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

Hu Yunpeng; Zhou Jiang; Cao Yonghui

doi:10.1515/math-2019-0061

Article Open Access

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

Hu Yunpeng , Zhou Jiang and Cao Yonghui

Published/Copyright: July 31, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

In the present paper, the authors investigate the two weight, weak-(p, q) type norm inequalities for a class of sublinear operators 𝓣_γ and their commutators [b, 𝓣_γ] 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.

Keywords: Sublinear operator; Two weight; Morrey space; Amalgam space; BMO space

MSC 2010: 42B25; 42B35

1 Introduction

As it is well-known, Muckenhoupt [1] characterized the weights ω by means of the Hardy-Littlewood maximal operator M. He showed that M is bounded on L^p(ω) if and only if ω satisfied the so-called A_p condition: there exists a constant C such that for all cube Q,

1|Q|∫Qω(x)dx1|Q|∫Qω(x)−p′/pdxp/p′≤C,

Muckenhoupt and Wheeden [2] showed that fractional integral operator I_α is bounded from L^p(ω) to L^q(ω) if and only if ω satisfied the so-called A_p,q condition: there exists a constant C such that for all cube Q,

1|Q|∫Qω(x)dx1|Q|∫Qω(x)−p′/qdxq/p′≤C,

These estimates are of interest on their own and they also have relevance to partial differential equations and quantum mechanics.

On the other hand, Saywer [3] characterized the two weight inequality. However, Saywer’s condition is often difficult to verify in practice, since it involves the maximal operator. Thus it is necessary to look for other simple sufficient conditions. The first attempt was made by Neugebauer [4] in 1983. He gave a sufficient condition closely in spirit to the classical A_p condtion: if weight (u, v) satisfied the so-called “power-bump” condition:

1|Q|∫Qv(x)rdx1/(rp)1|Q|∫Qu(x)−rp′/pdx1/(rp′)≤C,

for some r > 1. Later, in 1995, Pérez [5] improved condition by

1|Q|∫Qv(x)rdx1/(rp)1|Q|∫Qu(x)−p′/pdx1/p′≤C,

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 satisfies 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 definitions. 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 𝓣_γ on weighted Morrey and Amalgam spaces. Finally, two weight norm inequality for sublinear operators high order commutator [b, 𝓣_γ]_m is considered in Section 4.

Throughout this paper all notation is standard or will be defined as needed. We have used the notation Q(y, r) to denote the cube centered at y and its side length r > 0. Given λ > 0, a cube Q(y, r), λ Q(y, r) stands for the cube concentric with Q(y, r) and having side length λ n times as long, that is λ Q(y, r) := Q(y, λ n r). 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) := ∫_Eω(x)dx. We also denote E^c := ℝⁿ \ E the complement of E. The class A_∞ is defined as union of the classes for 1 0 such that for any cube Q ⊂ ℝⁿ, ω(2Q) ≤ 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.

2 Some preliminaries

2.1 Sublinear operators and their commutators

In this paper, we cosider a class of linear or sublinear operator, which satisfies that given 0 ≤ γ < n, for any f ∈ L¹(ℝⁿ) with compact support and x ∉ supp f,

|Jγf(x)|≤C∫Rn|f(y)||x−y|n−γdy,x∈Rn (2.1)

The condition (2.1) was first introduced by Soria and Weiss in [6] (γ = 0). It is easy to see that (2.1) is satisfied by many integral operators in harmonic analysis. When γ = 0, such as the Hardy-Littewood maximal operator, Calderón-Zygmund singular integral operators, Bochner-Riesz operators at the critical index and so on. When 0 < γ < n, such as the fractional maximal operator, Riesz potential operators and fractional oscillatory singular integrals and so on.

Given 0 ≤ γ < n. Let m ≥ 1. b is a locally integrable function on ℝⁿ, and suppose that the m order commutator [b, 𝓣_γ]_m stands for a linear or a sublinear operator, which satisfies that for any f ∈ L¹(ℝⁿ) with compact support and x ∉ supp f,

|[b,Jγ]mf(x)|≤C∫Rn|b(x)−b(y)|m|f(y)||x−y|n−γdy,x∈Rn (2.2)

Observe that [b, 𝓣_γ]₀ = 𝓣, [b, 𝓣_γ]₁ = [b, 𝓣_γ] and [b, 𝓣_γ]_m = [b, [b, 𝓣_γ]_m−1].

2.2 Morrey spaces

The well-know Morrey space introduced in [7] to investigate the local behavior of solutions to second order elliptic partial differential equations, and presented in various books, see [8, 9, 10, 11].The Morrey space is a properly wider space than the lebesgue space when 0 < q < p < ∞ (cf. [12]) and this space works well with the fractional integral operators, see [13, 14, 15, 16]. For the maximal operator in Morrey spaces we refer to [17, 18], while the Calderon-Zygmund type singular operators is known from [19, 20, 21, 22].

Definition 2.1

Let 1 < p < ∞, 0 ≤ λ < 1. The classical Morrey space L^p,λ to be the subset of all L^p(ℝⁿ) locally integrable functions f on ℝⁿ so that

∥f∥Lp,λ=supQ∋x1|Q|λ∫Q|f(x)|pdx1/p<∞.

In particular, L^p,0 = L^p and L^p,1 = L^∞.

In 2009, Komori[19] introduced the weighted morrey spaces, and gave the definitions as follows.

Definition 2.2

Let 1 < p < ∞, 0 ≤ λ < 1 and u, v be two weights. The classical Morrey space L^p,λ(u, v) to be the subset of all L^p(ℝⁿ) locally integrable functions f on ℝⁿ so that

∥f∥Lp,λ(u,v)=supQ∋x1v(Q)λ∫Q|f(x)|pu(x)dx1/p<∞.

We are now ready for the definition of weak Morrey spaces.

Definition 2.3

Let 1 < p < ∞, 0 ≤ λ < 1 and ω be a weight. The weighted weak Morrey space WL^p,λ(ω) to be the subset of all L^p(ℝⁿ) locally integrable functions f on ℝⁿ so that

∥f∥WLp,λ(ω)=supQ∋xsupδ>01ω(Q)λ/pδ⋅ω({x∈Q:|f(x)|>δ})1/p<∞.

2.3 BMO spaces

Definition 2.4

[23] Let q ≥ 1, the space BMO(ℝⁿ) to be the subset of all locally integrable functions f on ℝⁿ so that

∥f∥BMOq:=supQ1|Q|∫Q|f(x)−fQ|qdx1/q<∞,

where f_Q denotes the mean value of f over Q, that is f_Q := 1|Q| ∫_Q f(x)dx.

Remark 2.5

By John-Nirenberg’s inequality, we have ∥f∥_BMO = ∥f∥_BMO^q for all q ≥ 1, so we denote by BMO simple.

2.4 Amalgam spaces

Let 1 ≤ p, q ≤ ∞, a measurable functions f ∈ Llocq (ℝⁿ) is said to be in the amalgam spaces (L^q, L^p)(ℝⁿ) of L^q(ℝⁿ) and L^p(ℝⁿ) and if ∥fχ_Q(y,r)∥_L^q(ℝⁿ) belongs to L^p(ℝⁿ), where χ_Q(y,r) is the characteristic function of the cube Q(y, r).

∥f∥(Lq,Lp)(Rn):=∫Rn∥fχQ(y,r)∥Lq(Rn)pdy1p

is a norm on (L^q, L^p)(ℝⁿ) under which it is a Banach space with the usual modification when p = ∞. These spaces were first 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(ℝⁿ) of (L^q, L^p)(ℝⁿ) in connection with the study of the continuity of the fractional maximal operator of Hardy-Littlewood and of the Fourier transformation in ℝⁿ, which consists of measurable functions f so that for 1 ≤ α ≤ ∞, 1 ≤ p, q < ∞,

∥f∥(Lq,Lp)α(Rn):=supr>0∫Rn(Q(y,r))1/α−1/q−1/p∥fχQ(y,r)∥Lq(Rn)pdy1p<∞

and a usual modification version for p = ∞ or q = ∞. As it was shown in [26] that the space (L^q, L^p)^α(ℝⁿ) 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 satisfied. By the definitions, it is clear (also see [27]) that (L^q, L^q)(ℝⁿ) = L^q(ℝⁿ), (L^q, L^∞)^α(ℝⁿ) = L^q,(nq/α) (ℝⁿ), where L^q,λ(ℝⁿ) with 1 ≤ q < ∞ and 0 < λ < n is the classical Morrey space.

Recently, Wang [28] studied the weighted version of these amalgam spaces.

Definition 2.6

Let u, v, ω be three weights on ℝⁿ and 1 ≤ 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

∥f∥(Lq,Lp)α(u,v,ω):=supr>0∫Rnv(Q(y,r))1/α−1/q−1/p∥fχQ(y,r)∥Lq(u)pω(y)dy1p<∞

and a usual modification version for p = ∞ or q = ∞, where L^q(u) is the weighted Lebesgue space.

It is easy to find that when λ = 1 − q/α and 1 ≤ q < α < ∞, the space (L^q(ω), L^∞)^α(ℝⁿ) is the weighted Morrey space L^q,λ(ω), which first introduced by Komori [19] in 2009. Next we introduce the new BMO type space, and our main results generalize the results in [28].

Definition 2.7

Let 1 ≤ q ≤ α ≤ p ≤ ∞. The space (BMO^q, L^p)(v, ω) is defined as the set of all locally integrable functions f satisfying ∥f∥_{(BMO^q,L^p)^α(v,ω)} < ∞, where

∥f∥(BMOq,Lp)α(v,ω):=supr>0∫Rn(v(Q(y,r))1/α−1/q−1/p∥(f−fQ(y,r))χQ(y,r)∥Lq(v)pω(y)dy1/p.

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 = ∞.

3 Sublinear operators

To prove our main results, we need the following Lemma.

Lemma 3.1

[19] If ω ∈ △₂, then there exists a constant A > 1 such that

ω(2Q)≥Aω(Q).

Remark 3.2

If γ > 0 and ω ∈ △₂, then there exists a constant C such that

∑j=1∞ω(Q)ω(2j+1Q)γ≤∑j=1∞ω(Q)Aj+1ω(Q)γ=∑j=1∞1Aj+1γ≤C.

Theorem 3.3

Let 1 1 and for all cubes Q ⊂ ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qv(x)rdx1/(rq)1|Q|∫Qu(x)−p′/pdx1/p′≤C (3.1)

Furthermore, we also suppose that 𝓣_γ satisfies the weak-(p, q) type inequality

δ⋅v({x∈Rn:|Jγf(x)|>δ})1/q≤C∫Rn|f(x)|pu(x)dx1/p,δ>0. (3.2)

If v ∈ △₂, then the sublinear operator 𝓣_γ is bounded from L^p,λ(u, v) to WL^q,λq/p(v).

Proof of Theorem 3.3

Let f ∈ L^p,λ(u, v) with 1 0,

1v(Q)qλ/p⋅1/qδ⋅v({x∈Q:|Jγ(f)|>δ})1/q≤1v(Q)λ/pδ⋅v({x∈Q:|Jγ(f1)|>δ/2})1/q+1v(Q)λ/pδ⋅v({x∈Q:|Jγ(f2)|>δ/2})1/q=:I+II.

For I, we recall that v ∈ △₂. By the assumption (3.2), we have

I≤Cv(Q)λ/p∫Rn|f1(x)|pu(x)dx1/p=Cv(Q)λ/p∫2Q|f(x)|pu(x)dx1/p≤Cv(2Q)v(Q)λ/p∥f∥Lp,λ(u,v)≤C∥f∥Lp,λ(u,v).

For the term II, observe that for x, x₀ ∈ Q and y ∈ (2Q)^c we have |x − y| ≈ |x₀ − y|. Thus, from Chebyshev’s inequality and Hölder’s inequality, we can obtain

II≤2v(Q)λ/p∫Q|Jγ(f2)(x)|qv(x)dx1/q≤2v(Q)λ/p∫Q|∫(2Q)c|f(y)||x−y|n−γdy|qv(x)dx1/q=2v(Q)λ/p∫Q|∑j=1∞∫2j+1Q∖2jQ|f(y)||x−y|n−γdy|qv(x)dx1/q≤2v(Q)λ/p∫Q|∑j=1∞∫2j+1Q∖2jQ|f(y)||x0−y|n−γdy|qv(x)dx1/q≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|f(y)|dy≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Qu(y)−p′/pdy1/p′∫2j+1Q|f(y)|pu(y)dy1/p≤C∥f∥Lp,λ(u,v)v(Q)1/q−λ/p∑j=1∞v(2j+1Q)λ/p|2j+1Q|1−γ/n∫2j+1Qu(y)−p′/pdy1/p′=C∥f∥Lp,λ(u,v)∑j=1∞v(Q)1/q−λ/pv(2j+1Q)1/q−λ/pv(2j+1Q)1/q|2j+1Q|1−γ/n∫2j+1Qu(y)−p′/pdy1/p′.

For any positive integer j, we apply Hölder’s inequality, (3.1) and Lemma 3.1 to get

≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)1/q−λ/pv(2j+1Q)1/q−λ/p×|2j+1Q|1/r′q|2j+1Q|1−γ/n∫2j+1Qv(y)rdy1/(rq)∫2j+1Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)1/q−λ/pv(2j+1Q)1/q−γ/p=C∥f∥Lp,λ(u,v).

Combining the above estimates for I and II, and then taking the supremum over all cubes Q ⊂ ℝⁿ and all δ > 0, we finish the proof of Theorem.□

Theorem 3.4

Let 1 1 and for all cubes Q ⊂ ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qv(x)rdx1/(rq)1|Q|∫Qu(x)−p′/pdx1/p′≤C (3.3)

Furthermore, we also suppose that 𝓣_γ satisfies the weak-(p, q) type inequality

δ⋅v({x∈Rn:|Jγf(x)|>δ})1/q≤C∫Rn|f(x)|pu(x)dx1/p,δ>0. (3.4)

If p ≤ α ≤ β < s < ∞ and v, ω ∈ △₂, then the sublinear operator 𝓣_γ is bounded from (L^p, L^s)^α(u, v, ω) to (L^q,∞, L^s)^β(v, ω) with 1/β = 1/α − (1/p − 1/q).

Proof of Theorem 3.4

Let f ∈ (L^p, L^s)^α(u, v, ω) with 1 < p ≤ α ≤ β < s < ∞ and v, ω ∈ △₂. Fix Q := Q(y, r) ⊂ ℝⁿ, we split f = f₁ + f₂ with f₁ = fχ_2Q, where 2Q := Q(y, 2 n r). Then, for any y ∈ ℝⁿ,

v(Q(y,r))1/β−1/q−1/s∥Jγ(f)χQ(y,r)∥Lq,∞(v)≤v(Q(y,r))1/β−1/q−1/s∥Jγ(f1)χQ(y,r)∥Lq,∞(v)+v(Q(y,r))1/β−1/q−1/s∥Jγ(f2)χQ(y,r)∥Lq,∞(v)=:I+II.

For I, according to assumption (3.4), we obtain

I≤v(Q(y,r))1/β−1/q−1/s∥Jγ(f1)∥Lq,∞(v)≤Cv(Q(y,r))1/β−1/q−1/s∫Rn|f1(x)|pu(x)dx1/p=Cv(Q(y,r))1/β−1/q−1/s∫Q(y,2nr)|f(x)|pu(x)dx1/p=Cv(Q(y,r))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u)=Cv(Q(y,r))v(Q(y,2nr))1/α−1/p−1/sv(Q(y,2nr))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u)≤Cv(Q(y,2nr))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u).

where we have used 1/β − 1/q − 1/s = 1/α − 1/p − 1/s, 1/α − 1/p − 1/s < 0 and v ∈ △₂. For the term II, observe that for x, x₀ ∈ Q and y ∈ (2Q)^c we have |x − y| ≈ |x₀ − y|. Thus, by Chebyshev’s inequality and Hölder’s inequality yields

II≤2v(Q(y,r))1/β−1/q−1/s∫Q(y,r)|Tγ(f2)(x)|qv(x)dx1/q≤2v(Q(y,r))1/β−1/q−1/s∫Q(y,r)|∫(Q(y,2nr))c|f(y)||x−y|n−αdy|qv(x)dx1/q=2v(Q(y,r))1/β−1/q−1/s∫Q(y,r)|∑j=1∞∫Q(y,2j+1nr)∖Q(y,2jnr)|f(y)||x−y|n−γdy|qv(x)dx1/q≤2v(Q(y,r))1/β−1/q−1/s∫Q(y,r)|∑j=1∞∫Q(y,2j+1nr)∖Q(y,2jnr)|f(y)||x0−y|n−γdy|qv(x)dx1/q≤Cv(Q(y,r))1/β−1/s∑j=1∞1|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)|f(z)|dz≤Cv(Q(y,r))1/β−1/s∑j=1∞1|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)|f(z)|pu(z)dz1/p×∫Q(y,2j+1nr)u(z)−p′/pdz1/p′≤C∑j=1∞v(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)×v(Q(y,2j+1nr))1/q|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)u(z)−p′/pdz1/p′.

A further application of Hölder’s inequality, (3.3) and Lemma 3.1, we have

≤C∑j=1∞v(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)×|Q(y,2j+1nr)|1/r′q|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)v(z)rdz1/(rq)∫Q(y,2j+1nr)u(z)−p′/pdz1/p′≤C∑j=1∞v(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u)≤C∑j=1∞v(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u).

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 finish the proof of Theorem.□

Theorem 3.5

Let 1 1 and for all cubes Q ⊂ ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qv(x)rdx1/(rq)1|Q|∫Qu(x)−p′/pdx1/p′≤C (3.5)

Furthermore, if λ = p/q, then the sublinear operator 𝓣_γ is bounded from L^p,λ(u, v) to BMO.

Proof of Theorem 3.5

Let f ∈ L^p,λ(u, v) with 1 < p ≤ q < ∞ and λ = p/q. Fix Q := Q(x₀, r) ⊂ ℝⁿ, we split f = f₁ + f₂ with f₁ = fχ_4Q, where 4Q := Q(x₀, 4 n r). Then,

1|Q|∫Q|Jγf(x)−(Jγf)Q|dx≤1|Q|∫Q|Jγf1(x)−(Jγf1)Q|dx+1|Q|∫Q|Jγf2(x)−(Jγf2)Q|dx=:I+II.

For I, it follows directly from Fubini’s theorem that

I≤2|Q|∫Q|Jγf1(x)|dx≤C|Q|∫Q∫4Q1|x−y|n−γ|f(y)|dydx=C|Q|∫4Q∫Q1|x−y|n−γdx|f(y)|dy.

By simple geometric observation, we have for any x ∈ Q and y ∈ 4Q,

|x−y|≤|x−x0|+|x0−y|≤3nr.

Using the transform x − y ↦ z and polar coordinates, we have

∫Q1|x−y|n−γdx≤∫|z|≤3nr1|z|n−γdz=ωn−1∫03nr1ρn−γρn−1dρ=ωn−11γ3nrγ. (3.6)

where ω_n−1 denote the measure of the unit sphere. Therefore,

I≤C|Q|1−γ/n∫4Q|f(y)|dy.

Notice that λ/p = 1/q. It follows from Hölder’s inequality and (3.5) that

I≤C|Q|1−γ/n∫4Q|f(y)|pu(y)dy1/p∫4Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)v(4Q)λ/p|Q|1−γ/n∫4Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)|4Q|1/(r′q)|4Q|1−γ/n∫4Qv(y)rdy1/(rq)∫4Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v).

For the term II, observe that for x, y ∈ Q and z ∈ (4Q)^c, we have |x − z| ≥ 2|x − y| and |x − z| ≈ |z − x₀|. Thus, we have that for any x ∈ Q,

|Jγf2(x)−(Jγf2)Q|=1|Q|∫Q(Jγf2(x)−Jγf2(y))dy≤1|Q|∫Q∫(4Q)c1|x−z|n−γ−1|y−z|n−γ|f(z)|dzdy≤C|Q|∫Q∫(4Q)c|x−y||x−z|n−γ+1|f(z)|dzdy≤C|Q|∫Q∫(4Q)cr|x0−z|n−γ+1|f(z)|dzdy≤C∫(4Q)cr|x0−z|n−γ+1|f(z)|dz≤C∑j=2∞12j1|2j+1Q|1−γ/n∫2j+1Q|f(z)|dz.

Futher, notice that λ/p = 1/q, by Hölder’s inequality and (3.5),

≤C∑j=1∞12j1|2j+1Q|1−γ/n∫2j+1Qu(z)−p′/pdz1/p′∫2j+1Q|f(y)|pu(z)dz1/p≤C∥f∥Lp,λ(u,v)∑j=1∞12jv(2j+1Q)λ/p|2j+1Q|1−γ/n∫2j+1Qu(z)−p′/pdz1/p′=C∥f∥Lp,λ(u,v)∑j=1∞12jv(2j+1Q)1/q|2j+1Q|1−γ/n∫2j+1Qu(z)−p′/pdz1/p′≤C∥f∥Lp,λ(u,v)∑j=1∞12j|2j+1Q|1/(r′q)|2j+1Q|1−γ/n∫2j+1Qv(z)rdz1/(rq)∫2j+1Qu(z)−p′/pdz1/p′≤C∥f∥Lp,λ(u,v).

Therefore,

II≤C∥f∥Lp,λ(u,v).

Combining the above estimates for I and II, and then taking the supremum over all cubes Q ⊂ ℝⁿ, we finish the proof of Theorem.□

Theorem 3.6

Let 1 1 and for all cubes Q ⊂ ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qv(x)rdx1/(rq)1|Q|∫Qu(x)−p′/pdx1/p′≤C (3.7)

Furthermore, we also suppose that 𝓣_γ satisfies the (p, q) type inequality

∫Rn|Jγf(x)|qv(x)dx1/q≤C∫Rn|f(x)|pu(x)dx1/p. (3.8)

If p ≤ α ≤ β < s < ∞, 1/s = 1/α − (1/p − 1/q) and v, ω ∈ △₂, then the sublinear operator 𝓣_γ is bounded from (L^p, L^s)^α(u, v, ω) to (BMO^q, L^s)^β(v, ω) with 1/β = 1/α − (1/p − 1/q).

Proof of Theorem 3.6

Let f ∈ (L^p, L^s)^α(u, v, ω) with 1 0,

v(Q(y,r))1/β−1/q−1/s∥(Jγf(x)−(Jγf)Q(y,r))χQ(y,r)∥Lq(v)≤v(Q(y,r))1/β−1/q−1/s∥(Jγf1(x)−(Jγf1)Q(y,r))χQ(y,r)∥Lq(v)+v(Q(y,r))1/β−1/q−1/s∥(Jγf2(x)−(Jγf2)Q(y,r))χQ(y,r)∥Lq(v)=:I+II.

For I, according to assumption (3.8), we obtain

where we have used 1/β − 1/q − 1/s = 1/α − 1/p − 1/s, 1/α − 1/p − 1/s < 0 and v ∈ △₂. For the term II, observe that for x, z ∈ Q(y, r) and η ∈ Q(y, 4 n r)^c, we have |x − η| ≥ 2|x − z| and |x − η| ≈ |η − y|. Thus, we have that for any x ∈ Q(y, r),

|Jγf2(x)−(Jγf2)Q(y,r)|=1|Q(y,r)|∫Q(y,r)(Jγf2(x)−Jγf2(z))dz≤C|Q(y,r)|∫Q(y,r)∫(Q(y,4nr))c1|x−η|n−γ−1|z−η|n−γ|f(η)|dηdz≤C|Q(y,r)|∫Q(y,r)∫(Q(y,4nr))c|x−z||x−η|n−γ+1|f(η)|dηdz≤C|Q(y,r)|∫Q(y,r)∫(Q(y,4nr))c|x−z||y−η|n−γ+1|f(η)|dηdz=C∫(Q(y,4nr))cr|y−η|n−γ+1|f(η)|dη≤C∑j=2∞12j1|Q(y,2j+1nr)|1−γ/n∫(Q(y,2j+1nr))|f(η)|dη.

Therefore,

II≤Cv(Q(y,r))1/β−1/s∑j=2∞12j1|Q(y,2j+1nr)|1−γ/n∫(Q(y,2j+1nr))|f(η)|dη.

A further application of Hölder’s inequality, (3.7) and Lemma 3.1, we have

≤Cv(Q(y,r))1/β−1/s∑j=2∞12j1|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)|f(η)|pu(η)dη1/p×∫Q(y,2j+1nr)u(η)−p′/pdη1/p′≤C∑j=2∞12jv(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)×v(Q(y,2j+1nr))1/q|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)u(η)−p′/pdη1/p′≤C∑j=2∞12jv(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)×|Q(y,2j+1nr)|1/r′q|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)v(η)rdη1/(rq)∫Q(y,2j+1nr)u(η)−p′/pdη1/p′≤C∑j=2∞12jv(Q(y,r))v(Q(y,2j+1nr))1/β−1/sv(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u)≤C∑j=2∞12jv(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u).

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 finish the proof of Theorem.□

4 High order commutators

4.1 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 suffices 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 : [0, +∞) → [0, +∞) is a continuous, convex and increasing function and satisfies Φ(0) = 0 and Φ(t) → +∞ as t → +∞. Given a Young function Φ, let E be a measurable set with |E| < ∞, define the Luxemburg norm of f over E as

∥f∥Φ,E:=infλ>0:1|E|∫EΦ|f(x)|λdx≤1.

In particular, when Φ = t^p, 1 < p < ∞, it is easy to check that

∥f∥Φ,Q=1|Q|∫Q|f(x)|pdx1/p.

that is, the Luxemburg norm coincides with the normalized L^p norm. For further details, we refer the reader to [29].

4.2 Boundedness of sublinear operator commutators

To prove our theorem we need the following Lemmas.

Lemma 4.1

[30] Let b be a function in BMO(ℝⁿ). Then

For every cube Q ⊂ ℝⁿ and for any positive integer j, then

|b2j+1Q−bQ|≤C(j+1)∥b∥BMO.
Let 1 < p < ∞. For every cube Q ⊂ ℝⁿ and for any ω ∈ A_∞. Then

∫Q|b(x)−bQ|pω(x)dx1/p≤Cω(Q)1/p∥b∥BMO.

Lemma 4.2

[29] Let 𝓐, 𝓑 and 𝓒 be Young functions such that for all t > 0,

A−1(t)B−1(t)≤C−1(t),

where 𝓐⁻¹(t) is the inverse function of 𝓐(t). Then for all functions f and g and all cubes Q ⊂ ℝⁿ, the generalized Hölder’s inequality

∥fg∥C,Q≤2∥f∥A,Q∥g∥B,Q.

Lemma 4.3

[31] If ω ∈ A_∞, then there exist δ > 0, C > 0 such that a measurable set E contained in a cube Q, the following inequality holds:

ω(E)ω(Q)≤C|E||Q|δ.

Now let us state our main results.

Theorem 4.4

Let m ≥ 1, 1 1 and for all cubes Q in ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qv(x)rdx1/rq∥u−1/p∥Φm,Q≤C, (4.1)

where Φ₁(t) = t^p′ (1 + log(e + t))^p′, Φ_m(t) = t^p′ (1 + log⁺ t)^mp′ (m = 2, 3, …).

Furthermore, we also suppose that [b, 𝓣_γ]_m satisfies the weak-(p, q) type inequality

δ⋅v({x∈Rn:|[b,Jα]m(f)(x)|>δ})1/q≤C∫Rn|f(x)|pu(x)dx1/p,δ>0. (4.2)

If v ∈ A_∞, then the sublinear operator higher order commutators [b, 𝓣_γ]_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 = 1. Let f ∈ L^p,λ(u, v) with 1 0,

1v(Q)qλ/p⋅1/qδ⋅v({x∈Q:|[b,Jγ](f)(x)|>δ})1/q≤1v(Q)λ/pδ⋅v({x∈Q:|[b,Jγ](f1)(x)|>δ/2})1/q+1v(Q)λ/pδ⋅v({x∈Q:|[b,Jγ](f2)(x)|>δ/2})1/q=:I+II.

For I, we notice that v ∈ △₂ (cf. [32]). according to assumption (4.2), we have

I≤Cv(Q)λ/p∫Rn|f1(x)|pu(x)dx1/p=Cv(Q)λ/p∫2Q|f(x)|pu(x)dx1/p≤Cv(2Q)v(Q)λ/p∥f∥Lp,λ(u,v)≤C∥f∥Lp,λ(u,v).

For the term II, notice that

|[b,Jγ](f2)(x)|≤C∫Rn|b(x)−b(y)||f2(y)||x−y|n−γdy≤C|b(x)−bQ|∫Rn|f2(y)||x−y|n−γdy+≤C∫Rn|b(y)−bQ||f2(y)||x−y|n−γdy=:ξ(x)+η(x).

Therefore,

II≤1v(Q)λ/pδ⋅v({x∈Q:ξ(x)>δ/4})1/q+1v(Q)λ/pδ⋅v({x∈Q:η(x)>δ/4})1/q=:II1+II2.

Since the condition (4.1) is stronger than the condition (3.1). By Chebyshev’s inequality, we obtain

II1≤4v(Q)λ/p∫Q|ξ(x)|qv(x)dx1/q≤Cv(Q)λ/p∫Q|b(x)−bQ|qv(x)dx1/q∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|f(y)|dy≤C∥b∥BMOv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|f(y)|dy≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|f(y)|pu(y)dy1/p∫2j+1Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)v(2j+1Q)1/q−λ/pv(2j+1Q)1/q|2j+1Q|1−γ/n∫2j+1Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)v(2j+1Q)1/q−λ/p×|2j+1Q|1/(r′q)|2j+1Q|1−γ/n∫2j+1Qv(y)rdy1/(rq)∫2j+1Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v).

where we have used Lemma 3.1. For the term II₂, observe that for x, x₀ ∈ Q and y ∈ (2Q)^c we have |x − y| ≈ |x₀ − y|. Thus, from Chebyshev’s inequality, we can obtain

II2≤4v(Q)λ/p∫Q|η(x)|qv(x)dx1/q≤Cv(Q)λ/p∫Q|∫(2Q)c|b(y)−bQ||f(y)||x−y|n−γdy|qv(x)dx1/q≤Cv(Q)λ/p∫Q|∑j=1∞∫2j+1Q∖2jQ|b(y)−bQ||f(y)||x−y|n−γdy|qv(x)dx1/q≤Cv(Q)λ/p∫Q|∑j=1∞∫2j+1Q∖2jQ|b(y)−bQ||f(y)||x0−y|n−γdy|qv(x)dx1/q≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b(y)−bQ||f(y)|dy≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b(y)−b2j+1Q||f(y)|dy+Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b2j+1Q−bQ||f(y)|dy=:II21+II22.

For II₂₁, putting 𝓒 = t^p′ is a Young function. We use Hölder’s inequality to get,

II21≤Cv(Q)1/q−λ/p∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b(y)−b2j+1Q|p′u(y)−p′/pdy1/p′×∫2j+1Q|f(y)|pu(y)dy1/p≤C∥f∥Lp,λ(u,v)v(Q)1/q−λ/p∑j=1∞v(2j+1Q)λ/p|2j+1Q|1−γ/n|2j+1Q|1/p′∥(b−b2j+1Q)u−1/p∥C,2j+1Q.

For 1 < p < ∞, we have

C−1(t)=t1/p′=t1/p′1+log+t(1+log+t)=:A−1(t)B−1(t),

and it is easy to see that 𝓐 ≈ t^p′(1 + log⁺ t)^p′, 𝓑 ≈ e^t − 1. ∥f∥_expL,Q denotes the mean Luxemburg norm of f on cube Q with Young function 𝓑 ≈ e^t − 1. By Lemma 4.2

∥(b−b2j+1Q)u−1/p∥C,2j+1Q≤C∥b−b2j+1Q∥expL,2j+1Q∥u−1/p∥A,2j+1Q≤C∥b∥BMO∥u−1/p∥A,2j+1Q,

where we have used the well-known fact that for any cube Q ⊂ ℝⁿ (cf. [30]),

∥b−bQ∥expL,Q≤C∥b∥BMO.

Therefore, we apply Hölder’s inequality, (4.1) and Lemma 3.1 to get

II21≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)v(2j+1Q)1/q−α/pv(2j+1Q)1/q|2j+1Q|1/p−γ/n∥b∥BMO∥u−1/p∥A,2j+1Q≤C∥f∥Lp,λ(u,v)∑j=1∞v(Q)v(2j+1Q)1/q−α/p×|2j+1Q|γ/n+1/q−1/p1|2j+1Q|∫2j+1Qv(y)rdy1/rq∥u−1/p∥A,2j+1Q≤C∥f∥Lp,λ(u,v).

For the term II₂₂, we make use of Lemma 4.1 and Hölder’s inequality, then

II22≤Cv(Q)1/q−λ/p∑j=1∞(j+1)∥b∥BMO|2j+1Q|1−γ/n∫2j+1Q|f(y)|dy≤Cv(Q)1/q−λ/p∑j=1∞(j+1)1|2j+1Q|1−γ/n∫2j+1Q|f(y)|pu(y)dy1/p∫2j+1Qu(y)−p′/pdy1/p′=C∥f∥Lp,λ(u,v)∑j=1∞(j+1)v(Q)v(2j+1Q)1/q−λ/pv(2j+1Q)1/q|2j+1Q|1−γ/n∫2j+1Qu(y)−p′/pdy1/p′.

Put 𝓒(t) and 𝓐(t) be the same as before. Obviously, 𝓒(t) ≤ 𝓐(t) for all t > 0, then for any cube Q ⊂ ℝⁿ, ∥f∥_𝓒,Q ≤ ∥f∥_𝓐,Q by definition, which implies that the condition (4.1) is stronger than the condition (4.1). From this and Hölder’s inequality yields

≤C∥f∥Lp,λ(u,v)∑j=1∞(j+1)v(Q)v(2j+1Q)1/q−α/p×|2j+1Q|1/r′q|2j+1Q|1−γ/n∫2j+1Qv(y)rdy1/rq∫2j+1Qu(y)−p′/pdy1/p′≤C∥f∥Lp,λ(u,v)∑j=1∞(j+1)v(Q)v(2j+1Q)1/q−α/p≤C∥f∥Lp,λ(u,v).

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 ⊂ ℝⁿ and all δ > 0, we finish the proof of Theorem.□

Theorem 4.5

Let m ≥ 1, 1 1 and for all cubes Q ⊂ ℝⁿ,

|Q|γ/n+1/q−1/p1|Q|∫Qω(x)rdx1/rq∥v−1/p∥Am,Q≤C, (4.3)

where 𝓐₁(t) = t^p′(1 + log(e + t))^p′, 𝓐_m(t) = t^p′(1 + log⁺t)^mp′ (m = 2, 3, …).

Furthermore, we also suppose that [b, 𝓣_γ]_m satisfies the weak-(p, q) type inequality

δ⋅v({x∈Rn:|Jγf(x)|>β})1/q≤C∫Rn|f(x)|pu(x)dx1/p,δ>0. (4.4)

If v ∈ △₂, ω ∈ A_∞, then the sublinear operator higher order commutators [b, 𝓣_γ]_m is bounded from (L^p, L^s)^α(u, v, ω) to (L^q,∞, L^s)^β(v, ω) with 1/β = 1/α − (1/p − 1/q).

Proof of Theorem 4.5

Because the method is similar, we only need to prove the case of m = 1. Let f ∈ (L^p, L^s)^α(u, v, ω) with 1 < p ≤ α < s ≤ ∞ and v ∈ △₂, ω ∈ A_∞. Fix Q := Q(y, r) ⊂ ℝⁿ, we split f = f₁ + f₂ with f₁ = fχ_4Q, where 4Q := Q(y, 4 n r). Then, for any y ∈ ℝⁿ,

v(Q(y,r))1/β−1/q−1/s∥[b,Jγ](f)χQ(y,r)∥Lq,∞(v)≤v(Q(y,r))1/β−1/q−1/s∥[b,Jγ](f1)χQ(y,r)∥Lq,∞(v)+v(Q(y,r))1/β−1/q−1/s∥[b,Jγ](f2)χQ(y,r)∥Lq,∞(v)=:I+II.

For I, notice that 1/β − 1/q − 1/s = 1/α − 1/p − 1/s, 1/α − 1/p − 1/s < 0 and v ∈ △₂. According to assumption (3.4), we have

I≤v(Q(y,r))1/β−1/q−1/s∥[b,Jγ](f1)∥Lq,∞(v)≤Cv(Q(y,r))1/β−1/q−1/s∫Rn|f1(x)|pu(x)dx1/p=Cv(Q(y,r))1/β−1/q−1/s∫Q(y,2nr)|f(x)|pu(x)dx1/p≤Cv(Q(y,2nr))1/β−1/q−1/s∥fχQ(y,2nr)∥Lp(u)≤v(Q(y,r))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u)=v(Q(y,r))v(Q(y,2nr))1/α−1/p−1/sv(Q(y,2nr))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u)≤Cv(Q(y,2nr))1/α−1/p−1/s∥fχQ(y,2nr)∥Lp(u).

For the term II, notice that

|[b,Jγ](f2)(x)|≤|b(x)−bQ(y,r)||Jγ(f2)(x)|+|Jγ((bQ(y,r)−b)f2)(x)|=:ξ(x)+η(x).

Therefore,

II≤v(Q(y,r))1/β−1/q−1/s∥ξ(⋅)χQ(y,r)∥Lq,∞(v)+v(Q(y,r))1/β−1/q−1/s∥η(⋅)χQ(y,r)∥Lq,∞(v)=:II1+II2.

For the term II₁, By Chebyshev’s inequality, Hölder’s inequality and v ∈ △₂ yields

For the term II₂, it is similar to Theorem 4.4. Together with Chebyshev’s inequality, we can obtain

II2≤4v(Q(y,r))1/β−1/q−1/s∫Q(y,r)|η(x)|qv(x)dx1/q≤Cv(Q(y,r))1/β−1/q−1/s∫Q|∫(2Q)c|b(y)−bQ||f(y)||x−y|n−γdy|qv(x)dx1/q≤Cv(Q(y,r))1/β−1/q−1/s∫Q|∫(2Q)c|b(y)−bQ||f(y)||x0−y|n−γdy|qv(x)dx1/q≤Cv(Q(y,r))1/β−1/s∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b(y)−bQ||f(y)|dy≤Cv(Q(y,r))1/β−1/s∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b(y)−b2j+1Q||f(y)|dy+Cv(Q(y,r))1/β−1/s∑j=1∞1|2j+1Q|1−γ/n∫2j+1Q|b2j+1Q−bQ||f(y)|dy=:II21+II22.

For II₂₁, putting 𝓒 = t^p′ is a Young function. It is similar to Theorem 4.4 and together with generalized Hölder’s inequality and Hölder’s inequality to get,

where we have used v ∈ A_∞. For the term II₂₂, we make use of Lemma 4.1 and Hölder’s inequality, then

It is similar to Theorem 4.4 and notice that the condition (4.3) is stronger than the condition (3.3). From these and Hölder’s inequality yields

≤C∑j=1∞v(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)(j+1)v(Q(y,r))v(Q(y,2j+1nr))1/β−1/s×|Q(y,2j+1nr)|1/r′q|Q(y,2j+1nr)|1−γ/n∫Q(y,2j+1nr)v(y)rdy1/rq∫Q(y,2j+1nr)u(y)−p′/pdy1/p′≤C∑j=1∞v(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)(j+1)v(Q(y,r))v(Q(y,2j+1nr))1/β−1/s.

Summing up all the above estimates and from the fact that 1/β − 1/q − 1/s = 1/α − 1/p − 1/s and v ∈ A_∞, we obtain

II≤C∑j=1∞v(Q(y,2j+1nr))1/β−1/q−1/s∥fχQ(y,2j+1nr)∥Lp(u)(j+1)v(Q(y,r))v(Q(y,2j+1nr))1/β−1/s=C∑j=1∞v(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u)(j+1)v(Q(y,r))v(Q(y,2j+1nr))1/β−1/s≤C∑j=1∞v(Q(y,2j+1nr))1/α−1/p−1/s∥fχQ(y,2j+1nr)∥Lp(u).

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 finish the proof of Theorem.□

Acknowledgement

The authors would like to thank the Referees and Editors for carefully reading the manuscript and making several useful suggestions.

This work is supported by the Natural Science Foundation of XinJiang Province (2016D01C044).

References

[1] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207-22610.1090/S0002-9947-1972-0293384-6Search in Google Scholar

[2] Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261-27410.1090/S0002-9947-1974-0340523-6Search in Google Scholar

[3] Sawyer E.T., A characterization of a two Weight norm inequality for maximal operators, Studia Math., 1982, 75, 1-1110.4064/sm-75-1-1-11Search in Google Scholar

[4] Neugebauer C.J., Inserting A_p weights, P. Am. Math. Soc., 1983, 87(4), 644-64810.1090/S0002-9939-1983-0687633-2Search in Google Scholar

[5] Pérez C., On sufficient conditions for the boundedness of the Hardy-Littlewood maximal function and the Hilbert transform, Studia Math., 1995, 60(19), 279-294Search in Google Scholar

[6] Soria F., Weiss G., A remark on singular integrals and power weights, Indiana Univ. Math. J., 1994, 43, 187-20410.1512/iumj.1994.43.43009Search in Google Scholar

[7] Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations, Amer. Math. Soc., 1938, 43, 126-16610.1090/S0002-9947-1938-1501936-8Search in Google Scholar

[8] Gilbarg D., Trudinger N.S., Elliptic partial differential equations of second order, 1983, Springer Verlag, Berlin, 2nd edition.Search in Google Scholar

[9] Giaquinta M., Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, 1983, Princeton Univ. Press.10.1515/9781400881628Search in Google Scholar

[10] Kufner A., John O., Fučik S., Function Spaces, Noordhoff International Publishing, 1977, 454 + XV pagesSearch in Google Scholar

[11] Taylor M.E., Tools for PDE: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials, 2000, volume 81 of Math. Surveys and Monogr, AMS, Providence, R.I.Search in Google Scholar

[12] Sawano Y., Tanaka H., The Fatou property of block spaces, J. Math. Sci. Univ. Tokyo, 2015, 22(1), 663-683Search in Google Scholar

[13] Sawano Y., Sugano S., Tanaka H., A note on generalized fractional integral operators on generalized Morrey spaces, Boundary Value Problems, 2009, Art. ID 835865, 18 pp.10.1155/2009/835865Search in Google Scholar

[14] Sawano Y., Sugano S., Tanaka H., Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc., 2011, 363, 6481-650310.1090/S0002-9947-2011-05294-3Search in Google Scholar

[15] Sawano Y., Sugano S., Tanaka H., Orlicz-Morrey spaces and fractional operators, Potential Anal., 2012, 36, 517-55610.1007/s11118-011-9239-8Search in Google Scholar

[16] Tanaka H., Gunawan H., The local trace inequality for potential type integral operators, Potential Anal., 2013, 38, 653-68310.1007/s11118-012-9291-zSearch in Google Scholar

[17] Adams D.R., Xiao J., Morrey spaces in harmonic analysis, Ark. Mat., 2012, 50, 201-23010.1007/s11512-010-0134-0Search in Google Scholar

[18] Chiarenza F., Frasca M., Morrey spaces and Hardy-Littlewood maximal function, Rend. Math., 1987, 7, 273-279Search in Google Scholar

[19] Komori Y., Shirai S., Weighted Morrey spaces and singular integral operator, Math. Nachr., 2009, 282(2), 219-23110.1002/mana.200610733Search in Google Scholar

[20] Peetre J., On convolution operators leaving 𝓛^p,λ spaces invariant, Annali di Mat. Pura ed Appl., 1966, 72(1), 295-30410.1007/BF02414340Search in Google Scholar

[21] Peetre J., On the theory of 𝓛^p,λ spaces, Function. Analysis, 1969, 4, 71-8710.1016/0022-1236(69)90022-6Search in Google Scholar

[22] Spanne S., Some function spaces defind by using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa, 1965, 19, 593-608Search in Google Scholar

[23] John F., Nirenberg L., On functions of bounded mean oscillation, Comm. Pure Appl. Math., 1961, 14, 415-42610.1007/978-1-4612-5412-6_36Search in Google Scholar

[24] Wiener N., On the representation of functions by trigonometrical integrals, Math. Z., 1926, 24, 575-61610.1007/BF01216799Search in Google Scholar

[25] Holland F., Harmonic analysis on amalgams of L^p and l^q, J. London Math. Soc., 1975, 10(2), 295-30510.1112/jlms/s2-10.3.295Search in Google Scholar

[26] Fofana I., Étude ďune classe ďespaces de fonctions contenant les espaces de Lorentz, Afr. Mat., 1988, 2, 29-50Search in Google Scholar

[27] Fofana I., Continuité de ľintégrale fractionnaire et espace (L^q,l^p)^α, CR Acad. Sci. Paris, 1989, 308(1), 525-527Search in Google Scholar

[28] Wang H., Two weight, weak type norm inequalities for a class of sublinear operators on weighted Morrey and Amalgam spaces, arXiv: 1712. 01269v1 [math.CA] 3 Dec, 2017.10.1155/2020/3673921Search in Google Scholar

[29] Musielak J., Orlicz spaces and Modular spaces, Lect. Notes Math., 1983, 1034(4), 1-21610.1007/BFb0072210Search in Google Scholar

[30] Pérez C., Endpoint estimates for commutators of singular integral operators, J. Funct. Anal., 1995, 128, 163-18510.1006/jfan.1995.1027Search in Google Scholar

[31] Garca-Cuerva J., Rubio de Francia J.L., Weighted norm inequalities and related topics, Math. Stud., 1985, 116, North-HollandSearch in Google Scholar

[32] Fefferman C., Muckenhoupt B., Two nonequivalent conditions for weight functions, P. Am. Math. Soc., 1974, 45(1), 99-10410.1090/S0002-9939-1974-0360952-XSearch in Google Scholar

Received: 2018-12-13

Accepted: 2019-05-15

Published Online: 2019-07-31

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0061

Keywords for this article

Sublinear operator; Two weight; Morrey space; Amalgam space; BMO space

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