Norm Inequalities in Generalized Morrey Spaces

Abstract

We prove that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel as well as commutators associated to these operators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on a wide family of function spaces.

1 Introduction

In the last decade, many works in classical harmonic analysis have been devoted to norm inequalities involving classical and non-classical operators in the setting of weighted Morrey spaces. The results obtained are most of the time extensions of well known analogues in the weighted Lebesgue spaces.

We equip the $n$-dimensional Euclidean space $\mathbb R^{n}$ with the Euclidean norm $\left| \cdot \right|$ and the Lebesgue measure $dx$. A weight is any positive measurable function $w$ which is locally integrable on $\mathbb R^{n}$. Let $w$ be a weight, $1\le q<\infty$ and $0<\kappa <1$. As Komori and Shirai in [12], we define the weighted Morrey space by

$$\begin{aligned} L^{q,\kappa }(w)=\left\{ f\in L^{q}_{loc}(w):\left\| f\right\| _{L^{q,\kappa }(w)}<\infty \right\} \end{aligned}$$

where

$$\begin{aligned} \left\| f\right\| _{L^{q,\kappa }(w)}:=\sup _{B}\left( \frac{1}{w(B)^{\kappa }}\int \limits _{B}\left| f(x)\right| ^{q}w(x)dx\right) ^{\frac{1}{q}}. \end{aligned}$$

(1.1)

The supremum is taken over all balls $B$ in $\mathbb R^{n}$, and $w(B)=\int \nolimits _{B}w(x)dx$. When $w\equiv 1$, we use the notation $\left| B\right|$ for the Lebesgue measure of $B$. These spaces can be viewed as extensions of weighted Lebesgue spaces $L^{q}(w)$,i.e., spaces that consist of measurable functions $f$ satisfying

$$\begin{aligned} \left\| f\right\| _{L^{q}(w)}:=\left( \int \limits _{\mathbb R^{n}}\left| f(x)\right| ^{q}w(x)dx\right) ^{\frac{1}{q}}<\infty . \end{aligned}$$

It has been proved by many authors (see [12],[16],[17]) that most of the operators which are bounded on a weighted Lebesgue space are also bounded in an appropriate weighted Morrey space. We are going to prove that these results are valid on a larger family of functional spaces including weighted Morrey spaces.

Let $w$ be a weight and $1\le q\le \alpha \le p\le \infty$. We define the space $(L^{q}(w),L^{p})^{\alpha }:=(L^{q}(w),L^{p})^{\alpha }(\mathbb R^{n})$ as the space of all measurable functions $f$ satisfying $\left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}<\infty$, where

$$\begin{aligned} \left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}:=\sup _{r>0}{} _{r}\left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}, \end{aligned}$$

with

$$\begin{aligned} _{r}\left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}:=\left[ \int \limits _{\mathbb R^{n}}\left( w(B(y,r))^{\frac{1}{\alpha }-\frac{1}{q}-\frac{1}{p}}\left\| f\chi _{B(y,r)}\right\| _{L^{q}(w)}\right) ^{p}dy\right] ^{\frac{1}{p}} \end{aligned}$$

(1.2)

for any $r>0$, with the usual modification when $p=\infty$. In the case $w\equiv 1$, we recover the space $(L^{q},L^{p})^{\alpha }$ defined in [9] by Fofana (see also [7, 8]). Condition $q\le \alpha \le p$ ensures that the space is non trivial. For $q<\alpha$ and $p=\infty$, the space $(L^{q}(w),L^{\infty })^{\alpha }(\mathbb R^{n})$ is the weighted Morrey space $L^{q,\kappa }(w)$, with $\kappa =\frac{1}{q}-\frac{1}{\alpha }$.

It is immediate that the spaces $(L^{q}(w),L^{p})^{\alpha }$ equipped with $\left\| \cdot \right\| _{(L^{q}(w),L^{p})^{\alpha }}$ are Banach spaces for $1\le q\le \alpha \le p\le \infty$. Let $1\le q_{1}\le q_{2}\le \alpha \le p\le \infty$. For any weight $w$, we have the following inclusion :

$$\begin{aligned} \left\| f\right\| _{(L^{q_{1}}(w),L^{p})^{\alpha }}\le \left\| f\right\| _{(L^{q_{2}}(w),L^{p})^{\alpha }} \end{aligned}$$

which comes from Hölder inequality. In the particular case where $w=1$, this family of spaces is an increasing family in $p$.i.e., $(L^{q},L^{p_{1}})^{\alpha }\subset (L^{q},L^{p_{2}})^{\alpha }$ whenever $1\le q\le \alpha \le p_{1}\le p_{2}$. To our knowledge, the similar inclusions for spaces with weight are still open problems.

We prove in this paper that Calderón-Zygmund operators, Marcinkiewicz operators, maximal operators associated to Bochner-Riesz operators, operators with rough kernel and associated commutators which are known to be bounded on weighted Morrey spaces under appropriate conditions, are bounded on this weighted spaces for appropriate weight. In fact, we prove that operators which are bounded on weighted Lebesgue spaces and satisfy some local pointwise control, are also bounded in our context. Operators fulfilling this conditions, include Littlewood Paley operators with rough kernels, whose control in this spaces was given by Wei and Tao in [18]. We point out that the paper of Wei and Tao was published while ours was already in the hands of the referee and available on arxiv.

An important fact here is that the proof is simple and is the same for all kinds of operators that have been considered.

This paper is organized as follows.

In the next section, we recall the definitions of the operators we are going to deal with, and state the main results. Section 3 is devoted to proofs.

Throughout the paper, the letter $C$ is used for non-negative constants independent of the relevant variables that may change from one occurrence to another. Constants with subscript, such as $C_{0}$, do not change in different occurrences. We propose the following abbreviation $\mathbf{A}\lesssim \mathbf{B}$ for the inequalities $\mathbf{A}\le C \mathbf{B}$, where $C$ is a positive constant independent of the main parameters. If we have $\mathbf{A}\lesssim \mathbf{B}$ and $\mathbf{B}\lesssim \mathbf{A}$ then we put $\mathbf{A}\cong \mathbf{B}.$

For $\lambda >0$ and a ball $B\subset \mathbb R^{n}$, we write $\lambda B$ for the ball with same center as $B$ and radius $\lambda$ times radius of $B$. We denote by $E^{c}$ the complement of $E$.

2 Definitions and Statement of the Main Results

A weight $w$ belongs to $\mathcal A_{q}$ for $1\le q<\infty$ if there exists a constant $C>0$ such that for all balls $B\subset \mathbb R^{n}$ we have

$$\begin{aligned} \left\{ \begin{array}{lll} \left( \frac{1}{\left| B\right| }\int \nolimits _{B}w(x)dx\right) \left( \frac{1}{\left| B\right| }\int \nolimits _{B}w^{\frac{-q'}{q}}(x)dx\right) ^{\frac{q}{q'}}\le C&{}\text { if }&{}q>1,\\ \frac{1}{\left| B\right| }\int \nolimits _{B}w(z)dz\le C\mathrm {ess }\inf \nolimits _{x\in B}w(x)&{}\text { if }&{}q=1. \end{array}\right. \end{aligned}$$

(2.1)

where $\frac{1}{q}+\frac{1}{q'}=1$. We put $\mathcal A_{\infty }=\cup _{q\ge 1}\mathcal A_{q}$.

It is known (see [10] Proposition 9.1.5 p. 679 and Theorem 9.2.2 p. 685) that for $w\in \mathcal A_{p}$ with $1\le p<\infty$:

• the measure $w(x)dx$ is a doubling measure: precisely for all $\lambda >1$ and all balls $B$ we have

  $$\begin{aligned} w(\lambda B)\lesssim \lambda ^{np}w(B), \end{aligned}$$

  (2.2)

• there exists $s>1$ such that for any ball $B\subset \mathbb R^{n}$, we have

  $$\begin{aligned} \left( \frac{1}{\left| B\right| }\int \limits _{B}w(x)^{s}dx\right) ^{\frac{1}{s}}\lesssim \frac{1}{\left| B\right| }\int \limits _{B}w(x)dx. \end{aligned}$$

  (2.3)

Hölder inequality and (2.3) lead to

$$\begin{aligned} \frac{w(E)}{w(B)}\lesssim \left( \frac{\left| E\right| }{\left| B\right| }\right) ^{\frac{s-1}{s}}, \end{aligned}$$

(2.4)

for any measurable subset $E$ of a ball $B$.

Let $T$ be a Calderón-Zygmund operator given by

$$\begin{aligned} Tf(x)=\mathrm {p.v}\int \limits _{\mathbb R^{n}}K(x-y)f(y)dy, \end{aligned}$$

where $K$ is of class $\mathcal C^{1}(\mathbb R^{n}{\setminus }\left\{ 0\right\} )$ with

$$\begin{aligned} \left| K(x)\right| \le \frac{C_{K}}{\left| x\right| ^{n}}\text { and }\left| \nabla K(x)\right| \le \frac{C_{K}}{\left| x\right| ^{n+1}}\text { for }\ x\ne 0. \end{aligned}$$

It is a classical result that the operator $T$ is bounded on $L^{q}(w)$ whenever $1<q<\infty$ and $w\in \mathcal A_{q}$, whereas for $q=1$ and $w\in \mathcal A_{1}$ we have the following weak type inequality

$$\begin{aligned} \left\| Tf\right\| _{L^{1,\infty }(w)}:=\sup _{\lambda >0}\lambda w\Big (\left\{ x\in \mathbb R^{n}:\left| Tf(x)\right| >\lambda \right\} \Big )\lesssim \left\| f\right\| _{L^{1}(w)}. \end{aligned}$$

(2.5)

These results can be found in [4] or [10]. Komori and Shirai extended them to weighted Morrey spaces in [12]. We prove in this paper that the results remain valid in our weighted spaces. For this purpose, we put for $r>0$

$$\begin{aligned} _{r}\left\| f\right\| _{(L^{1,\infty }(w),L^{p})^{\alpha }}:=\left[ \int \limits _{\mathbb R^{n}}\left( w(B(y,r))^{\frac{1}{\alpha }-1-\frac{1}{p}}\left\| f\chi _{B(y,r)}\right\| _{L^{1,\infty }(w)}\right) ^{p}dy\right] ^{\frac{1}{p}}, \end{aligned}$$

and

$$\begin{aligned} \left\| f\right\| _{(L^{1,\infty }(w),L^{p})^{\alpha }}:=\sup _{r>0}{} _{r}\left\| f\right\| _{(L^{1,\infty }(w),L^{p})^{\alpha }}. \end{aligned}$$

(2.6)

We have the following result :

Theorem 2.1

If $1<q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q}$, then the Calderón-Zygmund operator $T$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

If $q=1$ and $w\in \mathcal A_{1}$, then we have

$$\begin{aligned} \left\| T f\right\| _{(L^{1,\infty }(w),L^{p})^{\alpha }}\lesssim \left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}. \end{aligned}$$

Remark that this result contains Theorem 3.3 in [12] as a particular case.

In the case $n\ge 2$, we denote by $\mathbb S^{n-1}$ the unit sphere in $\mathbb R^{n}$ equipped with the normalized Lebesgue measure $d\sigma$. For any $\Omega \in L^{\theta }(\mathbb S^{n-1})$ with $1<\theta \le \infty$, homogeneous of degree zero and such that

$$\begin{aligned} \int \limits _{\mathbb S^{n-1}}\Omega (x')d\sigma (x')=0, \end{aligned}$$

where $x'=x/\left| x\right|$ for any $x\ne 0$, we define the homogeneous singular integral operator $T_{\Omega }$ by

$$\begin{aligned} T_{\Omega }f(x)=\mathrm {p.v}\int \limits _{\mathbb R^{n}}\frac{\Omega (y')}{\left| y\right| ^{n}}f(x-y)dy, \end{aligned}$$

and the Marcinkiewicz integral of higher dimension $\mu _{\Omega }$ by

$$\begin{aligned} \mu _{\Omega }(f)(x)=\left( \int \limits _{0}^{\infty }\left| \int \limits _{\left| x-y\right| \le t}\frac{\Omega (x-y)}{\left| x-y\right| ^{n-1}}f(y)dy\right| ^{2}\frac{dt}{t^{3}}\right) ^{\frac{1}{2}}. \end{aligned}$$

Duoandikoetxea in [4] proved that for $\Omega \in L^{\theta }(\mathbb S^{n-1})$ and $1<\theta <\infty$, if $\theta '\le q<\infty$ and $w\in \mathcal A_{q/\theta '}$ then the operator $T_{\Omega }$ is bounded on $L^{q}(w)$. We prove the following extension of Theorem 2 in [17].

Theorem 2.2

Let $\Omega \in L^{\theta }(\mathbb S^{n-1})$ with $1<\theta <\infty$. Then for $\theta '\le q\le \alpha < p\le \infty$ and $w\in \mathcal A_{q/\theta '}$, the operator $T_{\Omega }$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

As far as Marcinkiewicz operators are concerned, it is proved in [2] that if $\Omega \in L^{\theta }(\mathbb S^{n-1})$ and $1<\theta \le \infty$, then for every $\theta '< q<\infty$ and $w\in \mathcal A_{q/\theta '}$, there exists $C>0$ such that

$$\begin{aligned} \left\| \mu _{\Omega }f\right\| _{L^{q}(w)}\le C\left\| f\right\| _{L^{q}(w)}. \end{aligned}$$

(2.7)

Theorem 2.3

Let $\Omega \in L^{\theta }(\mathbb {S}^{n-1})$ with $1<\theta \le \infty$. Then for $\theta '< q\le \alpha < p\le \infty$ and $w\in \mathcal A_{q/\theta '}$, the operator $\mu _{\Omega }$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

We can find a similar result in Theorem 1.1 of [18]. This result for the limit case $p=\infty$, corresponds to Theorem 4 in [17].

We also define the Bochner-Riesz operators of order $\delta >0$ in terms of Fourier transforms by

$$\begin{aligned} \left( T^{\delta }_{R}f\right) \widehat{}(\xi )=\left( 1-\frac{\left| \xi \right| ^{2}}{R^{2}}\right) ^{\delta }_{+}\hat{f}(\xi ), \end{aligned}$$

where $\hat{f}$ denote the Fourier transform of $f$. These operators can be expressed as convolution operators by the formula

$$\begin{aligned} T^{\delta }_{R}f(x)=(f*\phi _{1/R})(x), \end{aligned}$$

(2.8)

where $\phi (x)=[(1-\left| \cdot \right| ^{2})^{\delta }_{+}]\,\check{}\;(x)$, and $\check{f}$ standing for the inverse Fourier transform of $f$. The associate maximal operator is defined by

$$\begin{aligned} T^{\delta }_{*}f(x)=\sup _{R>0}\left| T^{\delta }_{R}f(x)\right| . \end{aligned}$$

Let $n\ge 2$. For $1<q<\infty$ and $w\in \mathcal A_{q}$, Shi and Sun showed in [14] that $T^{(n-1)/2}_{*}$ is bounded on $L^{q}_{w}$. In the limit case $q=1$ we have in [15] a weak type inequality when $w\in \mathcal A_{1}$, i.e.,

$$\begin{aligned} \left\| T^{(n-1)/2}_{1}f\right\| _{L^{1,\infty }(w)}\lesssim \left\| f\right\| _{L^{1}(w)}. \end{aligned}$$

(2.9)

Putting together (2.9) and the fact that for a fixed $R>0$

$$\begin{aligned} T^{(n-1)/2}_{R}f(x)=(\phi *f_{R})_{1/R}(x), \end{aligned}$$

this implies (see [16]) that the weak-type inequality (2.9) is satisfied for any $R>0$.

Theorem 2.4

Let $1\le q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q}$.

• (1) If $q>1$ and $\delta =\frac{n-1}{2}$, then $T^{\delta }_{*}$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

• (2) If $q=1$ and $\delta =\frac{n-1}{2}$ then for any $R>0$,

  $$\begin{aligned} \left\| T^{\delta }_{R}f\right\| _{(L^{1,\infty }(w),L^{p})^{^{\alpha }}}\lesssim \left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }}. \end{aligned}$$

This result contains Theorems 1 and 2 of [16].

We also have results about commutators generated by those operators.

We recall that for a linear operator $\mathcal T$ and a locally integrable function $b$, the commutator operator is defined by

$$\begin{aligned} \left[ b,\mathcal T\right] f(x)=b(x)\mathcal T f(x)-\mathcal T (bf)(x). \end{aligned}$$

In [13], it is proved that for the Calderón-Zygmund operator $T$ and $b\in BMO$, i.e., the space consisting of locally integrable functions satisfying $\left\| b\right\| _{BMO}<\infty$, where

$$\begin{aligned} \left\| b\right\| _{BMO}:=\sup _{B:\text { ball}}\frac{1}{\left| B\right| }\int \limits _{B}\left| b(x)-b_{B}\right| dx, \end{aligned}$$

with $b_{B}=\frac{1}{\left| B\right| }\int \nolimits _{B}b(z)dz$, the commutators $\left[ b,T\right]$ are bounded in the weighted Lebesgue space $L^{q}(w)$ whenever $1<q<\infty$ and $w\in \mathcal A_{q}$. More precisely, there exists $C>0$ such that

$$\begin{aligned} \left\| \left[ b,T\right] f\right\| _{L^{q}(w)}\le C\left\| b\right\| _{BMO}\left\| f\right\| _{L^{q}(w)}, \end{aligned}$$

for all $f\in L^{q}(w)$.

Theorem 2.5

Let $b\in BMO$ and $T$ be a Calderón-Zygmund operator. If $1<q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q}$, then the operator $\left[ b,T\right]$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

For $b\in BMO$, the boundedness of $\left[ b,T_{\Omega }\right]$ on $L^{q}(w)$ when $\Omega \in L^{\theta }(\mathbb S^{n-1})$, $1<\theta <\infty$, $\theta '<q<\infty$ and $w\in \mathcal A_{q/\theta '}$ and the one of $\left[ b,T^{\delta }_{R}\right]$ on $L^{q}(w)$ for $1<q<\infty$ and $w\in \mathcal A_{q}$ are just consequences of the well-known boundedness criterion for commutators of linear operators obtained by Alvarez et al in [1]. We deduce from this, the following.

Theorem 2.6

Let $\Omega \in L^{\theta }(\mathbb S^{n-1})$ with $1<\theta <\infty$ and $b\in BMO$. For every $\theta '<q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q/\theta '}$, the commutator $\left[ b,T_{\Omega }\right]$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

Theorem 2.7

Let $\Omega \in L^{\theta }(\mathbb S^{n-1})$ with $1<\theta <\infty$ and $b\in BMO$. For every $\theta '<q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q/\theta '}$, the commutator $\left[ b,\mu _{\Omega }\right]$ is bounded on $(L^{q}(w),L^{p})^{\alpha }$.

The commutators of Marcinkiewicz operators $\mu _{\Omega }$ and a locally integrable function $b$ can be defined by

$$\begin{aligned} \left[ b,\mu _{\Omega }\right] (f)(x)=\left( \int \limits _{0}^{\infty }\left| \int \limits _{\left| x-y\right| \le t}\frac{\Omega (x-y)}{\left| x-y\right| ^{n-1}}[b(x)-b(y)]f(y)dy\right| ^{2}\frac{dt}{t^{3}}\right) ^{\frac{1}{2}}. \end{aligned}$$

Notice that $\left[ b,\mu _{\Omega }\right] (f)(x)=\mu _{\Omega }[(b(x)-b)f](x)$. For $\Omega \in L^{\theta }(\mathbb S^{n-1})$, $1<\theta \le \infty$, the boundedness of $\left[ b,\mu _{\Omega }\right]$ on $L^{q}(w)$ when $b\in BMO$, $\theta '<q<\infty$ and $w\in \mathcal A_{q/\theta '}$ was established in [2]. We have the following result.

Theorem 2.8

Let $1<q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q}$. If $\delta \ge \frac{n-1}{2}$ and $b\in BMO$ then the linear commutators $\left[ b,T^{\delta }_{R}\right]$ are bounded on $(L^{q}(w),L^{p})^{\alpha }$.

The case $p=\infty$ in Theorem 2.5 was proved by Komori and Shirai (see Theorem 3.4 [12]), while the same case in Theorems 2.6, 2.7 and 2.8 are done by Wang in [17] and [16].

3 Proof of the Main Results

The following lemma will be the cornerstone in the proofs of our theorems. The results established in [18] can also be viewed as consequences.

Lemma 3.1

Let $1\le s\le q<\infty$, $w\in \mathcal A_{q/s}$ and $\mathcal T:L^{q}_{\mathrm {loc}}(w)\rightarrow L^{q}_{\mathrm {loc}}(w)$ a sub linear operator which satisfies the following property : for all balls $B\subset \mathbb R^{n}$

$$\begin{aligned} \mathcal T(f\chi _{(2B)^{c}})(x)\lesssim \sum ^{\infty }_{k=1}k\left( \frac{1}{\left| 2^{k+1}B\right| }\int \limits _{2^{k+1}B}\left| f(z)\right| ^{s}dz\right) ^{\frac{1}{s}}\text { a.e. on }B. \end{aligned}$$

(3.1)

Then

• (1) if $q>1$ and $\mathcal T$ is bounded on $L^{q}(w)$, then it is also bounded on $(L^{q}(w),L^{p})^{\alpha }$, for $q\le \alpha <p\le \infty$,

• (2) if for all $\lambda >0$

  $$\begin{aligned} w\left( \left\{ x\in \mathbb R^{n}:\left| \mathcal Tf(x)\right| >\lambda \right\} \right) \lesssim \frac{1}{\lambda }\int \limits _{\mathbb R^{n}}\left| f(y)\right| w(y)dy, \end{aligned}$$

  (3.2)

  then for $1\le \alpha <p\le \infty$, $\mathcal T$ is bounded from $(L^{1}(w),L^{p})^{\alpha }$ to $(L^{1,\infty }(w),L^{p})^{\alpha }$.

The proof is partially inspired by [5]. The same arguments are used in [3] (see also [6]), to prove norm inequalities involving Riesz potentials and integral operators satisfying the hypothesis of Theorem 2.1 of [5] in the context of $(L^{q},L^{p})^{\alpha }(\mathbb R^{n})$ spaces.

Proof

Let $1\le q\le \alpha <p\le \infty$ and $f\in (L^{q}(w),L^{p})^{\alpha }$. We fix $y\in \mathbb R^{n}$ and $r>0$. For almost every $x\in B(y,r)$, we have

$$\begin{aligned} \left| \mathcal Tf(x)\right|&\le \left| \mathcal T(f\chi _{B(y,2r)})(x)\right| +\left| \mathcal T(f\chi _{B(y,2r)^{c}})(x)\right| \\&\lesssim \left| \mathcal T(f\chi _{B(y,2r)})(x)\right| +\sum ^{\infty }_{k=1}k\left( \frac{1}{\left| 2^{k+1}B\right| }\int \limits _{2^{k+1}B}\left| f(z)\right| ^{s}dz\right) ^{\frac{1}{s}} \end{aligned}$$

according to (3.1).

• If $q=s$ then $w\in \mathcal A_{1}$ implies that

  $$\begin{aligned} \left( \!\frac{1}{\left| B\right| }\int \limits _{B}\left| f(z)\right| ^{q}dz\!\right) ^{\frac{1}{q}}\!=\left( \!\frac{1}{w(B)}\int \limits _{B}\frac{w(B)}{\left| B\right| }\left| f(z)\right| ^{q}dz\!\right) ^{\frac{1}{q}}\!\lesssim \frac{1}{w(B)^{\frac{1}{q}}}\left\| f\chi _{B}\right\| _{L^{q}(w)} \end{aligned}$$

  for all balls $B\subset \mathbb R^{n}$.

• If $s<q$ then Hölder inequality and $\mathcal A_{q/s}$ characterization yield

  $$\begin{aligned} \left( \frac{1}{\left| B\right| }\int \limits _{B}\left| f(z)\right| ^{s}dz\right) ^{\frac{1}{s}}&\le \left[ \frac{1}{\left| B\right| }\left( \int \limits _{B}\left| f(z)\right| ^{q}w(z)dz\right) ^{\frac{s}{q}}\left( \int \limits _{B}w(z)^{-\frac{s}{q-s}}dz\right) ^{\frac{q-s}{q}}\right] ^{\frac{1}{s}}\\&\lesssim \frac{1}{w(B)^{\frac{1}{q}}}\left\| f\chi _{B}\right\| _{L^{q}(w)}. \end{aligned}$$

It comes that

$$\begin{aligned} \left| \mathcal Tf(x)\right| \lesssim \left| \mathcal T(f\chi _{B(y,2r)})(x)\right| +\sum ^{\infty }_{k=1}\frac{k}{w(B(y,2^{k+1}r))^{\frac{1}{q}}}\big \Vert f\chi _{B(y,2^{k+1}r)}\big \Vert _{L^{q}(w)}\,\,\, \end{aligned}$$

(3.3)

for almost every $x\in B(y,r)$.

First case $q>1$.

Taking the $L^{q}(w)$-norm on the ball $B(y,r)$ of both sides of (3.3) we obtain

$$\begin{aligned} \begin{array}{lll} \left\| \mathcal Tf\chi _{B(y,r)}\right\| _{L^{q}(w)}&{}\lesssim &{}\left\| f\chi _{B(y,2r)}\right\| _{L^{q}(w)}\\ &{}+&{}\sum \limits _{k=1}^{\infty } k\left\| f\chi _{B(y,2^{k+1}r)}\right\| _{L^{q}(w)}\left( \frac{w(B(y,r))}{w(B(y,2^{k+1}r))}\right) ^{\frac{1}{q}} \end{array}. \end{aligned}$$

(3.4)

according to the boundedness of $\mathcal T$ on $L^{q}(w)$.

Hence, multiplying both sides of (3.4) by $w(B)^{\frac{1}{\alpha }-\frac{1}{q}-\frac{1}{p}}$ we obtain

$$\begin{aligned}&w(B(y,r))^{\frac{1}{\alpha }-\frac{1}{q}-\frac{1}{p}}\left\| \mathcal Tf\chi _{B(y,r)}\right\| _{L^{q}(w)}\nonumber \\&\quad \lesssim \sum ^{\infty }_{k=0}\frac{k}{2^{kn\frac{s-1}{s}(\frac{1}{\alpha }-\frac{1}{p})}}w(B(y,2^{k+1}r))^{\frac{1}{\alpha }-\frac{1}{q}-\frac{1}{p}}\left\| f\chi _{B(y,2^{k+1}r)}\right\| _{L^{q}(w)}, \end{aligned}$$

(3.5)

for some $s>1$, according to Relations (2.2) and (2.4). Since (3.5) holds for every $y\in \mathbb R^{n}$, this leads to

$$\begin{aligned} _{r}\left\| \mathcal Tf\right\| _{(L^{q}(w),L^{p})^{\alpha }}\lesssim \left( 1+\sum ^{\infty }_{k=1}\frac{ k}{2^{kn\frac{s-1}{s}(\frac{1}{\alpha }-\frac{1}{p})}}\right) \left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }},\ r>0. \end{aligned}$$

The expected result follows from taking the supremum over all $r>0$, since $\sum ^{\infty }_{k=1}\frac{k}{2^{kn\frac{s-1}{s}(\frac{1}{\alpha }-\frac{1}{p})}}<\infty$.

Second case $q=1$.

For $\lambda >0$, we have

$$\begin{aligned} \begin{aligned}&w\Big (\left\{ x\in B(y,r):\left| \mathcal T f(x)\right| >\lambda \right\} \Big )\\&\ \ \ \ \ \ \ \ \ \ \ \ \ \lesssim \frac{1}{\lambda }\left( \left\| f\chi _{B(y,2r)}\right\| _{L^{1}(w)}+\sum ^{\infty }_{k=1}\frac{kw(B)}{w(B(y,2^{k+1}r))}\big \Vert f\chi _{B(y,2^{k+1}r)}\big \Vert _{L^{1}(w)}\right) \end{aligned} \end{aligned}$$

according to (3.3). That is,

$$\begin{aligned} \left\| \mathcal T f\chi _{B(y,r)}\right\| _{L^{1,\infty }(w)}\!\lesssim \! \left\| f\chi _{B(y,2r)}\right\| _{L^{1}(w)}\!+\!\sum ^{\infty }_{k=1}\frac{kw(B)}{w(B(y,2^{k+1}r))}\big \Vert f\chi _{B(y,2^{k+1}r)}\big \Vert _{L^{1}(w)}. \end{aligned}$$

Multiplying both sides by $w(B(y,r))^{\frac{1}{\alpha }-1-\frac{1}{p}}$, we conclude as in the case $q>1$.$\square$

An immediate application of the above lemma is the following weighted version of Theorem 2.1 in [5].

Proposition 3.2

Let $1< q\le \alpha <p\le \infty$. Assume that $\mathcal T$ is a sublinear operator satisfying the property that for any $f\in L^{1}$ with compact support and $x\notin \mathrm {supp }f$

$$\begin{aligned} \left| \mathcal Tf(x)\right| \lesssim \int \limits _{\mathbb R^{n}}\frac{\left| f(y)\right| }{\left| x-y\right| ^{n}}dy. \end{aligned}$$

(3.6)

• (1) If for $q>1$ and $w\in \mathcal A_{q}$ the operator $\mathcal T$ is bounded on $L^{q}(w)$ then it is also bounded on $(L^{q}(w),L^{p})^{\alpha }$.

• (2) If for $w\in \mathcal A_{1}$ we have the weak type estimate

  $$\begin{aligned} \left\| \mathcal Tf\right\| _{L^{1,\infty }(w)}\lesssim \left\| f\right\| _{L^{1}(w)}, \end{aligned}$$

  then we have

  $$\begin{aligned} \left\| \mathcal Tf\right\| _{(L^{1,\infty }(w),L^{p})^{\alpha }}\lesssim \left\| f\right\| _{(L^{1}(w),L^{p})^{\alpha }}. \end{aligned}$$

Proof

Let $B(y,r)$ be a ball in $\mathbb R^{n}$. For $x,z\in \mathbb R^{n}$, we have

$$\begin{aligned} x\in B(y,r)\text { and }z\notin B(y,2r)\Rightarrow \left| y-z\right| \le 2\left| z-x\right| \le 3\left| y-z\right| . \end{aligned}$$

(3.7)

Thus for $x\in B(y,r)$, we have

$$\begin{aligned} \left| \mathcal T(f\chi _{(B(y,2r))^{c}})(x)\right|&\lesssim \int \limits _{\mathbb R^{n}}\frac{\left| f\chi _{(B(y,2r))^{c}}(z)\right| }{\left| x-z\right| ^{n}}dz \lesssim \sum ^{\infty }_{k=1}\int \limits _{2^{k}r\le \left| y-z\right| <2^{k+1}r}\frac{\left| f(z)\right| }{\left| x-z\right| ^{n}}dz\\&\lesssim \sum ^{\infty }_{k=1}\frac{1}{\left| B(y,2^{k+1}r)\right| }\int \limits _{B(y,2^{k+1}r){\setminus }B(y,2^{k}r)}\left| f(z)\right| dz. \end{aligned}$$

The conclusion follows from Lemma 3.1. $\square$

The proof of Theorem 2.1 is straightforward from Proposition 3.2. Notice that this result is valid for non translation invariant CZ operators.

For Theorems 2.2, 2.3 and 2.4, we just have to prove that Hypothesis (3.1) of Lemma 3.1 is fulfilled to conclude.

Proof of Theorem 2.2 Let $B=B(y,r)$ be a ball of $\mathbb R^{n}$. For $x\in B(y,r)$ we have

$$\begin{aligned} \left| T_{\Omega }(f\chi _{(2B)^{c}})(x)\right|&\le \sum ^{\infty }_{k=1}\left( \,\,\int \limits _{2^{k+1}B{\setminus } 2^{k}B}\left| \Omega ((x-z)')\right| ^{\theta }dz\right) ^{\frac{1}{\theta }}\\&\times \,\left( \,\,\int \limits _{2^{k+1}B{\setminus } 2^{k}B}(\frac{\left| f(z)\right| }{\left| x-z\right| ^{n}})^{\theta '}dy\right) ^{\frac{1}{\theta '}} \end{aligned}$$

by Hölder Inequality. From (3.7), it comes that for $x\in B$ and $z\in 2^{k+1}B{\setminus }2^{k}B$, we have $2^{k-1}r\le \left| x-z\right| \le 2^{k+2}r$. Thus, for $x\in B(y,r)$ and any positive integer $k$, we have

$$\begin{aligned} \left( \,\,\int \limits _{2^{k+1}B{\setminus }2^{k}B}\left| \Omega ((x-z)')\right| ^{\theta }dz\right) ^{\frac{1}{\theta }}\lesssim \left\| \Omega \right\| _{L^{\theta }(\mathbb S^{n-1})}\left| 2^{k+1}B\right| ^{\frac{1}{\theta }}, \end{aligned}$$

(3.8)

and

$$\begin{aligned} \left( \,\,\int \limits _{2^{k+1}B{\setminus }2^{k}B}(\frac{\left| f(z)\right| }{\left| x-z\right| ^{n}})^{\theta '}dy\right) ^{\frac{1}{\theta '}}\lesssim \frac{1}{\left| 2^{k+1}B\right| }\left( \,\,\int \limits _{2^{k+1}B}\left| f(z)\right| ^{\theta '}dz\right) ^{\frac{1}{\theta '}}. \end{aligned}$$

(3.9)

Therefore, for any ball $B$ in $\mathbb R^{n}$, we have

$$\begin{aligned} \left| T_{\Omega }(f\chi _{(2B)^{c}})(x)\right| \lesssim \sum ^{\infty }_{k=1}\left( \frac{1}{\left| 2^{k+1}B\right| }\int \limits _{2^{k+1}B}\left| f(z)\right| ^{\theta '}dz\right) ^{\frac{1}{\theta '}}, \end{aligned}$$

for all $x\in B$, which ends the proof. $\square$

Proof of Theorem 2.3 Put $g=f\chi _{(2B)^{c}}$ where $B$ is a ball in $\mathbb R^{n}$. For $x\in B$ and $t>0$ we have

$$\begin{aligned} \left\{ z:\left| x-z\right| \le t\right\} \cap (2^{k+1}B{\setminus }2^{k}B)\ne \emptyset \Rightarrow t\ge 2^{k-1}r, \end{aligned}$$

(3.10)

for any positive integer $k$. Therefore,

$$\begin{aligned} \left| \mu _{\Omega }g(x)\right|&= \left( \int \limits ^{\infty }_{0}\left| \int \limits _{(2B)^{c}\cap \left\{ z:\left| x-z\right| \le t\right\} }\frac{\Omega (x-z)}{\left| x-z\right| ^{n-1}}f(z)dz\right| ^{2}\frac{dt}{t^{3}}\right) ^{\frac{1}{2}}\\&\le \sum ^{\infty }_{k=1}\left( \int \limits _{2^{k+1}B{\setminus }2^{k}B}\frac{\left| \Omega (x-z)\right| }{\left| x-z\right| ^{n-1}}f(z)dz\right) \left( \int \limits ^{\infty }_{2^{k-1}r}\frac{dt}{t^{3}}\right) ^{\frac{1}{2}}\\&\lesssim \sum ^{\infty }_{k=1}\frac{1}{\left| 2^{k+1}B\right| ^{1/n}}\int \limits _{2^{k+1}B{\setminus }2^{k}B}\frac{\left| \Omega (x-z)\right| }{\left| x-z\right| ^{n-1}}f(z)dz, \end{aligned}$$

for all $x\in B$. We end as in the proof of Theorem 2.2. $\square$

Proof of Theorem 2.4 As in the above proof, we put $g=f\chi _{(2B)^{c}}$, where $B$ is any ball in $\mathbb R^{n}$. Since for $R>0$, $T^{\delta }_{R}(g)(x)=\left| g*\phi _{1/R}(x)\right|$ with $\phi (x)=\left[ (1-\left| \cdot \right| ^{2})^{\delta }_{+}\right] \check{}(x)$, we have

$$\begin{aligned} \left| (g*\phi _{1/R})(x)\right| \le R^{n}\int \limits _{\mathbb R^{n}}\frac{\left| g(z)\right| }{(R\left| x-z\right| )^{n}}dz=\int \limits _{(2B)^{c}}\frac{\left| f(z)\right| }{\left| x-z\right| ^{n}}dz. \end{aligned}$$

for $x\in B$, where we use the fact that $\left| \phi (x)\right| \lesssim \frac{1}{(1+\left| x\right| )^{\frac{n+1}{2}+\delta }}$ for $\delta \ge \frac{n-1}{2}$. The Assertions (1) and (2) follow from Proposition 3.2. $\square$

For the results involving commutators, we need the following properties of $BMO$ (see [11]). For $b\in BMO$, $1<q<\infty$ and $w\in \mathcal A_{\infty }$ we have

$$\begin{aligned} \left\| b\right\| _{BMO}\cong \sup _{B:\text { ball}}\left( \frac{1}{\left| B\right| }\int \limits _{B}\left| b(x)-b_{B}\right| ^{q}dx\right) ^{\frac{1}{q}}, \end{aligned}$$

(3.11)

and for all balls $B$

$$\begin{aligned} \left( \frac{1}{w(B)}\int \limits _{B}\left| b(x)-b_{B}\right| ^{q}w(x)dx\right) ^{\frac{1}{q}}\lesssim \left\| b\right\| _{BMO}. \end{aligned}$$

(3.12)

Let $b\in BMO$ and $B$ a ball in $\mathbb R^{n}$. For all nonnegative integers $k$, we have

$$\begin{aligned} \left| b_{2^{k+1}B}-b_{B}\right| \lesssim (k+1)\left\| b\right\| _{BMO}. \end{aligned}$$

(3.13)

We also need the following lemma which is just an application of Hölder Inequality, the definition of $\mathcal A_{q}$ weights and Estimation (3.12). The proof is omitted.

Lemma 3.3

Let $1\le s<q<\infty$. For $b\in BMO$ and $w\in \mathcal A_{q/s}$, we have

$$\begin{aligned} \left( \,\,\int \limits _{2B{\setminus }B}\left| b(z)-b_{2B}\right| ^{s}\left| f(z)\right| ^{s}dz\right) ^{\frac{1}{s}}\lesssim \left\| b\right\| _{BMO}\left| 2B\right| ^{\frac{1}{s}}w(2B)^{-\frac{1}{q}}\left\| f\chi _{2B}\right\| _{L^{q}(w)} \end{aligned}$$

for all balls $B$ and $f\in L^{q}_{\mathrm {loc}}$.

Theorems 2.5, 2.6, 2.7 and 2.8 are immediate from the following weighted version of Theorem 2.2 in [5].

Proposition 3.4

Let $1\le \theta <q\le \alpha <p\le \infty$ and $w\in \mathcal A_{q/\theta }$. Assume $T$ is a sublinear operator which fulfills conditions (3.1) with $s=\theta$ and admits a commutator with any locally integrable function $b$, satisfying

$$\begin{aligned} \left[ b,T\right] (f)(x)=T[(b(x)-b)f](x). \end{aligned}$$

(3.14)

If $\left[ b,T\right]$ is bounded on $L^{q}(w)$, then $\left[ b,T\right]$ is also bounded on $(L^{q}(w),L^{p})^{\alpha }$.

Proof

Fix a ball $B=B(y,r)$ in $\mathbb R^{n}$. We have for all $x\in B(y,r)$

$$\begin{aligned}&\left| \left[ b,T\right] (f)(x)\right| \lesssim \left| \left[ b,T\right] (f\chi _{2B})(x)\right| +\left| b(x)-b_{B}\right| \left| T(f\chi _{(2B)^{c}})(x)\right| \\&\quad +\,\left| T[(b_{B}-b)f\chi _{(2B)^{c}}](x)\right| . \end{aligned}$$

Thus by the $L^{q}(w)$-boundedness of $\left[ b,T\right]$, Relations (3.1) and (3.12), we have

$$\begin{aligned} \left\| \left[ b,T\right] f\chi _{B}\right\| _{L^{q}(w)}&\lesssim \left\| f\chi _{2B}\right\| _{L^{q}(w)} \!+\!\left\| b\right\| _{BMO}\sum ^{\infty }_{k=1}k\left( \frac{w(B)}{w(2^{k+1}B)}\right) ^{\frac{1}{q}}\left\| f\chi _{2^{k+1}B}\right\| _{L^{q}(w)}\\&+\,w(B)^{\frac{1}{q}} \sum ^{\infty }_{k=1}k\left( \frac{1}{\left| 2^{k+1}B\right| }\int \limits _{2^{k+1}B{\setminus }2^{k}B}\left| b_{B}\!-\!b(z)\right| ^{\theta }\left| f(z)\right| ^{\theta }dz\right) ^{\frac{1}{\theta }}. \end{aligned}$$

But then it comes from (3.13) and Lemma 3.3 that

$$\begin{aligned} \left( \,\,\int \limits _{2^{k+1}B}\left| b_{B}-b(z)\right| ^{\theta }\left| f(z)\right| ^{\theta }dz\right) ^{\frac{1}{\theta }}\lesssim k\left\| b\right\| _{BMO}\frac{\left| 2^{k+1}B\right| ^{\frac{1}{\theta }}}{w(2^{k+1}B)^{\frac{1}{q}}}\left\| f\chi _{2^{k+1}B}\right\| _{L^{q}(w)}. \end{aligned}$$

Hence, we have

$$\begin{aligned}&\left\| \left[ b,T\right] f\chi _{B(y,r)}\right\| _{L^{q}(w)}\lesssim \left\| f\chi _{B(y,2r)}\right\| _{L^{q}(w)}+\left\| b\right\| _{BMO}\sum \limits _{k=1}^{\infty } \nonumber \\&\quad k^{2}\left( \frac{w(B(y,r))}{w(B(y,2^{k+1}r))}\right) ^{\frac{1}{q}}\left\| f\chi _{B(y,2^{k+1}r)}\right\| _{L^{q}(w)},\qquad \end{aligned}$$

(3.15)

for all $y\in \mathbb R^{n}$. Therefore, multiplying both sides of (3.15) by $w(B(y,r))^{\frac{1}{\alpha }-\frac{1}{q}-\frac{1}{p}}$, Estimates (2.2) and (2.4) and the $L^{p}$-norm allow us to obtain

$$\begin{aligned} \;_{r}\left\| \left[ b,T\right] f\right\| _{(L^{q}(w),L^{p})^{\alpha }}\lesssim \left\| b\right\| _{BMO}(1+\sum ^{\infty }_{k=1}\frac{k^{2}}{2^{nk\frac{s-1}{s}(\frac{1}{\alpha }-\frac{1}{p})}})\left\| f\right\| _{(L^{q}(w),L^{p})^{\alpha }},\nonumber \\ \end{aligned}$$

(3.16)

for some constant $s>1$ and all $r>0$. We end the proof by taking the supremum over all $r>0$. $\square$