1 Introduction

Given positive locally integrable functions (called weights)

$$\begin{aligned} w:(0,\infty ) \rightarrow (0,\infty ) \qquad \text { and } \qquad u: {\mathbb {R}}^n \rightarrow (0,\infty ), \end{aligned}$$

the main goal of this paper is to prove boundedness of important operators in harmonic analysis on weighted classical Lorentz spaces \(\Lambda ^p_u(w)\), \(p> 0\), defined by those measurable functions f such that

$$\begin{aligned} ||f||_{\Lambda ^p_u(w)} := \left( \int _0^{\infty }f_u^*(t)^pw(t)\,dt \right) ^{\frac{1}{p}} < \infty , \end{aligned}$$

(see [18]) where \(f_u^*\) is the decreasing rearrangement of f defined by

$$\begin{aligned} f_u^*(t) := \inf \{y>0:\lambda _f^u(y)\le t\}, \qquad t > 0, \end{aligned}$$

with

$$\begin{aligned} \lambda _f^u(y):=u(\{|f|>y\}) = \int _{\{|f|>y\}} u(x)\,dx, \qquad y > 0. \end{aligned}$$

This will lead to the unification of classical results about the boundedness of operators on different weighted settings, like the Lebesgue spaces \(L^p(u):= \Lambda ^p_u(1)\) or the classical Lorentz spaces \(\Lambda ^p(w):= \Lambda ^p_1(w)\), resulting into a general framework involving both theories.

Boundedness on weighted classical Lorentz spaces have been already studied for some operators. For instance, let us just mention [3], including the case of the Hilbert transform and the Hilbert maximal operator, and [17] that contains the case of Calderón-Zygmund operators and commutators. Further, we should also consider [18], where the authors study inter alia the characterization of the Hardy-Littlewood maximal operator.

Here, by means of the theory of weighted extrapolation, we present new estimates for operators such as Calderón-Zygmund operators \(T_K\), sparse operators \(A_{{\mathcal {S}}}\) or the Bochner-Riesz operator at the critical index \(B_{\frac{n-1}{2}}\), among many others.

Now, the examples mentioned above share one important property: they all satisfy that, for some (and hence for all) \(p_0 > 1\),

$$\begin{aligned} T: L^{p_0}(v) \longrightarrow L^{p_0}(v), \qquad \forall v \in A_{p_0}, \end{aligned}$$
(1)

is bounded, where for a given exponent \(p > 1\), \(A_p\) is the class of Muckenhoupt weights defined by those weights v that satisfy

$$\begin{aligned} ||v||_{A_p}:= \sup _{Q} \left( \frac{1}{|Q|} \int _Q v (x) dx\right) \left( \frac{1}{|Q|} \int _Q v (x) ^{\frac{1}{1-p}}dx \right) ^{p-1} <\infty , \end{aligned}$$

with the supremum being taken overall cubes \(Q \subseteq {\mathbb {R}^n}\). Further, those \(A_p\) weights characterize the boundedness on \(L^p(v)\) of the Hardy-Littlewood maximal operator M defined by

$$\begin{aligned} Mf(x)=\sup _{Q\ni x}\frac{1}{|Q|}\int _Q |f(y)|dy, \qquad f \in L^1_\text {loc}({\mathbb {R}^n}),\, x \in {\mathbb {R}}^n, \end{aligned}$$

where the supremum is taken overall cubes \(Q \subseteq {\mathbb {R}}^n\) containing x. Moreover, the definition of \(A_p\) can be extended to \(p = 1\), and we say that \(v\in A_1\) if there exists a positive constant C such that

$$\begin{aligned} Mv(x)\le C v(x), \qquad \text {a.e. } x\in {\mathbb {R}}^n, \end{aligned}$$

and \(\left\Vert v\right\Vert _{A_1}\) is set to be the infimum of such constants C. We shall point here that the \(A_p\) class of weights increases with the exponent (that is, \(A_p \subseteq A_q\), for \(p \le q\)) so it is natural to define the class of weights

$$\begin{aligned} A_\infty := \bigcup _{p \ge 1} A_p. \end{aligned}$$

Throughout this paper, an operator T satisfying (1) for some \(p_0\) and with norm constant less than or equal to \(\varphi (\Vert v \Vert _{A_{p_0}})\), and \(\varphi : [1,\infty ) \rightarrow (0,\infty )\) being a nondecreasing function, will be called a Rubio de Francia operator [42]. Those operators satisfy the well-known Rubio de Francia extrapolation theorem.

Theorem 1.1

[29] Given a Rubio de Francia operator T for some exponent \(p_0 \ge 1\). Then, for all \(p_1 > 1\),

$$\begin{aligned} T:L^{p_1}(v) \rightarrow L^{p_1}(v), \qquad \forall v \in A_{p_1}, \end{aligned}$$

with norm constant less than or equal to \(\Phi (\Vert v \Vert _{A_{p_1}})\) where

$$\begin{aligned} \Phi (r) = C_1\varphi \left( C_2r^{\max \left( 1 , \frac{p_0 - 1}{p_1-1}\right) }\right) , \qquad r \ge 1. \end{aligned}$$

A Banach function norm \(\rho \) is a mapping \(\rho :\mathcal {M}^{+} \rightarrow [0,\infty ]\), with \(\mathcal {M}^{+}\) being the set of positive measurable functions, such that the following properties hold:

  1. (i)

    \(\rho (f) = 0 \Leftrightarrow f = 0\) a.e.;

  2. (ii)

    \(\rho (af) = a\rho (f)\), for \(a \ge 0\);

  3. (iii)

    \(\rho (f + g) \le \rho (f) + \rho (g)\);

  4. (iv)

    if \(0 \le f \le g\) a.e., then \(\rho (f)\le \rho (g)\);

  5. (v)

    if a.e., then ;

  6. (vi)

    if E is a measurable set such that \(|E| < \infty \), then \(\rho (\chi _E) < \infty \) and \(\int _Ef\,dx \le C_E\rho (f)\) for some constant \(0< C_E < \infty \), depending on E and \(\rho \), but independent of f, where \(\chi _E\) represents the characteristic function of E.

The collection \(\mathbb {X} = \mathbb {X}(\rho )\) defined by

$$\begin{aligned} \mathbb {X} = \{f \in \mathcal {M} : \left\Vert f\right\Vert _{\mathbb {X}} := \rho (|f|) < \infty \} \end{aligned}$$

is called a Banach function space. Besides, by means of a function norm \(\rho \), we can define its associate norm \(\rho ':\mathcal {M}^{+} \rightarrow [0,\infty ]\) by

$$\begin{aligned} \rho '(f) = \sup \left\{ \int _{{\mathbb {R}}^n} f(x)g(x)\,dx : g \in \mathcal {M}^{+}, \rho (g) \le 1\right\} , \end{aligned}$$

which is itself a function norm. This allows us to define the associate space of \(\mathbb {X} = \mathbb {X}(\rho )\) to be the Banach function space \(\mathbb {X}' = \mathbb {X}(\rho ')\) (see [10, Ch. 1-Theorem 2.2]).

A function norm \(\rho \) is called rearrangement invariant (r.i. in short) if \(\rho (f) = \rho (g)\) for every pair of functions f and g that satisfy \(\lambda _f(y) = \lambda _g(y)\) for every \(y > 0\). In this case, we say that \(\mathbb {X} = \mathbb {X}(\rho )\) is a r.i. Banach function space (see [10, 27] for more details on those spaces). For instance, since the decreasing rearrangement on \((0,\infty )\) of f satisfy \(\lambda _{f^*}(y) = \lambda _f(y)\) for every \(y > 0\), in fact it can be obtained a representation of \(\mathbb {X}\) on \((\mathbb {R}^{+}, dt)\) (see [10, Ch. 2-Theorem 4.10]) as follows: there exists a r.i. Banach function space \(\overline{\mathbb {X}}\) over \((\mathbb {R}^{+}, dt)\) such that \(f \in \mathbb {X}\) if and only if \(f^{*} \in \overline{\mathbb {X}}\) with

$$\begin{aligned} \left\Vert f\right\Vert _{\mathbb {X}} = \left\Vert f^{*}\right\Vert _{\overline{\mathbb {X}}} := \sup _{\left\Vert g\right\Vert _{\mathbb {X}'}\le 1}\int _0^{\infty }f^{*}(t)g^{*}(t)\,dt. \end{aligned}$$

Moreover, when restricted to a r.i. Banach function space \(\mathbb {X}\) it is possible to define a weighted version of \(\mathbb {X}\) as

$$\begin{aligned} \mathbb {X}(u) = \{f \in \mathcal {M} : \left\Vert f\right\Vert _{\mathbb {X}(u)} :=\left\Vert f^{*}_u\right\Vert _{\overline{\mathbb {X}}} < \infty \}, \end{aligned}$$

for some weight u.

Now, as a consequence of [26, Theorem 4.10], it is known that the Rubio de Francia operators also satisfy boundedness over the weighted version of a r.i. Banach function spaces \(\mathbb {X}\) where, for the sake of simplicity, here we state a different version of that theorem consisting on introducing the maximal operator M and its dual induced by a weight \(u \in A_\infty \) defined as

$$\begin{aligned} M_u'f(x) = \frac{M(fu)(x)}{u(x)}, \qquad f \in L^1_{\text {loc}}({\mathbb {R}^n}), \, x \in {\mathbb {R}^n}, \end{aligned}$$
(2)

and where we have kept track of the involved norm constants.

Theorem 1.2

Given a Rubio de Francia operator T for some exponent \(p_0 > 1\). Let \(\mathbb {X}\) be a r.i. Banach function space and let \(u \in A_\infty \) such that

$$\begin{aligned} M:\mathbb {X}(u) \rightarrow \mathbb {X}(u) \qquad \text { and } \qquad M'_u:\mathbb {X}(u)' \rightarrow \mathbb {X}(u)', \end{aligned}$$
(3)

where \(\mathbb {X}(u)'\) is the associate space of \(\mathbb {X}(u)\) defined by those functions f that satisfy

$$\begin{aligned} \Vert f \Vert _{\mathbb {X}(u)'} := \sup _{\Vert g \Vert _{\mathbb {X}(u)} \le 1} \left| \int _{{\mathbb {R}^n}} f(x)g(x)u(x)\,dx\right| < \infty . \end{aligned}$$

Then,

$$\begin{aligned} T: \mathbb {X}(u) \rightarrow \mathbb {X}(u), \end{aligned}$$

with constant less than or equal to \(C_1\varphi \left( C_2\left\Vert M_u'\right\Vert _{\mathbb {X}(u)'}\left\Vert M\right\Vert _{\mathbb {X}(u)}^{p_0 - 1}\right) \), where \(\left\Vert M\right\Vert _{\mathbb {X}(u)}\) and \(\left\Vert M_u'\right\Vert _{\mathbb {X}(u)'}\) represent the norm constants of (3) respectively.

The previous result is very useful to prove the boundedness of operators for which condition (1) has been widely studied while this is not the case in other contexts such as, for example, weighted classical Lorentz spaces or more generally r.i. Banach function spaces. However, in order to get estimates over \(\mathbb {X}(u)\) we first must study whether \(\mathbb {X}\) is a r.i. Banach function space and when (3) holds.

For our goal, we will consider \(\mathbb {X}= \Lambda ^p(w)\), for \(p > 0\) and w being a weight in \(\mathbb {R}^{+}\), so that \(\mathbb {X}(u) = \Lambda ^p_u(w)\) and \(\mathbb {X}(u)' = (\Lambda ^p_u(w))'\) is defined by

$$\begin{aligned} \Vert f \Vert _{(\Lambda ^p_u(w))'} := \sup _{\Vert g \Vert _{\Lambda ^p_u(w)} \le 1} \left| \int _{{\mathbb {R}^n}} f(x)g(x)u(x)\,dx\right| < \infty . \end{aligned}$$

As a first approach, let us consider the particular case \(u = 1\):

i) Ariño and Muckenhoupt [6] proved that for \(p > 1\),

$$\begin{aligned} M:\Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w) \qquad \iff \qquad w\in B_p, \end{aligned}$$
(4)

where

$$\begin{aligned} w \in B_p \qquad \iff \qquad ||w||_{B_p} := \sup _{t>0}\dfrac{\int _0^\infty w(r) \min \big ( 1, \frac{t^p}{r^p}\big )\,dr}{W(t)} < \infty , \end{aligned}$$

with \(W(t) = \int _0^t w(r)\,dr\). This class of weights has been widely studied (see for instance [7, 8, 13, 39, 40]) and now it is known that the same result in (4) holds for every \(p>0\) (see [19]). Further, although for \(0< p < 1\), \(\Lambda ^p(w)\) is never a Banach function space (see [45, Remark 3.2]) for \(p \ge 1\), \(\Lambda ^p(w)\) is a Banach function space when \(w \in B_p\), and the reciprocal is also true whenever \(p > 1\) (see [15, 43]).

ii) The characterization of the boundedness of M on \((\Lambda ^p(w))'\) (see [8]) is a consequence of the Lorentz-Shimogaki theorem (see, for instance, [10, Ch.3 p. 154]). In particular, if \(w \in B_p\) then

$$\begin{aligned} M:(\Lambda ^{p}(w))' \longrightarrow (\Lambda ^{p}(w))' \qquad \iff \qquad w\in B_\infty ^*, \end{aligned}$$
(5)

where [39]

$$\begin{aligned} w \in B_\infty ^* \qquad \iff \qquad ||w||_{B_\infty ^*} := \sup _{t>0} \dfrac{1}{W(t)}\int _0^t\frac{W(r)}{r}\,dr < \infty . \end{aligned}$$

From (i) and (ii), we can conclude that for every Rubio de Francia operator T and \(p \ge 1\),

$$\begin{aligned} w\in B_p\cap B_\infty ^*\quad \implies \quad T: \Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w), \end{aligned}$$

and it is sharp since it is known [43] that

$$\begin{aligned} w\in B_p\cap B_\infty ^*\quad \iff \quad H: \Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w), \end{aligned}$$

with H being the Hilbert transform, which is a Rubio de Francia operator defined as

$$\begin{aligned} Hf(x) = \frac{1}{\pi }\lim _{\varepsilon \rightarrow 0^{+}}\int _{|x - y| > \varepsilon }\frac{f(y)}{x - y}\,dy, \qquad f \in \mathcal C^\infty _c\left( \mathbb {R} \right) ,\, x\in \mathbb {R}, \end{aligned}$$
(6)

whenever this limit exists almost everywhere and where \(\mathcal C_c^\infty ({\mathbb {R}})\) is the set of infinitely differentiable functions with compact support.

Now, take a general \(u \in A_\infty \). In [18, Theorem 3.3.5] it was characterized the weighted strong-type boundedness on \(\Lambda ^{p}_{u}\left( w \right) \) of the Hardy-Littlewood maximal operator for every \(p > 0\) by

$$\begin{aligned} M:\Lambda ^{p}_{u}\left( w \right) \rightarrow \Lambda ^{p}_{u}\left( w \right) \qquad \Longleftrightarrow \qquad w \in B_p(u), \end{aligned}$$
(7)

where we say that \(w \in B_p(u)\) if there exists some \(\varepsilon > 0\) such that

$$\begin{aligned} \sup _{E_j \subseteq Q_j, \, \forall 1 \le j \le J} \left( \inf _{1 \le j \le J} \frac{|E_j|}{|Q_j|}\right) \frac{W\left( u\left( \bigcup _{j = 1}^JQ_j \right) \right) ^\frac{1}{p-\varepsilon }}{W\left( u\left( \bigcup _{j = 1}^JE_j \right) \right) ^\frac{1}{p-\varepsilon }} < \infty , \end{aligned}$$
(8)

where the supremum is taken over every finite family of cubes \(\{Q_j\}_{j = 1}^J \subseteq {\mathbb {R}^n}\). (We refer the reader to [1, 18] for more details on this class of weights.)

  1. (i)

    If \(u = 1\), due to the \(B_p\) class of weights satisfy the \(p - \varepsilon \) property [18, 40] (that is, for \(w \in B_p\) there exists some \(\varepsilon > 0\) such that \(w \in B_{p-\varepsilon }\)) we have that (8) is equivalent to \(w \in B_p\).

  2. (ii)

    If \(w = 1\) and \(p > 1\), then (8) is equivalent to that there exists some \(0< q < p\) such that

    $$\begin{aligned} \frac{u\left( \bigcup _{j = 1}^JQ_j \right) }{u\left( \bigcup _{j = 1}^J E_j \right) } \lesssim \max _{1\le j\le J}\left( \frac{|Q_j|}{|E_j|} \right) ^q, \end{aligned}$$

    which agrees with \(u \in A_p\) (see, for instance, [22, 33]).

Moreover, \(B_p(u) \subseteq B_p\) for every \(p > 0\) (see [18, Corollary 3.3.4]) so, for \(p \ge 1\) and \(w \in B_p(u)\) then \(\Lambda ^p(w)\) is a r.i. Banach function space.

Therefore, together with (7), if we were able to see when \(M'_u\) is bounded over \((\Lambda ^p_u(w))'\), we could make use of Theorem 1.2 to obtain estimates for a Rubio de Francia operator T on \(\Lambda ^p_u(w)\), at least for \(p \ge 1\) and \(w \in B_p(u)\). Indeed, our main result on this paper says more:

Theorem 1.3

Given a Rubio de Francia operator T for some exponent \(p_0 > 1\). Let \(u \in A_\infty \) and \(p > 0\). Then,

$$\begin{aligned} T: \Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u) \cap B_\infty ^*. \end{aligned}$$

Now, in [4, Theorem 5.5] it was shown that for \(p > 1\),

$$\begin{aligned} H:\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w) \qquad \iff \qquad w \in B_p(u) \cap {B_{\infty }^{*}}, \end{aligned}$$
(9)

while for \(0 < p \le 1\) it is just known that

$$\begin{aligned} w \in B_p(u) \cap {B_{\infty }^{*}}\qquad \Longrightarrow \qquad H:\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w). \end{aligned}$$

Remark 1.4

At least for \(p > 1\), by means of the Hilbert transform (see (9)) the condition \(B_p(u) \cap {B_{\infty }^{*}}\) on the weight w of Theorem 1.3 is sharp in the sense that it can not be found a greater class for w.

Now, let \(\Lambda ^{p,\infty }_u(w)\), \(p > 0\), be the space defined by those measurable functions f such that

$$\begin{aligned} \left\Vert f\right\Vert _{\Lambda ^{p,\infty }_u(w)}= \sup _{t > 0} t W(\lambda _f^u(t) )^\frac{1}{p} < \infty , \end{aligned}$$

and let \(L^{p,\infty }(u) := \Lambda ^{p,\infty }_u(1)\), \(p \ge 1\). In [20] the authors consider as hypothesis weighted weak-type estimates instead of (1); that is, for some \(p_0 \ge 1\),

$$\begin{aligned} T:L^{p_0}(v) \rightarrow L^{p_0,\infty }(v),\qquad \forall v \in A_{p_0}, \end{aligned}$$
(10)

and then try to find conditions on p and w for which

$$\begin{aligned} T:\Lambda ^p(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

holds, so it could be interesting to study the previous to weighted classical Lorentz spaces for a weight \(u \in A_\infty \). However, this is just a consequence of the weighted strong-type extrapolation settled in Theorem 1.3, and the precise result is the following.

Corollary 1.5

If an operator T satisfies (10) for some \(p_0 \ge 1\) and with constant less than or equal to \(\varphi (\left\Vert v\right\Vert _{A_{p_0}})\), where \(\varphi \) is a nonnegative nondecreasing function on \([1, \infty )\), then, for \(p > 0\) and \(u \in A_\infty \),

$$\begin{aligned} T:\Lambda ^p_u(w) \rightarrow \Lambda ^{p,\infty }_u(w), \qquad \forall w \in B_p(u)\cap B_{\infty }^{*}. \end{aligned}$$

Again, for \(p > 1\) and by means of the Hilbert transform (see [2, Theorem 1.1]) the condition \(B_p(u) \cap {B_{\infty }^{*}}\) on the weight w of Corollary 1.5 is sharp in the sense that it can not be found a greater class for w.

The paper is organized as follows: in Sect. 2 we study the boundedness of \(M'_u\) over the associate space of \(\Lambda ^p_u(w)\). The proofs of Theorem 1.3 and Corollary 1.5 will be given in Sect. 3, and Sect. 4 will be devoted to applying our results to the boundedness of Calderón-Zygmund operators, sparse operators, the Bochner-Riesz operator and intrinsic square functions.

As usual, we shall use the symbol \(A\lesssim B\) to indicate that there exists a universal positive constant C, independent of all important parameters, such that \(A\le C B\). When \(A\lesssim B\) and \(B\lesssim A\), we will write \(A\approx B\).

2 Boundedness on the Associate Space of \(\Lambda ^p_u(w)\)

Given \(u \in A_\infty \), our goal is to see for which conditions on p and the weight w

$$\begin{aligned} M_u':(\Lambda ^p_u(w))' \rightarrow (\Lambda ^p_u(w))' \end{aligned}$$
(11)

holds, where \(M'_u\) is defined in (2).

  1. (I)

    If \(u = 1\) and \(w \in B_p\), then (11) is equivalent to \(w \in B_\infty ^*\) (see (5)).

  2. (II)

    If \(w = 1\) and \(p > 1\), then (11) is equivalent to \(M_u': L^{\frac{p}{p-1}}(u) \rightarrow L^{\frac{p}{p-1}}(u)\), which in turn remains true whenever \(u \in A_p\).

In order to see the general setting (see Theorem 2.6), it is necessary to establish a few technical findings initially. In an effort to ensure that the paper is self-contained, we include these results within its contents. First, for a nonnegative measurable function h let

$$\begin{aligned} Qh(t) := \int _t^\infty h(r)\,\frac{dr}{r}, \qquad t > 0. \end{aligned}$$

Therefore, by carefully monitoring the constants involved in [5, Theorem 4] (or even [40, Theorem 3.3]), the following result holds.

Proposition 2.1

For every \(p > 0\),

$$\begin{aligned} Q:L^p_{\text {dec}}(w) \rightarrow L^p(w) \qquad \Longleftrightarrow \qquad w \in {B_{\infty }^{*}}\end{aligned}$$

with \(\left\Vert Q\right\Vert _{L^p_{\text {dec}}(w) \rightarrow L^p(w)} \le c_{n,p} \left\Vert w\right\Vert _{{B_{\infty }^{*}}}\) and where \(L^p_{\text {dec}}(w)\) is the set of measurable decreasing functions f satisfying \(\left\Vert f\right\Vert _{L^p(w)} < \infty \).

We also need some technical results related with the Fefferman-Stein maximal operator, which is defined, for every \(h \in L^1_{\text {loc}}({\mathbb {R}^n})\), by

$$\begin{aligned} M^\#h(x) = \sup _{Q\ni x} \frac{1}{|Q|} \int _Q\left| h(y) - \frac{1}{|Q|} \int _Q h(z)\,dz\right| \,dy, \qquad x \in {\mathbb {R}^n}, \end{aligned}$$

where the supremum is taken over all cubes \(Q\subseteq {\mathbb {R}^n}\) containing the point x (see [30, 41]). Hence, by carefully considering the constants involved in [9, Corollary 4.3 (a)], the following result can be established.

Proposition 2.2

There exists a nondecreasing function \(\varphi _q\) on \([1,\infty )\) such that for every \(t > 0\),

$$\begin{aligned} f^*_u(t)&\le \varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) Q\left( \left( M^\#f \right) _u^* \right) (t) + \lim _{r \rightarrow \infty } f_u^*(r). \end{aligned}$$

And, as a consequence, the next result follows.

Proposition 2.3

Given \(u \in A_q\), \(q > 1\), and \(w \in {B_{\infty }^{*}}\). If \(\lim _{r\rightarrow \infty } f^*_u(r) = 0\), then

$$\begin{aligned} \left\Vert f\right\Vert _{\Lambda _u^p(w)} \le c_{n,p} \varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{B_\infty ^*} \left\Vert M^\#f\right\Vert _{\Lambda _u^p(w)}, \end{aligned}$$

where \(\varphi _q\) is a nonnegative nondecreasing function on \([1,\infty )\).

Proof

By means of Proposition 2.2, there exists a nondecreasing function \(\varphi _q\) on \([1,\infty )\) such that for every \(t > 0\),

$$\begin{aligned} f^*_u(t)&\le \varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) Q\left( \left( M^\#f \right) _u^* \right) (t). \end{aligned}$$

Therefore, taking into account Proposition 2.1,

$$\begin{aligned} \left\Vert f\right\Vert _{\Lambda ^p_u(w)}&\le \varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert Q\left( \left( M^\# f \right) _u^* \right) \right\Vert _{L^p(w)} \\ {}&\le c_{n,p} \varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{{B_{\infty }^{*}}} \left\Vert M^\# f\right\Vert _{\Lambda ^p_u(w)}. \end{aligned}$$

\(\square \)

Furthermore, the Fefferman-Stein maximal operator also satisfy the following statement.

Proposition 2.4

[28] For every \(f \in L^1_{\text {loc}}({\mathbb {R}^n})\), there exists a linear operator \(L_f\) such that

$$\begin{aligned} Mf(x) \approx L_f(|f|)(x), \qquad \text {a.e. } \, x \in {\mathbb {R}^n}. \end{aligned}$$

Moreover, the adjoint of \(L_f\), \(\widetilde{L_f}\), satisfies that for every \(g \in L^1_{\text {loc}}({\mathbb {R}^n})\),

$$\begin{aligned} M^\# \left( \widetilde{L_f}(|g|) \right) (x) \lesssim Mg(x), \qquad \text {a.e. }\, x \in {\mathbb {R}^n}. \end{aligned}$$
(12)

Additionally, we will utilize the following result related with weights properties.

Proposition 2.5

Given \(u \in A_q\), \(q > 1\), and \(w \in B_p(u)\), \(p > 0\). Let, for \(N \in \mathbb {N}\), \(u_N = \min (u, N)\). Then,

  1. (i)

    \(u_N \in A_q\) with

    $$\begin{aligned} \left\Vert u_N\right\Vert _{A_q} \le 2^{q + 1}\left\Vert u\right\Vert _{A_q}, \end{aligned}$$
  2. (ii)

    and there exists some constant \(C_{u,w} > 0\) independent of N such that

    $$\begin{aligned} \left\Vert M\right\Vert _{\Lambda ^p_{u_N}(w)} \le C_{u,w}. \end{aligned}$$

Proof

First, taking an arbitrary cube \(Q \subseteq {\mathbb {R}^n}\),

$$\begin{aligned}&\left( \frac{1}{|Q|} \int _Q u_N (x) dx\right) \left( \frac{1}{|Q|} \int _Q u_N (x) ^{\frac{1}{1-q}}dx \right) ^{q-1} \\ {}&= \frac{1}{|Q|^q}\left( \int _{Q \cap \{x:u(x)< N\}} u (x) dx + N\int _{Q \cap \{x:u(x) \ge N\}}dx\right) \\ {}&\times \left( \int _{Q \cap \{x:u(x) < N\}} u (x) ^{\frac{1}{1-q}} dx + N^{\frac{1}{1-q}}\int _{Q \cap \{x:u(x) \ge N\}}dx\right) ^{q-1} \\ {}&\le 2^q\left\Vert u\right\Vert _{A_q} + 2 \le 2^{q + 1}\left\Vert u\right\Vert _{A_q}, \end{aligned}$$

from which (i) follows by taking the supremum overall cubes Q.

To see (ii), let \(f \in \Lambda ^p_{u_N}(w)\) and write

$$\begin{aligned} f = f\chi _{\{x \, : \, u(x) < N\}} + f\chi _{\{x \, : \, u(x) \ge N\}}:= f_1 + f_2. \end{aligned}$$

Then,

$$\begin{aligned} \left\Vert Mf\right\Vert _{\Lambda ^p_{u_N}(w)}&\le C\left( \left\Vert Mf_1\right\Vert _{\Lambda ^p_{u_N}(w)} + \left\Vert Mf_2\right\Vert _{\Lambda ^p_{u_N}(w)}\right) \\ {}&\le C\left( \left\Vert Mf_1\right\Vert _{\Lambda ^p_u(w)} + N\left\Vert Mf_2\right\Vert _{\Lambda ^p(w)}\right) \\ {}&\le C\left( \left\Vert M\right\Vert _{\Lambda ^p_u(w)}\left\Vert f_1\right\Vert _{\Lambda ^p_u(w)} + N\left\Vert M\right\Vert _{\Lambda ^p(w)}\left\Vert f_2\right\Vert _{\Lambda ^p(w)}\right) , \end{aligned}$$

where in the last estimate we are using that \(w \in B_p(u) \subset B_p\). Now observe that for every \(y > 0\),

$$\begin{aligned} u(\{x \, : \, f_1(x)> y\}) \le u_N(\{x \, : \, f(x) > y\}) \end{aligned}$$

and

$$\begin{aligned} N|\{x \, : \, f_2(x)> y\}| \le u_N(\{x \, : \, f(x) > y\}), \end{aligned}$$

so that \(\left\Vert f_1\right\Vert _{\Lambda ^p_u(w)} \le \left\Vert f\right\Vert _{\Lambda ^p_{u_N}(w)}\) and \(N\left\Vert f_2\right\Vert _{\Lambda ^p(w)} \le \left\Vert f\right\Vert _{\Lambda ^p_{u_N}(w)}\). Therefore, the results follows by taking \(C_{u,w} = C\left( \left\Vert M\right\Vert _{\Lambda ^p_u(w)}+ \left\Vert M\right\Vert _{\Lambda ^p(w)}\right) \). \(\square \)

Finally, with the previous results at hand, we are able to find conditions so that (11) holds.

Theorem 2.6

Given \(u \in A_{\infty }\). For every \(p > 0\),

$$\begin{aligned} M'_u:\left( \Lambda ^{p}_{u}\left( w \right) \right) ' \rightarrow \left( \Lambda ^{p}_{u}\left( w \right) \right) ', \qquad \forall w \in B_p(u) \cap {B_{\infty }^{*}}. \end{aligned}$$
(13)

Proof

Since \(w \in B_p(u) \subseteq B_p\) we have that \((\Lambda ^p_u(w))'\ne \{0\}\) (see [18]) and, by definition of associate space,

$$\begin{aligned} \left\Vert \frac{M(fu)}{u}\right\Vert _{\left( \Lambda ^{p}_{u}\left( w \right) \right) '} = \sup _{\left\Vert h\right\Vert _{\Lambda ^p_u(w)} \le 1} \int _{\mathbb {R}^n}M(f u)(x) h(x) \,dx, \end{aligned}$$
(14)

where we can assume that the supremum is taken over all nonnegative functions h satisfying that \(\left\Vert h\right\Vert _{\Lambda ^p_u(w)} \le 1\). We will see that, for such a function h,

$$\begin{aligned} \int _{\mathbb {R}^n}M(f u)(x) h(x) \,dx \le C_{n,p,q,u,w}\left\Vert f\right\Vert _{(\Lambda ^p_u(w))'} \left\Vert h\right\Vert _{\Lambda ^p_u(w)}, \end{aligned}$$
(15)

from which together with (14), the boundedness of (13) will follow.

First, we observe that h can be chosen to be in \(L^1({\mathbb {R}^n})\). Otherwise, we can take \(h_k = \chi _{{\mathbb {B}}(0,k)}h \in L^1({\mathbb {R}^n})\) (with \({\mathbb {B}}(0,k)\) being the ball of center 0 and radius k) and by the monotone convergence theorem,

$$\begin{aligned} \int _{\mathbb {R}^n}M(f u)(x) h(x) \,dx = \lim _{k \rightarrow \infty }\int _{\mathbb {R}^n}M(f u)(x) h_k(x) \,dx, \end{aligned}$$

so together with \(\left\Vert h_k\right\Vert _{\Lambda ^p_u(w)} \le \left\Vert h\right\Vert _{\Lambda ^p_u(w)}\), we would obtain (15).

Hence, take such a nonnegative function h in (14) satisfying \(h \in L^1({\mathbb {R}^n})\). Further, take \(q > 1\) such that \(u \in A_q\) (since \(u \in A_\infty \)). Then, by Proposition 2.4,

$$\begin{aligned} \int _{\mathbb {R}^n}M(f u)(x) h(x) \,dx&\le C_n \int _{\mathbb {R}^n}|f(x)| \widetilde{L_{fu}}(h)(x) u(x) \,dx \\ {}&\le C_n \left\Vert f\right\Vert _{\left( \Lambda ^{p}_{u}\left( w \right) \right) '} \left\Vert \widetilde{L_{fu}}(h)\right\Vert _{\Lambda ^p_u(w)}, \end{aligned}$$

and if we are able to see that

$$\begin{aligned} \lim _{t \rightarrow \infty } (\widetilde{L_{fu}}(h))_u^*(t) = 0, \end{aligned}$$
(16)

by virtue of Proposition 2.3 and estimate (12) we will deduce that

$$\begin{aligned} \left\Vert \widetilde{L_{fu}}(h)\right\Vert _{\Lambda ^p_u(w)}&\le c_{n,p}\varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{{B_{\infty }^{*}}} \left\Vert M^\#\left( \widetilde{L_{fu}}(h) \right) \right\Vert _{\Lambda ^p_u(w)} \\&\le {\tilde{c}}_{n,p}\varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{{B_{\infty }^{*}}} \left\Vert Mh \right\Vert _{\Lambda ^p_u(w)} \\ {}&\le \tilde{c}_{n,p}\varphi _q\left( \left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{{B_{\infty }^{*}}} \left\Vert M\right\Vert _{\Lambda ^p_u(w)}\left\Vert h\right\Vert _{\Lambda ^p_u(w)}, \end{aligned}$$

where \(\left\Vert M\right\Vert _{\Lambda ^p_u(w)} < \infty \) since \(w \in B_p(u)\).

Thus, it all reduces to show that (16) holds. Now, if we assume that u is bounded, that is \(u \le C_u\) for some positive constant \(C_u\), then it is just a consequence of that by construction it is known that \(\widetilde{L_{fu}}(h)\) is bounded in \(L^1({\mathbb {R}^n})\) (see [28]) so for every \(t > 0\),

$$\begin{aligned} u\left( \left\{ x \in {\mathbb {R}^n}: |\widetilde{L_{fu}}(h)|> t \right\} \right)&\le C_u \left| \left\{ x \in {\mathbb {R}^n}: |\widetilde{L_{fu}}(h)| > t \right\} \right| \\ {}&\le \frac{C_u}{t} \left\Vert \widetilde{L_{fu}}\right\Vert _{L^1({\mathbb {R}^n}) \rightarrow L^{1,\infty }({\mathbb {R}^n})}\left\Vert h\right\Vert _{L^1({\mathbb {R}^n})}, \end{aligned}$$

and hence,

$$\begin{aligned} (\widetilde{L_{fu}}(h))_u^*(t) \le \frac{C_u}{t} \left\Vert \widetilde{L_{fu}}\right\Vert _{L^1({\mathbb {R}^n}) \rightarrow L^{1,\infty }({\mathbb {R}^n})}\left\Vert h\right\Vert _{L^1({\mathbb {R}^n})} \xrightarrow [t\rightarrow \infty ]{} 0. \end{aligned}$$

Finally, if u is not bounded, taking \(N \in \mathbb {N}\), we just have to observe that \(u_N = \min (u,N)\) is a bounded weight that satisfy (see Proposition 2.5)

$$\begin{aligned} \left\Vert u_N\right\Vert _{A_q} \le 2^{q + 1}\left\Vert u\right\Vert _{A_q}, \qquad \left\Vert M\right\Vert _{\Lambda ^p_{u_N}(w)} \le C_{u,w} \end{aligned}$$

and \(\left\Vert h\right\Vert _{\Lambda ^p_{u_N}(w)} \le 1\), so that by the monotone convergence theorem,

$$\begin{aligned} \int _{\mathbb {R}^n}M(f u)(x) h(x) \,dx&= \lim _{N \rightarrow \infty } \int _{\mathbb {R}^n}M(f u_N)(x) h(x) \,dx \\&\le {\tilde{c}}_{n,p}C_{u,w} \varphi _q\left( 2^{q + 1}\left\Vert u\right\Vert _{A_q}\right) \left\Vert w\right\Vert _{{B_{\infty }^{*}}}\left\Vert f\right\Vert _{(\Lambda ^p_u(w))'} \left\Vert h\right\Vert _{\Lambda ^p_u(w)}. \end{aligned}$$

\(\square \)

3 Proof of Main Results

Proof of Theorem 1.3

Suppose first that \(p \ge 1\). Since \(w \in B_p(u) \subseteq B_p\), then \(\Lambda ^p(w)\) is a r.i. Banach function space, and so the result follows by means of Theorems 1.2 and 2.6, together with (7).

Now, assume that \(0< p < 1\). Since \(w \in B_p(u)\), there exists some \(0< \varepsilon < p\) such that \(w \in B_{p - \varepsilon }(u)\), and we can define

$$\begin{aligned} Rf(x) = \sum _{k = 0}^{\infty }\frac{M^kf(x)}{\left( 2\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}\right) ^k} \end{aligned}$$

to be a version of the function resulting from the Rubio de Francia Algorithm [42], where \(M^0 = Id\) is the identity operator and \(M^k = M \underbrace{\circ \cdots \circ }^k M\) is the k-fold composition of M with itself. Then,

  1. (1)

    \(f \le \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^\frac{p - \varepsilon }{p}\),

  2. (2)

    \(R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \in A_1\) with \(\left\Vert R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right\Vert _{A_1} \le 2\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}\),

  3. (3)

    and

    $$\begin{aligned} \left\Vert \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^\frac{p - \varepsilon }{p}\right\Vert _{\Lambda ^p_u(w)} \le \left( \frac{2^{p - \varepsilon }}{2^{p - \varepsilon } - 1}\right) ^\frac{1}{p} \left\Vert f\right\Vert _{\Lambda ^p_u(w)}. \end{aligned}$$

Certainly, since \(B_p(u) \subseteq B_p \subseteq B_1\) then \(\Lambda ^1(w)\) is also a r.i. Banach function space and property (3) can be established as follows:

$$\begin{aligned} \begin{aligned} \left\Vert \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^\frac{p - \varepsilon }{p}\right\Vert _{\Lambda ^p_u(w)}&= \left\Vert \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^{p - \varepsilon }\right\Vert _{\Lambda ^1_u(w)}^\frac{1}{p} \\ {}&\le \left\Vert \sum _{k = 0}^{\infty }\frac{M^k\left( |f|^{\frac{p}{p - \varepsilon }}\right) (x)^{p - \varepsilon }}{\left( 2\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}\right) ^{k({p - \varepsilon })}}\right\Vert _{\Lambda ^1_u(w)}^\frac{1}{p} \\ {}&\le \left( \sum _{k = 0}^{\infty }\frac{\left\Vert M^k\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right\Vert _{\Lambda _u^{p - \varepsilon }(w)}^{p - \varepsilon }}{\left( 2\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}\right) ^{k({p - \varepsilon })}} \right) ^\frac{1}{p} \\&\le \left\Vert f\right\Vert _{\Lambda ^p_u(w)} \left( \sum _{k = 0}^{\infty }\frac{1}{2^{k({p - \varepsilon })}} \right) ^\frac{1}{p} = \left( \frac{2^{p - \varepsilon }}{2^{p - \varepsilon } - 1}\right) ^\frac{1}{p} \left\Vert f\right\Vert _{\Lambda ^p_u(w)}. \end{aligned} \end{aligned}$$

What’s more, by means of Theorem 2.6 we can define for an arbitrary nonnegative function \(h \in (\Lambda _u^1(w))'\),

$$\begin{aligned} Sh(x) = \sum _{k = 0}^{\infty }\frac{(M'_u)^kh(x)}{\left( 2\left\Vert M'_u\right\Vert _{(\Lambda _u^1(w))'}\right) ^k} \end{aligned}$$

so that

  1. (1)’

    \(h \le S h\),

  2. (2)’

    \((Sh) u \in A_1\) with \(\left\Vert (Sh) u\right\Vert _{A_1} \le 2\left\Vert M'_u\right\Vert _{(\Lambda _u^1(w))'}\),

  3. (3)’

    and

    $$\begin{aligned} \left\Vert Sh\right\Vert _{(\Lambda ^1_u(w))'} \le 2 \left\Vert h\right\Vert _{(\Lambda ^1_u(w))'}. \end{aligned}$$

Now, \(\Lambda ^1(w)\) is a Banach function space so that for any nonnegative function g (see [10, Ch. 1 Theorem 2.7]),

$$\begin{aligned} \begin{aligned} \left\Vert g\right\Vert _{\Lambda _u^p(w)}^p&= \left\Vert g^p\right\Vert _{\Lambda _u^{1}(w)} = \sup _{\left\Vert h\right\Vert _{\left( \Lambda _u^{1}(w)\right) '} \le 1} \int _{\mathbb {R}^n}g(y)^p h(y)u(y)\,dy. \end{aligned} \end{aligned}$$
(17)

Then, taking such a nonnegative function h in (17), by Fubini,

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^n}g(y)^p h(y)u(y)\,dy&= \int _{0}^{\infty } \int _{\{g(y)^p> x\}}h(y)u(y)\,dy\,dx \\ {}&= \int _{0}^{\infty } \int _{\left\{ g(y)^p> x, \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y) \right] ^{p - \varepsilon }> \gamma x \right\} }h(y)u(y)\,dy\,dx \\ {}&+ \int _{0}^{\infty } \int _{\left\{ g(y)^p >x, \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y) \right] ^{p - \varepsilon } \le \gamma x \right\} }h(y)u(y)\,dy\,dx \\ {}&:= I_1(h) + I_2(h). \end{aligned} \end{aligned}$$

Let’s work with \(I_1(h)\) and \(I_2(h)\) separately. First, by Fubini again,

$$\begin{aligned} \begin{aligned} I_1(h)&\le \int _{0}^{\infty } \int _{\left\{ \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y) \right] ^{p - \varepsilon } > \gamma x \right\} }h(y)u(y)\,dy\,dx \\ {}&= \frac{1}{\gamma } \int _{\mathbb {R}^n}\left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y) \right] ^{p - \varepsilon }h(y)u(y)\,dy, \end{aligned} \end{aligned}$$

and taking the supremum in \(0 \le h \in \left( \Lambda _u^{1}(w)\right) '\) we have using property (3) that

$$\begin{aligned} \begin{aligned} \sup _{\left\Vert h\right\Vert _{\left( \Lambda _u^{1}(w)\right) '} {\le } 1} \frac{1}{\gamma }\int _{\mathbb {R}^n}\left[ R\left( |f|^{\frac{p}{p {-} \varepsilon }}\right) (y) \right] ^{p - \varepsilon }h(y)u(y)\,dy&{=} \frac{1}{\gamma }\left\Vert \left[ R\left( |f|^{\frac{p}{p {-} \varepsilon }}\right) \right] ^{p {-} \varepsilon }\right\Vert _{\Lambda _u^{1}(w)} \\ {}&\le \frac{1}{\gamma } \left( \frac{2^{p - \varepsilon }}{2^{p {-} \varepsilon } - 1}\right) \left\Vert f\right\Vert _{\Lambda _u^{p}(w)}^p. \end{aligned} \end{aligned}$$

Now let’s work with \(I_2(h)\). Observe that since \(p_0 \ge 1\), then \(\frac{p(1 - p)}{p_0 - p} \le p.\) Hence, if we take

$$\begin{aligned} \varepsilon < \frac{p(1 - p)}{p_0 - p} \qquad \text { and } \qquad p_1 := \frac{p(1 - p + \varepsilon )}{\varepsilon }, \end{aligned}$$

then, \(p_1 \ge p_0\), \(p_1 > 1\) and \(1 - p_1 = (p - \varepsilon )(1 - p_1/p)\). Therefore, by [29, Lemma 2.1]

$$\begin{aligned} v = \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^{(p - \varepsilon )(1-p_1/p)}(Sh) u = \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right] ^{1-p_1}(Sh) u\in A_{p_1} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned} \left\Vert v\right\Vert _{A_{p_1}} \le \left\Vert (Sh)u\right\Vert _{A_1}\left\Vert R\left( |f|^{\frac{p}{p - \varepsilon }}\right) \right\Vert ^{p_1 - 1}_{A_1}&\le 2^{p_1 }\left\Vert M'_u\right\Vert _{(\Lambda ^1_u(w))'}\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}^{p_1 - 1} \end{aligned} \end{aligned}$$

and, due Theorem 1.1, if we let \(g = |Tf|\),

$$\begin{aligned} \begin{aligned} I_2(h)&\le \gamma ^{p_1/p - 1} \int _{0}^{\infty }x^{p_1/p - 1} \int _{\{|Tf(y)|^p>x\}} \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y)\right] ^{(p - \varepsilon )(1-p_1/p)}\\&\quad Sh(y)u(y)\,dy\,dx \\ {}&=\gamma ^{p_1/p - 1}\frac{p}{p_1} \int _{\mathbb {R}^n}|Tf(y)|^{p_1} \left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y)\right] ^{1 - p_1}Sh(y)u(y)\,dy\\ {}&\le \gamma ^{p_1/p - 1} \frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1} \int _{\mathbb {R}^n}f(y)^{p_1}\left[ R\left( |f|^{\frac{p}{p - \varepsilon }}\right) (y)\right] ^{1 - p_1}Sh(y)u(y)\,dy \\ {}&\le \gamma ^{p_1/p - 1} \frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1} \int _{\mathbb {R}^n}f(y)^pSh(y)u(y)\,dy \\ {}&\le \gamma ^{p_1/p - 1} \frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1} \left\Vert f^p\right\Vert _{\Lambda _u^1(w)}\left\Vert Sh\right\Vert _{(\Lambda ^1_u(w))'} \\ {}&\le 2\gamma ^{p_1/p - 1} \frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1} \left\Vert f\right\Vert _{\Lambda _u^p(w)}^p\left\Vert h\right\Vert _{(\Lambda ^1_u(w))'}. \end{aligned} \end{aligned}$$

Therefore,

$$\begin{aligned} \sup _{\left\Vert h\right\Vert _{\left( \Lambda _u^{1}(w)\right) '} \le 1} I_2(h) \le 2\gamma ^{p_1/p - 1}\frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1} \left\Vert f\right\Vert _{\Lambda _u^p(w)}^p. \end{aligned}$$

Thus,

$$\begin{aligned} \left\Vert Tf\right\Vert _{\Lambda _u^p(w)}^p \le \max \left( \frac{1}{\gamma }\frac{2^{p - \varepsilon }}{2^{p - \varepsilon } - 1}, 2\gamma ^{p_1/p - 1}\frac{p\Phi \left( \left\Vert v\right\Vert _{A_{p_1}}\right) ^{p_1}}{p_1}\right) \left\Vert f\right\Vert _{\Lambda _u^p(w)}^p, \end{aligned}$$

and taking the infimum in \(\gamma >0\),

$$\begin{aligned}&\left\Vert Tf\right\Vert _{\Lambda _u^p(w)} \lesssim \varphi \left( C2^{\frac{p(1 - p + \varepsilon )}{\varepsilon }}\left\Vert M'_u\right\Vert _{(\Lambda ^1_u(w))'}\left\Vert M\right\Vert _{\Lambda _u^{p - \varepsilon }(w)}^{\frac{(1 - p)(p - \varepsilon )}{\varepsilon }}\right) \left\Vert f\right\Vert _{\Lambda _u^p(w)}. \end{aligned}$$

\(\square \)

Proof of Corollary 1.5

Observe that

$$\begin{aligned} \left\Vert Tf\right\Vert _{L^{p_0,\infty }(v)} \le \varphi \left( \left\Vert v\right\Vert _{A_{p_0}}\right) \left\Vert f\right\Vert _{L^{p_0}(v)}, \qquad \forall v \in A_{p_0}, \end{aligned}$$

implies that for every \(y > 0\),

$$\begin{aligned} \left\Vert y\chi _{\{|Tf| > y\}}\right\Vert _{L^{p_0}(v)} \le \varphi \left( \left\Vert v\right\Vert _{A_{p_0}}\right) \left\Vert f\right\Vert _{L^{p_0}(v)}, \qquad \forall v \in A_{p_0}. \end{aligned}$$

Fix \(y > 0\) and define \(T_yf := y\chi _{\{|T f| > y\}}\). Hence, by means of Theorem 1.3 we obtain that, for every \(p > 0\) and \(u \in A_\infty \),

$$\begin{aligned} \left\Vert y\chi _{\{|Tf| > y\}}\right\Vert _{\Lambda ^p_u(w)} = \left\Vert T_yf\right\Vert _{\Lambda ^p_u(w)} \le C_{n,p_0,p,u,w}\left\Vert f\right\Vert _{\Lambda ^p_u(w)}, \qquad \forall w \in B_p(u)\cap {B_{\infty }^{*}}. \end{aligned}$$

Therefore, taking the supremum over all \(y > 0\), we obtain the desired result.

4 Examples and Applications

4.1 Calderón-Zygmund Operators and Commutators

A function K(x, y) on \(\mathbb {R}^n \times \mathbb {R}^n \setminus \{(x,x)\,:\, x \in {\mathbb {R}}^n\}\) is called a standard kernel if there exists \(A, \delta > 0\) satisfying the size condition

$$\begin{aligned} |K(x,y)| \le \frac{A}{|x - y|^n}, \end{aligned}$$

and the regularity conditions

$$\begin{aligned} \left| K(x,y) - K(x', y)\right| \le \frac{A|x - x'|^\delta }{(|x - y| + |x' - y|)^{n + \delta }}, \end{aligned}$$

when \(|x - x'| \le \frac{1}{2} \max (|x - y|, |x' - y|)\) and

$$\begin{aligned} \left| K(x,y) - K(x, y')\right| \le \frac{A|y - y'|^\delta }{(|x - y| + |x - y'|)^{n + \delta }}, \end{aligned}$$

when \(|y - y'| \le \frac{1}{2} \max (|x - y|, |x - y'|)\). Now, given a standard Kernel K, \(T_K\) is called a Calderón-Zygmund operator associated with K if it is defined on the class of Schwartz functions \({\mathcal {S}}({\mathbb {R}}^n)\) (that is, the space of functions in \({\mathcal {C}}^\infty _c({\mathbb {R}}^n)\) that decreases rapidly), which admits a bounded extension on \(L^2({\mathbb {R}}^n)\)

$$\begin{aligned} \left\Vert T_K\right\Vert _{L^2({\mathbb {R}}^n)} \le C\left\Vert f\right\Vert _{L^2({\mathbb {R}}^n)}, \qquad \forall f \in L^2({\mathbb {R}}^n), \end{aligned}$$

and

$$\begin{aligned} T_K f(x) = \int _{{\mathbb {R}}^n} K(x,y) f(y)\,dy, \qquad \forall f \in {\mathcal {C}}^\infty _c({\mathbb {R}}^n) \text { and } x \notin \text {supp}(f), \end{aligned}$$

for \({\mathcal {C}}^\infty _c({\mathbb {R}}^n)\) being the space of infinitely differentiable functions with compact support (see [31, Ch. 4] for more details on these operators). Also, for a locally integrable function b, the commutator of \(T_K\) and b is defined as

$$\begin{aligned}{}[b, T_K](f) = bT_K(f) - T_K(bf). \end{aligned}$$

For a general Calderón-Zygmund operator \(T_K\), it is well known that it is bounded on \(L^p(R^n)\), for all \(p > 1\) (see [31, Theorem 4.2.2]). Moreover, R.R. Coifman and C. Fefferman [24] proved that \(T_K\) is bounded on the weighted Lebesgue space \(L^p(v)\) for \(v \in A_p\) and \(p > 1\). Indeed, T.P. Hytönen proved in [32] that

$$\begin{aligned} \left\Vert T_Kf\right\Vert _{L^2(v)} \lesssim \left\Vert v\right\Vert _{A_2}\left\Vert f\right\Vert _{L^2(v)}, \qquad \forall v \in A_2. \end{aligned}$$
(18)

For the commutator, R.R. Coifman, R. Rochberg and G. Weiss [25] proved that \([b,T_K]\) is bounded on \(L^p(\mathbb R^n)\), \(p > 1\), if b is a \(\text {BMO}({\mathbb {R}}^n)\) function, where the space of functions of bounded mean oscillation \(\text {BMO}({\mathbb {R}^n})\) is defined by

$$\begin{aligned} \text {BMO}({\mathbb {R}^n}) = \left\{ b \in L^1_{\text {loc}}({\mathbb {R}^n}) : \left\Vert f\right\Vert _{\text {BMO}({\mathbb {R}^n})} = \sup _{Q} \frac{1}{|Q|}\int _Q\left| b(y) - b_Q\right| \,dy < \infty \right\} , \end{aligned}$$

where the supremum is taken over all cubes \(Q \in {\mathbb {R}^n}\) with sides parallel to the axes and \(b_Q = \frac{1}{|Q|}\int _Q b(x)\,dx\). An analogue of (18) for the commutator \([b, T_K]\) with a \(\text {BMO}({\mathbb {R}}^n)\) function b is due to D. Chung, C. Pereyra and C. Pérez (see [21]), who proved that

$$\begin{aligned} \left\Vert [b, T_K]f\right\Vert _{L^2(v)} \lesssim \left\Vert b\right\Vert _{\text {BMO}} \left\Vert v\right\Vert _{A_2}^2\left\Vert f\right\Vert _{L^2(v)}, \qquad \forall v \in A_2. \end{aligned}$$

Therefore, as a consequence of Theorem 1.3:

Corollary 4.1

Let K be a standard kernel. Given \(u \in A_\infty \) and \(p > 0\),

$$\begin{aligned} T_K:\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u)\cap B_\infty ^* \end{aligned}$$
(19)

and, for \(b \in \text {BMO}(\mathbb {R}^n)\),

$$\begin{aligned}{}[b,T_K]:\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u)\cap B_\infty ^*. \end{aligned}$$
(20)

Remark 4.2

This extends the results on [17, Theorems 1.1 and 1.2] where the authors saw (19) and (20) for \(p > 1\). Moreover, we recall that an example of a Calderón-Zygmund operator is the Hilbert transform (see (6) for its definition).

4.2 Sparse Operators

These operators have become very popular due to their role in the often-called \(A_2\) conjecture consisting in proving, for instance, (18). This result was first obtained by T.P. Hytönen [32] and then simplified by Lerner [34, 35], who proved that the norm of a Calderón-Zygmund operator in a Banach function space \({\mathbb {X}}\) is dominated by the supremum of the norm in \({\mathbb {X}}\) of all the possible sparse operators, and then proved that every sparse operator is bounded in \(L^2(v)\) for every weight \(v \in A_2\) with sharp constant.

Let us give the precise definition. First, a general dyadic grid \(\mathcal {D}\) is a collection of cubes in \({\mathbb {R}^n}\) satisfying the following properties:

  1. (i)

    For any cube \(Q \in \mathcal {D}\), its side length is \(2^k\) for some \(k \in \mathbb {Z}\).

  2. (ii)

    Every two cubes in \(\mathcal {D}\) are either disjoint or one is wholly contained in the other.

  3. (iii)

    If \(\mathcal {D}_k \subseteq \mathcal {D}\) is the subfamily of cubes formed by the cubes of exactly side length \(2^k\), \(k \in \mathbb {Z}\), then \(\mathcal {D}_k\) form a partition of \({\mathbb {R}^n}\).

Hence, let \(0< \eta < 1\) and let \(\mathcal {D}\) be a family of dyadic cubes. A collection of cubes \(\mathcal {S} \subseteq \mathcal {D}\) is called \(\eta \)-sparse if one can choose pairwise disjoint measurable sets \(E_Q \subseteq Q\) with \(|E_Q| \ge \eta |Q|\), where \(Q\in \mathcal {S}\) (see [30, 37] for more details). Hence, given a \(\eta \)-sparse family of cubes \(\mathcal {S} \subseteq \mathcal {D}\) , the sparse operator \(\mathcal {A}_{\mathcal {S}}\) corresponding to the family \(\mathcal {S}\) is defined by

$$\begin{aligned} \mathcal {A}_{\mathcal {S}}f(x) = \sum _{Q\in \mathcal {S}}\frac{1}{|Q|}\int _Q|f(y)|\,dy\chi _Q(x), \qquad x \in {\mathbb {R}^n}. \end{aligned}$$

Therefore, as a consequence of Theorem 1.3:

Corollary 4.3

Given \(u \in A_\infty \) and \(p > 0\),

$$\begin{aligned} A_{\mathcal {S}}:\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u)\cap B_\infty ^*. \end{aligned}$$

4.3 Bochner-Riesz Operator

Let \(n > 1\) and

$$\begin{aligned} \hat{f}(\xi ) = \int _{{\mathbb {R}}^n}f(x)e^{-2\pi i x\cdot \xi }\,dx, \qquad \xi \in {\mathbb {R}}^n, \end{aligned}$$

be the Fourier transform of a function \(f \in L^2\left( {\mathbb {R}}^n \right) \). For \(\lambda > 0\), the Bochner-Riesz operator is defined as

These operators were first introduced by Bochner in [11] and, since then, they have been widely studied (see [12, 14, 23, 31, 46]).

When \(\lambda > \frac{n - 1}{2}\), it is well known that \(B_\lambda \) is controlled by the Hardy-Littlewood maximal operator M. As a consequence, all weighted inequalities for M are also satisfied by \(B_{\lambda }\). The value \(\lambda = \frac{n - 1}{2}\) is called the critical index. In this case, Shi and Sun [44] proved that \(B_{\frac{n-1}{2}}\) is bounded in \(L^p(v)\) for every \(p > 1\) and \(v \in A_p\). Moreover, in [38, Theorem 1.6] the authors obtained the following quantitative result.

Proposition 4.4

Let \(n > 1\). Then,

$$\begin{aligned} B_\frac{n-1}{2}: L^2(v)\rightarrow L^2(v), \qquad \forall v \in A_2, \end{aligned}$$

with constant less than or equal to \(C\left\Vert v\right\Vert ^2_{A_2}\).

Therefore, as a consequence of Theorem 1.3:

Corollary 4.5

Let \(\lambda \ge \frac{n-1}{2}\). Given \(u \in A_\infty \) and \(p > 0\),

$$\begin{aligned} B_\lambda :\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u)\cap B_\infty ^*. \end{aligned}$$

4.4 Intrinsic Square Functions

For \(0 < \alpha \le 1\), let \(\mathcal {C}_{\alpha }\) be the family of functions \(\phi \) supported in \({\mathbb {B}}(0,1)\) (the n-th dimensional open ball of center 0 and radius 1) such that

$$\begin{aligned} \int _{{\mathbb {B}}(0,1)} \phi (x)\,dx = 0 \quad \text { and }\quad |\phi (x) - \phi (x')| < |x - x'|^{\alpha }, \quad \forall x, x' \in {\mathbb {R}^n}. \end{aligned}$$

Then, given \(f \in L^1_{\text {loc}}({\mathbb {R}^n})\), set

$$\begin{aligned} A_\alpha f(y,t) = \sup _{\phi \in \mathcal {C}^{\alpha }}|(\phi _t*f)(y)|, \qquad (y,t) \in \mathbb {R}^{n + 1}_+, \end{aligned}$$

where we are using \(\phi _t\) to denote the usual \(L^1({\mathbb {R}^n})\) dilatation of \(\phi \); that is \(\phi _t(x) = t^{-n}\phi \left( \frac{x}{t} \right) \).

The intrinsic square function (of order \(\alpha \)) introduced by M. Wilson in [47] is defined by

$$\begin{aligned} G_{\alpha }f(x) = \left( \int _{\Gamma _{\alpha }(x)} |A_{\alpha }(f)(y,t)|^2 \frac{dydt}{t^{n + 1}} \right) ^\frac{1}{2}, \qquad x \in {\mathbb {R}^n}, \end{aligned}$$

with \(\Gamma _{\alpha }(x) = \{ (y,t) : |x - y| < \alpha t \}\). In [36] it was proved that

$$\begin{aligned} \left\Vert G_{\alpha }f\right\Vert _{L^3(v)} \lesssim \left\Vert v\right\Vert _{A_3}^{\frac{1}{2}}\left\Vert f\right\Vert _{L^3(v)}, \qquad \forall v \in A_3. \end{aligned}$$

Therefore, as a consequence of Theorem 1.3:

Corollary 4.6

Let \(0 < \alpha \le 1\). Given \(u \in A_\infty \) and \(p > 0\),

$$\begin{aligned} G_\alpha :\Lambda ^p_u(w) \rightarrow \Lambda ^p_u(w), \qquad \forall w \in B_p(u)\cap B_\infty ^*. \end{aligned}$$

Remark 4.7

In [47] was proved that \(G_\alpha \) dominates pointwise (modulo constant) operators such as the Lusin area integral, the Littlewood-Paley g-function and the continuous square function (see also [16]). Therefore, analogous results as in Corollary 4.6 can be derived for those operators as well.