Boundedness of Sparse and Rough Operators on Weighted Lorentz Spaces

Abstract

We present new estimates in the setting of weighted Lorentz spaces for important operators in Harmonic Analysis such as sparse operators, Bochner–Riesz at the critical index, Hörmander multipliers and rough singular integrals among others.

1 Introduction

The main purpose of this paper is to prove boundedness of important operators in harmonic analysis in weighted Lorentz spaces $\Lambda ^{p}(w)$, $0< p < \infty$, defined by

$$\begin{aligned} \Lambda ^{p}(w) = \left\{ f \in {\mathcal {M}}({\mathbb {R}}^n) : ||f||_{\Lambda ^{p}(w)} = \left( \int _0^{\infty }f^{*}(t)^pw(t)\,dt \right) ^{\frac{1}{p}} < \infty \right\} . \end{aligned}$$

In particular, we shall present new estimates for operators such as the Bochner–Riesz operator at the critical index $B_{\frac{n-1}{2}}$, the rough operators $T_\Omega$ or the sparse operators $A_{{\mathcal {S}}}$ among many others.

There are many results in the literature about boundedness of operators in $\Lambda ^p(w)$ or even in rearrangement invariant (r.i.) spaces but they do not usually include the above ones. Let us just mention the classical paper [42], including the case of the Hilbert transform and Riesz transforms, and a very recent one [26] which contains a rather complete list of papers on this topic, among which we should mention [1, 10, 24].

Now, the three examples mentioned above have one important property in common: they all satisfy that, for some (and hence for all) $1< p_0 < \infty$, and for every $v \in A_{p_0}$,

$$\begin{aligned} T: L^{p_0}(v) \longrightarrow L^{p_0}(v) \end{aligned}$$

(1.1)

is bounded, where $A_{p_0}$ is the class of Muckenhoupt weights defined in Sect. 2 (see [38]). Let us just mention here that these weights $A_{p_0}$ characterize the boundedness on $L^{p_0}(v)$ of the Hardy–Littlewood maximal operator

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

where the supremum is taken overall cubes $Q \subseteq {\mathbb {R}}^n$ containing x. An operator T satisfying (1.1) will be called a Rubio de Francia operator [41]. Such operators satisfy the following result:

Theorem 1.1

[25] If T is a Rubio de Francia operator and ${\mathbb {X}}$ is a Banach function space such that $M:{\mathbb {X}} \rightarrow {\mathbb {X}}$ and $M:{\mathbb {X}}' \rightarrow {\mathbb {X}}'$, with ${\mathbb {X}}'$ being the associated space of ${\mathbb {X}}$, then

$$\begin{aligned} T:{\mathbb {X}} \longrightarrow {\mathbb {X}} \end{aligned}$$

is bounded.

This result is very useful to prove the boundedness of operators for which condition (1.1) has been widely studied while this is not the case in other contexts such as, for example, of weighted Lorentz spaces or more generally r.i. spaces. Let us explain, as an example, the case of the sparse operators $A_{{\mathcal {S}}}$ introduced by Lerner in [33]. These operators have become very useful since they dominate many operators such as Calderón–Zygmund operators. Independently of the fact that the decreasing rearrangement of an sparse operator $A_{{\mathcal {S}}}$ has not been estimated, the above theorem implies, for example, that if we consider the weighted Lorentz spaces, then if

$$\begin{aligned} M:\Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w), \qquad M: (\Lambda ^{p}(w))' \longrightarrow (\Lambda ^{p}(w))' \end{aligned}$$

(1.2)

are bounded operators, we have that

$$\begin{aligned} A_{{\mathcal {S}}}:\Lambda ^p(w) \longrightarrow \Lambda ^p(w) \end{aligned}$$

is bounded, where

$$\begin{aligned} (\Lambda ^p(w))'= & {} \left\{ f \in {\mathcal {M}}({\mathbb {R}}^n)\,:\, \Vert f\Vert _{(\Lambda ^p(w))'} \right. \\= & {} \left. \sup _{ g \in \Lambda ^p(w)}\frac{1}{\Vert g \Vert _{\Lambda ^p(w)}} \int _{{\mathbb {R}}^n} \left| f(x)g(x)\right| \,dx < \infty \right\} \end{aligned}$$

is the associated space of $\Lambda ^p(w)$. Now, on many occasions the difficulty to apply Theorem 1.1 to a concrete space ${\mathbb {X}}$ is precisely to characterize w so that (1.2) holds. In the case of ${\mathbb {X}}=\Lambda ^{p}(w)$ this has already been done, along the years, in several papers:

i) The boundedness of M in $\Lambda ^{p}(w)$ was first characterized for $p>1$, by Ariño and Muckenhoupt [4] in 1990, who showed

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

(1.3)

where

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

This class of weights has been widely studied (see for instance [11, 14, 39, 40, 44]) and now it is known that the same result holds for every $p>0$. Moreover, from (1.3), it can be easily seen that if $p>1$ and $w \in B_p$, then $\Lambda ^p(w)$ is a Banach function space. Besides, the reciprocal is also known to be true (see [42]). In addition, when $w \in B_1$, $\Lambda ^{1}(w)$ is a Banach function space as well (see [16]).

ii) To characterize the boundedness M on $(\Lambda ^p(w))'$ we can use the following. Given a r.i. Banach function space ${\mathbb {X}}$ on ${\mathbb {R}}^n$, the Lorentz–Shimogaki theorem (see [7, Chap.3 p. 154]) asserts that

$$\begin{aligned} M : {\mathbb {X}} \longrightarrow {\mathbb {X}} \end{aligned}$$

is bounded if and only if the upper Boyd index $\alpha _{\mathbb {X}} < 1$. Therefore, since $\alpha _{{\mathbb {X}}'} = 1 - \beta _{\mathbb {X}}$ ([7]),

$$\begin{aligned} M : {\mathbb {X}}' \longrightarrow {\mathbb {X}}' \end{aligned}$$

is bounded if and only if the lower Boyd index $\beta _{\mathbb {X}} > 0$ and it is known [2] that when $W(t)=\int _0^t w(r)\,dr$ and ${\mathbb {X}}=\Lambda ^p(w)$, this holds if and only if $w \in B^*_\infty$; that is,

$$\begin{aligned} \sup _{t>0} \dfrac{1}{W(t)}\int _0^t\frac{W(r)}{r}\,dr <\infty . \end{aligned}$$

From all the above results, we can conclude that if $A_{{\mathcal {S}}}$ is an sparse operator and $p \ge 1$,

$$\begin{aligned} w\in B_p\cap B_\infty ^*\quad \implies \quad A_{{\mathcal {S}}}: \Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w). \end{aligned}$$

In fact, the same result is true for every Rubio de Francia operator and it is sharp since it is known [42] 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 the Hilbert transform, which is a Rubio de Francia operator. Now, up to now, we have considered $p\ge 1$ since Theorem 1.1 concerns the case of Banach function spaces. But, what can we say if $0<p<1$? Moreover, in the case $p=1$, it is known that

$$\begin{aligned} A_{{\mathcal {S}}}:L^1\longrightarrow L^{1,\infty } \end{aligned}$$

is bounded, and this case is not covered by Theorem 1.1. To study these two cases are the purposes of this paper, since they will have as a consequence new estimates for all the above mentioned operators (see also [19] for related results). Now, it is known that (1.1) does not imply, in general, the weak boundedness of T from $L^1$ into $L^{1, \infty }$, but using the results in [20] we now know that we can arrive to the endpoint $p=1$ if we assume a slightly stronger condition on T which is satisfied by the sparse operators among many others. This condition reads:

$$\begin{aligned} T: L^{p_0, 1}(v) \longrightarrow L^{p_0, \infty }(v) \end{aligned}$$

(1.4)

need to be bounded for every $v\in A_{p_0}^{{\mathcal {R}}}$, a slightly bigger class than $A_{p_0}$ defined by

$$\begin{aligned} \Vert v\Vert _{A_{p_0}^{{\mathcal {R}}}} =\sup _{Q} \frac{1}{|Q|}\Vert \chi _Q \Vert _{L^{p_0}(v)} \Vert \chi _Q v^{-1}\Vert _{L^{p_0',\infty }(v)} <\infty . \end{aligned}$$

This class was introduced in [23, 31] where it was proved that

$$\begin{aligned} M:L^{p_0,1}(v) \longrightarrow L^{p_0, \infty }(v) \quad \iff \quad v\in A_{p_0}^{{\mathcal {R}}}. \end{aligned}$$

Operators satisfying (1.4) will be called restricted weak type Rubio de Francia operators or RWT-Rubio de Francia operators, in short.

So a question we want to answer is the following: If T is a RWT-Rubio de Francia operator, what conditions do we need on w in order to have that some restricted weak type boundedness on weighted Lorentz spaces

$$\begin{aligned} T:\Lambda ^{p,1}(w) \longrightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

holds? At this point we have to recall that if $w=1$, then $w\in B^*_\infty$ and although $w\notin B_1$ it belongs to a slightly bigger class $B_1^{{\mathcal {R}}}$ defined in Sect. 2. Also, we have the following extrapolation result [20]: Let T be a sublinear operator satisfying that, for some $p_0 > 1$ and every $v\in {A}_{p_0}^{{\mathcal {R}}}$,

$$\begin{aligned} T:L^{p_0,1}(v) \rightarrow L^{p_0,\infty }(v) \end{aligned}$$

is bounded, with constant less than or equal to $\varphi _{p_0}(||{v}||_{\hat{A}_{p_0}})$, with $\varphi _{p_0}$ an increasing function on $(0,\infty )$. Then,

$$\begin{aligned} T:L^{1, \frac{1}{p_0}} \rightarrow L^{1, \infty } \end{aligned}$$

is bounded. Hence, if T is a RWT-Rubio de Francia operator, it is natural to expect that

$$\begin{aligned} w\in B_1^{{\mathcal {R}}}\cap B^*_\infty \quad \implies \quad T:\Lambda ^{1, q}(w) \longrightarrow \Lambda ^{1,\infty }(w), \qquad \forall \, 0<q<1. \end{aligned}$$

In fact, our main theorems show this result holds true.

Theorem 1.2

Let T be an operator satisfying that, for every $v \in A_1$,

$$\begin{aligned} T:L^{1}(v) \rightarrow L^{1, \infty }(v) \end{aligned}$$

is bounded, with constant less than or equal to $\varphi (||{v}||_{A_1})$, with $\varphi$ an increasing function on $(0, \infty )$. Then, for every $0< p < \infty$ and every $w \in B_{p}^{{\mathcal {R}}}\cap B_{\infty }^{*}$,

$$\begin{aligned} T:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p, \infty }(w) \end{aligned}$$

is bounded with constant less than or equal to $C_1||w||_{B_{p}^{{\mathcal {R}}}}\varphi \left( C_2 ||w||_{B_\infty ^*} \right)$ for some positive constants $C_1$, $C_2$ independent of w.

Theorem 1.3

Let T be an operator satisfying that, for some $1< p_0 < \infty$ and every $v \in A_{p_0}^{\mathcal {R}}$,

$$\begin{aligned} T:L^{p_0,1}(v) \rightarrow L^{p_0, \infty }(v) \end{aligned}$$

is bounded, with constant less than or equal to $\varphi (||{v}||_{A_{p_0}^{\mathcal {R}}})$, with $\varphi$ an increasing function on $(0, \infty )$. Then, for every $0< p < \infty$, every $w \in B_{p}^{{\mathcal {R}}}\cap B_{\infty }^{*}$ and every $0< q < 1$,

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

is bounded with constant less than or equal to

$$\begin{aligned} \Phi _{p_0, p, q} (||{w}||_{B_p^{{\mathcal {R}}}}, ||w||_{B_\infty ^*} ) \le \frac{C_1}{1-q} ||w||_{B_p^{{\mathcal {R}}}}\varphi \left( C_2 ||w||_{B_\infty ^*}^{\max \left( \frac{1}{p_0}, q\right) } \right) , \end{aligned}$$

for some positives constants $C_1$, $C_2$ independent of w.

The paper is organized as follows: in Sect. 2 we present some technical lemmas and previous results which shall be used later and also the statements of our main results. Their proof will be given in Sect. 3, and Sect. 4 will be devoted to apply our results to the boundedness not only of sparse operators but also of Bochner–Riesz operator at the critical index, Hörmander multipliers, and many others.

As usual, we write $A \lesssim B$ if there exists an universal constant $C>0$ which may depends on the exponents but is independent of A and B, such that $A\le C B$. If $A\lesssim B$ and $B\lesssim A$, then we write $A\approx B$.

2 Definitions, Previous Results and Some Technical Lemmas

Let us recall the definition of the spaces which are going to be important for us. Given $0< p, q < \infty$, $L^{p,q}$ is the Lorentz space of measurable functions such that

$$\begin{aligned} ||f||_{L^{p,q}} =\bigg ( p\int _0^\infty y^q \lambda _f (y)^{\frac{q}{p}} \,\frac{dy}{y} \bigg )^{\frac{1}{q}} =\bigg (\int _0^\infty f^*(t)^q t^{\frac{q}{p}-1} dt\bigg )^{\frac{1}{q}} <\infty , \end{aligned}$$

and $L^{p,\infty }$ is the Lorentz space of measurable functions such that

$$\begin{aligned} ||f||_{L^{p,\infty } }= \sup _{y>0}y \lambda _f (y)^{\frac{1}{p}}= \sup _{t>0}f ^*(t)t^{\frac{1}{p}} <\infty , \end{aligned}$$

where $f ^*$ is the decreasing rearrangement of f defined by

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

For further information about these notions and related topics see [7]. These spaces are, in fact, a particular case of the so called weighted Lorentz spaces $\Lambda ^{p,q}(w)$ defined for $0< p,q < \infty$ by

$$\begin{aligned} ||f||_{\Lambda ^{p,q}(w)}= \bigg ( p\int _0^\infty y^qW(\lambda _f (y))^{\frac{q}{p}} \frac{dy}{y} \bigg )^{\frac{1}{q}} = \bigg ( \int _0^\infty f^*(t)^q W(t)^{\frac{q}{p} -1}w(t) dt \bigg )^{\frac{1}{q}}, \end{aligned}$$

and for $q = \infty$,

$$\begin{aligned} ||f||_{\Lambda ^{p,\infty }(w)} = \sup _{y> 0} y W(\lambda _f(y))^{\frac{1}{p}} = \sup _{t>0}f^*(t)W(t)^\frac{1}{p}, \end{aligned}$$

where w is a positive locally integrable function defined on $(0, \infty )$ and $W(t)=\int _0^t w(r)\,dr$. We should emphasize here that for $0< q < \infty$,

$$\begin{aligned} \Lambda ^{p,q}(w)=\Lambda ^q({\widetilde{w}}), \qquad {\widetilde{W}}(t) \approx W(t)^{\frac{q}{p}}, \end{aligned}$$

(2.1)

and, similar, $\Lambda ^{p,\infty }(w)=\Lambda ^{1,\infty }({\widetilde{w}})$, with ${\widetilde{W}}(t) \approx W(t)^{\frac{1}{p}}$. Besides, these spaces satisfy the embeddings

$$\begin{aligned} \Lambda ^{p,q_0}(w) \hookrightarrow \Lambda ^{p,q_1}(w) \end{aligned}$$

(2.2)

continuously for $0 < q_0 \le q_1 \le \infty$. For more details on these spaces, we refer to [18].

Let us consider the Hardy–Littlewood maximal operator M, defined for locally integrable functions on ${\mathbb {R}}^n$ by

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

where the supremum is taken over all cubes $Q\subseteq {\mathbb {R}}^n$ containing x. It is known [38] that for every $1<p<\infty$,

$$\begin{aligned} M:L^p(v) \longrightarrow L^p(v) \quad \iff \quad v\in A_p, \end{aligned}$$

where $v\in A_p$ if v is a positive and locally integrable function such that

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

Moreover, if $1\le p<\infty$,

$$\begin{aligned} M:L^p(v) \longrightarrow L^{p, \infty }(v) \quad \iff \quad v\in A_p, \end{aligned}$$

where $v\in A_1$ if

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

and the infimum of all such constants C in the above inequality is denoted by $||v||_{A_1}$. Also, in the context of restricted weak type inequalities the following result was proved in [23, 31]:

$$\begin{aligned} M:L^{p,1}(v) \longrightarrow L^{p, \infty }(v) \quad \iff \quad v\in A_p^{{\mathcal {R}}}, \end{aligned}$$

where a weight $v\in A_p^{{\mathcal {R}}}$ if

$$\begin{aligned} \Vert v\Vert _{A_p^{{\mathcal {R}}}} =\sup _{Q} \frac{1}{|Q|}\Vert \chi _Q \Vert _{L^{p}(v)} \Vert \chi _Q v^{-1}\Vert _{L^{p',\infty }(v)} <\infty . \end{aligned}$$

Moreover,

$$\begin{aligned} \Vert M\Vert _{L^{p,1}(v)\rightarrow L^{p,\infty }(v)} \lesssim \Vert v\Vert _{A_p^{{\mathcal {R}}}}. \end{aligned}$$

Now, given $p\ge 1$, it was proved in [15, 20] that $\widehat{A}_p \subseteq A_p^{{\mathcal {R}}}$, where

$$\begin{aligned} \widehat{A}_p:=\{0<v\in L^1_{loc} : \, \exists h \in L^1_{loc}, u \in A_1; \, v=(Mh)^{1-p}u \}, \end{aligned}$$

with

$$\begin{aligned} \left\| v \right\| _{A_p^{\mathcal {R}}} \lesssim \left\| v \right\| _{\widehat{A}_p}:= \inf ||u||_{A_1}^{\frac{1}{p}}, \end{aligned}$$

where the infimum is taken over all $u\in A_1$ for which there exists $h \in L^1_{loc}$ such that $v=(Mh)^{1-p}u$. Observe that $A_1^{{\mathcal {R}}}= \widehat{A}_1=A_1$.

Also, the following extrapolation result is going to be important in our main theorems.

Theorem 2.1

[20] Let T be an operator satisfying that, for some $1< p_0 < \infty$ and every $v \in {A}_{p_0}^{{\mathcal {R}}}$,

$$\begin{aligned} T:L^{p_0,1}(v) \rightarrow L^{p_0, \infty }(v) \end{aligned}$$

is bounded, with constant less than or equal to $\varphi _{p_0}(||{v}||_{A_{p_0}^{\mathcal {R}}})$, with $\varphi _{p_0}$ an increasing function on $(0, \infty )$. Then, for every $1 < p \le p_0$ and every $v \in \widehat{A}_{p}$

$$\begin{aligned} T:L^{p,1}(v) \rightarrow L^{p, \infty }(v) \end{aligned}$$

is bounded with constant

$$\begin{aligned} \varphi _{p_0, p}(t) \lesssim t^{1-\frac{p}{p_0}} \varphi _{p_0}\left( C t^{\frac{p}{p_0}}\right) . \end{aligned}$$

Concerning boundedness of the Hardy–Littlewood maximal operator on weighted Lorentz spaces we have, as mentioned in the introduction, that for every $p>0$,

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

Now, in order to characterize the boundedness of

$$\begin{aligned} M:\Lambda ^{p, 1}(w) \longrightarrow \Lambda ^{p, \infty }(w), \end{aligned}$$

by (2.1), $\Lambda ^{p,1}(w)=\Lambda ^1({\widetilde{w}})$ where ${\widetilde{w}}$ is such that its primitive ${\widetilde{W}}(t) \approx W(t)^{\frac{1}{p}}$ and hence, since $(Mf)^*(t)\approx f^{**}(t):= \frac{1}{t} \int _0^t f^*(r)\,dr$, this boundedness is equivalent to the embedding

$$\begin{aligned} \Lambda ^1({{\tilde{w}}}) \subset \Gamma ^{p, \infty }(w) := \left\{ f \in {\mathcal {M}}({\mathbb {R}}^n) \,:\, \Vert f^{**} \Vert _{\Lambda ^{p,\infty }(w)} < \infty \right\} , \end{aligned}$$

which is known [17] to be characterized by the condition $w\in B_p^{{\mathcal {R}}}$ defined by

$$\begin{aligned} ||w||_{B_p^{{\mathcal {R}}}}=\sup _{0< s\le t<\infty } \dfrac{sW(t)^{\frac{1}{p}}}{tW(s)^{\frac{1}{p}}}<\infty . \end{aligned}$$

(2.3)

In fact, $||M||_{\Lambda ^{p, 1}(w) \longrightarrow \Lambda ^{p, \infty }(w)}\le ||w||_{B_p^{{\mathcal {R}}}}$. Also,

$$\begin{aligned} \Vert g\Vert _{(\Lambda ^{p,1}(w))'} =\sup _{f \in \Lambda ^1({{\tilde{w}}})}\frac{\int _{{\mathbb {R}}^n} \left| f(x)g(x)\right| \,dx}{\int _0^\infty f^*(t){\widetilde{w}}(t)dt }=\sup _{t>0}\frac{\int _0^t g^*(r)\,dr}{\int _0^t {\widetilde{w}}(r)dr } \approx \sup _{t>0}\frac{\int _0^t g^*(r)\,dr}{W(t)^{\frac{1}{p}} }, \end{aligned}$$

where in the second equality we have used [18, Corollary 1.2.12]. Hence

$$\begin{aligned} ||M f||_{(\Lambda ^{p, 1}(w) )'}&\approx \sup _{t> 0}\frac{\int _0^t \frac{1}{r} \int _0^r f^*(s) ds dt}{W(t)^{\frac{1}{p}} } \lesssim ||f||_{(\Lambda ^{p, 1}(w) )'} \left( \sup _{t>0}\frac{\int _0^t W(r)^{\frac{1}{p}} \frac{dr}{r}}{W(t)^{\frac{1}{p}} }\right) , \end{aligned}$$

and consequently

$$\begin{aligned} ||M||_{(\Lambda ^{p, 1}(w) )'} := ||M||_{(\Lambda ^{p, 1}(w) )' \rightarrow (\Lambda ^{p, 1}(w) )'} \lesssim \sup _{t>0} \frac{\int _0^t W(r)^{\frac{1}{p}} \frac{dr}{r} }{W(t)^{\frac{1}{p}}} . \end{aligned}$$

(2.4)

Now, the last expression is finite if and only if $w\in B^*_\infty$ (see [3]). So let us recall several facts on this class of weights and give an expression for the quantity $||{w}||_{B_{\infty }^{*}}$ which is going to be important for our purposes.

Definition 2.2

A weight $w\in B^*_\infty$ if and only if

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

Now, many equivalent definitions have appeared in the literature and, as a consequence, we have the following result [3]:

Lemma 2.3

For $0< p < \infty$,

$$\begin{aligned} ||M||_{(\Lambda ^{p, 1}(w) )'} \lesssim ||w||_{B^*_\infty }. \end{aligned}$$

Proof

In virtue of estimate (2.4), it would be enough to see that

$$\begin{aligned} \sup _{t>0} \frac{\int _0^t W(r)^{\frac{1}{p}} \frac{dr}{r} }{W(t)^{\frac{1}{p}}} \lesssim ||w||_{B^*_\infty }. \end{aligned}$$

If $0<p\le 1$, then the result is immediate, since for every $t > 0$,

$$\begin{aligned} \int _0^t W(r)^{\frac{1}{p}} \frac{dr}{r} \le W(t)^{\frac{1}{p} - 1}\int _0^t \frac{W(r)}{r}\,dr \le W(t)^{\frac{1}{p}} ||w||_{B^*_\infty }. \end{aligned}$$

So let us consider the case $p > 1$. First, integrating by parts,

$$\begin{aligned} \begin{aligned} \int _0^t\frac{W(r)^{\frac{1}{p}}}{r}\,dr =&\lim _{s \rightarrow 0} \log \left( \frac{t}{s}\right) W(s)^\frac{1}{p} + \frac{1}{p}\int _0^t \log \left( \frac{t}{r}\right) W(r)^{\frac{1}{p} - 1}w(r)\,dr \\ \le&\left( \int _0^t \left( \log \frac{t}{r}\right) ^p w(r) \,dr\right) ^{\frac{1}{p}} \\&+ \frac{1}{p}\int _0^t \log \left( \frac{t}{r}\right) W(r)^{\frac{1}{p} - 1}w(r)\,dr, \end{aligned} \end{aligned}$$

(2.5)

an observe that by means of the Hölder’s inequality,

$$\begin{aligned} \begin{aligned}&\int _0^t \log \left( \frac{t}{r}\right) W(r)^{\frac{1}{p} - 1}w(r)\,dr\\&\quad \le \left( \int _0^t\Big (\log \frac{t}{r} \Big )^{2p}w(r)\,dr \right) ^{\frac{1}{2p}}\left( \int _0^tW(r)^{\frac{1}{2p - 1} - 1}w(r)\,dr \right) ^{\frac{2p - 1}{2p}} \\&\quad \le \left( 2p - 1\right) ^{\frac{2p - 1}{2p}} \left( \int _0^t\Big (\log \frac{t}{r} \Big )^{2p}w(r)\,dr \right) ^{\frac{1}{2p}} W(t)^{\frac{1}{2p}}. \end{aligned} \end{aligned}$$

(2.6)

We claim that for every $q \ge 2$,

$$\begin{aligned} \int _0^t\Big (\log \frac{t}{r} \Big )^{q}w(r)\,dr \le q(q-1)\cdots (q - [q] + 1) \Vert w \Vert _{B^*_\infty }^q W(t), \end{aligned}$$

(2.7)

from which putting together (2.5) and (2.6), we will conclude that

$$\begin{aligned} \int _0^t\frac{W(r)^{\frac{1}{p}}}{r}\,dr&\lesssim \left( \int _0^t\Big (\log \frac{t}{r} \Big )^{2p}w(r)\,dr \right) ^{\frac{1}{2p}} W(t)^{\frac{1}{2p}} \lesssim \Vert w \Vert _{B^*_\infty } W(t)^\frac{1}{p}. \end{aligned}$$

So let us see (2.7). Observe that since $q \ge 2$, using Fubini twice we get

$$\begin{aligned} \int _0^t\Big (\log \frac{t}{r} \Big )^{q}w(r)\,dr= & {} q \int _0^t \int _r^t \Big (\log \frac{t}{s} \Big )^{q-1} \frac{ds}{s} w(r)\,dr \\= & {} q \int _0^t \Big (\log \frac{t}{s} \Big )^{q-1} W(s)\, \frac{ds}{s} \\= & {} q (q-1) \int _0^t \Big (\log \frac{t}{r} \Big )^{q-2} \int _0^r W(s)\, \frac{ds}{s}\, \frac{dr}{r} \\\le & {} q (q-1) ||w||_{B^*_\infty } \int _0^t \Big (\log \frac{t}{r} \Big )^{q-2} W(r) \, \frac{dr}{r}. \end{aligned}$$

Hence, iterating in that way we obtain

$$\begin{aligned}&\int _0^t\Big (\log \frac{t}{r} \Big )^{q}w(r)\,dr \\&\quad \le q(q-1)\cdots (q - [q] + 1) \Vert w \Vert _{B^*_\infty }^{[q] - 1} \int _0^t \Big (\log \frac{t}{r} \Big )^{q-[q]} W(r) \frac{dr}{r}, \end{aligned}$$

with [q] being the integer part of q. Finally,

$$\begin{aligned} \int _0^t\Big (\log \frac{t}{r} \Big )^{q - [q]}\frac{W(r)}{r} \,dr\le & {} \left( \int _0^t \log \frac{t}{r} \, \frac{W(r)}{r}\, dr \right) ^{q-[q]} \left( \int _0^t \frac{W(r)}{r}\, dr \right) ^{1 + [q] - q} \\\le & {} \left( \int _0^t \int _0^u \frac{W(s)}{s} ds \frac{du}{u} \right) ^{q-[q]} \left( ||w||_{B^*_\infty } W(t) \right) ^{1 + [q] - q} \\\le & {} \left( \Vert w \Vert _{B^*_\infty }^2 W(t)\right) ^{q-[q]} ||w||_{B^*_\infty }^{1 + [q] - q} W(t)^{1 + [q] -q} \\= & {} ||w||_{B^*_\infty }^{1 + q - [q]} W(t), \end{aligned}$$

and the result follows. $\square$

3 Proof of Theorems 1.2 and 1.3

Proof of Theorem 1.2

Since $w \in B_\infty ^*$, we have that M is bounded in $(\Lambda ^{p,1}(w))'$, and hence we can define, for every nonnegative function $h \in \left( \Lambda ^{p,1}(w) \right) '$,

$$\begin{aligned} Rh(x) = \sum _{k = 0}^{\infty }\frac{M^kh(x)}{\left( 2||M||_{{(\Lambda ^{p,1}(w)})'} \right) ^k} \end{aligned}$$

to be the function resulting from the Rubio de Francia algorithm [41]. Then, $h(x)\le Rh(x)$ a.e., $Rh \in A_1$ with

$$\begin{aligned} ||Rh||_{A_1} \lesssim ||M||_{{(\Lambda ^{p,1}(w)})'} \lesssim ||w||_{B^*_\infty } \end{aligned}$$

by Lemma 2.3, and

$$\begin{aligned} \Vert Rh||_{(\Lambda ^{p,1}(w))'} \le 2||h||_{({\Lambda ^{p,1}(w)})'}. \end{aligned}$$

Let $y > 0$ and set $F := F_{Tf,y} = \{x \in {\mathbb {R}}^n \, : \, |Tf(x)| > y\}$. Then, $v = R(\chi _F) \in A_1$ and,

$$\begin{aligned} \lambda _{Tf}(y)\le & {} \int _{\{ |Tf(x)|> y \}}R\left( \chi _{F} \right) (x) \,dx \le \frac{ \varphi \left( ||v||_{A_1} \right) }{y} \int _{{\mathbb {R}}^n} |f(x)| R\left( \chi _{F} \right) (x)\,dx\\= & {} \frac{\varphi \left( ||v||_{A_1} \right) }{y} \int _0^\infty \left[ \int _{\{|f(x)| > z\}} R(\chi _F)(x)\,dx \right] \,dz. \end{aligned}$$

Now, in the inner integral we apply the definition of associate space to obtain that

$$\begin{aligned} \int _{\{|f(x)|> z\}} R(\chi _F)(x)\,dx \le ||\chi _{\{|f|>z\}}||_{\Lambda ^{p, 1}(w)} ||R(\chi _F)||_{(\Lambda ^{p, 1}(w))'}, \end{aligned}$$

and hence,

$$\begin{aligned} \lambda _{Tf}(y)\lesssim & {} \frac{\varphi \left( ||v||_{A_1} \right) }{y} ||R(\chi _F)||_{(\Lambda ^{p, 1}(w))'} ||f||_{\Lambda ^{p,1}(w)}\\\lesssim & {} \frac{\varphi \left( ||v||_{A_1} \right) }{y} ||\chi _F||_{(\Lambda ^{p, 1}(w))'} ||f||_{\Lambda ^{p,1}(w)}. \end{aligned}$$

Now, since $w \in B_p^{{\mathcal {R}}}$, we have by (2.3) that

$$\begin{aligned} ||{\chi _F}||_{ {(\Lambda ^{p,1}(w)})'} \lesssim ||w||_{B_{p}^{{\mathcal {R}}}}\frac{\lambda _{Tf}(y)}{W(\lambda _{Tf}(y))^{\frac{1}{p}}}, \end{aligned}$$

(3.1)

and hence,

$$\begin{aligned} \sup _{y > 0} y W(\lambda _{Tf}( y))^{\frac{1}{p}} \lesssim ||w||_{B_{p}^{{\mathcal {R}}}} \varphi \big (C ||w||_{B^*_\infty }\big )||f||_{\Lambda ^{p,1}(w)} \end{aligned}$$

as we wanted to see. $\square$

Remark 3.1

(1) We observe that, since

$$\begin{aligned} \Lambda ^{p,1}(w) = \Lambda ^1({\widetilde{w}}), \quad \text{ and }\quad \Lambda ^{p,\infty }(w) = \Lambda ^{1,\infty }({\widetilde{w}}), \end{aligned}$$

with ${\widetilde{w}}$ such that ${\widetilde{W}}(t) \approx W(t)^{\frac{1}{p}}$ (see (2.1)), and we also have that

$$\begin{aligned} w\in B_p^{{\mathcal {R}}}\cap B^*_\infty \quad \iff \quad {\widetilde{w}}\in B_1^{{\mathcal {R}}}\cap B^*_\infty , \end{aligned}$$

the condition

$$\begin{aligned} w\in B_p^{{\mathcal {R}}} \cap B^*_\infty \implies T: \Lambda ^{p,1}(w) \longrightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is equivalent to

$$\begin{aligned} w\in B_1^{{\mathcal {R}}} \cap B^*_\infty \implies T: \Lambda ^{1}(w) \longrightarrow \Lambda ^{1,\infty }(w). \end{aligned}$$

(2) Using interpolation theory of weighted Lorentz spaces (see for instance [18, Theorem 2.6.5]) we can deduce that if T is a quasi-linear operator satisfying the hypothesis of Theorem 1.2, then

$$\begin{aligned} w\in B_1 \cap B^*_\infty \implies T: \Lambda ^{1}(w) \longrightarrow \Lambda ^{1}(w), \end{aligned}$$

and similarly, for every $0<p<\infty$,

$$\begin{aligned} w\in B_p \cap B^*_\infty \implies T: \Lambda ^{p}(w) \longrightarrow \Lambda ^{p}(w). \end{aligned}$$

Proof of Theorem 1.3

Let $y > 0$ and set $F := F_{Tf,y} = \{x \in {\mathbb {R}}^n \, : \, |Tf(x) |> y\}$. Then, $v = (Mf)^{1 - p_0}R(\chi _F) \in \widehat{A}_{p_0} \subset A_{p_0}^{\mathcal {R}}$ and, for every $\gamma >0$,

$$\begin{aligned}&\lambda _{Tf}(y) \le \lambda _{Mf}(\gamma y) + \int _{\{ |Tf(x)|> y, \, Mf(x) \le \gamma y\}}R\left( \chi _{F} \right) (x) \,dx\\&\quad \le \lambda _{Mf}(\gamma y) + \gamma ^{p_0 - 1}\frac{y^{p_0}}{y}\int _{F}Mf(x)^{1 - p_0}R\left( \chi _{F} \right) (x)\,dx \\&\quad \lesssim \lambda _{Mf}(\gamma y) + \frac{\gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}{y}||f||^{p_0}_{L^{p_0,1}\left( Mf^{1 - p_0}R\left( \chi _{F} \right) \right) } \\&\quad =\lambda _{Mf}(\gamma y) + \frac{\gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}{y} \\&\qquad \times \left( p_0 \int _0^\infty \left[ \int _{\{|f(x) |> z\}} Mf(x)^{1 - p_0}R(\chi _F)(x)\,dx \right] ^{\frac{1}{p_0}}\,dz \right) ^{p_0} \\&\quad \lesssim \lambda _{Mf}(\gamma y) + \frac{\gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}{y}\\&\qquad \left( \int _0^\infty z^{\frac{1}{p_0} - 1}\left[ \int _{\{|f(x)| > z\}} R(\chi _F)(x)\,dx \right] ^{\frac{1}{p_0}}\,dz \right) ^{p_0}, \end{aligned}$$

where in the last estimate we have used that for every $x \in \{|f(x)| > z\}$,

$$\begin{aligned} Mf(x)^{1 - p_0} \le |f(x)|^{1 - p_0} \le z^{1 - p_0}. \end{aligned}$$

Now, in the inner integral we apply the definition of associate space to obtain that

$$\begin{aligned} \int _{\{|f(x)|> z\}} R(\chi _F)(x)\,dx \le ||\chi _{\{|f|>z\}}||_{\Lambda ^{p, 1}(w)} ||R(\chi _F)||_{(\Lambda ^{p, 1}(w))'}, \end{aligned}$$

and hence, using (3.1) and the boundedness of the operator R,

$$\begin{aligned} \lambda _{Tf}(y)\lesssim & {} \lambda _{Mf}(\gamma y) + {p^{1 - p_0}}\frac{\gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}{y} ||f||_{\Lambda ^{p, \frac{1}{p_0}}(w)}||{R\left( \chi _{F} \right) }||_{ (\Lambda ^{p,1}(w))'} \\\lesssim & {} \lambda _{Mf}(\gamma y) + { p^{1 - p_0}}\frac{\gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}{y} ||f||_{\Lambda ^{p, \frac{1}{p_0}}(w)}||w||_{B_{p}^{{\mathcal {R}}}}\frac{\lambda _{Tf}(y)}{W(\lambda _{Tf}(y))^{\frac{1}{p}}}, \end{aligned}$$

so that using that $w \in B_p^{{\mathcal {R}}}$ and the continuous embedding $\Lambda ^{p,\frac{1}{p_0}}(w) \subset \Lambda ^{p,1}(w)$ (see (2.2)),

$$\begin{aligned} \sup _{y > 0} yW(\lambda _{Mf}(\gamma y))^{\frac{1}{p}}= & {} \dfrac{1}{\gamma } ||{Mf}||_{\Lambda ^{p,\infty }(w)} \\\lesssim & {} \dfrac{||w||_{B_{p}^{{\mathcal {R}}}} }{\gamma } ||f||_{\Lambda ^{p,1}(w)} \\\lesssim & {} \dfrac{||w||_{B_{p}^{{\mathcal {R}}}}}{\gamma } ||f||_{\Lambda ^{p,\frac{1}{p_0}}(w)}. \end{aligned}$$

Therefore, we conclude

$$\begin{aligned} ||{Tf}||_{\Lambda ^{p, \infty }(w)} \lesssim \max \left( {\dfrac{1}{\gamma }, \gamma ^{p_0 - 1}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ^{p_0}}\right) ||w||_{B_{p}^{{\mathcal {R}}}} ||f||_{\Lambda ^{p, \frac{1}{p_0}}(w)}, \end{aligned}$$

and taking the infimum in $\gamma > 0$,

$$\begin{aligned} ||{Tf}||_{\Lambda ^{p, \infty }(w)} \lesssim ||w||_{B_{p}^{{\mathcal {R}}}}\varphi \left( ||v||_{\widehat{A}_{p_0}} \right) ||f||_{\Lambda ^{p, \frac{1}{p_0}}(w)}. \end{aligned}$$

Now,

$$\begin{aligned} ||v||_{\widehat{A}_{p_0}} \lesssim ||{R\left( \chi _F \right) }||_{A_1}^{1/{p_0}} \lesssim ||w||_{B_\infty ^*} ^{1/{p_0}}. \end{aligned}$$

Therefore,

$$\begin{aligned} ||{Tf}||_{\Lambda ^{p, \infty }(w)} \lesssim ||w||_{B_{p}^{{\mathcal {R}}}}\varphi \left( C ||w||_{B_\infty ^*} ^{1/{p_0}} \right) ||f||_{\Lambda ^{p, \frac{1}{p_0}}(w)}. \end{aligned}$$

Observe that if $0 < q\le \frac{1}{p_0}$, by the continuous embeddings of the weighted Lorentz spaces (see (2.2)), then

$$\begin{aligned} \Lambda ^{p, q}(w)\hookrightarrow \Lambda ^{p, \frac{1}{p_0}}(w) \end{aligned}$$

and the result follows. Otherwise, take $\frac{1}{p_0}< q < 1$. Hence, by Theorem  2.1,

$$\begin{aligned} T:L^{\frac{1}{q},1}(v) \rightarrow L^{\frac{1}{q}, \infty }(v) \end{aligned}$$

is bounded for every $v \in \widehat{A}_{\frac{1}{q}}$ with constant less than or equal to $\displaystyle \varphi _{p_0, \frac{1}{q}}\left( ||v||_{\widehat{A}_{\frac{1}{q} }} \right)$. Therefore, applying the first part of the proof we have that

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

is bounded with constant less than or equal to $\Phi _{p_0, p, q}\left( ||w||_{B_p^{{\mathcal {R}}}}, ||w||_{B_\infty ^*}\right)$. $\square$

4 Boundedness of Operators on Weighted Lorentz Spaces

The main result of this section is the following:

Theorem 4.1

Let T be any of the following operators:

• (i) Sparse operators $A_{{\mathcal {S}}}$.

• (ii) Bochner–Riesz at the critical index $B_{\frac{n-1}{2}}$.

• (iii) Radial Fourier multipliers satisfying (4.2).

• (iv) Hörmander multipliers with $m \in HC(s,k)$.

• (v) Rough singular integrals.

Then, for every $p>0$ and every $w\in B_p^{\mathcal {R}} \cap B^*_\infty$

$$\begin{aligned} T: \Lambda ^{p,1}(w) \longrightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

(4.1)

is bounded.

Let us give the definition of all the above mentioned operators, some references related to them and, when possible, an estimate for the norm of the operator T in (4.1).

4.1 Sparse Operators

These operators have become very popular due to their role in the so called $A_2$ conjecture consisting in proving that if T is a Calderón–Zygmund operator then

$$\begin{aligned} ||Tf||_{L^2(v)} \lesssim ||v||_{A_2}||f||_{L^2(v)}. \end{aligned}$$

This result was first obtained by Hytönen [29] and then simplified by Lerner [33, 34], 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 on $L^2(v)$ for every weight $v \in A_2$ with sharp constant. Let us give the precise definition. A general dyadic grid ${\mathcal {D}}$ is a collection of cubes in ${\mathbb {R}}^n$ satisfying the following properties:

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

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

• (iii) Given $x \in {\mathbb {R}}^n$, for every $k \in {\mathbb {Z}}$ there is only one cube in ${\mathcal {D}}$ of side length $2^k$ containing it.

Let $0< \eta < 1$, a collection of cubes ${\mathcal {S}} \subset {\mathcal {D}}$ is called $\eta$-sparse if one can choose pairwise disjoint measurable sets $E_Q \subset Q$ with $|E_Q| \ge \eta |Q|$, where $Q\in {\mathcal {S}}$.

Definition 4.2

Given a sparse family of cubes ${\mathcal {S}} \subset {\mathcal {D}}$, the sparse operator is defined by

$$\begin{aligned} A_{{\mathcal {S}}}f(x) = \sum _{Q\in {\mathcal {S}}}\left( \frac{1}{|Q|}\int _Qf(y)\,dy \right) \chi _Q(x). \end{aligned}$$

Proposition 4.3

[30] For every $v \in A_{1}$, it holds that

$$\begin{aligned} || A_{{\mathcal {S}}}f||_{L^{1,\infty }(v)} \lesssim ||v||_{A_1} (1+ \log ||v||_{A_1}) ||f||_{L^1(v)}. \end{aligned}$$

In [35], this same bound was proved for a Calderón–Zygmund operator and in [36] the authors have proved that it is sharp. Hence, using the domination property of the sparse operators, we conclude that the bound in the above proposition is also sharp. We thank A.K. Lerner for this information.

Therefore, as a consequence of Theorem 1.2, we get the following result.

Corollary 4.4

Let $0< p < \infty$ and $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$. Then,

$$\begin{aligned} A_{{\mathcal {S}}}:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is bounded with constant controlled by

$$\begin{aligned} C_1 ||w||_{B_p^{{\mathcal {R}}}}||w||_{B^*_\infty } \left( 1+ \log \left( C_2||w||_{B^*_\infty } \right) \right) , \end{aligned}$$

with $C_1, C_2$ constants independent of w.

4.2 The Bochner–Riesz Operator

Let

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

be the Fourier transform of an integrable function $f \in L^1\left( {\mathbb {R}}^n \right)$ and let $a_{+} = \max \{a,0\}$ denote the positive part of $a \in {\mathbb {R}}$. Given $\lambda > 0$, the Bochner–Riesz operator $B_{\lambda }$ on ${\mathbb {R}}^n$ is defined by

$$\begin{aligned} \widehat{B_{\lambda }f}(\xi ) = \left( 1 - |\xi |^2 \right) ^{\lambda }_{+}\hat{f}(\xi ). \end{aligned}$$

They were first introduced by Bochner in [8] and, since then, they have been widely studied. The case $\lambda = 0$ corresponds to the so-called ball multiplier, which is known to be unbounded on $L^p({\mathbb {R}}^n)$ if $n \ge 2$ and $p \ne 2$ ([27]). It is known that when $\lambda > \frac{n - 1}{2}$, $B_\lambda f$ is controlled by the Hardy–Littlewood maximal function Mf, and hence $B_\lambda$ satisfies the same weighted estimates as M. We will focus on the value $\lambda = \frac{n - 1}{2}$, which is called the critical index. In this case, Christ [22] showed that $B_{\frac{n-1}{2}}$ is bounded from $L^1({\mathbb {R}}^n)$ to $L^{1,\infty }({\mathbb {R}}^n)$, and although we do not have the control of $B_{\frac{n-1}{2}}$ by M, Shi and Sun proved in [43] that $B_{\frac{n-1}{2}}$ is bounded on $L^p(v)$ for every $v \in A_p$ when $1< p < \infty$. The corresponding weak-type inequality for $p = 1$ was obtained by Vargas in [45], where she proved that $B_{\frac{n-1}{2}}$ is bounded from $L^1(v)$ to $L^{1,\infty }(v)$ for every $v \in A_1$.

Proposition 4.5

[37] For every $n > 1$,

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

is bounded with constant less than or equal to $||v||^2_{A_1}\log _2(C||v||_{A_1} + 1)$.

Therefore, as a consequence of Theorem 1.2, we get the following result.

Corollary 4.6

For every $n > 1$, $0< p < \infty$ and $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$,

$$\begin{aligned} B_{\frac{n-1}{2}}:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is bounded with constant controlled by

$$\begin{aligned} C_1||w||_{B_p^{{\mathcal {R}}}}||w||_{B^*_\infty }^2 (1+ \log (C_2||w||_{B^*_\infty } ) ), \end{aligned}$$

with $C_1, C_2$ constants independent of w.

4.3 Radial Multipliers

Given a multiplier m, we say that $T_m$ is a radial Fourier multiplier if $\widehat{T_mf}(\xi ) = m(|\xi |){\hat{f}}(\xi )$.

Definition 4.7

Given $0 \le \delta < 1$ and $r > 0$, we define the truncated fractional integral of order $1 - \delta$ of a locally integrable function f on ${\mathbb {R}}$ by

$$\begin{aligned} I_r^{1 - \delta }f(t):=\left\{ \begin{array}{ll} \dfrac{1}{\Gamma (1 - \delta )}\int _{-r}^r(s - t)_{+}^{-\delta }f(s)\,ds, &{} \quad t < r, \\ 0, &{} \quad t \ge r, \end{array}\right. \qquad t \in {\mathbb {R}}, \end{aligned}$$

with $\Gamma$ being the Gamma function. Moreover, if $\alpha = [\alpha ] + \delta > 0$, with $[\alpha ]$ being its integer part and $\delta$ its fractional part, we define the fractional derivative of f of order $\alpha$ by

$$\begin{aligned} D^\alpha f(t)=- \left( \dfrac{d}{dt}\right) ^{[\alpha ]}\lim _{r \rightarrow \infty }\frac{d}{dt}I_r^{1 - \delta }f(t), \end{aligned}$$

whenever the right-hand side exists.

Let $AC_{\text {loc}}$ be the space of functions which are absolutely continuous on every compact subset of $(0,\infty )$.

Proposition 4.8

([13]) Fix $n \ge 2$ and $\alpha = \frac{n+1}{2}$. Let m be a bounded, continuous function on $(0,\infty )$ which vanishes at infinity and satisfies that

$$\begin{aligned} D^{\alpha - j}m\in AC_{\text {loc}} \forall j = 1,\dots ,[\alpha ]. \end{aligned}$$

Then, if $D^{\alpha }m$ exists and

$$\begin{aligned} \Phi (t) = t^{\alpha - 1}D^{\alpha }m(t) \in L^{1}(0,\infty ), \end{aligned}$$

(4.2)

the operator $T_m$ defined by

$$\begin{aligned} \widehat{T_mf}(\xi ) = m(|\xi |^2)\widehat{f}(\xi ), \xi \in {\mathbb {R}}^n, \end{aligned}$$

satisfies that

$$\begin{aligned} T_m:L^{1}(v)\rightarrow L^{1,\infty }(v) \end{aligned}$$

is bounded for every $v \in A_1$ with constant controlled by $C||\Phi ||_{L^1(0,\infty )}||u||_{A_1}^5$.

Therefore, as a consequence of Theorem 1.2, we get the following result.

Corollary 4.9

Let $0< p < \infty$ and $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$. Then if m satisfies the hypothesis of Proposition 4.8 we have that

$$\begin{aligned} T_m:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is bounded with constant controlled by

$$\begin{aligned} C||\Phi ||_{L^1(0,\infty )} ||w||_{B_p^{{\mathcal {R}}}}||w||_{B^*_\infty }^5, \end{aligned}$$

with C independent of w.

4.4 Fourier Multipliers of Hörmander Type

Let us use the standard notation $|\alpha | = \alpha _1 + \cdots + \alpha _n$ for a multi-index $\alpha = (\alpha _1, \dots , \alpha _n) \in {\mathbb {N}}^n$ and if $x \in {\mathbb {R}}^n$,

$$\begin{aligned} \left( \frac{\partial }{\partial x} \right) ^{\alpha } = \frac{\partial ^{\alpha _1}}{\partial x_1^{\alpha _1}} \cdots \frac{\partial ^{\alpha _n}}{\partial x_n^{\alpha _n}}. \end{aligned}$$

Then, the Hörmander condition for a multiplier m is the following.

Definition 4.10

Let $k \in {\mathbb {N}}$ such that $k > n/2$ and let $m:{\mathbb {R}}^n \rightarrow {\mathbb {R}}$ be a bounded function of ${\mathcal {C}}^k$ class on ${\mathbb {R}}^n\setminus \{0\}$. Given $1 < s \le 2$, we say that m satisfies the Hörmander condition respect s and k, and denote it by $m \in HC(s,k)$, if

$$\begin{aligned} \sup _{r > 0} \left( r^{2|\alpha | - n}\int _{r< |x|< 2r} \left| \left( \frac{\partial }{\partial x} \right) ^{\alpha } m(x) \right| ^s\,dx \right) ^{1/s} < \infty , |\alpha | \le k. \end{aligned}$$

The Fourier multipliers operators of Hörmander type are those defined by

$$\begin{aligned} \widehat{T_mf}(\xi ) = m(\xi )\hat{f}(\xi ), \xi \in {\mathbb {R}}^n, \end{aligned}$$

where $m \in HC(s,k)$ for $k > n/2$ and $1 < s \le 2$. The classical Hörmander theorem (see for example [28, Theorem 5.2.7]) says that when $s = 2$, in the unweighted case, $T_m$ is bounded on $L^p({\mathbb {R}}^n)$ for $1< p < \infty$ and satisfies the weak-type inequality

$$\begin{aligned} T_m:L^1({\mathbb {R}}^n) \rightarrow L^{1,\infty }({\mathbb {R}}^n) \end{aligned}$$

whenever $m \in HC(2,k)$. The generalization of the condition to $1< s < 2$ was introduced by Calderón and Torchinsky in [12], where the authors see that in order to the classical result to be true for $m\in HC(s,k)$ is needed that $k > n/s$.

Proposition 4.11

[32, Theorem 1] Let $1 < s \le 2$ and $m \in HC(s,n)$. Then, for every $v \in A_1$,

$$\begin{aligned} T_m : L^1(v) \rightarrow L^{1, \infty }(v) \end{aligned}$$

is bounded with constant less than or equal to $\varphi (||v||_{A_1})$ with $\varphi$ being an increasing function in $(0,\infty )$.

Therefore, as a consequence of Theorem 1.2, we get the following result.

Corollary 4.12

Let $0< p < \infty$ and $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$. Then

$$\begin{aligned} T_m:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is bounded with constant controlled by

$$\begin{aligned} C_1||w||_{B_p^{{\mathcal {R}}}} \, \varphi (C_2||w||_{B^*_\infty }), \end{aligned}$$

with $C_1$ and $C_2$ independent of w.

At this point we have to say that although the property of $\varphi$ being increasing is known, the sharp expression for such $\varphi$ is unknown.

4.5 Rough Singular Integrals

Definition 4.13

Let $\Sigma ^{n-1} = \left\{ x \in {\mathbb {R}}^n \, : \, |x| = 1 \right\}$ and given $\Omega \in L^{\infty }(\Sigma ^{n-1})$ such that

$$\begin{aligned} \int _{\Sigma ^{n-1}}\Omega (x)\,dx = 0. \end{aligned}$$

The rough singular integral is defined by

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

where $y' = \dfrac{y}{|y|}$. In [37], the authors obtained the following result:

Proposition 4.14

For every $v\in A_1$,

$$\begin{aligned} T_{\Omega }:L^1(v)\rightarrow L^{1,\infty }(v) \end{aligned}$$

is bounded with constant controlled by $||v||^2_{A_1}\log _2(||v||_{A_1} + 1)$.

Therefore, as a consequence of Theorem 1.2, we get the following result.

Corollary 4.15

Let $0< p < \infty$ and $\Omega \in L^\infty (\Sigma ^{n-1})$ such that $\int _{\Sigma ^{n-1}}\Omega = 0$. If $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$, then

$$\begin{aligned} T_{\Omega }:\Lambda ^{p,1}(w) \rightarrow \Lambda ^{p,\infty }(w) \end{aligned}$$

is bounded with constant controlled by

$$\begin{aligned} C_1||w||_{B_p^{{\mathcal {R}}}}||w||_{B^*_\infty }^2 \left( 1+ \log \left( C_2||w||_{B^*_\infty } \right) \right) , \end{aligned}$$

with $C_1, C_2$ constants independent of w.

4.6 Assani Operator

There is a very interesting operator which satisfies the hypothesis of Theorem 1.3 but it is not of weak type (1, 1) and hence we cannot apply Theorem 1.2. This operator is related with the Return time theorem of Bourgain [9], and we refer to [5] for a very interesting review on the topic and also to [6], where the following related operator was introduced:

$$\begin{aligned} Af(x)=\left\| \frac{f(\cdot )\chi _{(0, x)}(\cdot )}{x-\cdot }\right\| _{L^{1, \infty }(0,1)}. \end{aligned}$$

Since one can easily check that $A\chi _E \le M\chi _E$, we have that A satisfies (1.4) and consequently, we can deduce that, for every $1< p < \infty$ and $v \in A_p^{{\mathcal {R}}}$,

$$\begin{aligned} ||Af||_{L^{p,\infty }(v)} \lesssim \frac{1}{p-1} ||v||_{A_p^{{\mathcal {R}}}} ||f||_{L^{p,1}(v)}. \end{aligned}$$

Therefore, as a consequence of Theorem 1.3, we get the following result.

Corollary 4.16

Let $0< p < \infty$, $0< q < 1$ and $w \in B_p^{{\mathcal {R}}}\cap B^*_\infty$, then

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

is bounded with constant less than or equal to

$$\begin{aligned} \frac{C}{(1-q)^2} ||w||_{B_p^{{\mathcal {R}}}} ||w||_{B^*_\infty }^q, \end{aligned}$$

with C independent of w.

At this point, we have to say that, in the case $w=1$, the boundedness of A in $L^{1,q}$ was obtained in [21] and, as a consequence, it was proved that the space $L^{1,q}$ satisfies the Return Time Property for the Tail, while this is not the case for $L^1$ (see [6]).

On the other hand, it is an interesting open question in the area whether the space $L\log \log \log L$ satisfies this property. Since

$$\begin{aligned} L\log \log \log L = \Lambda ^1 \left( 1+\log \left( 1+ \log \left( 1+ \log \frac{1}{t} \right) \right) \right) , \end{aligned}$$

it will be very interesting to study, for which weights w, the Assani operator A is bounded on $\Lambda ^1(w)$.