theorems to these classical integral operators.

Notations and Definitions
2.1. Weights. A nonnegative function w defined on R n will be called a weight if it is locally integrable. Qðx 0 , ℓÞ will denote the cube centered at x 0 and has side length ℓ > 0; all cubes are assumed to have their sides parallel to the coordinate axes. Given a cube Q = Qðx 0 , ℓÞ and λ > 0, λQ stands for the cube concentric with Q having side length λ ffiffiffi n p times as long, i.e., λQðx 0 , ℓÞ ≔ Qðx 0 , λ ffiffiffi n p ℓÞ. For any given weight w and any Lebesgue measurable set E of R n , we denote the characteristic function of E by χ E , the Lebesgue measure of E by jEj, and the weighted measure of E by wðEÞ, where wðEÞ ≔ Ð E w ðxÞ dx. We also denote E c ≔ R n \ E the complement of E. Given a weight w, we say that w satisfies the doubling condition, if there exists a universal constant C > 0 such that for any cube Q ⊂ R n , we have When w satisfies condition (28), we denote w ∈ Δ 2 for brevity. A weight w is said to belong to Muckenhoupt's class holds for every cube Q ⊂ R n . The class A ∞ is defined as the union of the A p classes for 1 < p < ∞, i.e., A ∞ = S 1<p<∞ A p . If w is an A ∞ weight, then we have w ∈ Δ 2 (see [12]). Moreover, this class A ∞ is characterized as the class of all weights 4 Abstract and Applied Analysis satisfying the following property: there exists a number δ > 0 and a finite constant C > 0 such that (see [12]) holds for every cube Q ⊂ R n and all measurable subsets E of Q. Given a weight w on R n and for 1 ≤ p < ∞, the weighted Lebesgue space L p ðwÞ is defined as the set of all measurable functions f such that We also define the weighted weak Lebesgue space 2.2. Orlicz Spaces and BMO. We next recall some basic facts about Orlicz spaces needed for the proofs of the main results. For further details, we refer the reader to [13]. 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 ⟶ +∞. Given a Young function A, we define the A-average of a function f over a cube Q by means of the following Luxemburg norm: In particular, when AðtÞ = t p , 1 < p < ∞, it is easy to check that that is, the Luxemburg norm coincides with the normalized L p norm.
The main examples that we are going to consider are AðtÞ = t p ½log ðe + tÞ p with 1 < p < ∞.
Let us now recall the definition of the space of BMOðR n Þ (see [14]). A locally integrable function b is said to be in where b Q denotes the mean value of b over Q, namely, and the supremum is taken over all cubes Q in R n . After identifying functions that differ by a constant, the space BMOðR n Þ becomes a Banach space. Throughout this paper, C always denotes a positive constant independent of the main parameters involved, but it may be different from line to line. We will use c 1 , ⋯, c 4 appearing in the first section of this paper to denote certain constants. 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 A and B such that

Proofs of Theorems 5 and 6
Proof of Theorem 5. Let f ∈ L p,κ ðν, wÞ with 1 < p < ∞ and 0 < κ < 1. For an arbitrary fixed cube Q = Qðx 0 , ℓÞ ⊂ R n , we set 2Q ≔ Qðx 0 , 2 ffiffiffi n p ℓÞ. Decompose f as where χ E denotes the characteristic function of the set E. Then, for any given σ > 0, we write We first consider the term I 1 . Using assumption (17) and the condition w ∈ Δ 2 , we get This is just our desired estimate. Let us estimate the second term I 2 . To this end, we observe that when x ∈ Q and y ∈ ð2QÞ c , one has jx − yj ≈ jx 0 − yj. We then decompose R n into a geometrically increasing sequence of concentric cubes and obtain the following pointwise estimate by condition (1).

Abstract and Applied Analysis
This pointwise estimate (40) together with Chebyshev's inequality implies that It follows directly from Hölder's inequality with exponent p > 1 that Moreover, for any positive integer j, we apply Hölder's inequality again with exponent r > 1 to get Thus, in view of (43), we conclude that The last inequality is obtained by the A p -type condition (16) on ðw, νÞ. Since w ∈ Δ 2 , we can easily check that there exists a reverse doubling constant D = DðwÞ > 1 independent of Q such that (see [3], Lemma 4.1) which implies that for any positive integer j, by iteration. Hence, where the last series is convergent since the reverse doubling constant D > 1 and 0 < κ < 1. This yields our desired estimate Summing up the above estimates for I 1 and I 2 , and then taking the supremum over all cubes Q ⊂ R n and all σ > 0, we finish the proof of Theorem 5.
Proof of Theorem 6. Let f ∈ L p,κ ðν, wÞ with 1 < p < ∞ and 0 < κ < 1. For an arbitrary fixed cube Q = Qðx 0 , ℓÞ in R n , we decompose f as where 2Q ≔ Qðx 0 , 2 ffiffiffi n p ℓÞ. For any given σ > 0, we then write Let us consider the first term I ′ 1 . Using assumption (19) and the condition w ∈ Δ 2 , we have 6 Abstract and Applied Analysis This is exactly what we want. We now deal with the second term I ′ 2 . Note that jx − yj ≈ jx 0 − yj, whenever x, x 0 ∈ Q and y ∈ ð2QÞ c . For 0 < γ < n and all x ∈ Q, using the standard technique and condition (2), we can see that This pointwise estimate (52) together with Chebyshev's inequality yields By using Hölder's inequality with exponent p > 1, we can deduce that Moreover, we apply estimate (43) to get The last inequality is obtained by the A p -type condition (18) on ðw, νÞ. Therefore, in view of (47), we find that Combining the above estimates for I ′ 1 and I ′ 2 , and then taking the supremum over all cubes Q ⊂ R n and all σ > 0, we complete the proof of Theorem 6.

Proofs of Theorems 7 and 8
For the results involving commutators, we need the following properties of BMOðR n Þ, which can be found in [15].

Lemma 14. Let b be a function in BMOðR n Þ.
(i) For every cube Q in R n and for any positive integer j, then (ii) Let 1 < p < ∞. For every cube Q in R n and for any w ∈ A ∞ , then Before proving our main theorems, we will also need a generalization of Hölder's inequality due to O'Neil [16].

Lemma 15. 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 all cubes Q in R n , We are now ready to give the proofs of Theorems 7 and 8.
Proof of Theorem 7. Let f ∈ L p,κ ðν, wÞ with 1 < p < ∞ and 0 < κ < 1. For any given cube Q = Qðx 0 , ℓÞ ⊂ R n , we split f as usual by 7 Abstract and Applied Analysis where 2Q ≔ Qðx 0 , 2 ffiffiffi n p ℓÞ. Then, for any given σ > 0, one Since w is an A ∞ weight, we know that w ∈ Δ 2 . By our assumption (21) and inequality (28), we have which is exactly what we want. For any x ∈ Q, from the size condition (3), we can easily see that Hence, we can further split J 2 into two parts as follows: For the term J 3 , it follows from the pointwise estimate (40) and Chebyshev's inequality that where in the last inequality, we have used the second part of Lemma 14 since w ∈ A ∞ . Repeating the arguments used in Theorem 5, we can also prove that Let us consider the term J 4 . Similar to the proof of (40), for any given x ∈ Q, we can obtain the following pointwise estimate as well.
This, together with Chebyshev's inequality, yields An application of Hölder's inequality leads to Abstract and Applied Analysis where CðtÞ = t p ′ is a Young function by (34). For 1 < p < ∞, it is immediate that the inverse function of CðtÞ is C −1 ðtÞ = t 1/p ′. Also, observe that the following identity is true: where Let khk exp L,Q denote the mean Luxemburg norm of h on cube Q with Young function BðtÞ ≈ exp ðtÞ − 1. Thus, by where in the last inequality, we have used the well-known fact that (see [15]) This is equivalent to the following inequality: which is just a corollary of the celebrated John-Nirenberg's inequality (see [14]). Consequently, Moreover, in view of (43), we can deduce that The last inequality is obtained by the A p -type condition (20) on ðw, νÞ and estimate (47). It remains to estimate the last term J 6 . Applying the first part of Lemma 14 and Hölder's inequality, we get Let CðtÞ and AðtÞ be the same as before. Obviously, CðtÞ ≤ AðtÞ for all t > 0, then for any cube Q in R n , one has k f k C,Q ≤ k f k A,Q by definition, which implies that condition (20) is stronger than condition (16). This fact together with (43) yields Moreover, by our hypothesis on w : w ∈ A ∞ and inequality (30), we compute where the last series is convergent since the exponent δð1 − κÞ/p is positive. This implies our desired estimate 9 Abstract and Applied Analysis Summing up all the above estimates, and then taking the supremum over all cubes Q ⊂ R n and all σ > 0, we conclude the proof of Theorem 7.
Proof of Theorem 8. Let f ∈ L p,κ ðν, wÞ with 1 < p < ∞ and 0 < κ < 1. For any given cube Q = Qðx 0 , ℓÞ ⊂ R n , as before, we set where 2Q ≔ Qðx 0 , 2 ffiffiffi n p ℓÞ. Then, for any given σ > 0, one Since w is an A ∞ weight, then we have w ∈ Δ 2 . From our assumption (23) and inequality (28), it follows that On the other hand, for any x ∈ Q, from the size condition (4), it then follows that Thus, we can further split J ′ 2 into two parts as follows: Using the pointwise estimate (52) and Chebyshev's inequality, we obtain that where the last inequality is due to w ∈ A ∞ and Lemma 14(ii). By using the same arguments as that of Theorem 6, we can also show that Similar to the proof of (52), for all x ∈ Q, we can show the following pointwise inequality as well.
This, together with Chebyshev's inequality, yields An application of Hölder's inequality leads to 10 Abstract and Applied Analysis where CðtÞ = t p ′ is a Young function. Let BðtÞ and AðtÞ be the same as in Theorem 7. In view of (73) and (43), we can deduce that Furthermore, by the A p -type condition (22) on ðw, νÞ and estimate (47), we obtain It remains to estimate the last term J ′ 6 . Making use of the first part of Lemma 14 and Hölder's inequality, we get It was pointed out in Theorem 7 that for any cube Q in R n , one has k f k C,Q ≤ k f k A,Q , where CðtÞ = t p ′ and AðtÞ ≈ t p ′ ½log ðe + tÞ p ′ . This implies that condition (22) is stronger than condition (20). Using this fact along with (43), we can see that where the last inequality follows from estimate (80). Summarizing the estimates derived above, and then taking the supremum over all cubes Q ⊂ R n and all σ > 0, we therefore conclude the proof of Theorem 8.