Weighted estimates for Bochner–Riesz operators on Lorentz spaces

Abstract

We present new estimates in the setting of weighted Lorentz spaces of operators satisfying a limited Rubio de Francia condition; namely $T$ is bounded on $L^{p}(v)$ for every $v$ in an strictly smaller class of weights than the Muckenhoupt class $A_p$. Important examples will be the Bochner–Riesz operators $BR_\lambda$ with $0<\lambda <{(n-1)}/2$, sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$, $1 < r < \infty$.

1. Introduction

Let

\[ \hat{f}(\xi) = \int_{\mathbb{R}^{n}}f(x){\rm e}^{{-}2\pi i x\cdot \xi}\,{\rm d}x, \quad \xi \in \mathbb{R}^{n}, \]

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

\[ \widehat{BR_\lambda f}(\xi) = \left( 1 - |\xi|^{2} \right)^{\lambda}_{+}\hat{f}(\xi), \quad \xi \in \mathbb{R}^{n}. \]

These operators were first introduced by Bochner in [6] and, since then, they have been widely studied (see [7–9, 14, 22, 27, 40, 42]). The case $\lambda = 0$ corresponds to the so-called disc multiplier, which is unbounded on $L^{p}(\mathbb {R}^{n})$ if $n \geq 2$ and $p \ne 2$ (see [20]).

When $\lambda > {(n - 1)}/2$, it is well known that $BR_\lambda$ is controlled by the Hardy–Littlewood maximal operator $M$, defined for every locally integrable function $f$ (i.e. $f \in L^{1}_{\text {loc}}(\mathbb {R}^{n})$) by

(1.1)\begin{equation} Mf(x) = \sup_{Q \ni x} \frac 1{|Q|} \int_Q |f(y)|\,{\rm d}y, \end{equation}

where the supremum is taken overall cubes $Q \subseteq \mathbb {R}^{n}$ that contains $x \in \mathbb {R}^{n}$. As a consequence, all weighted inequalities for $M$ are also satisfied by $BR_\lambda$.

The value $\lambda = {n - 1}/{2}$ is called the critical index. In this case, Shi and Sun [38] proved that $BR_{{(n-1)}/2}$ is bounded on $L^{p}(v)$ for every $1 < p < \infty$ and $v \in A_p$. The weak-type inequality for $p = 1$ was first settled by Christ [16], who showed that $BR_{{(n-1)}/2}$ is bounded from $L^{1}(\mathbb {R}^{n})$ to $L^{1,\infty }(\mathbb {R}^{n})$, and the corresponding weighted weak-type inequality was obtained by Vargas in [42], where she proved that, indeed, $BR_{{(n-1)}/2}$ is bounded from $L^{1}(v)$ to $L^{1,\infty }(v)$ for every $v \in A_1$.

Below the critical index, that is $0 < \lambda < {(n-1)}/2$, $BR_\lambda$ is not bounded on $L^{p}({{\mathbb {R}^{n}}})$ for the whole range $1 < p < \infty$. For instance, in dimension $n=2$, Carleson and Sjölin [8] proved that $BR_\lambda$ is bounded on $L^{p}({{\mathbb {R}^{n}}})$ if and only if $p > 1$ and

\[ \lambda > \max\left( {2\left|\frac{1}{p} - \frac{1}{2} \right| - \frac{1}{2} , 0 } \right), \]

or equivalently, $0 < \lambda < \frac 12$ and

\[ \frac 4{3 + 2\lambda} < p < \frac 4{1 -2\lambda}. \]

Moreover, Seeger [37] showed that the corresponding weak-type inequality at the endpoint $BR_\lambda : L^{4/{(3 + 2\lambda )}}({{\mathbb {R}^{n}}}) \rightarrow L^{4/{(3 + 2\lambda )},\, \infty }({{\mathbb {R}^{n}}})$ also holds.

For higher dimensions, it is already well known that $BR_\lambda$ is not bounded on $L^{p}(\mathbb {R}^{n})$ for $p \leq {2n}/{(n + 1 + 2\lambda )}$ or $p \geq {2n}/{(n - 1 - 2\lambda )}$ (see for instance [18, theorem 8.15] or [22, proposition 5.2.3]). Furthermore, it was conjectured the following:

Conjecture 1.1 (Bochner–Riesz conjecture)

$BR_\lambda$ is bounded on $L^{p}(\mathbb {R}^{n})$ if $p > 1$ and

\[ \lambda >\lambda(n,p) = \max\left( {n\left|\frac{1}{p} - \frac{1}{2}\right| - \frac{1}{2} , 0 } \right), \]

or equivalently, $0 < \lambda < {(n-1)}/2$ and

\[ \frac{2n}{n + 1 + 2\lambda} < p < \frac{2n}{n - 1 - 2\lambda}. \]

This has an equivalent formulation (see [29]): let $\chi _{[-1/4, 1/4]} \leq \phi \leq \chi _{[-1/2,1/2]}$ be a Schwartz function, and set $S_\tau$, $\tau > 0$, to be the Fourier multiplier having as a symbol $\phi ((|\xi | - 1)/\tau )$, $\xi \in \mathbb {R}^{n}$.

Conjecture 1.2 For $p > 1$ such that $n\big |\frac 1p - \frac 12 \big | < \frac 12$,

\[ S_\tau: L^{p}({{\mathbb{R}^{n}}}) \rightarrow L^{p}({{\mathbb{R}^{n}}}), \quad 0 < \tau < 1, \]

with constant controlled by $C_\varepsilon \tau ^{-\varepsilon }$ for all $0 < \varepsilon < 1$.

However, the Bochner–Riesz conjecture only has been partially answered and the best results known are currently due to Guo, Oh, Wang, Wu and Zhang (see [23]) where the authors used polynomial partitioning. Before them, the best known results followed from the work of [24] (see also [43]). We thank the referee for this information.

Further, under the condition ${(n - 1)}/{(2(n + 1))} < \lambda < {(n-1)}/2$, the corresponding weak-type inequality at the endpoint $BR_\lambda : L^{{2n}/{(n + 1 + 2\lambda )}} \rightarrow L^{{2n}/{(n + 1 + 2\lambda )},\, \infty }$ was settled by Christ [15] and extended to $\lambda = {(n - 1)}/{(2(n + 1))}$ by Tao [40], while it remains unknown for the range $0 < \lambda < {(n - 1)}/{(2(n + 1))}$. This is the so-called endpoint Bochner–Riesz conjecture (see [41]) and we observe that if it holds for some $\lambda$, then by duality and interpolation the Bochner–Riesz conjecture holds for such $\lambda$ as well.

In two recent papers [27, 29], new weighted estimates for $BR_\lambda$ have been proved using the fact that the Bochner–Riesz operators can be dominated by sparse type operators. As far as we know, these are the best weighted estimates known together with the results for the $(2,2)$-strong type inequality in [9, 14]. We summarize them here in a suitable way for the following main results of this paper. To do so, first we need the following notion.

Definition 1.3 Given $0 \leq \alpha,\, \beta \leq 1$ and $1 \leq p < \infty$, let us define the class of weights

\[ A_{p;(\alpha, \beta)} = \{0< v\in L^{1}_{\text{\rm loc}}(\mathbb{R}^{n}) : v = v_0^{\alpha} v_1^{\beta(1-p)}, v_j \in A_1\}, \]

where

(1.2)\begin{equation} v \in A_1 \quad \iff \quad ||v||_{A_1}= \left\Vert \frac{Mv}v\right\Vert_\infty<\infty. \end{equation}

Remark 1.4 $A_{p;(\alpha,\beta )} \subseteq A_p$ (see for instance [9]) where $A_p$ is the Muckenhoupt class of weights (see (2.1)). Besides, we observe that

\[ v\in A_{p;(\alpha, \beta)} \quad\implies\quad v\in A_{1+\beta(p-1)}. \]

Let us start stating the result for dimension $n=2$ where the conjecture has completely been solved.

Proposition 1.5 Let $n=2$ and $0<\lambda <\frac 12$.

1. (i) [9, 29] For every $\frac 43 \leq q_0 \leq 4$,

   \[ BR_\lambda: L^{q_0}(v) \longrightarrow L^{q_0}(v), \quad \forall v \in A_{q_0;\left( { {2\lambda}, {2\lambda}} \right)}. \]

2. (ii) [27, corollary 1.3]

   \begin{align*} & BR_\lambda: L^{4/{(3 + 2\lambda)}}\left(v\right) \rightarrow L^{4/{(3 + 2\lambda)}, \infty}\left(v\right), \quad \forall v\nonumber\\ & \quad \in A_{4/{(3 + 2\lambda)};\left( {{2\lambda}/{(3 + 2\lambda)}, 0 } \right)} \hspace{2mm} (\text{i.e. } v^{{(3 + 2\lambda)}/{2\lambda}} \in A_1). \end{align*}

3. (iii) In particular, in this case, the operator norm is bounded by $c(n,\,\lambda ) \left \lVert {v}\right \rVert ^{{(7 + 4\lambda )}/4}_{A_1}$.

Further, for $n > 2$ we have the following result.

Proposition 1.6 Let $0 < \lambda < {(n-1)}/2$.

1. (i) [9, 29] If $q_0 = 2$ or $q_0$ is such that ${2n}/{(n+1)} < q_0 < 2$ and conjecture 1.2 holds with $p = q_0,$

   \[ BR_\lambda: L^{q_0}(v) \longrightarrow L^{q_0}(v), \quad \forall v \in A_{q_0;\left( {{2\lambda}/{(n-1)},{2\lambda}/{(n-1)}} \right)}. \]

2. (ii) And, by duality,

   \[ BR_\lambda: L^{q_0'}(v) \longrightarrow L^{q_0'}(v), \quad \forall v \in A_{q_0';\left( {{2\lambda}/{(n-1)},{2\lambda}/{(n-1)}} \right)}. \]

3. (iii) [14] If ${(n-1)}/{(2(n+1))} < \lambda < {(n-1)}/2,$ then

   \[ BR_\lambda: L^{2}(v) \longrightarrow L^{2}(v), \quad \forall v \in A_{2;\left( {{(1 + 2\lambda)}/n,0} \right)}\cup A_{2; \left( {0, {(1 + 2\lambda)}/n} \right)}. \]

Now, set $\mathcal {M}(\mathbb {R}^{n})$ to be the set of all measurable functions on $\mathbb {R}^{n}$ and let us consider for a positive and locally integrable function $w$ in $\mathbb {R}^{+}$ (i.e. $w\in L^{1}_{\text {loc}}(\mathbb {R}^{+})$) the primitive $W(t)=\int _0^{t} w(r){\rm d}r$. Given a sublinear operator $T$ defined over the function space

\[ \Lambda^{1}(w) = \left\{ f \in \mathcal{M}(\mathbb{R}^{n}) : ||f||_{\Lambda^{1}(w)} = \int_0^{\infty}f^{*}(t)w(t)\,{\rm d}t < \infty \right\}, \]

we define the class

\[ \mathcal{W}(T)=\Big\{0 \leq w\in L^{1}_{\text{loc}}(\mathbb{R}^{+}): \left\lVert{w}\right\rVert_{\mathcal{W}(T)} = \sup_{f \in \Lambda^{1}(w)}\dfrac{\sup_{y > 0}\, y\, W(\mu_{Tf}(y))}{\int_0^{\infty} f^{*}(t) w(t) {\rm d}t} < \infty\Big\}, \]

where $\mu _f$ and $f^{*}$ are, respectively, the distribution function and the decreasing rearrangement of $f \in \mathcal {M}(\mathbb {R}^{n})$ defined by

(1.3)\begin{equation} \mu_f(t):= |\{|f|>t\}|, \quad f ^{*}(t):=\inf\{y>0:\mu_f(y)\leq t\}, \quad t > 0. \end{equation}

Then, one can immediately see that

\[ w(t)\!=\!t^{{( 1 - n+2\lambda)}/{2n}}\in \mathcal{W}(BR_\lambda)\quad \!\!\implies\!\!\quad \text{ Bochner-{-}Riesz conjecture holds for } BR_\lambda. \]

The original motivation of this paper was to study the class $\mathcal {W}(T)$ not only for the Bochner–Riesz operators but also for other interesting operators in harmonic analysis having in common some properties connected with weighted estimates; namely, for some $1 \leq p < \infty$,

(1.4)\begin{equation} T: L^{p}(v) \longrightarrow L^{p,\infty}(v), \end{equation}

for every positive and locally integrable function $v$ in $\mathbb {R}^{n}$ belonging in $A_{p;(\alpha,\beta )}$ for some $0 \leq \alpha,\,\beta \leq 1$. In particular, and as a consequence of our main results, we will obtain boundedness on weighted Lorentz spaces $\Lambda ^{p}(w)$, $0 < p < \infty$, defined by

\[ \Lambda^{p}(w) = \left\{ f \in \mathcal{M}(\mathbb{R}^{n}) : ||f||_{\Lambda^{p}(w)} = \left(\int_0^{\infty}f^{*}(t)^{p}w(t)\,{\rm d}t \right)^{ 1/p} < \infty \right\}. \]

Weighted type inequalities as in (1.4) are also satisfied by other operators such as the Hörmander Fourier multipliers $m\in M(s,\, l)$ with $l< n$ (see [28]) among many others. And this is the starting point of this paper. We want to use the information about weighted estimates on Lebesgue spaces to study the class $\mathcal {W}(T)$ and, as a consequence, to obtain boundedness on weighted Lorentz spaces. To this end, we need to adapt the ideas in [4] where the case $\alpha =\beta =1$ was studied and applications to the case $\lambda ={(n-1)}/2$ (for the Bochner–Riesz operator) and the case $l = n$ (for the Hörmander multiplier) were considered. In particular, the following result was proved: if $T$ satisfies that $T:L^{1}(v) \rightarrow L^{1, \infty }(v)$, for every $v \in A_1$, with constant less than or equal to $\varphi (||{v}||_{A_1})$, with $\varphi$ an increasing function on $(0,\, \infty )$, then (see (2.3) for the definition of the class $B_p^{{\mathcal{R}}}$, and definition 2.2 for $B^{*}_\infty$)

(1.5)\begin{equation} B_{1}^{\mathcal{R}}\cap B_{\infty}^{*}\subset \mathcal{W}(T)\quad \text{ with }\quad ||w||_{\mathcal{W}(T)}\leq C_1 ||w||_{B_{1}^{\mathcal{R}}}\varphi\left(C_2 ||w||_{B_{\infty}^{*}} \right) \end{equation}

for some universal constants $C_1,\, C_2>0$.

We write $A \lesssim B$, where $A$, $B$ are nonnegative quantities, if there exists a positive constant $C>0$, independent of $A$ and $B$, such that $A\leq C B$. If $A\lesssim B$ and $B\lesssim A$, then we write $A\approx B$.

Definition 1.7 We say that $f$ is quasi-decreasing, and we denote it by $f \approx \downarrow$, if $f(t) \lesssim f(s)$, $0< s\le t$. If $-f$ is quasi-decreasing we say that $f$ is quasi-increasing and we denote it by $f \approx \uparrow$.

Hence, the inclusion in (1.5) can be reinterpreted (see (2.3) and (2.7)) to the existence of some $\varepsilon > 0$ so that if

\[ \frac{W(t)}t \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. This motivates the next theorems that allows us to generalize the results in [4] to operators satisfying (1.4). Indeed, (1.5) turns out to be the case $p_0 = \alpha = 1$ of our second main result (see theorem 1.11). Our first main result is the following (see (2.2) for the definition of the class $B_p$, and definition 2.3 for $B^{*}_p$):

Theorem 1.8 Let $1 < p_0 < \infty,$ $0 \leq \alpha \leq 1,$ $0 < \beta \leq 1$ and let $T$ be a sublinear operator such that

\[ T:L^{p_0}(v)\longrightarrow L^{p_0,\infty}(v), \quad \forall v \in A_{p_0;(\alpha, \beta)}, \]

with constant less than or equal to $\varphi (||{v}||_{A_{1+\beta (p_0-1)}}),$ with $\varphi$ an increasing function on $(0,\, \infty )$. Set

\[ p_{+} = \frac{p_0}{1 - \alpha}, \hspace{7mm} p_{-}' = \frac{p_0'}{1-\beta}. \]

Then, $B_{{1}/{p_-}}\cap B_{ {p_+}}^{*} \subset \mathcal {W}(T),$ or equivalently, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{1/{p_-} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{1/{p_+}+\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. Moreover, it holds that $\left \lVert {w}\right \rVert _{\mathcal {W}(T)} \leq \Phi (\left \lVert {w}\right \rVert _{B_{{1}/{p_-}}},\, \left \lVert {w}\right \rVert _{B_{ {p_+}}^{*}}),$ where $\Phi$ is an increasing function in each variable depending on $\varphi$.

By taking $T=M\circ M$ with $M$ the Hardy–Littlewood maximal operator, it is clear that the above result is false with $\varepsilon =0$ since $T$ satisfies the hypothesis of our theorem with $\alpha =1$ and $\beta =1$ but, although $w = 1$ satisfies the hypothesis, $T$ is not of weak type $(1,\,1)$.

Remark 1.9 Recall that a weight $v$ belongs to the reverse Hölder class $RH_q$, $1 < q < \infty$, if for every measurable cube $Q\subset \mathbb {R}^{n}$,

\[ \left(\frac{1}{|Q|}\int_Qv^{q}\right)^{1/q} \lesssim \frac{1}{|Q|}\int_Qv. \]

It is known that $v \in A_p \cap RH_q$ if and only if $v^{q} \in A_{\tilde p}$ with $\tilde p = q(p - 1) + 1$ (see for instance [26, (P6)]) so that when $\alpha > 0$,

\[ A_{p_0;(\alpha, \beta)} = A_{{p_0}/{p_-}} \cap RH_{\left( {{p_+}/{p_0}} \right)'}, \]

which keeps some symmetry with the class of weights $B_{1/{p_-}}\cap B_{p_+}^{*}$.

As a consequence of theorem 1.8, we have the following result:

Corollary 1.10 Let $0 < p < \infty$. Under the hypotheses of theorem 1.8, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{p/{p_-} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{ p/{p_+}+\varepsilon}}\approx\uparrow, \]

we have that

\[ T:\Lambda^{p}(w) \longrightarrow \Lambda^{p}(w). \]

Our next main result reads as follows:

Theorem 1.11 Let $1 \leq p_0 < \infty,$ $0 < \alpha \leq 1,$ and let $T$ be a sublinear operator such that

\[ T:L^{p_0,1}(v^{\alpha}) \longrightarrow L^{p_0,\infty}(v^{\alpha}), \quad \forall v \in A_1, \]

with constant less than or equal to $\varphi ( \left \lVert {v}\right \rVert _{A_1}),$ where $\varphi$ is an increasing function. Then, $B_{1/{p_0}}^{{\mathcal{R}}}\cap B_{{p_0}/{(1-\alpha )}}^{*} \subset \mathcal {W}(T),\,$ or equivalently, if there exists some $\varepsilon > 0$ so that

\[ \frac{W(t)}{t^{1/{p_0}}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{{(1 - \alpha)}/{p_0}+\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T)$. Moreover,

\[ \left\lVert{w}\right\rVert_{\mathcal{W}(T)} \lesssim \left\lVert{w}\right\rVert_{B_{1/{p_0}}^{\mathcal{R}}}^{1/{p_0}} \left\{ \begin{array}{@{}cc} \varphi\left( { C \left\lVert{w}\right\rVert_{B^{*}_{{p_0}/{(1-\alpha)}}}^{{p_0}/\alpha} } \right), & 0 < \alpha < 1, \\ \varphi\left( {C \left\lVert{w}\right\rVert_{B_\infty^{*}}} \right), & \alpha = 1. \end{array}\right. \]

Corollary 1.12 Let $0 < p < \infty$ and let us assume the hypotheses of theorem 1.11.

(a) If

\[ \frac{W(t)}{t^{p/{p_0}}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon > 0 :\,\frac{W(t)}{ t^{{ (p(1-\alpha))}/{p_0}+\varepsilon}}\approx\uparrow, \]

then

\[ T:\Lambda^{p,1}(w) \longrightarrow \Lambda^{p,\infty}(w). \]

(b) If

\[ \exists \varepsilon > 0 : \frac{W(t)}{t^{p/{p_0}-\varepsilon}} \approx\downarrow \quad\mbox{and}\quad \,\frac{W(t)}{ t^{{ (p(1-\alpha))}/{p_0}+\varepsilon}}\approx\uparrow, \]

then

\[ T:\Lambda^{p}(w) \longrightarrow \Lambda^{p}(w). \]

Remark 1.13 Although theorems 1.8 and 1.11 are stated with $T$ being a sublinear operator, which is enough for the applications that we present in § 4, they can be proved without this hypothesis. In fact, we only need this condition when applying interpolation in corollaries 1.10 and 1.12.

The paper is organized as follows. In § 2 we present some definitions, previous results and technical lemmas which shall be used later on, while § 3 contains our main results. The last section (see § 4) will be devoted to obtain estimates in the setting of weighted Lorentz spaces for the Bochner–Riesz operators in the critical range, among others, such as sparse operators, Hörmander multipliers with a limited regularity condition and rough operators with $\Omega \in L^{r}(\Sigma )$.

2. Definitions, previous results and some technical lemmas

Given $0 < p,\, q < \infty$, $w$ a positive locally integrable function defined on $\mathbb {R}^{+}$ and $W(t)=\int _0^{t} w(r)\,{\rm d}r$, the weighted Lorentz space $\Lambda ^{p,q}(w)$ is defined by the condition $||f||_{\Lambda ^{p,q}(w)} < \infty$ where

\[ ||f||_{\Lambda^{p,q}(w)}= \left\{ \begin{array}{@{}cc} \displaystyle \bigg( \int_0^{\infty} f^{*}(t)^{q} W(t)^{q/p -1}w(t)\, {\rm d}t \bigg)^{1/q}, & q < \infty, \\ \displaystyle \sup_{y>0} y\, W(\mu_f(y))^{1/p}, & q = \infty, \end{array} \right. \]

where $\mu _f$ and $f^{*}$ are, respectively, the distribution function and the decreasing rearrangement of $f$ defined in (1.3). Observe that if $w=1$, $\Lambda ^{p,q}(w)$ coincides with the more usual case $L^{p,q}$. We should also emphasize here that for $0 < q < \infty$,

\[ \Lambda^{p,q}(w)=\Lambda^{q}(\widetilde w), \quad \widetilde W(t)\approx W(t)^{q/p} \quad \text{(or } \tilde{w}(t) \approx W(t)^{q/p - 1}w(t)\text{)}, \]

and, similar, $\Lambda ^{p,\infty }(w) = \Lambda ^{1,\infty }(\tilde w)$, with $\tilde W(t) \approx W(t)^{1/p}$ $\text {(or } \tilde {w}(t) \approx W(t)^{1/p - 1}w(t)\text {)}$. Besides, these spaces satisfy the embeddings

\[ \Lambda^{p,q_0}(w)\hookrightarrow \Lambda^{p,q_1}(w) \]

continuously for $0 < q_0 \leq q_1 \leq \infty$. For further information about these notions and related topics see [5, 12].

We observe here that,

\[ w\in \mathcal{W}(T)\quad\iff\quad T:\Lambda^{1}(w)\longrightarrow \Lambda^{1, \infty}(w), \]

and

\[ ||w||_{\mathcal{W}(T)}= ||T||_{\Lambda^{1}(w)\to \Lambda^{1, \infty}(w)} = \sup_{f \in \Lambda^{1}(w)}\frac{\left\lVert{Tf}\right\rVert_{\Lambda^{1,\infty}(w)}}{\left\lVert{f}\right\rVert_{\Lambda^{1}(w)}}. \]

Let us consider the Hardy–Littlewood maximal operator $M$, defined in (1.1). It is known [34] that for every $1< p<\infty$,

\[ M:L^{p}(v) \longrightarrow L^{p}(v) \quad\iff\quad v\in A_p, \]

where

(2.1)\begin{equation} v \in A_p \quad \iff \quad ||v||_{A_p}= \sup_{Q} \left(\frac 1{|Q|} \int_Q v (x) {\rm d}x\right) \left(\frac 1{|Q|} \int_Q v (x) ^{ 1/{(1-p)}}{\rm d}x \right)^{p-1} <\infty. \end{equation}

Moreover, by a weighted norm inequality due to Fefferman and Stein in [21], it was seen that, for $1 \leq p < \infty$, the $A_p$ class also characterizes the weak-type boundedness

\[ M:L^{p}(v) \rightarrow L^{p,\infty}(v) \]

(see (1.2) for the definition of the class $A_1$).

Concerning the boundedness of $M$ on weighted Lorentz spaces we have (see [3]) that for every $p>0$,

\[ M:\Lambda^{p}(w) \longrightarrow \Lambda^{p}(w) \iff w\in B_p, \]

where

(2.2)\begin{equation} ||w||_{B_p}= \sup_{t > 0}\frac{ \int_0^{\infty} \min (1, t/r)^{p} w(r) {\rm d}r }{W(t)}<\infty, \end{equation}

and $\left \lVert {M}\right \rVert _{\Lambda ^{p}(w)} := \left \lVert {M}\right \rVert _{\Lambda ^{p}(w) \rightarrow \Lambda ^{p}(w)}\lesssim \left \lVert {w}\right \rVert _{B_p}$. Also, since for $\tilde {W}(t) \approx W(t)^{1/p}$,

\[ M:\Lambda^{p, 1}(w) \longrightarrow \Lambda^{p, \infty}(w) \quad \iff \quad M: \Lambda^{1}(\tilde{w}) \rightarrow \Lambda^{1,\infty} (\tilde{w}), \]

we have that [10, 13]

\[ M:\Lambda^{p, 1}(w) \longrightarrow \Lambda^{p, \infty}(w) \quad \iff \quad w\in B_p^{\mathcal{R}}, \]

where $w\in B_p^{\mathcal{R}}$ is defined by

(2.3)\begin{equation} ||w||_{B_p^{\mathcal{R}}}=\sup_{0 < s\le t<\infty} \dfrac{sW(t)^{{1/p}}}{tW(s)^{{1/p}}}<\infty. \end{equation}

Further, $||M||_{\Lambda ^{p, 1}(w) \longrightarrow \Lambda ^{p, \infty }(w)}\lesssim ||w||_{B_p^{{\mathcal{R}}}}$ and, for $0 < q < p < \infty$, it is easy to see that $B_q^{\mathcal {R}} \subsetneq B_p \subsetneq B_p^{\mathcal {R}}$ with

\[ \left\lVert{w}\right\rVert_{B_p^{\mathcal{R}}} \lesssim \left\lVert{w}\right\rVert_{B_p}^{{1/p}} \leq \frac p{p-q} \left\lVert{w}\right\rVert_{B_q^{\mathcal{R}}}^{q/p}. \]

Indeed, it is known that [3, 35] for $0 < p < \infty$,

(2.4)\begin{equation} w\in B_p \iff \exists \varepsilon>0 : w \in B_{p - \varepsilon}^{\mathcal{R}} \iff \exists \varepsilon>0 : \frac{W(t)}{t^{p-\varepsilon}} \approx\downarrow. \end{equation}

Finally, we say that $w \in \Delta _2$ if there exists $C > 0$ such that $W(2t) \leq C W(t)$, for all $t > 0$. Clearly, it holds that, for every $0 < p < \infty$, $B_p^{{\mathcal{R}}} \subsetneq \Delta _2$. Besides, for every $0 < p,\,q < \infty$, if $w \in \Delta _2$, then $\Lambda ^{p,q}(w)$ is a quasi-Banach r.i. function space (see [12]).

Let us recall now, that given a quasi-Banach r.i. space $\mathbb {X}$, the associate space $\mathbb {X}'$ is defined by the condition

\[ ||f||_{\mathbb{X}'} = \sup_{||g||_\mathbb{X}\le 1} \int_0^{\infty} f^{*}(t) g^{*}(t) {\rm d}t <\infty. \]

Lemma 2.1 Let $0 < p \leq 1$. If $w\in B_p^{{\mathcal{R}}},$ then for every measurable set $E \subseteq \mathbb {R}^{n},$

\[ ||\chi_E||_{(\Lambda^{p}(w))'} \leq \left\lVert{w}\right\rVert_{B_p^{\mathcal{R}}} \frac{|E|}{W(E)^{{1/p}}}. \]

Proof. It is known [12, theorem 2.4.7] that

\[ ||h||_{\left(\Lambda^{p}(w)\right)' }= \sup_{t>0}\frac{\int_0^{t} h^{*}(r)\,{\rm d}r}{W(t)^{1/p}}, \]

and hence

\[ ||\chi_{E}||_{(\Lambda^{ p}(w))'} =\sup_{t>0}\frac{\min(t,|E|)}{W(t)^{1/p}} \le ||w||_{B_p^{\mathcal{R}}} \frac{|E|} {W(|E|)^{{1/p}}}. \]

To prove our main results we need several estimates on the boundedness of the operators $M_\alpha f = M( f^{1/\alpha })^{\alpha }$, $0 < \alpha \leq 1$, on the corresponding associate spaces $(\Lambda ^{p}(w))'$. To this end, the following classes of weights are going to be fundamental.

Definition 2.2 [36]

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

\[ ||w||_{B^{*}_\infty}=\sup_{t>0} \dfrac{1}{W(t)}\int_0^{t}\frac{W(r)}{r}\,{\rm d}r <\infty. \]

Definition 2.3 [30, 36]

Let $0 < p < \infty$. A weight $w \in B_p^{*}$ if and only if

\[ ||w||_{B_p^{*}} = \sup_{t > 0}\frac{1}{W(t)}\int_0^{t}\left( \frac{t}{r}\right)^{{1/p}}w(r)\,{\rm d}r < \infty. \]

It is known that [30, 33] for every $0 < p < \infty$,

(2.5)\begin{equation} w\in B_p^{*} \iff \exists\varepsilon>0: \frac{W(t)}{t^{{1/p}+\varepsilon} }\approx\uparrow. \end{equation}

Besides, direct from the definition, for $0 < q \leq p < \infty$,

\[ B_{q}^{*} \subseteq B_{p}^{*} \subseteq B^{*}_\infty, \]

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

(2.6)\begin{equation} B_p^{*} = \bigcup_{0 < q < p} B_q^{*}. \end{equation}

Indeed, (2.5) also holds for $p = \infty$ as the following result shows. Consequently, (2.6) is also true for this exponent.

Lemma 2.4

(2.7)\begin{equation} w\in B_\infty^{*} \iff \exists \varepsilon > 0: \frac{W(t)}{t^{\varepsilon}} \approx\uparrow. \end{equation}

Proof. Since $B_p^{*} \subseteq B_\infty ^{*}$, by (2.5), it is enough to see that if $w \in B_\infty ^{*}$, then there exists some $p > 0$ such that $w \in B_p^{*}$.

Let $\overline {W}:(0,\,\infty ) \rightarrow (0,\,\infty )$ be defined by

\[ \overline{W}(t) := \sup\left\{ \frac{W(r)}{W(s)} : 0 < r \leq t s \right\} = \sup_{x \in [0,\infty)}\frac{W(t x)}{W(x)}, \]

where the last equality follows since $W$ is increasing. It was seen [1, lemma 2.6] that $w \in B_\infty ^{*}$ is equivalent to the existence of some $t_0 \in (0,\,1)$ such that $\overline {W}(t_0) < 1$. What's more, arguing similar as in [1, lemma 2.7], it can be seen that $w \in B_p^{*}$ is equivalent to the existence of some $t_1 \in (0,\,1)$ such that $\overline {W}(t_1) < t_1^{{1/p}}$. Therefore, given $w \in B_\infty ^{*}$ and taking $p$ large enough, it holds that $\overline {W}(t_0) < 1$ implies $\overline {W}(t_0) < t_0^{{1/p}}$.

Hence, from lemma 2.4, equivalences (2.4) and (2.5), and the definition of $B_p^{\mathcal {R}}$, it follows immediately that for every $0 < p < \infty$, $0 < q \leq \infty$,

\[ w\in B_{1/p}\cap B_q^{*}\quad \iff\quad \exists \varepsilon>0: \frac{W(t)}{t^{1/p - \varepsilon}}\approx \downarrow \text{ and } \frac{W(t)}{t^{1/q+\varepsilon} }\approx\uparrow, \]

and

\[ w\in B_{{1/p}}^{\mathcal{R}}\cap B_{ q}^{*}\quad\iff\quad \frac{W(t)}{t^{{1/p}}}\approx \downarrow \ \mbox{ and } \ \exists \varepsilon>0: \frac{W(t)}{t^{1/q+\varepsilon} }\approx\uparrow. \]

Example 2.5 (a) Let $0 < p < \infty$ and take $w(t) = t^{\gamma - 1}$, $t > 0$. Then,

\[ w \in B_p \quad \iff \quad \gamma < p\text{ and } w \in B_p^{{\mathcal{R}}} \ \iff\ \gamma\le p \]

with $\left \lVert {w}\right \rVert _{B_p} = \frac p{p - \gamma }$ and $\left \lVert {w}\right \rVert _{B_p^{R}} = 1$. Moreover,

\[ w \in B_p^{*} \quad\iff\quad \gamma > \frac 1p \text{ and } w \in B_\infty^{*} \ \iff\ \gamma > 0 \]

with $\left \lVert {w}\right \rVert _{B_p^{*}} = \frac \gamma {\gamma - 1/p}$ and $\left \lVert {w}\right \rVert _{B_\infty ^{*}} = \frac 1\gamma$.

(b) Let $1 \leq p < q < \infty$ and set, for every $m \in \mathbb {N}$,

\[ w_m(t) = \left( 1 + \log_+\dfrac 1t \right)^{m}t^{1/p - 1}, \quad t > 0. \]

By induction on $m$, it is easy to see that $w_m\in B_p^{\mathcal {R}}\cap B^{*}_q$ with

\[ \left\lVert{w_m}\right\rVert_{B_{1/p}^{\mathcal{R}}} = 1 \quad \text{ and }\quad \left\lVert{w_m}\right\rVert_{B^{*}_q} \lesssim \left[ {\left( {\frac{pq}{q-p}} \right) \left( {\frac{m+1}m} \right)} \right]^{m}(m + 1)!. \]

(c) Similarly, let $1 \leq p < q < \infty$ and set, for every $m \in \mathbb {N}$,

\[ \tilde w_m(t) = \left( 1 + \log_+ t\right)^{{-}m}t^{1/p - 1}, \quad t > 0. \]

Then, $\tilde w_m\in B_p^{\mathcal {R}}\cap B^{*}_q$ with

\[ \left\lVert{\tilde w_m}\right\rVert_{B_{{1/p}}^{\mathcal{R}}} \lesssim 1 \quad \text{ and } \quad \left\lVert{\tilde w_m}\right\rVert_{B^{*}_q} \lesssim \left[\frac{pq}{q-p}\right]^{m} (m+1)^{m+1}. \]

From the above results we are now ready to estimate the boundedness of $M_\alpha$, $0 < \alpha \leq 1$, on the associate spaces $\left (\Lambda ^{p}\left ( {w} \right )\right )'$ for $0 < p \leq 1$. First, the case $\alpha = 1$:

Proposition 2.6 [2, 4]

For every $0 < p < \infty,$

\[ ||M||_{(\Lambda^{p}(w) )'} \lesssim ||w||_{B^{*}_\infty} \quad \text{ and } \quad ||M||_{(\Lambda^{p,1}(w) )'} \lesssim ||w||_{B^{*}_\infty}. \]

For $0 < \alpha < 1$, we need this technical lemma:

Lemma 2.7 Let $0 < p \leq 1,$ $0 < \alpha < 1$ and

\[ V(t) = \inf_{s > 0} \left\{ \max\left( { \frac ts, 1 } \right)W(s)^{1/p} \right\}. \]

If

\[ A_V := \sup_{t>0}\frac{1}{V(t)}\int_{0}^{t}\left( \frac{1}{s} \int_0^{s} \left[ {\frac{V(r)}{r}} \right]^{1/\alpha} \, {\rm d}r \right)^{\alpha}\,{\rm d}s < \infty, \]

then $||M_\alpha f ||_{(\Lambda ^{p}(w))'} \lesssim A_V ||f||_{(\Lambda ^{p}(w))'}$. Moreover, if $M_{\alpha }: \left (\Lambda ^{p}\left ( {w} \right )\right )'\rightarrow \left (\Lambda ^{p}\left ( {w} \right )\right )',$ and $w \in \Delta _2,$ then $A_V < \infty$.

Proof. Since

\[ ||h||_{(\Lambda^{p}(w))'} = \sup_{t > 0}\frac{1}{W(t)^{1/p}}\int_0^{t}h^{*}(r)\,{\rm d}r \]

and $\int _0^{t}h^{*}(r)\,{\rm d}r$ is a concave function, we have that, for every $t > 0$,

\[ \int_0^{t}h^{*}(r)\,{\rm d}r \le ||h||_{(\Lambda^{p}(w))'} W(t)^{1/p} \iff \int_0^{t}h^{*}(r)\,{\rm d}r \le ||h||_{(\Lambda^{p}(w))'} V(t), \]

and hence

\[ ||h||_{(\Lambda^{p}(w))'} = \sup_{t > 0}\frac{1}{V(t)}\int_0^{t}h^{*}(r)\,{\rm d}r = ||h||_{(\Lambda^{p}({\rm d}V))'}. \]

First assume that $A_V < \infty$. Then,

\begin{align*} ||M_\alpha f||_{(\Lambda^{p}(w))'} & =\sup_{t>0}\frac{1}{V(t)}\int_{0}^{t}\left( (M(f^{1/\alpha}))^{*}(r) \right)^{\alpha}\,{\rm d}r \\ & \approx \sup_{t>0}\frac{1}{V(t)}\int_{0}^{t}\left( \frac{1}r\int_0^{r} f^{*}(s)^{1/\alpha} \, {\rm d}s \right)^{\alpha}\,{\rm d}r \\ & \leq \sup_{t>0}\frac{1}{V(t)}\int_{0}^{t}\left( \frac{1}r \int_0^{r} f^{**}(s)^{1/\alpha} \, {\rm d}s \right)^{\alpha}\,{\rm d}r \\ & \leq A_V ||f||_{(\Lambda^{p}({\rm d}V))'} = A_V ||f||_{(\Lambda^{p}(w))'}. \end{align*}

Conversely, the first observation is that since $Mf\le M(f^{1/\alpha } )^{\alpha }$, the boundedness of $M_\alpha$ implies the boundedness of $M$ and hence, since $w \in \Delta _2$ (so that $(\Lambda ^{p}(w))'$ is a Banach space [12]) we already know that the derivative of $V$ must satisfy ${\rm d}V \in B^{*}_\infty$ (see, for instance, [4]).

Now, since $V$ is concave, the function $\displaystyle f(r) = {V(r)}/{r}$ is decreasing and hence, if we take $g(x) = f(C_n|x|^{n})$, $x \in \mathbb {R}^{n}$, where $C_n$ is the volume of the unit sphere in $\mathbb {R}^{n}$, we have that $g^{*} = f$ and thus

\begin{align*} A_V & =\sup_{t>0}\frac{1}{V(t)}\int_{0}^{t}\left( \frac{1}{s} \int_0^{s} \left( \frac{V(r)}{r}\right)^{1/\alpha} \, {\rm d}r \right)^{\alpha}\,{\rm d}s \approx \left\lVert{\left( M(g^{1/\alpha})\right)^{\alpha}}\right\rVert_{(\Lambda^{p}(w))'} \\ & \lesssim||g||_{(\Lambda^{p}(w))'} = \sup_{t>0}\frac{1}{V(t)}\int_{0}^{t} \frac{V(r)}{r}\, {\rm d}r \leq ||{\rm d}V||_{B^{*}_\infty} < \infty. \end{align*}

Lemma 2.8 Let $0 < p \leq 1$ and $0 < \alpha < 1$. If $w \in B_{{1}/{((1 - \alpha )p)}}^{*},$ then

\[ ||M_\alpha f ||_{(\Lambda^{p}(w))'} \lesssim \dfrac{\alpha}{(1 - \alpha)^{{(p + 1)}/{p}}}\left\lVert{w}\right\rVert_{B_{{1}/{((1 - \alpha)p)}}^{*}}^{{1/p}} ||f||_{(\Lambda^{p}(w))'}. \]

Proof. Observe that, by lemma 2.7, it is enough to see that

\[ A_V \lesssim \dfrac{\alpha}{(1 - \alpha)^{{(p + 1)}/{p}}}\left\lVert{w}\right\rVert_{B_{{1}/{)(1 - \alpha)p)}}^{*}}^{{1/p}}, \]

and we have seen in the proof of lemma 2.7 that, in particular,

\[ A_V \approx \sup_{t > 0}\frac{1}{W(t)^{1/p}}\int_0^{t}\left( {\frac 1s \int_0^{s} \left[ {\frac{V(r)}{r}} \right]^{1/\alpha}\,{\rm d}r } \right)^{\alpha}\,{\rm d}s. \]

Now take $s > 0$. By the Minkowski's inequality applied with the exponent ${1}/{\alpha } > 1$,

\begin{align*} \left(\frac{1}s \int_{0}^{s}\left[ {\frac{V(r)}r} \right]^{1/\alpha}\,{\rm d}r\right)^{\alpha} & = \left(\frac{1}s\int_{0}^{s}\left( { \int_0^{{V(r)}/r}\,{\rm d}x } \right)^{1/\alpha}\,{\rm d}r\right)^{\alpha} \\ & \leq \frac{1}{s^{\alpha}}\int_{0}^{\infty}\left( { \int_0^{\infty} \chi_{(0,s)}(r)\chi_{{0, {V(r)}/r}}(x)\,{\rm d}r } \right)^{\alpha}\,{\rm d}x \\ & = \frac{1}{s^{\alpha}}\int_{0}^{\infty} \mu_{\chi_{(0,s)}\frac V{\boldsymbol{\cdot}}}^{\alpha}(x) \,{\rm d}x = \frac{\alpha}{s^{\alpha}}\int_{0}^{s} \frac{V(r)}r \,\frac{{\rm d}r}{r^{1 - \alpha }}. \end{align*}

Therefore, using that $V(t) \leq W(t)^{1/p}$ and the Minkowski's inequality again with exponent $1/ p \geq 1$, we obtain that, for every $t > 0$,

\begin{align*} & \frac{1}{W(t)^{1/p}}\int_{0}^{t}\left( \frac{1}{s} \int_0^{s} \left[ {\frac{V(r)}{r}} \right]^{1/\alpha} \, {\rm d}r \right)^{\alpha}\,{\rm d}s \leq \frac{1}{W(t)^{1/p}}\int_{0}^{t}\frac{\alpha}{s^{\alpha}}\int_{0}^{s} \frac{W(r)^{1/p}}{r} \,\frac{{\rm d}r}{r^{1 - \alpha }}\,{\rm d}s \\ & \quad\leq \frac{\alpha}{1 - \alpha} \left( { \frac{t^{1 - \alpha}}{W(t)^{1/p}} \int_0^{t} \frac{W(r)^{1/p}}{r}\,\frac{{\rm d}r}{r^{1 - \alpha}} } \right) \leq \left( {\frac{1}{(1 - \alpha)W(t)} \int_0^{t} {\frac{t}{r}}^{(1 - \alpha)p} w(r)\,{\rm d}r } \right)^{{1/p}} \\ & \quad\leq \frac 1{(1 - \alpha)^{{1/p}}} \left\lVert{w}\right\rVert_{B_{{1}/{((1 - \alpha)p)}}^{*}}^{{1/p}}, \end{align*}

and the result follows.

Lemma 2.9 Let $0 < p \leq 1$ and $0 < \alpha < 1$. If $w \in B_{\frac 1{(1 - \alpha )p}}^{*},$ then

\[ \left\lVert{M_{\alpha}f}\right\rVert_{\left(\Lambda^{p,1}\left( {w} \right)\right)'} \lesssim \dfrac \alpha{(1 - \alpha)^{{(1 + 3p)}/p}} \left\lVert{w}\right\rVert_{B^{*}_{{1}/{((1 - \alpha)p)}}}^{1/ p} \left\lVert{f}\right\rVert_{\left(\Lambda^{p,1}\left( {w} \right)\right)'}. \]

Proof. Recall that $\Lambda ^{p,\,1}\left ( {w} \right ) = \Lambda^{1}\left( \tilde {w} \right)$ with $\tilde {w} = W^{1/p - 1}w$. Therefore, by lemma 2.8

\[ \left\lVert{M_{\alpha}}\right\rVert_{\left(\Lambda^{p,1}\left( {w} \right)\right)'} = \left\lVert{M_{\alpha}}\right\rVert_{\left(\Lambda^{1}\left( \tilde{w} \right)\right)'} \leq \frac{\alpha}{(1 - \alpha)^{2}} \left\lVert{\tilde{w}}\right\rVert_{B^{*}_{{1}/{(1 - \alpha)}}}. \]

Now, integrating by parts and using the Minkowski's inequality we obtain that, for every $t > 0$,

\begin{align*} \int_0^{t} \left( {\frac{t}{r}} \right)^{1 - \alpha}\tilde{w}(r)\,{\rm d}r & = \int_0^{t} \left( {\frac{t}{r}} \right)^{1 - \alpha}W(r)^{1/p-1}w(r)\,{\rm d}r \\ & \leq pW(t)^{1/p} + p\frac{t^{1 - \alpha}}{1 - \alpha}\int_0^{t}\frac{W(r)^{1/p}}{r^{1 - \alpha}}\,\frac{{\rm d}r}{r} \\ & \leq pW(t)^{1/p} +\frac{p}{(1 - \alpha)^{1+{1/p}}} \left( { \int_0^{t} {\frac{t}{r}}^{(1 - \alpha)p} w(r)\,{\rm d}r } \right)^{{1/p}} \\ & \leq pW(t)^{1/p} + \frac{p}{(1 - \alpha)^{1+{1/p}}} \left\lVert{w}\right\rVert_{B^{*}_{1/{((1 - \alpha)p)}}}^{{1/p}}W(t)^{1/p}, \end{align*}

and the result follows taking the supremum on $t$.

To end this section, we present an extrapolation theorem that we shall need for our purposes (see [9, theorem 2.7] and [19, theorem 7.1]).

Definition 2.10 Given $1 \leq p_0 < \infty$ and $0 \leq \alpha,\, \beta \leq 1$, set

\[ p_{+} = \frac{p_0}{1 - \alpha}, \quad p_{-}' = \frac{p_0'}{1-\beta}, \]

and, for every $p \in (p_{-},\, p_{+})$, take $0 \leq \alpha (p),\, \beta (p) \leq 1$ such that

\[ p_{+} = \frac{p}{1 - \alpha(p)}, \quad p_{-}' = \frac{p'}{1-\beta(p)}. \]

Theorem 2.11 Assume that for some family of pairs of functions $(f,\,g),$ for some $1 \leq p_0 < \infty$ and $0 \leq \alpha,\, \beta \leq 1,$ $\alpha$ and $\beta$ not both identically zero, and for all $v \in A_{p_0;(\alpha, \beta )}$

\[ ||g||_{L^{p_0,\infty}(v)} \leq \varphi(||v||_{A_{p_0}}) ||f||_{L^{p_0}(v)}, \]

where $\varphi$ is an increasing function in $(0,\,\infty )$. Then, for every $p_- < q < p_0$ and for every $v \in A_{q;(\alpha (q), \beta (q))}$,

(2.8)\begin{equation} ||g||_{L^{q,\infty}(v)} \lesssim \varphi\left(C \left\lVert v^{{1}/{\alpha(q)}}\right\rVert_{A_{r_q}}^{\frac{\alpha(p)(p_0 - p_-)}{(q - p_-)}} \right) ||f||_{L^{q}(v)} \end{equation}

with $r_q = 1 + \frac {\beta (q)}{\alpha (q)}(q-1)$.

3. Proof of our main results

Proof of theorem 1.8. Assume first that $\alpha < 1$ (i.e. $p_+ < \infty$) and let $w \in B_{1/{p_-}} \cap B_{p_+}^{*}$. We want to see that $w \in \mathcal {W}(T)$, or equivalently,

\[ ||Tf||_{\Lambda^{1,\infty}(w)} \lesssim ||f||_{\Lambda^{1}(w)}. \]

Since we have that $w \in B_{1/{p_-} }$, we can define

\[ R_{1/{p_-}}f(x)^{p_-} := R\left(f^{p_-}\right)(x) = \sum_{k = 0}^{\infty}\frac{M^{k}\left(f^{p_-}\right)(x)}{\left(2||M||_{\Lambda^{1/{p_-} }(w)}\right)^{k}}. \]

Then,

1. (i) $f(x)\leq R_{1/{p_-}}f(x)$ a.e. $x \in {{\mathbb {R}^{n}}}$,

2. (ii) $R(f^{p_-}) \in A_1$ with $\left \lVert {R(f^{p_-})}\right \rVert _{A_1} \leq 2\left \lVert {M}\right \rVert _{\Lambda ^ {1/{p_-}}(w)}$,

3. (iii) and, using the definition of $B_{1/{p_-}}$ together with that $\Lambda ^{ 1/{p_-}} (w) \subset \Lambda ^{1/{p_-},\,\infty }(w)$ and $\Lambda ^{1/{p_-},\,\infty }(w)$ is a Banach space (see [39]),

   \[ ||R_{1/{p_-}} f||_{\Lambda^{1,\infty}(w)} \leq \left(\sum_{k = 0}^{\infty} \frac{ || M^{k} (f^{p_-}) ||_{\Lambda^{1/{p_-},\infty }(w)} }{\left( 2||M||_{\Lambda^{1/{p_-} }(w)} \right)^{k} } \right)^{1/{p_-}} \leq 2^{1/{p_-}} ||f||_{\Lambda^{1}(w)}. \]

Moreover, using (2.6) we have that

\[ B_{ {p_+}}^{*} = B_{{p_-}/{(1 - \alpha(p_-))}}^{*} = \bigcup_{p_-{<} q < \, p_0}B_{{p_-}/{(1 - \alpha(q))}}^{*}, \]

with

\[ \alpha(q) = 1 - \frac{q}{p_+} \text{ and }\alpha(p_-) = 1 - \frac{p_-}{p_+}, \]

and hence, we have that there exists some $p_- < q < p_0 \leq p_+$ such that $w \in B_{{p_-}/{(1 - \alpha (q))}}^{*}$ and $0 < \alpha (q) < 1$, $0 < \beta (q) \leq 1$. Besides, by theorem 2.11, we obtain that (2.8) holds with $g = Tf$. Hence, bylemma 2.8, since $w \in B_{{p_-}/{(1-\alpha (q)) }}^{*}$, we can define, for an arbitrary nonnegative function $h \in (\Lambda ^{1/{p_-} }(w))'$,

(3.1)\begin{equation} S_{\alpha(q)}h(x)^{1/{\alpha(q)}} := S\left(h^{1/{\alpha(q)}}\right)(x) = \sum_{k = 0}^{\infty}\frac{M^{k}(h^{1/{\alpha(q)}})(x)}{\left(2|| M_{\alpha(q)}||_{\left(\Lambda^{1/{p_-} }(w)\right)'}\right)^{k/\alpha(q)}}, \end{equation}

so that

1. (i) $h(x) \leq S_{\alpha (q)}h(x)$ a.e. $x \in {{\mathbb {R}^{n}}}$,

2. (ii) $S(h^{1/{\alpha (q)}}) \in A_1$ with $\left \lVert {S(h^{1/{\alpha (q)}})}\right \rVert _{A_1}^{\alpha (q)} \leq 2\left \lVert {M_{\alpha (q)}}\right \rVert _{(\Lambda ^{1/{p_-}}(w))'}$,

3. (iii) and, since $(\Lambda ^{1/{p_-}}(w))'$ is a Banach space (see [12]),

   \[ \lVert S_{\alpha(q)}h\rVert_{\left(\Lambda^{1/{p_-} }(w)\right)'} \leq \sum_{k = 0}^{\infty} \frac{ \left\lVert{ M_{\alpha(q)}^{k} h }\right\rVert_{\left(\Lambda^{1/{p_-}}\left( w \right)\right)' }}{\left( 2||M_{\alpha(q)}||_{\left(\Lambda^{1/{p_-}}\left( w \right)\right)'} \right)^{k} } \leq 2||h||_{\left(\Lambda^{1/{p_-} }(w)\right)'}. \]

Let $y > 0$ and observe that since $q - p_- = \beta (q)(q - 1)p_-$,

\[ v = R_{1/{p_-}}f(x)^{p_-{-} q} S_{\alpha(q)}\left( \chi_{\{|Tf| > y\}} \right) = \left[R\left( f^{p_-} \right) \right]^{\beta(q)(1 - q)}\left[ S\left( \chi_{\{|Tf| > y\}} \right) \right]^{\alpha(q)}, \]

and $v \in A_{q; (\alpha (q), \beta (q))}$. Hence, for every $\gamma > 0$ and since we know that (2.8) holds with $g = Tf$,

\begin{align*} \mu_{Tf}(y) & \leq \mu_{R_{1/{p_-}} f}(\gamma y) + \int_{\left\{ |Tf| > y, R_{1/{p_-}}f \leq \gamma y \right\}}\,{\rm d}x \\ & \leq\mu_{R_{1/{p_-}} f}(\gamma y) + \frac{(\gamma y)^{q}}{(\gamma y)^{p_-}}\int_{\left\{ |Tf| > y \right\}}R_{1/{p_-}}f(x)^{p_-{-} q} S_{\alpha(q)}\left( \chi_{\{|Tf|> y\}} \right)(x)\,{\rm d}x \\ & \lesssim\mu_{R_{1/{p_-}} f}(\gamma y) + \frac{\left[\gamma\varphi\left(C \left\lVert v^{{1}/{\alpha(q)}}\right\rVert_{A_{r_q}}^{\alpha(q) \left( {\frac{p_0 - p_-}{q - p_-}} \right)} \right)\right]^{q}}{(\gamma y)^{p_-}}\nonumber\\ & \quad \int_{\mathbb{R}^{n}}f(x)^{p_-} S_{\alpha(q)}\left( \chi_{\{Tf > y\}} \right)(x)\,{\rm d}x. \end{align*}

Now, by the Hölder's inequality,

\begin{align*} \int_{\mathbb{R}^{n}}f(x)^{p_-} S_{\alpha(q)}\left( \chi_{\{|Tf| > y\}} \right)(x)\,{\rm d}x & \leq ||f||_{\Lambda^{1}(w)}^{p_-}||S_{\alpha(q)}\left( \chi_{\{|Tf| > y\}}\right)||_{\left(\Lambda^{1/{p_-} }(w)\right)'} \\ & \lesssim||f||_{\Lambda^{1}(w)}^{p_-} ||\chi_{\{|Tf| > y\}}||_{\left(\Lambda^{1/{p_-} }(w)\right)'} \\ & \lesssim ||f||_{\Lambda^{1}(w)}^{p_-} ||w||_{B_{1/{p_-} }}^{p_-} \frac{\mu_{Tf}(y)}{W(\mu_{Tf}(y))^{p_-}}, \end{align*}

where in the last estimate we have used lemma 2.1. Therefore, by the properties of $R_{{1}/{p_-}}$,

\begin{align*} & yW (\mu_{Tf}(y))\nonumber\\ & \quad\lesssim \max \left( \frac 1\gamma , \gamma^{{q}/{p_-} - 1}||w||_{B_{{1}/{p_-}}}\varphi\left(C \left\lVert v^{{1}/{\alpha(q)}}\right\rVert_{A_{r_q}}^{\alpha(q) \left( {\frac{p_0 - p_-}{q - p_-}} \right)} \right)^{q/p_-} \right) ||f||_{\Lambda^{1}(w)}. \end{align*}

Thus, taking the supremum in $y > 0$ and the infimum in $\gamma > 0$,

\[ ||Tf||_{\Lambda^{1,\infty}(w)} \lesssim ||w||_{ B_{ 1/ {p_-} } }^{ {p_-}/q} \varphi\left(C \left\lVert v^{{1}/{\alpha(q)}}\right\rVert_{A_{r_q}}^{\alpha(q) \left( {\frac{p_0 - p_-}{q - p_-}} \right)} \right) ||f||_{\Lambda^{1}(w)}. \]

Finally, since

\[ v = (S_{\alpha(q)}h) \left(R_{1/{p_-}}f\right)^{p_-{-} q} = \left[S(h^{1/\alpha(q)})R(f^{p_-})^{1 - r_q}\right]^{\alpha(q)}, \]

then (see [19, lemma 2.1])

\begin{align*} \left\lVert{v^{{1}/{\alpha(q)}}}\right\rVert_{A_{r_q}} & \leq \left\lVert{S(h^{1/{\alpha(q)}})}\right\rVert_{A_1}\left\lVert{R(f^{p_-})}\right\rVert_{A_1}^{r_q - 1} \\ & \leq 2^{1/{\alpha(q)} + r_q - 1}\left\lVert{M_{\alpha(q)}}\right\rVert_{\left(\Lambda^{{1}/{p_-}}(w)\right)'}^{1/{\alpha(q)}}\left\lVert{M}\right\rVert_{\Lambda^{{1}/{p_-}}(w)}^{r_q - 1}, \end{align*}

so that

\[ ||Tf||_{\Lambda^{1,\infty}(w)} \leq C_w||f||_{\Lambda^{1}(w)}, \]

with

\[ C_w \lesssim ||w||_{ B_{ 1 /{p_-} } }^{ {p_-}/{q}} \varphi\left(C \left[\frac{ \alpha(q)}{(1 - \alpha(q))^{ {1+p_-}}} \left\lVert{w}\right\rVert_{B^{*}_{{p_-}/{(1 - \alpha(q))}}}^ {p_-}\right]^{\frac{p_0 - p_-}{q - p_-}} \left\lVert{w}\right\rVert_{B_{1/{p_-}}}^{{(p_0 - p_-)}/{p_-}} \right). \]

If $\alpha = 1$, then $p_+ = \infty$ and $w \in B_{1/{p_-}} \cap B^{*}_\infty$. So, arguing identically as before but with $q = p_0$ and $\alpha (q) = 1$, in addition to use proposition 2.6 instead of lemma 2.8 to define (3.1), we get that

\[ ||Tf||_{\Lambda^{1,\infty}(w)} \leq C_w||f||_{\Lambda^{1}(w)}, \]

with

\[ C_w \lesssim ||w||_{ B_{ 1/ {p_-} } }^{{p_-}/{p_0}} \varphi\left( \left\lVert{w}\right\rVert_{B^{*}_{\infty}} \left\lVert{w}\right\rVert_{B_{1/{p_-}}}^{ {(p_0 - p_-)}/{p_-}} \right). \]

Proof of corollary 1.10. Let $\tilde {w}:=W^{1/p-1}w$. Hence, observe that

\begin{align*} w\in B_{p/{p_-}} \cap B_{{p_+}/{p}}^{*}& \Longleftrightarrow \exists \varepsilon > 0: \, \frac{W(t)}{t^{p/{p_-} - \varepsilon}} \approx\downarrow \text{ and } \frac{W(t)}{ t^{p/{p_+}+\varepsilon}}\approx\uparrow \\ & \Longleftrightarrow \quad \tilde{w} \in B_{1/{p_-}}\cap B_{ {p_+}}^{*}. \end{align*}

Therefore, if $w\in B_{p/{p_-}} \cap B_{{p_+}/{p}}^{*}$, by theorem 1.8, we have that

\[ T:\Lambda^{1}(\tilde{w}) \longrightarrow \Lambda^{1, \infty}(\tilde{w}), \]

and hence

\[ T:\Lambda^{p,1}(w) \longrightarrow \Lambda^{p, \infty}(w). \]

Now if $w\in B_{p/{p_-}} \cap B_{{p_+}/{p}}^{*}$, there exists some $\delta > 0$ such that $w\in B_{{(p- \delta )}/{p_-}} \cap B_{{p_+}/{(p - \delta )}}^{*}$ and $w\in B_{{(p+\delta )}/{p_-}} \cap B_{{p_+}/{(p+\delta )}}^{*}$. Hence,

\[ T:\Lambda^{p - \delta,1}(w) \rightarrow \Lambda^{p - \delta,\infty}(w) \text{ and }\quad T:\Lambda^{p + \delta,1}(w) \rightarrow \Lambda^{p + \delta,\infty}(w), \]

so that the result follows by interpolation on Lorentz spaces [12, theorem 2.6.5].

Proof of theorem 1.11. The proof essentially follows the same steps as the proof of theorem 1.8 with few modifications. For the sake of completeness and the convenience of the reader, we have added a proof of it here.

Let $w \in B^{\mathcal {R}}_{1/{p_0}} \cap B_{{p_0}/{(1 - \alpha )}}^{*}$. First, by lemma 2.9 (when $0 < \alpha < 1$) and proposition 2.6 (when $\alpha = 1$) we can define, for an arbitrary nonnegative function $h \in (\Lambda ^{1/{p_0},\, 1}(w))'$,

\[ R_{\alpha}h(x)^{1/\alpha} := R\left(h^{1/\alpha}\right)(x) = \sum_{k = 0}^{\infty}\frac{M^{k}(h^{1/\alpha})(x)}{\left(2\left\lVert{M_{\alpha}}\right\rVert_{\left(\Lambda^{1/{p_0}, 1}(w)\right)'}\right)^{k/\alpha}}. \]

Then,

1. (i) $h(x) \leq R_{\alpha }h(x)$ a.e. $x\in {{\mathbb {R}^{n}}}$,

2. (ii) $R_{\alpha }h \in A_1$ with $\left \lVert {R(h^{1/\alpha })}\right \rVert _{A_1}^{\alpha } \leq 2\left \lVert {M_\alpha }\right \rVert _{(\Lambda ^{1/{p_0},\,1}(w))'}$,

3. (iii) and, since $(\Lambda ^{1/{p_0},\, 1}(w))'$ is a Banach space (see [12]),

   (3.2)\begin{equation} \lVert R_{\alpha}h\rVert_{\left(\Lambda^{1/{p_0}, 1}(w)\right)'} \leq 2 \left\lVert{h}\right\rVert_{\left(\Lambda^{1/{p_0}, 1}(w)\right)'}. \end{equation}

Let $y > 0$ and observe that $v = R( \chi _{\{Tf > y\}} ) \in A_1$. Then, we have

\begin{align*} \mu_{Tf}(y) & \leq \int_{\left\{ |Tf| > y \right\}} R_{\alpha}\left( \chi_{\{|Tf|> y\}} \right)(x)\,{\rm d}x \leq \frac{\varphi\left( { \left\lVert{v}\right\rVert_{A_1} } \right)^{p_0}}{y^{p_0}} \left\lVert{f}\right\rVert_{L^{p_0, 1}(v^{\alpha})}^{p_0} \\ & \approx \frac{\varphi\left( { \left\lVert{v}\right\rVert_{A_1} } \right)^{p_0}}{y^{p_0}} \left( {\int_0^{\infty} \left[ {\int_{\{f > z\}} R_{\alpha}{\chi_{\{Tf > y\}}}(x)\,{\rm d}x } \right]^{{1}/{p_0}}\,{\rm d}z } \right)^{p_0} \\ & \lesssim \frac{\varphi\left( { \left\lVert{v}\right\rVert_{A_1} } \right)^{p_0}}{y^{p_0}}\left\lVert{f}\right\rVert_{\Lambda^{1}\left( {w} \right)}^{p_0}\left\lVert{\chi_{\{Tf > y\}}}\right\rVert_{\left(\Lambda^{1/{p_0}, 1}\left( {w} \right)\right)'}, \end{align*}

where in the last estimate we have used the Hölder's inequality and the property (3.2) of $R_\alpha$. Now, since $\Lambda ^{1/{p_0},\, 1}(w) = \Lambda ^{1}(\tilde w)$ with $\tilde {W}(t)\approx W(t)^{p_0}$, by lemma 2.1

\[ \mu_{Tf}(y) \lesssim\frac{\varphi\left( { \left\lVert{v}\right\rVert_{A_1} } \right)^{p_0}}{y^{p_0}}\left\lVert{f}\right\rVert_{\Lambda^{ 1}\left( {w} \right)}^{p_0}\left\lVert{w}\right\rVert_{B_{1/{p_0}}^{{\mathcal{R}}}} \frac{\mu_{Tf}(y)}{W(\mu_{Tf}(y))^{p_0}}. \]

Thus,

\[ \left\lVert{Tf}\right\rVert_{\Lambda^{1,\infty}(w)} \lesssim \left\lVert{w}\right\rVert_{B_{ 1/{p_0}}^{\mathcal{R}}}^{1/{p_0}}\varphi\left( { \left\lVert{v}\right\rVert_{A_1} } \right)\left\lVert{f}\right\rVert_{\Lambda^{ 1}\left( {w} \right)}, \]

and since

\[ \left\lVert{v}\right\rVert_{A_1} \leq \left(2\left\lVert{M_{\alpha}}\right\rVert_{\left(\Lambda^{1/{p_0}, 1}\left( {w} \right)\right)'}\right)^{1/{\alpha}} \lesssim \left\{ \begin{array}{@{}cc} \left\lVert{w}\right\rVert_{B^{*}_{{p_0}/{(1-\alpha)}}}^{{p_0}/{\alpha }}, & 0 < \alpha < 1, \\ \left\lVert{w}\right\rVert_{B_\infty^{*}}, & \alpha = 1, \end{array}\right. \]

the result follows.

Proof of corollary 1.12. The proof follows the same pattern as in corollary 1.10.

4. Applications to the boundedness of operators

4.1 Bochner–Riesz $BR_\lambda$

Corollary 4.1 Let $n = 2,$ $0 < \lambda < \frac 12$ and $\frac 43 \leq q_0 \leq 4$. If there exists some $\varepsilon >0$ so that

\[ \frac{W(t)}{ t^{{(1+2\lambda(q_0 - 1))}/{q_0} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{{(1 -2\lambda)}/{q_0}+\varepsilon}}\approx\uparrow \]

then $w\in \mathcal {W}(BR_\lambda ).$

Proof. By proposition 1.5 (i), we have that $p_+=\frac {q_0}{1-2\lambda } \mbox { and } p_-= \frac {q_0}{1+2\lambda (q_0 - 1)}$ and the result follows by theorem 1.8.

Corollary 4.2 Let $n = 2$ and $0 < \lambda < \frac 12$. If

(4.1)\begin{equation} \frac{W(t)}{ t^{{(3+2\lambda)}/4}} \approx \downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{3/4+\varepsilon}}\approx\uparrow \end{equation}

then $w\in \mathcal {W}(BR_\lambda ) \text { with } \left \lVert {w}\right \rVert _{\mathcal {W}(BR_\lambda )} \lesssim \left \lVert {w}\right \rVert _{B^{{\mathcal{R}}_{{(3+2\lambda )}/4}}}^{{(3+2\lambda )}/4} \left \lVert {w}\right \rVert _{B^{*}_{ 4/3}}^{{(7+4\lambda )}/{2\lambda }}.$

Proof. By proposition 1.5 (ii), we have that $p_0= 4/{(3 + 2\lambda )} \mbox { and } \alpha = {2\lambda }/{(3 + 2\lambda )}$ and the result follows by theorem 1.11.

Remark 4.3 As a first example of corollary 4.2, if we let $w_\lambda (t) = t^{{(2\lambda - 1)}/4 }$, which clearly satisfies (4.1), then we conclude that $w_\lambda \in \mathcal {W}(BR_\lambda )$; that is, for this power weight, $BR_\lambda : \Lambda ^{1}(w_\lambda ) \rightarrow \Lambda ^{1,\infty }(w_\lambda )$, or equivalently,

\[ BR_\lambda: L^{4/{(3 + 2\lambda)},1} \longrightarrow L^{4/{(3 + 2\lambda)},\infty} \]

and, as expected, we obtain that the Bochner–Riesz conjecture holds for $BR_\lambda$ when $n = 2$. Further, we also obtain spaces for which the boundedness of $BR_\lambda$ was not previously known. For instance, taking $w_m$ and $\tilde w_m$ as in examples 2.5 (b) and (c), we have the inclusions

\[ \left\{\begin{array}{@{}l} \Lambda^{4/{(3 + 2\lambda)},1}(w_m) \hookrightarrow L^{4/{(3 + 2\lambda)},1} \hookrightarrow \Lambda^{4/{(3 + 2\lambda)},1}(\tilde w_m) \\ \Lambda^{4/{(3 + 2\lambda)},\infty}(w_m) \hookrightarrow L^{4/{(3 + 2\lambda)},\infty} \hookrightarrow \Lambda^{ 4/{(3 + 2\lambda)},\infty}(\tilde w_m), \end{array}\right. \]

and the boundedness of

\begin{align*} & BR_\lambda: \Lambda^{4/{(3 + 2\lambda)},1}(w_m)\longrightarrow \Lambda^{4/{(3 + 2\lambda)},\infty}(w_m),\nonumber\\ & \quad BR_\lambda: \Lambda^{4/{(3 + 2\lambda)},1}(\tilde w_m)\longrightarrow \Lambda^{ 4/{(3 + 2\lambda)},\infty}(\tilde w_m). \end{align*}

Corollary 4.4 Let $n > 2,$ $0 < \lambda < {(n-1)}/2$ and $q_0 = 2$ or let $q_0$ be such that ${2n}/{(n+1)} < q_0 < 2$ and conjecture 1.2 holds with $p = q_0$. If $q \in \{q_0,\, q_0'\}$ and there exists some $\varepsilon >0$ so that

\[ \frac{W(t)}{ t^{{(n - 1+2\lambda(q - 1))}/{(q(n-1))} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{{(n - 1 -2\lambda)}/{(q(n-1))}+\varepsilon}}\approx\uparrow \]

then $w\in \mathcal {W}(BR_\lambda ).$

Proof. By proposition 1.6 (i), we have that $p_+=\frac {q(n-1)}{n - 1 - 2\lambda }\mbox { and } p_-= \frac {q(n-1)}{n - 1 + 2\lambda (q-1)}$ for $q \in \{q_0,\, q_0'\}$, and the result follows by theorem 1.8.

Corollary 4.5 Let $n > 2$ and $\frac {n-1}{2(n+1)} < \lambda < {(n-1)}/2$. If

\[ \frac{W(t)}{ t^{1/2}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{{(n - 1 -2\lambda)}/{2n}+\varepsilon}}\approx\uparrow \]

or

\[ \exists \varepsilon>0: \frac{W(t)}{ t^{{(n + 1 + 2\lambda)}/{2n} - \varepsilon}} \approx\downarrow \quad\mbox{and}\quad \frac{W(t)}{ t^{1/2+\varepsilon}}\approx\uparrow \]

then $w\in \mathcal {W}(BR_\lambda ).$

Proof. By proposition 1.6 (ii), we have that either $p_0= 2 \mbox { and } \alpha = {(1 + 2\lambda )}/n$ and the result follows by theorem 1.11, or $p_+= 2 \mbox { and } p_-= {2n}/{(n + 1 + 2\lambda )}$ and the result follows by theorem 1.8.

4.2 Sparse operators

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

\[ ||Tf||_{L^{2}(v)} \lesssim ||v||_{A_2}||f||_{L^{2}(v)}, \quad \forall v \in A_2. \]

This result was first obtained by Hytönen [25] (see also [31, 32]). 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:

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) For every $k \in \mathbb {Z}$ and given $x \in \mathbb {R}^{n}$, 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, for every $Q\in \mathcal {S}$, there exist pairwise disjoint measurable sets $E_Q \subset Q$ with $|E_Q| \geq \eta |Q|$.

Definition 4.6 Let $1 \leq r < \infty$. Given a sparse family of cubes $\mathcal {S} \subset \mathcal {D}$, the sparse operator is defined by

\[ \mathcal{A}_{r,\mathcal{S}}f(x) = \sum_{Q\in \mathcal{S}}\left(\frac{1}{|Q|}\int_Qf^{r}(y)\,{\rm d}y\right)^{1/r} \chi_Q(x), \quad x \in \mathbb{R}^{n}. \]

The boundedness of $A_{1, \mathcal {S}}f$ over the Lorentz spaces $\Lambda ^{p}(w)$ was settled in [4]. Here we deal with the case $1 < r < \infty$. To do so, first we present the following restricted weak-type inequality for the sparse operator $\mathcal {A}_{r, \mathcal {S}}$. The proof follows by duality using the same ideas as in [11, theorem 4.1] with the obvious modifications.

Proposition 4.7 Given $1 < r < \infty$ and a weight $v \in A_1$. The sparse operator $\mathcal {A}_{r,\mathcal {S}}$ satisfies the weak-type estimate

\[ \left\lVert{\mathcal{A}_{r,\mathcal{S}}f}\right\rVert_{L^{r,\infty}(v)} \lesssim \left\lVert{v}\right\rVert_{A_1}^{2} \left\lVert{f}\right\rVert_{L^{r,1}(v)}. \]

Consequently, by theorem 1.11 and corollary 1.12:

Corollary 4.8 If

\[ \frac{W(t)}{ t^{1/r}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\,\frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then $w\in \mathcal {W}(\mathcal {A}_{r,\mathcal {S}})$ with $\left \lVert {w}\right \rVert _{\mathcal {W}(A_{r,\mathcal {S}})} \lesssim \left \lVert {w}\right \rVert _{B_{1/r}^{{\mathcal{R}}}}^{1/r} \left \lVert {w}\right \rVert _{B^{*}_\infty }^{2}.$ Moreover, if for $0 < p < \infty$,

\[ \frac{W(t)}{ t^{p/r}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then

\[ \mathcal{A}_{r,\mathcal{S}}: \Lambda^{p,1}(w) \rightarrow \Lambda^{p,\infty}(w). \]

4.3 Hörmander multipliers

Let $m$ be a bounded function on $\mathbb {R}^{n}$ and consider the Fourier multiplier $T_m$ defined by

\[ \widehat{T_mf}(\xi)=m(\xi)\hat f(\xi),\quad \xi \in {{\mathbb{R}^{n}}}. \]

Set $s>1$ and let $\gamma =(\gamma _1,\, \ldots,\, \gamma _n)$ a multi-index of nonnegative integers with $|\gamma |=\gamma _1+\cdots +\gamma _n$. It is said that $m\in M(s,\, l)$ if

\[ \sup_{R>0} \left(R^{s|\gamma|-n} \int_{R<|x|<2R} |D^{\gamma} m(x)|^{s} {\rm d}x \right)^{ 1/s} <\infty, \quad \forall |\gamma|\le l. \]

Proposition 4.9 [28]

Let $1 < s \leq 2,$ $n/s < l < n$ and $m \in M(s,\,l)$. Then:

1. (i) If $v \in A_1,$

   \[ T_m:L^{1}(v^{ l/n}) \rightarrow L^{1,\infty}(v^{ l/n}), \]
   with constant less than or equal to $\varphi (\left \lVert {v}\right \rVert _{A_1}).$ with $\varphi$ an increasing function.

2. (ii) If $v \in A_1,$

   \[ T_m:L^{ n/l}(v) \rightarrow L^{ n/l, \infty}(v), \]
   with constant less than or equal to ${\varphi }(\left \lVert {v}\right \rVert _{A_1}),$ with ${\varphi }$ an increasing function.

Consequently, by theorem 1.11 and corollary 1.12:

Corollary 4.10 Let $1 < s \leq 2,$ $n/s < l < n$ and $m \in M(s,\,l)$. If

\[ \frac{W(t)}t \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{{(n - l)}/n + \varepsilon}}\approx\uparrow \text{ or } \frac{W(t)}{ t^{ l/n}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\,\frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T_m).$ Moreover, if for $0 < p < \infty,$

\[ \frac{W(t)}{t^{p}} \approx\downarrow \!\quad\mbox{and}\quad\! \exists \varepsilon\!>\!0:\, \frac{W(t)}{ t^{{(p(n - l))}/n + \varepsilon}}\approx\uparrow \, \text{ or } \, \frac{W(t)}{ t^{{pl}/n}} \approx\downarrow \!\quad\mbox{and}\quad\! \exists \varepsilon\!>\!0:\, \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then

\[ T_m: \Lambda^{p,1}(w) \rightarrow \Lambda^{p,\infty}(w). \]

4.4 Rough operators

Let $\Sigma =\Sigma _{n-1}=\{x\in \mathbb {R}^{n}: |x|=1\}$ and, for $1 < r < \infty$, take $\Omega \in L^{r}(\Sigma )$ to be a positive function homogeneous of degree zero such that $\int _\Sigma \Omega = 0$. Let us consider the rough operator

\[ T_\Omega f(x)=p.v. \int_{\mathbb{R}^{n} }\frac{\Omega(y')}{|y|^{n}} f(x-y) {\rm d}y = \lim_{\varepsilon > 0} \int_{|y| > \varepsilon}\frac{\Omega(y')}{|y|^{n}} f(x-y) {\rm d}y, \]

whenever this limit exists almost everywhere and where $y'= y/{|y|}$, $y\neq 0$.

Proposition 4.11 [17]

If $v \in A_1$,

\[ T_{\Omega}:L^{r'}(v) \rightarrow L^{r',\infty}(v), \]

with constant less than or equal to $\varphi (\left \lVert {v}\right \rVert _{A_1}),$ with $\varphi$ an increasing function.

Consequently, by theorem 1.11 and corollary 1.12:

Corollary 4.12 If

\[ \frac{W(t)}{ t^{1/{r'}}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T_\Omega ).$ Moreover, for $0 < p < \infty,$ if

\[ \frac{W(t)}{ t^{ p/{r'}}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{\varepsilon}}\approx\uparrow, \]

then

\[ T_\Omega: \Lambda^{p,1}(w) \rightarrow \Lambda^{p,\infty}(w). \]

Also, if we assume that $\Omega$ satisfies the $L^{r}$-Dini condition, that is

\[ \int_0^{1} \omega_r(\delta)\frac{d\delta}\delta<{+}\infty, \quad \omega_r(\delta)=\sup_{|\rho|<\delta}\left(\int_\Sigma |\Omega(\rho x)-\Omega(x)|^{r} {\rm d}\sigma\right)^{1/r}, \]

with $\rho$ any rotation of $\Sigma$ and $|\rho | = \sup _{x \in \Sigma }|\rho x - x|$, we also have the following result:

Proposition 4.13 [28]

If $v \in A_1,$

\[ T_{\Omega}:L^{1}(v^{1/{r'}}) \rightarrow L^{1,\infty}(v^{1/{r'}}), \]

with constant less than or equal to $\varphi (\left \lVert {v}\right \rVert _{A_1}),$ with $\varphi$ an increasing function.

Consequently, by theorem 1.11 and corollary 1.12:

Corollary 4.14 If

\[ \frac{W(t)}t \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{ 1/r + \varepsilon}}\approx\uparrow, \]

then $w \in \mathcal {W}(T_\Omega ).$ Moreover, for $0 < p < \infty,$ if

\[ \frac{W(t)}{t^{p}} \approx\downarrow \quad\mbox{and}\quad \exists \varepsilon>0:\, \frac{W(t)}{ t^{ p/r + \varepsilon}}\approx\uparrow , \]

then

\[ T_\Omega: \Lambda^{p,1}(w) \rightarrow \Lambda^{p,\infty}(w). \]

Remark 4.15 We should finally mention that although the property of being increasing of $\varphi$ in propositions 4.9, 4.11 and 4.13 is known, the sharp expression for such function is unknown.