Two weight estimates for a class of (p, q) type sublinear operators and their commutators

Abstract In the present paper, the authors investigate the two weight, weak-(p, q) type norm inequalities for a class of sublinear operators 𝓣γ and their commutators [b, 𝓣γ] on weighted Morrey and Amalgam spaces. What should be stressed is that we introduce the new BMO type space and our results generalize known results before.


Introduction
As it is well-known, Muckenhoupt [1]  Muckenhoupt and Wheeden [2] showed that fractional integral operator Iα is bounded from L p (ω) to L q (ω) if and only if ω satis ed the so-called Ap,q condition: there exists a constant C such that for all cube Q, These estimates are of interest on their own and they also have relevance to partial di erential equations and quantum mechanics.
On the other hand, Saywer [3] characterized the two weight inequality. However, Saywer's condition is often di cult to verify in practice, since it involves the maximal operator. Thus it is necessary to look for other simple su cient conditions. The rst attempt was made by Neugebauer [4] in 1983. He gave a su cient condition closely in spirit to the classical Ap condtion: if weight (u, v) satis ed the so-called " power-bump " condition: |Q|ˆQ v(x) r dx /(rp) |Q|ˆQ u(x) −rp /p dx /(rp ) for some r > . Later, in 1995, Pérez [5] improved condition by A long-standing problem in harmonic analysis has been to characterize the weights governing weighted norm inequalities for classical operators. The purpose of this paper is to give (p, q)-type two weight norm inequalities for class of sublinear operators and their commutators by using the pair of weights (u, v) which satis es a Muckenhoupt condition with a " power-bump " and " Orlicz-bump " on the weight v.
Precisely, this paper is organized as follows.
In Section 2 contains some de nitions. The next Section 3, we give some basic lemmas and investigate the two weight, weak-(p, q) type norm inequalities for a class of sublinear operators Tγ on weighted Morrey and Amalgam spaces. Finally, two weight norm inequality for sublinear operators high order commutator [b, Tγ]m is considered in Section 4. Throughout this paper all notation is standard or will be de ned as needed. We have used the notation Q(y, r) to denote the cube centered at y and its side length r > . Given λ > ,a cube Q(y, r),λQ(y, r) stands for the cube concentric with Q(y, r) and having side length λ √ n times as long, that is λQ(y, r) := Q(y, λ √ nr). Given a lebesgue measurable set E, χ E will denote the characteristic function of E, |E| is the Lesbesgue measure of E and weighted mrasure of E by ω(E), where ω(E) :=´E ω(x)dx. We also denote E c := R n \E the complement of E. The class A∞ is de ned as union of the classes for < p < ∞. Given a weight ω, we say that ω ∈ , if there exists a constant C > such that for any cube Q ⊂ R n , ω( Q) ≤ Cω(Q). By the way, the letter C will be used for various constants that may vary from line to line but remains independent of the main variables.

Some preliminaries . Sublinear operators and their commutators
In this paper, we cosider a class of linear or sublinear operator, which satis es that given ≤ γ < n, for any f ∈ L (R n ) with compact support and x ∉ supp f , The condition (2.1) was rst introduced by Soria and Weiss in [6] (γ = ). It is easy to see that (2.1) is satis ed by many integral operators in harmonic analysis. When γ = , such as the Hardy-Littewood maximal operator, Calderón-Zygmund singular integral operators, Bochner-Riesz operators at the critical index and so on. When < γ < n, such as the fractional maximal operator, Riesz potential operators and fractional oscillatory singular integrals and so on.
Given ≤ γ < n. Let m ≥ . b is a locally integrable function on R n , and suppose that the m order commutator [b, Tγ]m stands for a linear or a sublinear operator, which satis es that for any f ∈ L (R n ) with compact support and x ∉ supp f ,
De nition 2.1. Let < p < ∞, ≤ λ < . The classical Morrey space L p,λ to be the subset of all L p (R n ) locally integrable functions f on R n so that In particular, L p, = L p and L p, = L ∞ .
In 2009, Komori [19] introduced the weighted morrey spaces, and gave the de nitions as follows.
De nition 2.2. Let < p < ∞, ≤ λ < and u, v be two weights. The classical Morrey space L p,λ (u, v) to be the subset of all L p (R n ) locally integrable functions f on R n so that We are now ready for the de nition of weak Morrey spaces.
De nition 2.3. Let < p < ∞, ≤ λ < and ω be a weight. The weighted weak Morrey space WL p,λ (ω) to be the subset of all L p (R n ) locally integrable functions f on R n so that

. BMO spaces
De nition 2.4. [23] Let q ≥ , the space BMO(R n ) to be the subset of all locally integrable functions f on R n so that where f Q denotes the mean value of f over Q, that is f Q := |Q|´Q f (x)dx.
Remark 2.5. By John-Nirenberg's inequality, we have f BMO = f BMO q for all q ≥ , so we denote by BMO simple.

. Amalgam spaces
Let ≤ p, q ≤ ∞, a measurable functions f ∈ L q loc (R n ) is said to be in the amalgam spaces (L q , L p )(R n ) of L q (R n ) and L p (R n ) and if fχ Q(y,r) L q (R n ) belongs to L p (R n ), where χ Q(y,r) is the characteristic function of the cube Q(y, r).
is a norm on (L q , L p )(R n ) under which it is a Banach space with the usual modi cation when p = ∞. These spaces were rst introduced by Winer [24] in 1926 and its systematic study goes back to the work of Holland [25].
In 1989, Fofana [26,27] considered the subspace (L q , L p ) α (R n ) of (L q , L p )(R n ) in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in R n , which consists of measurable functions f so that for ≤ α ≤ ∞, ≤ p, q < ∞, f (L q ,L p ) α (R n ) := sup r> ˆR n (Q(y, r)) /α− /q− /p fχ Q(y,r) L q (R n ) p dy p < ∞ and a usual modi cation version for p = ∞ or q = ∞. As it was shown in [26] that the space (L q , L p ) α (R n ) is non-trivial if and only if q ≤ α ≤ p. Thus in the remaning of this paper we will always assume that this condition q ≤ α ≤ p is satis ed. By the de nitions, it is clear (also see [27]) that (L q , L q )(R n ) = L q (R n ), (L q , L ∞ ) α (R n ) = L q,(nq/α) (R n ), where L q,λ (R n ) with ≤ q < ∞ and < λ < n is the classical Morrey space.
Recently, Wang [28] studied the weighted version of these amalgam spaces.
De nition 2.6. Let u, v, ω be three weights on R n and ≤ q ≤ α ≤ p ≤ ∞, the weighted amalgam spaces (L q , L p ) α (u, v, ω) as the space of all measurable functions f so that and a usual modi cation version for p = ∞ or q = ∞, where L q (u) is the weighted Lebesgue space.
It is easy to nd that when λ = − q/α and ≤ q < α < ∞, the space (L q (ω), L ∞ ) α (R n ) is the weighted Morrey space L q,λ (ω), which rst introduced by Komori [19] in 2009. Next we introduce the new BMO type space, and our main results generalize the results in [28].
where the f Q(y,r) denote the mean value of f on Q(y, r). It is clear that the space goes back to the classical BMO space when α = ∞.

Sublinear operators
To prove our main results, we need the following Lemma.
Lemma 3.1. [19] If ω ∈ , then there exists a constant A > such that

Remark 3.2.
If γ > and ω ∈ , then there exists a constant C such that Theorem 3.3. Let < p ≤ q < ∞, < λ < p/q and Tγ satisfy (2.1) with ≤ γ < n. Given a pair of weights (u, v), suppose that for some r > and for all cubes Q ⊂ R n , Furthermore, we also suppose that Tγ satis es the weak-(p, q) type inequality If v ∈ , then the sublinear operator Tγ is bounded from L p,λ (u, v) to WL q,λq/p (v).
Then, for any given δ > , For I, we recall that v ∈ . By the assumption (3.2), we have For the term II, observe that for x, x ∈ Q and y ∈ ( Q) c we have |x − y| ≈ |x − y|. Thus, from Chebyshev's inequality and Hölder's inequality, we can obtain For any positive integer j, we apply Hölder's inequality, (3.1) and Lemma 3.1 to get Combining the above estimates for I and II, and then taking the supremum over all cubes Q ⊂ R n and all δ > , we nish the proof of Theorem.
Theorem 3.4. Let < p ≤ q < ∞ and Tγ satisfy (2.1) with ≤ γ < n. Given a pair of weights (u, v), suppose that for some r > and for all cubes Q ⊂ R n , Furthermore, we also suppose that Tγ satis es the weak-(p, q) type inequality For I, according to assumption (3.4), we obtain where we have used /β − /q − /s = /α − /p − /s, /α − /p − /s < and v ∈ . For the term II, observe that for x, x ∈ Q and y ∈ ( Q) c we have |x − y| ≈ |x − y|. Thus, by Chebyshev's inequality and Hölder's inequality yields A further application of Hölder's inequality, (3.3) and Lemma 3.1, we have which is desired inequality. Combining the above estimates for I and II, and taking the L s (ω)-norm of both sides of with respect to the variable y, we nish the proof of Theorem.
Theorem 3.5. Let < p ≤ q < ∞ and Tγ satisfy (2.1) with ≤ γ < n. Given a pair of weights (u, v), suppose that for some r > and for all cubes Q ⊂ R n , Furthermore, if λ = p/q, then the sublinear operator Tγ is bounded from L p,λ (u, v) to BMO.
For I, it follows directly from Fubini's theorem that By simple geometric observation, we have for any x ∈ Q and y ∈ Q, Using the transform x − y → z and polar coordinates, we havê where ω n− denote the measure of the unit sphere. Therefore, Notice that λ/p = /q. It follows from Hölder's inequality and (3.5) that For the term II, observe that for x, y ∈ Q and z ∈ ( Q) c , we have |x − z| ≥ |x − y| and |x − z| ≈ |z − x |. Thus, we have that for any x ∈ Q, Futher, notice that λ/p = /q, by Hölder's inequality and (3.5), Therefore, Combining the above estimates for I and II, and then taking the supremum over all cubes Q ⊂ R n , we nish the proof of Theorem. Theorem 3.6. Let < p ≤ q < ∞ and Tγ satisfy (2.1) with ≤ γ < n. Given a pair of weights (u, v), suppose that for some r > and for all cubes Q ⊂ R n , Furthermore, we also suppose that Tγ satis es the (p, q) type inequality ˆR If p ≤ α ≤ β < s < ∞, /s = /α − ( /p − /q) and v, ω ∈ , then the sublinear operator Tγ is bounded from (L p , L s ) α (u, v, ω) to (BMO q , L s ) β (v, ω) with /β = /α − ( /p − /q).
Proof of Theorem 3.6. Let f ∈ (L p , L s ) α (u, v, ω) with < p ≤ α ≤ β < s < ∞ and v, ω ∈ . Fix Q := Q(y, r) ⊂ For I, according to assumption (3.8), we obtain where we have used /β − /q − /s = /α − /p − /s, /α − /p − /s < and v ∈ . For the term II, observe that for x, z ∈ Q(y, r) and η ∈ Q(y, √ nr) c , we have |x − η| ≥ |x − z| and |x − η| ≈ |η − y|. Thus, we have that for any x ∈ Q(y, r), A further application of Hölder's inequality, (3.7) and Lemma 3.1, we have ≤ Cv (Q(y, r) which is desired inequality. Combining the above estimates for I and II, and taking the L s (ω)-norm of both sides of with respect to the variable y, we nish the proof of Theorem.

High order commutators . Orlicz spaces
Since commutators have a greater degree of " singularity ", we need a slightly stronger condition. Roughly, we need to " bump " the right hand term as well, but it su ces to do so in the scale of Orlicz spaces, so it is called as " Orlicz bump ". Next, we recall some basic facts about Orlicz spaces. Let Φ be a Young function, that is to say, Φ : [ , +∞) → [ , +∞) is a continuous, convex and increasing function and satis es Φ( ) = and Φ(t) → +∞ as t → +∞. Given a Young function Φ, let E be a measurable set with |E| < ∞, de ne the Luxemburg norm of f over E as In particular, when Φ = t p , < p < ∞, it is easy to check that that is, the Luxemburg norm coincides with the normalized L p norm. For further details, we refer the reader to [29].

. Boundedness of sublinear operator commutators
To prove our theorem we need the following Lemmas. (ii) Let < p < ∞. For every cube Q ⊂ R n and for any ω ∈ A∞. Then

Lemma 4.2. [29] Let A, B and C be Young functions such that for all t > ,
Furthermore, we also suppose that [b, Tγ]m satis es the weak-(p, q) type inequality If v ∈ A∞, then the sublinear operator higher order commutators [b, Tγ]m is bounded from L p,λ (u, v) to WL q,λq/p (v).

Proof of Theorem 4.4.
Because the method is similar, we only need to prove the case of m = . Let f ∈ L p,λ (u, v) with < p ≤ q < ∞ and < λ < p/q.
√ nr). Then, for any given δ > , For I, we notice that v ∈ (cf . [32]). according to assumption (4.2), we have For the term II, notice that Therefore, Since the condition (4.1) is stronger than the condition (3.1). By Chebyshev's inequality, we obtain where we have used Lemma 3.1. For the term II , observe that for x, x ∈ Q and y ∈ ( Q) c we have |x − y| ≈ |x − y|. Thus, from Chebyshev's inequality, we can obtain For II , putting C = t p is a Young function. We use Hölder's inequality to get, and it is easy to see that A ≈ t p ( + log + t) p , B ≈ e t − . f expL,Q denotes the mean Luxemburg norm of f on cube Q with Young function B ≈ e t − . By Lemma 4.2 where we have used the well-known fact that for any cube Q ⊂ R n (cf . [30]), Therefore, we apply Hölder's inequality, (4.1) and Lemma 3.1 to get For the term II , we make use of Lemma 4.1 and Hölder's inequality, then Put C(t) and A(t) be the same as before. Obviously, C(t) ≤ A(t) for all t > , then for any cube Q ⊂ R n , f C,Q ≤ f A,Q by de nition, which implies that the condition (4.1) is stronger than the condition (3.1). From this and Hölder's inequality yields where in the last inequality we have used Lemma 4.3.
Combining the above estimates for I and II, and then taking the supremum over all cubes Q ⊂ R n and all δ > , we nish the proof of Theorem. Theorem 4.5. Let m ≥ , < p ≤ q < ∞, < λ < p/q, b ∈ BMO and [b, Tγ]m satisfy (2.1) with ≤ γ < n. Given a pair of weights (u, v), suppose that for some r > and for all cubes Q ⊂ R n , where A (t) = t p + log(e + t) p , Am(t) = t p + log + t mp (m = , , ...).
For the term II , By Chebyshev's inequality ,Hölder's inequality and v ∈ yields For the term II , it is similar to Theorem 4.4. Together with Chebyshev's inequality, we can obtain  which is desired inequality. Combining the above estimates for I and II, and taking the L s (ω)-norm of both sides of with respect to the variable y, we nish the proof of Theorem.