1. Introduction and main result

One-sided theory of weights was begun by Sawyer in [Reference Sawyer35] where he provided the characterization of the two weighted inequalities for the one-sided maximal functions

\[ M^+f(x)=\sup_{h>0}\frac{1}{h}\int_{x}^{x+h}|f(y)|\,{\rm d}y\quad M^{-}f(x)=\sup_{h>0}\frac{1}{h}\int_{x-h}^{x}|f(y)|\,{\rm d}y. \]

In some sense it is a somehow curious fact that the results in [Reference Sawyer35] appeared more than a decade later than the characterization of the one weight inequalities for the maximal function due to Muckenhoupt [Reference Muckenhoupt29] if one bears in mind that actually the maximal operators studied by Hardy and Littlewood [Reference Hardy and Littlewood10] were $M^+$ and $M^{-}$.

Since Sawyer's work a number of papers such as [Reference Aimar, Forzani and Martín-Reyes1, Reference Forzani, Martín-Reyes and Ombrosi7, Reference Gurka, Martín-Reyes, Ortega, Pick, Sarrión and de la Torre9, Reference Lorente and Martín-Reyes20, Reference Lorente and Martín-Reyes21, Reference Martín-Reyes23–Reference Martín-Reyes, Pick and de la Torre26, Reference Martín-Reyes and de la Torre28, Reference Ombrosi31, Reference Ombrosi and de Rosa32] and even more that we will cite throughout this paper were devoted to develop the one-sided theory. However, at this point, we believe it is worth mentioning some papers which have expanded the field of one-sided estimates in the last years. Kinnunen and Saari [Reference Kinnunen and Saari13, Reference Kinnunen and Saari14] studied parabolic Muckenhoupt conditions in connection with PDEs and more recently Hytönen and Rosén devoted their work [Reference Hytönen and Rosén11] to causal sparse domination motivated by maximal regularity estimates for elliptic PDEs, obtaining results related to one-sided weighted estimates for singular integrals.

A well-known estimate in theory of weights that was settled by Coifman and Fefferman, says that if $w\in A_{\infty }$ then, for every $p\in (0,\infty )$ we have that

(1.1)\begin{equation} \|Tf\|_{L^{p}(w)}\leq c_{p,n,w}\|Mf\|_{L^{p}(w)}\end{equation}

where $T$ stands for any Calderón–Zygmund operator.

Although $w\in A_{\infty }$ is sufficient for (1.1) to hold, it turns out to not to be necessary. Muckenhoupt [Reference Muckenhoupt30] showed that if (1.1) holds for the Hilbert transform for $p>1$ then there exist $c,\varepsilon >0$ such that for every cube $Q$ and every measurable subset $E\subset Q$,

\[ w(E)\leq c\left(\frac{|E|}{|Q|}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M\chi_{Q}\right)^{p}w, \]

namely $w\in C_{p}$. Later on, Sawyer [Reference Sawyer34] showed that if $1< p< q<\infty$ and $w\in C_{q}$, then (1.1) holds.

Yabuta [Reference Yabuta37] provided a different approach to the question. He showed that if $1< p< q<\infty$ and $w\in C_{q}$

(1.2)\begin{equation} \|Mf\|_{L^{p}(w)}\leq c_{p,n,w}\|M^{\sharp}f\|_{L^{p}(w)}\end{equation}

and also that if such an inequality holds then $w\in C_{p}$. An alternative proof of this estimate and a slight generalization of the $C_{p}$ condition was studied by Lerner in [Reference Lerner15].

In the last years some advances have been made in the study of this kind of questions. Lerner [Reference Lerner16] fully characterized the weak type version of (1.1). Sawyer's result has been extended to the full range in [Reference Cejas, Li, Pérez and Rivera-Ríos6] and also quantitative estimates in terms of a suitable $C_{p}$ constant and further operators, such as rough singular integrals, have been explored in [Reference Canto4, Reference Canto, Li, Roncal and Tapiola5].

In the one-sided setting we are aware of just one work in this direction in which Riveros and de la Torre [Reference Riveros and de la Torre33] introduced the one-sided version of the $C_{p}$ condition, which reads as follows. We say that $w\in C_{p}^+$ if there exist $\varepsilon >0$ and $C>0$ such that for any $a< b< c$ with $c-b< b-a$ and any measurable set $E\subset (a,b)$, the following holds

(1.3)\begin{equation} \int_{E}w\leq C\left(\frac{|E|}{c-b}\right)^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{(a,c)}\right)^{p}w.\end{equation}

The main result in that work was the following one-sided counterpart of [Reference Sawyer34].

Theorem 1.1 [Reference Riveros and de la Torre33, theorem 1]

Let $1< p< q<\infty$. If $w\in C_{q}^+$ and $(T^+)^{*}$ is a maximal Calderón–Zygmund one-sided singular integral, then

(1.4)\begin{equation} \int_{\mathbb{R}}|(T^+)^{*}f|^{p}w\leq C\int_{\mathbb{R}}(M^+f)^{p}w.\end{equation}

Observe that in [Reference Riveros and de la Torre33], additionally the authors assume that the integral in the right-hand side of (1.3) is finite. Note that if the right-hand side of the condition $C_{q}^+$ is not finite, then the same happens to $\int _{\mathbb {R}}(M^+f)^{p}w$ and hence the inequality is trivial.

Note that the $C_{p}^+$ class is defined in terms of the one-sided maximal operator $M^+$. We will review the definitions of $M^+$ and the remainder of the one-sided operators studied in this paper in § 2. We would like to observe as well that the assumption $c-b< b-a$ can be dropped. Assume that $b-a\leq c-b$. Let $\bar {a}< a$ such that $a-\bar {a}=c-a$. Note, that then, $b\in (\bar {a},c)$, and $c-b< b-\bar {a}.$ On the other hand observe that $M^+\chi _{(\bar {a},c)}\simeq M^+\chi _{(a,c)}$ and hence the $C_{p}^+$ condition would hold just with a larger constant $C$ but without the restriction $c-b< b-a$.

The purpose of this paper is to provide a one-sided counterpart of Yabuta's characterization (1.2) and to derive a number of new results relying upon it. The precise statement of our theorem is the following.

Theorem 1.2 Let $1< p< q<\infty$. If $f\in L^{p_{0}}(\mathbb {R})$ for some $1< p_{0}<\infty$ and $w\in C_{q}^+$ then

\[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)}, \]

provided the left-hand side of the estimate is finite. Conversely if the preceding estimate holds, then $w\in C_{p}^+.$

We would like to note that the corresponding counterpart for $T^{-}$ operators holds as well. However, here and throughout the remainder of this paper we will just deal with the case of $T^+$ operators.

Exploiting the approach in [Reference Cejas, Li, Pérez and Rivera-Ríos6] we shall derive a number of consequences of this result. Among them we will recover the result for one-sided Calderón–Zygmund singular integrals due to de la Torre and Riveros that we stated above. We present those results in § 3.

The remainder of the paper is organized as follows. We devote § 2 to gather some results and definitions that will be useful throughout the remainder of the work. In § 3 we present the applications of theorem 1.2, namely counterparts of (1.4) for some other one-sided operators, and even for one-sided Calderón–Zygmund singular integrals themselves. Section 4 is devoted to the proof of theorem 1.2. Additionally we provide an appendix settling a suitable Cotlar inequality that we have not been able to find in the literature and that will be useful for us to recover and generalize [Reference Riveros and de la Torre33, theorem 1].

2. Preliminaries and definitions

We recall that the one-sided maximal function $M^+$ is defined, as we noted in the introduction, as

\[ M^+f(x)=\sup_{h>0}\frac{1}{h}\int_{x}^{x+h}|f| \]

and the sharp maximal function $M^{\sharp,+}$, that was introduced in [Reference Martín-Reyes and de la Torre27], as

\[ M^{\sharp,+}f(x)=\sup_{h>0}\frac{1}{h}\int_{x}^{x+h}\left(f(y)-\frac{1}{h}\int_{x+h}^{x+2h}f\right)^+\,{\rm d}y. \]

Another class of operators that we will be dealing with and that have already appeared in the previous section are one-sided singular integral operators that were introduced in [Reference Aimar, Forzani and Martín-Reyes1]. We say that a function $K\in L_{\text {loc}}^{1}(\mathbb {R}\setminus \{0\})$ is a Calderón–Zygmund kernel if the following properties hold.

1. (1) There exists a finite constant $B_{1}$ such that

   \[ \left|\int_{\varepsilon<|x|< N}K(x)\,{\rm d}x\right|\leq B_{1} \]
   for all $0<\varepsilon < N$. Furthermore, $\lim _{\varepsilon \rightarrow 0^+}\int _{\varepsilon <|x|< N}K(x)\,{\rm d}x$ exists.

2. (2) There exists a constant $B_{2}$ such that

   \[ |K(x)|\leq\frac{B_{2}}{|x|} \]
   for all $x\not =0$.

3. (3) There exists a finite constant $B_{3}$ such that

   (2.1)\begin{equation} |K(x-y)-K(x)|\leq B_{3}\frac{|y|}{|x|^{2}}\end{equation}
   for all $x$ and $y$ with $|x|>2|y|>0$.

We say that $T^+$ is a one-sided Calderón–Zygmund singular integral if

(2.2)\begin{equation} T^+f(x)=\lim_{\varepsilon\rightarrow0}\int_{x+\varepsilon}^{\infty}K(x-y)f(y)\,{\rm d}y\end{equation}

where $K$ is a Calderón-Zygmund kernel with support in $\mathbb {R}^{-}$.

We would like to emphasize that this kind of operators are Calderón–Zygmund operators, and hence they have all the usual properties of operators in that class, but with the extra feature that $K$ is supported in $\mathbb {R}^{-}$. Examples of such operators are provided in [Reference Aimar, Forzani and Martín-Reyes1].

Replacing (2.1) by some other smoothness conditions we obtain some more operators. For instance, we may assume that there exist numbers $c_{r},C_{r}>0$ such that for any $y\in \mathbb {R}$ and $R>c_{r}|y|$,

(2.3)\begin{equation} \sum_{m=1}^{\infty}2^{m}R\left(\frac{1}{2^{m}R}\int_{2^{m}R<|x|\leq2^{m+1}R}|K(x-y)-K(x)|^{r}\,{\rm d}x\right)^{\frac{1}{r}}\leq C_{r}\end{equation}

if $1\leq r<\infty$ and

\[ \sum_{m=1}^{\infty}2^{m}R\sup_{2^{m}R<|x|\leq2^{m+1}R}|K(x-y)-K(x)|\leq C_{\infty} \]

if $r=\infty$. If $K$ satisfies an $L^{r}$-Hörmander condition we say that $K\in \mathcal {H}_{r}$. This yields that we may define an operator $T^+$ exactly as we did in (2.2), but with $K$ satisfying (2.3) instead of (2.1). We may go even further. Let us recall first the notion of Orlicz average. Let $A:[0,\infty )\rightarrow [0,\infty )$ a Young function, namely a convex function such that $A(0)=0$, $A(1)=1$ and $\lim _{t\rightarrow \infty }A(t)=\infty$. Given a measurable set $E$ we define the average of $f$ over $E$ with respect to $A$ as

\[ \|f\|_{A,E}=\inf\left\{ \lambda>0\,:\,\frac{1}{|E|}\int_{E}A\left(\frac{|f(x)|}{\lambda}\right){\rm d}x\leq1\right\} . \]

Relying upon that definition we may define the maximal function $M_{A}^+$ as follows

\[ M_{A}^+f(x)=\sup_{h>0}\|f\|_{A,[x,x+h]}. \]

It is also worth mentioning that we can define a function associated to $A$, that we call $\overline {A}$, which turns out to be a Young function as well and satisfies the following inequalities

\[ t\leq\overline{A}^{{-}1}(t)A^{{-}1}(t)\leq2t. \]

Furthermore it can be shown that if $A_{1},A_{2},\ldots,A_{n}$ are Young functions such that

\[ A_{1}^{{-}1}(t)\ldots A_{n}^{{-}1}(t)\lesssim t \]

then

\[ \frac{1}{|E|}\int_{E}|f_{1}\cdots f_{n}|\lesssim\|f_{1}\|_{A_{1},E}\cdots\|f_{n}\|_{A_{n},E}. \]

Coming back to the previous discussion, as we mentioned above, we may define a class of kernels generalizing the $L^{r}$-Hörmander condition. Given a Young function $A$ we say that it satisfies an $L^{A}$-Hörmander condition if there exist $c_{A},C_{A}\geq 1$ such that for any $x\in \mathbb {R}$ and $R>c_{A}|x|$

(2.4)\begin{equation} \sum_{m=1}^{\infty}2^{m}R\|(K(x-\cdot)-K(-\cdot))\chi_{2^{m}R<|\cdot|\leq2^{m+1}R}\|_{A,B(0,2^{m+1}R)}\leq C_{A}.\end{equation}

If $K$ satisfies this condition we say that $K\in \mathcal {H}_{A}$. In order to be able to deal with commutators we introduce another condition. We say that $K\in \mathcal {H}_{A,k}$ if there exist $c_{A,k},C_{A,k}\geq 1$ such that for any $x\in \mathbb {R}$ and $R>c_{A,k}|x|$

\[ \sum_{m=1}^{\infty}2^{m}Rm^{k}\|(K(x-\cdot)-K(-\cdot))\chi_{2^{m}R<|\cdot|\leq2^{m+1}R}\|_{A,B(0,2^{m+1}R)}\leq C_{A,k}. \]

In both cases, whether $K\in \mathcal {H}_{A}$ or $K\in \mathcal {H}_{A,k}$, we may define a singular integral operator exactly as we did in (2.2). Those classes of kernels were introduced and studied in [Reference Lorente, Riveros and de la Torre17, Reference Lorente, Martell, Riveros and de la Torre19].

We would like to end recalling that we may define a maximal version of any of the singular integral operators that we have just presented in this section as follows

\[ (T^+)^{*}f(x)=\sup_{\varepsilon>0}\left|T_{\varepsilon}^+f(x)\right|=\sup_{\varepsilon>0}\left|\int_{\varepsilon+x}^{\infty}K(x-y)f(y)\,{\rm d}y\right|. \]

3. Corollaries of the main theorem

As we announced in the previous section we will derive a number of applications of theorem 1.2. Our use of that theorem will rely upon the following lemma.

Lemma 3.1 Let $0< p<\infty$. If $\delta \in (0,p)$ then, if $w\in C_{\rho }^+$ with $\rho >\frac {p}{\delta }$, we have that

\[ \|M_{\delta}^+f\|_{L^{p}(w)}\lesssim\|M_{\delta}^{\sharp,+}f\|_{L^{p}(w)} \]

where $M_{\delta }^+(f)=(M^+(|f|^{\delta }))^{\frac {1}{\delta }}$ and $M_{\delta }^{\sharp,+}(f)=(M^{\sharp,+}(|f|^{\delta }))^{\frac {1}{\delta }}$.

Proof. Observe that, since $\delta \in (0,p)$ we have that $\frac {p}{\delta }>1$. Taking that into account and the fact that $w\in C_{\rho }^+$ we have by theorem 1.2 that

\begin{align*} \|M_{\delta}^+f\|_{L^{p}(w)}^{p} & =\int_{\mathbb{R}}M^+(|f|^{\delta})^{\frac{p}{\delta}}(x)w(x)\,{\rm d}x\\ & \lesssim\int_{\mathbb{R}}M^{\sharp,+}(|f|^{\delta})^{\frac{p}{\delta}}(x)w(x)\,{\rm d}x\\ & =\|M_{\delta}^{\sharp,+}f\|_{L^{p}(w)}^{p} \end{align*}

and we are done.

3.1 Singular integral operators, $L^{A}$-Hörmander operators and their commutators

In this section we present our results for singular integral operators, $L^{A}$-Hörmander operators and their commutators. We will provide some full arguments here, that we shall omit for the remainder of the corollaries since they will be analogous to the ones provided here.

We begin recalling that for one-sided Calderón–Zygmund singular integrals $T^+$ it was shown in [Reference Lorente and Riveros22, lemma 1] that for $0<\delta <1$,

(3.1)\begin{equation} M_{\delta}^{\sharp,+}(T^+f)\lesssim M^+f.\end{equation}

Using that estimate we can derive the following result.

Theorem 3.2 Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+.$ Then

\[ \|T^+f\|_{L^{p}(w)}\lesssim\|M^+f\|_{L^{p}(w)}. \]

Proof. Let $\delta \in (0,1)$ such that $1<\frac {p}{\delta }<\max \{p,1\}+\varepsilon$. For that choice of $\delta$, taking into account (3.1) and lemma 3.1 with $\rho =\max \{p,1\}+\varepsilon$, we have that

\begin{align*} \|T^+f\|_{L^{p}(w)} & \lesssim\|M_{\delta}^+(T^+f)\|_{L^{p}(w)}\lesssim\|M_{\delta}^{\sharp,+}(T^+f)\|_{L^{p}(w)}\\ & \leq\|M^+f\|_{L^{p}(w)} \end{align*}

and we are done.

Recall that the commutator of a linear operator $T$ and a locally integrable function $b$ is defined as

\[ [b,T]f(x)=b(x)Tf(x)-T(bf)(x). \]

The iterated commutator $T_{b}^{k}$ consists precisely in iterating the commutator.

\[ T_{b}^{k}f(x)=[b,T_{b}^{k-1}]f(x) \]

where $T_{b}^{0}f(x)=Tf(x)$. For the commutator and the iterated commutator of $b\in BMO$ and a Calderón–Zygmund one-sided singular integral $T^+$ again in [Reference Lorente and Riveros22, lemma 1], it was shown that for $0<\delta <\gamma <1$, we have that

(3.2)\begin{equation} M_{\delta}^{\sharp,+}((T^+)_{b}^{k}f)(x)\lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}M_{\gamma}^+((T^+)_{b}^{j})f(x)+\|b\|_{BMO}^{k}(M^+)^{k+1}f(x).\end{equation}

Relying upon that pointwise estimate we have the following theorem.

Theorem 3.3 Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+.$ Then

\[ \|(T^+)_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|(M^+)^{k+1}f\|_{L^{p}(w)}. \]

Proof. Observe that it suffices to show that for $\delta _{1}\in (0,1)$ such that $1<\frac {p}{\delta _{1}}<\max \{p,1\}+\varepsilon$ the following holds

(3.3)\begin{equation} \|M_{\delta_{1}}^+(T^+)_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|(M^+)^{k+1}f\|_{L^{p}(w)}.\end{equation}

We proceed by induction. Assume first that $k=1$. Let $0<\delta _{1}<\delta _{2}<1$ such that $1<\frac {p}{\delta _{i}}<\max \{p,1\}+\varepsilon$. We have that, taking into account (3.2) and lemma 3.1,

\begin{align*} \|M_{\delta_{1}}^+(T^+)_{b}^{1}f\|_{L^{p}(w)} & \lesssim\left\Vert M_{\delta_{1}}^{\sharp,+}\left((T^+)_{b}^{1}f\right)\right\Vert _{L^{p}(w)}\\ & \lesssim\|b\|_{BMO}\|M_{\delta_{2}}^+(T^+f)\|_{L^{p}(w)}+\|b\|_{BMO}\|(M^+)^{2}f\|_{L^{p}(w)}\\ & \lesssim\|b\|_{BMO}\|M^+f\|_{L^{p}(w)}+\|b\|_{BMO}\|(M^+)^{2}f\|_{L^{p}(w)}\\ & \lesssim\|b\|_{BMO}\|(M^+)^{2}f\|_{L^{p}(w)} \end{align*}

where the estimate for $\|M_{\delta _{2}}(T^+f)\|_{L^{p}(w)}$ follows by the same argument provided in the proof of theorem 3.2.

Assume now that (3.3) holds for $1,2,\ldots k-1$. Let $0<\delta _{1}<\delta _{2}<1$ such that $1<\frac {p}{\delta _{i}}<\max \{p,1\}+\varepsilon$. Then, again by (3.2) and lemma 3.1,

\begin{align*} \|M_{\delta_{1}}^+(T^+)_{b}^{k}f\|_{L^{p}(w)} & \lesssim\left\Vert M_{\delta_{1}}^{\sharp,+}\left((T^+)_{b}^{k}f\right)\right\Vert _{L^{p}(w)}\\ & \lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}\|M_{\delta_{2}}^+((T^+)_{b}^{j})f(x)\|_{L^{p}(w)}\\ & \quad +\|b\|_{BMO}^{k}\|(M^+)^{k+1}f\|_{L^{p}(w)}.\\ & \lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k}\|(M^+)^{j}f(x)\|_{L^{p}(w)}+\|b\|_{BMO}^{k}\|(M^+)^{k+1}f\|_{L^{p}(w)}\\ & \lesssim\|b\|_{BMO}^{k}\|(M^+)^{k+1}f\|_{L^{p}(w)} \end{align*}

and we are done.

With analogous arguments relying upon the corresponding pointwise sharp inequality we may settle the following result. We recall that if $\bar {A}$ is a Young function and $T^+$ is an operator associated to a kernel $K\in H_{\bar {A}}$ with support in $(-\infty,0)$, then as it was established in [Reference Lorente, Riveros and de la Torre17, p. 505]

\[ M_{\delta}^{\sharp,+}((T^+f)(x)\lesssim M_{A}^+f(x) \]

and if, $A$ and $B$ are Young functions, $\overline {C}^{-1}(t)={\rm e}^{t^{\frac {1}{k}}}$ with $k$ a positive integer such that $A^{-1}(t)B^{-1}(t)\overline {C}^{-1}(t)\leq t$ for $t\geq 1$ and $K\in H_{B}\cap H_{\bar {A},k}$ then, for $0<\delta <\gamma <1$,

(3.4)\begin{equation} M_{\delta}^{\sharp,+}((T^+)_{b}^{k}f)(x)\lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}M_{\gamma}^+((T^+)_{b}^{j})f(x)+\|b\|_{BMO}^{k}M_{A}^+f(x).\end{equation}

We remit the reader to [Reference Lorente, Martell, Riveros and de la Torre19, § 5.3].

Arguing as above, we have the following results.

Theorem 3.4 Let $A$ be a Young function and assume that $K\in \mathcal {H}_{\overline {A}}$. Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+.$ Then

\[ \|T^+f\|_{L^{p}(w)}\lesssim\|M_{A}^+f\|_{L^{p}(w)}. \]

Theorem 3.5 Let $k$ be a positive integer and assume that $A$ and $B$ are Young functions, and $A^{-1}(t)B^{-1}(t)\overline {C}^{-1}(t)\leq t$ for $t\geq 1$ where $\overline {C}^{-1}(t)={\rm e}^{t^{\frac {1}{k}}}$. Assume also that $K\in \mathcal {H}_{B}\cap \mathcal {H}_{\bar {A},k}$ and that $b\in BMO$. Then, if $0< p<\infty$, $\varepsilon >0$ and $w\in C_{\max \{p,1\}+\varepsilon }^+$, we have that

\[ \|(T^+)_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|M_{A}^+f\|_{L^{p}(w)}. \]

We would like to end the section providing a result for maximal singular integral operators.

Theorem 3.6 Let $A$ be a Young function and assume that $K\in \mathcal {H}_{\overline {A}}$. Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+.$ Then

\[ \|(T^+)^{*}f\|_{L^{p}(w)}\lesssim\|M_{A}^+f\|_{L^{p}(w)}. \]

Before settling this result, observe that if $K$ satisfies (2.1) in particular $K\in \mathcal {H}_{\infty }$, and consequently, the preceding result recovers the main result in [Reference Riveros and de la Torre33].

Proof of theorem 3.6 Observe that by the Cotlar type inequality in theorem A.1, we have that for any $\delta \in (0,1)$.

\[ \|(T^+)^{*}f\|_{L^{p}(w)}\lesssim\|M_{\delta}^+(T^+f)\|_{L^{p}(w)}+\|M_{A}^+f\|_{L^{p}(w)} \]

and hence it suffices to deal with the first term. An analogous argument to the one provided to settle theorem 3.2, choosing a suitable $\delta$, shows that

\[ \|M_{\delta}^+(T^+f)\|_{L^{p}(w)}\lesssim\|M_{A}^+f\|_{L^{p}(w)} \]

and we are done.

3.2 The differential transform operator

Given $\{v_{j}\}\in \ell ^{\infty }$ we define

\[ T^+f(x)=\sum_{j\in\mathbb{Z}}v_{j}(D_{j}f(x)-D_{j-1}f(x)),\quad D_{j}f(x)=\frac{1}{2^{j}}\int_{x}^{x+2^{j}}f(t)\,{\rm d}t. \]

As the authors point out in [Reference Lorente, Martell, Riveros and de la Torre19] this operator, that was previously studied in [Reference Bernardis, Lorente, Martín-Reyes, Martínez, de la Torre and Torrea3, Reference Jones and Rosenblatt12], arises when studying the rate of convergence of the averages $D_{j}f$. Note that $D_{j}f\rightarrow f$ a.e. when $j\rightarrow -\infty$ and that $D_{j}f\rightarrow 0$ when $j\rightarrow \infty$ for appropriate $f$.

Observe that $T^+$ is a one-sided singular integral since $T^+f=K*f$ for $K$ supported on $(-\infty,0)$ and defined as

\[ K(x)=\sum_{j\in\mathbb{Z}}v_{j}\left(\frac{1}{2^{j}}\chi_{({-}2^{j},0)}(x)-\frac{1}{2^{j-1}}\chi_{({-}2^{j-1},0)}(x)\right). \]

As it was stated in [Reference Lorente, Martell, Riveros and de la Torre19, remark 4.11], it is possible to show that $K\in \mathcal {H}_{A,k}$ with $A(t)=\exp \left (\frac {t^{\frac {1}{1+k}}}{(\log t)^{\frac {1+\varepsilon }{1+k}}}\right )$, and hence by (3.4) we have that if $b\in BMO$ and $k$ is a non-negative integer, for $0<\delta <\gamma <1$,

\begin{align*} M_{\delta}^{\sharp,+}((T^+)_{b}^{k}f)(x)& \lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}M_{\gamma}^+((T^+)_{b}^{j})f(x)\\ & \quad +\|b\|_{BMO}^{k}M_{L(\log L)^{1+k}(\log\log L)^{1+\varepsilon}}^+f(x). \end{align*}

where the first term is interpreted as $0$ if $k=0$. Then, arguing as in the preceding section we have the following result.

Theorem 3.7 Let $k$ be a non-negative integer. Then, if $0< p<\infty$, $\varepsilon >0$ and $w\in C_{\max \{p,1\}+\varepsilon }^+$, we have that

\[ \|(T^+)_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|M_{L(\log L)^{1+k}(\log\log L)^{1+\varepsilon}}^+f\|_{L^{p}(w)}. \]

3.3 The one-sided discrete square function and its commutator

We recall that the one-sided discrete square function is defined as follows. If $f$ is locally integrable in $\mathbb {R}$ and $s>0$ we consider the averages

\[ A_{s}f(x)=\frac{1}{s}\int_{x}^{x+s}f(y)\,{\rm d}y. \]

Hence the one-sided discrete square function of $f$ is given by

\[ S^+f(x)=\left(\sum_{n\in\mathbb{Z}}|A_{2^{n}}f(x)-A_{2^{n-1}}f(x)|^{2}\right)^{\frac{1}{2}}. \]

This operator was studied in [Reference de la Torre and Torrea36] and [Reference Lorente, Riveros and de la Torre17]. In [Reference Lorente, Riveros and de la Torre18] the authors deal with the following operator

\[ \mathcal{O}^+f(x)=\left(\sum_{n\in\mathbb{Z}}\sup_{s\in[2^{n},2^{n+1})}|A_{2^{n}}f(x)-A_{s}f(x)|^{2}\right)^{\frac{1}{2}}, \]

which dominates pointwise $S^+f$, and show [Reference Lorente, Riveros and de la Torre18, Eq. (3.1) in p. 581] that if $0<\delta <1$

\[ M_{\delta}^{\sharp,+}(\mathcal{O}^+f)(x)\lesssim M_{L\log L}^+f(x). \]

This fact allows them to settle the corresponding Coifman–Fefferman estimate. Here, arguing as we did to settle theorem 3.2, we have the following result.

Theorem 3.8 Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+.$ Then

\[ \|Gf\|_{L^{p}(w)}\lesssim\|M_{L\log L}^+f\|_{L^{p}(w)} \]

where $G$ stands either for $S^+$ or for $\mathcal {O}^+$.

Further assuming that $b\in BMO$ the authors also study the commutators associated to the operators above. In [Reference Lorente, Riveros and de la Torre18, lemma 4.6] it is shown that, for $0<\delta <\gamma <1$,

\[ M_{\delta}^{\sharp,+}((\mathcal{O}^+)_{b}^{k}f)(x)\lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}M_{\gamma}^+((\mathcal{O}^+)_{b}^{j})f(x)+M_{L(\log L)^{1+k}}^+f(x). \]

As a consequence we can derive the following result.

Theorem 3.9 Let $0< p<\infty$ and $\varepsilon >0$ and assume that $w\in C_{\max \{p,1\}+\varepsilon }^+$ and that $b\in BMO$. If $k$ is a positive integer, then

\[ \|G{}_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|M_{L(\log L)^{1+k}}^+f\|_{L^{p}(w)} \]

where $G$ stands either for $S^+$ or for $\mathcal {O}^+$.

3.4 Riemann–Liouville and Weyl fractional integral operators and their commutators

We recall that given $0<\alpha <1$ and locally integrable functions $f$ and $b$, the Weyl fractional integral and its commutators are defined as

\begin{align*} I_{\alpha}^+f(x) & =\int_{x}^{\infty}\frac{f(y)}{(y-x)^{1-\alpha}}\,{\rm d}y\quad\text{and}\\ (I_{\alpha}^+)_{b}^{k}f(x) & =\int_{x}^{\infty}(b(x)-b(y))^{k}\frac{f(y)}{(y-x)^{1-\alpha}}\,{\rm d}y \end{align*}

respectively. Analogously we define the Riemann–Liouville fractional integral and its commutators as

\begin{align*} I_{\alpha}^{-}f(x) & =\int_{-\infty}^{x}\frac{f(y)}{(x-y)^{1-\alpha}}\,{\rm d}y\quad\text{and}\\ (I_{\alpha}^{-})_{b}^{k}f(x) & =\int_{-\infty}^{x}(b(x)-b(y))^{k}\frac{f(y)}{(x-y)^{1-\alpha}}\,{\rm d}y. \end{align*}

In [Reference Bernardis and Lorente2, lemma 4.1] it was shown that for every non-negative $k$, if $b\in BMO$ and $0<\delta <\gamma <1$,

\[ M_{\delta}^{\sharp,+}((I_{\alpha}^+)_{b}^{k}f)(x)\lesssim\sum_{j=0}^{k-1}\|b\|_{BMO}^{k-j}M_{\gamma}^+((I_{\alpha}^+)_{b}^{j})f(x)+\|b\|_{BMO}^{k}M_{(\alpha),L(\log L)^{k}}^+f(x) \]

where

\[ M_{(\alpha),L(\log L)^{k}}^+f(x)=\sup_{h>0}h^{\alpha}\|f\|_{L(\log L)^{k},(x,x+h)} \]

and the first term in the right-hand side is interpreted as $0$ if $k=0$. Relying upon that $M_{\delta }^{\sharp,+}$ estimate and arguing as in the proofs of theorems 3.2 and 3.3 it is possible to settle the following result.

Theorem 3.10 Let $k$ be a non-negative integer. Then, if $0< p<\infty$, $\varepsilon >0$ and $w\in C_{\max \{p,1\}+\varepsilon }^+$ we have that

\[ \|(I_{\alpha}^+)_{b}^{k}f\|_{L^{p}(w)}\lesssim\|b\|_{BMO}^{k}\|M_{(\alpha),L(\log L)^{k}}^+\|_{L^{p}(w)}. \]

4. Proof of theorem 1.2

4.1 Sufficiency

By a standard approximation argument it is enough to settle the result assuming that $f\in L_{c}^{\infty }$.

To settle the sufficiency part in theorem 1.2 we need to borrow two lemmas from [Reference Riveros and de la Torre33]. The first one is the following.

Lemma 4.1 [Reference Riveros and de la Torre33, lemma 1]

Assume that $w\in C_{q}^+$ with $1< q<\infty$. Then, for any $\delta >0$ there exists $c(\delta )$ such that for any disjoint family of intervals $J_{j}$ contained in $I=(a,b)$ we have that

\[ \int_{I}\sum_{j}(M^+\chi_{J_{j}})^{q}w\leq c(\delta)w(I)+\delta\int_{\mathbb{R}}(M^+\chi_{I})^{q}w \]

and

\[ \int_{\mathbb{R}}\sum_{j}(M^+\chi_{J_{j}})^{q}w\lesssim\int_{\mathbb{R}}(M^+\chi_{I})^{q}w. \]

To state the next lemma we need to define a new operator, $M_{p,q}^+$. Let $f$ be an non-negative measurable function. Let us consider

\[ \Omega_{k}=\left\{ x\in\mathbb{R}\,:\,f(x)>2^{k}\right\} =\bigcup_{i}I_{i}^{k} \]

where $I_{i}^{k}$ are the connected components of $\Omega _{k}$. Then

\[ (M_{p,q}^+f(x))^{p}=\sum_{i,k}2^{pk}(M^+(\chi_{I_{i}^{k}})(x))^{q}. \]

Having this definition at our disposal we present the second lemma we borrow from [Reference Riveros and de la Torre33].

Lemma 4.2 [Reference Riveros and de la Torre33, lemma 2]

Let $1< p< q<\infty,$ $w\in C_{q}^+$ and $f$ non-negative, bounded and of compact support. Then

\[ \int_{\mathbb{R}}M_{p,q}^+(M^+f)w\lesssim\int_{\mathbb{R}}(M^+f)^{p}w. \]

Having those lemmas at our disposal we are in the position to settle theorem 1.2.

Proof of theorem 1.2 Let $\Omega _{k}=\{x\,:\,M^+f(x)>2\cdot 2^{k}\}=\bigcup _{j}J_{j}^{k}$ where $J_{j}^{k}$ are the connected components of $\Omega _{k}$. Let us fix $(a,b)=J_{j}^{k}$. We partition $(a,b)$ as follows. Let $x_{0}=a$ and choose $x_{i+1}$ such that $x_{i+1}-x_{i}=b-x_{i+1}$ and let $I_{i}^{k}=(x_{i},x_{i+1})$. By the good-$\lambda$ inequality established in [Reference Martín-Reyes and de la Torre27, theorem 4], we have that

\[ |E_{i}^{k}|=\left|\left\{ x\in I_{i}^{k}\,:\,M^+f(x)>2^{k+1},M^{+,\sharp}f(x)\leq\gamma2^{k}\right\} \right|\leq C\gamma|I_{i+1}^{k}|\quad0<\gamma<1. \]

From the $C_{q}^+$ condition it follows that

\[ w(E_{i}^{k})\leq C\gamma^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{I_{i}^{k}\cup I_{i+1}^{k}}\right)^{q}w. \]

Summing on $i$ and taking into account lemma 4.1 we have that

\begin{align*} & w\left(\left\{ x\in J_{j}^{k}\,:\,M^+f(x)>2^{k+1},M^{\sharp,+}f(x)\leq\gamma2^{k}\right\} \right)\\ & \leq C\gamma^{\varepsilon}\sum_{i}\int_{\mathbb{R}}\left(M^+\chi_{I_{i}^{k}\cup I_{i+1}^{k}}\right)^{q}w\lesssim C\gamma^{\varepsilon}\int_{\mathbb{R}}\left(M^+\chi_{J_{j}^{k}}\right)^{q}w. \end{align*}

Now summing over all $j$,

\begin{align*} & w\left(\left\{ x\in\Omega_{k}\,:\,M^+f(x)>2^{k+1},M^{+,\sharp}f(x)\leq\gamma2^{k}\right\} \right)\\ & \leq C\gamma^{\varepsilon}\sum_{j}\int_{\mathbb{R}}\left(M^+\chi_{J_{j}^{k}}\right)^{q}w. \end{align*}

Having those estimates at our disposal we can argue as follows.

\begin{align*} & \int_{\mathbb{R}}(M^+f(x))^{p}w(x)\,{\rm d}x\lesssim\sum_{k\in\mathbb{Z}}2^{kp}w(\Omega_{k})\\ & \leq \sum_{k\in\mathbb{Z}}2^{kp}w\left(\left\{ x\in\Omega_{k}\,:\,M^+f(x)>2^{k+1},M^{+,\sharp}f(x)\leq\gamma2^{k}\right\} \right)\\ & \quad +\sum_{k\in\mathbb{Z}}2^{kp}w\left(\left\{ x\in\mathbb{R}\,:\,M^{+,\sharp}f(x)>\gamma2^{k}\right\} \right)\\ & \leq C\gamma^{\varepsilon}\sum_{k\in\mathbb{Z}}2^{kp}\sum_{j}\int_{\mathbb{R}}\left(M^+\chi_{J_{j}^{k}}\right)^{q}w+\sum_{k\in\mathbb{Z}}2^{kp}w\left(\left\{ x\in\mathbb{R}\,:\,M^{+,\sharp}f(x)>\gamma2^{k}\right\} \right)\\ & \leq C\gamma^{\varepsilon}\int_{\mathbb{R}}M_{p,q}^+(M^+f)(x)w(x)\,{\rm d}x+c_{\gamma}\int_{\mathbb{R}}(M^{+,\sharp}f(x))^{p}w(x)\,{\rm d}x\\ & \leq C\gamma^{\varepsilon}\int_{\mathbb{R}}(M^+f(x)){}^{p}w(x)dx+c_{\gamma}\int_{\mathbb{R}}(M^{+,\sharp}f(x))^{p}w(x)\,{\rm d}x \end{align*}

where in the last step we have used lemma 4.2.

Observe that if $\int _{\mathbb {R}}(M^+f(x)){}^{p}w(x)\,{\rm d}x<\infty$, choosing $\gamma$ small enough the desired estimate follows. We end the proof observing that for $f\in L_{c}^{\infty }$

\[ \int_{\mathbb{R}}\left(M^{+,\sharp}f(x)\right){}^{p}w(x)\,{\rm d}x<\infty \]

implies

\[ \int_{\mathbb{R}}\left(M^+f(x)\right){}^{p}w(x)\,{\rm d}x<\infty. \]

Indeed, note that since $f\in L_{c}^{\infty }$ we may assume that $\operatorname {supp} f\subset [a,b]$. For $x>b$ we have that $M^{+,\sharp }f(x)=M^+f(x)=0$, and for $x\rightarrow -\infty$, $M^+f(x)\simeq M^{+,\sharp }f(x)\simeq \frac {1}{|x|}.$

4.2 Necessity

The proof of the necessity will rely upon the following lemma, which is a one-sided version of some of the results in [Reference Muckenhoupt30], namely theorem 4.1, lemma 5.1 and the proof of theorem 1.2 and of [Reference Sawyer34, lemma 1].

Lemma 4.3 If for every $a< b< c$ with $c-b< b-a$ and $E\subset (a,b)$

(4.1)\begin{equation} w(E)\lesssim\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\int_{\mathbb{R}}M^+(\chi_{(a,c)})^{p}w(x)\,{\rm d}x\end{equation}

then $w\in C_{p}^+$.

Before settling the lemma we show how to derive from it the necessity in theorem 1.2.

Proof of the necessity in theorem 1.2 Assume that for a certain $1< p<\infty$ and a weight $w$,

\[ \|M^+f\|_{L^{p}(w)}\lesssim\|M^{\sharp,+}f\|_{L^{p}(w)}. \]

Let $I=(a,c)$ an interval. Let $a< b< c$. Assume that $E$ is a measurable set contained in $(a,b)$. Let us define

\begin{align*} f(x) & =\log^+\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})(x)\right)+\chi_{I}(x)\\ & =g(x)+\chi_{I}(x). \end{align*}

Analogously as in [Reference Sawyer34, theorem A] and [Reference Yabuta37] we have that

(4.2)\begin{align} & \frac{1}{|(a,c)|}\int_{(a,c)}\log^+\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right)\lesssim1 \end{align}

(4.3)\begin{align} & \|f\|_{BMO^+}\lesssim1 \end{align}

(4.4)\begin{align} & f(x)=\log^+\left(\frac{|(b,c)|}{|E|}\right)+1\quad\text{a.e.}\quad x\in E. \end{align}

where $\|f\|_{BMO^+}=\|M^{\sharp,+}f\|_{L^{\infty }}$. Note (4.4) readily follows from the definition of $f$. Assume by now that (4.2) and (4.3) hold as well. We shall show that this is the case at the end of this proof. We continue as follows.

Observe that $M^{-}(\chi _{E})\leq \frac {|E|}{\operatorname {dist}(x,E)}$ for $x\not \in E$. In particular, if $x\geq c$ we have that

\[ M^{-}(\chi_{E})(x)\leq\frac{|E|}{\operatorname{dist}(x,E)}\leq\frac{|E|}{x-b}\leq\frac{|E|}{c-b}. \]

This yields $g(x)=0$ if $x\geq c$. On the other hand, if $x\leq a$ then also $M^{-}(\chi _{E})(x)=0$. And consequently $g(x)=0$. Hence $\operatorname {supp} f\subset (a,c)$. Now we observe that if $x>c$ we have that $M^+f(x)=0$ and $M^{\sharp,+}f(x)=0$. If $x< c$ we have two cases. If $x\in (a-|I|,c)$ then, by (4.3)

\[ M^{\sharp,+}f(x)\lesssim\|f\|_{BMO^+}\lesssim M^+(\chi_{(a,c)})(x). \]

If $x< a-|I|$ then $M^+\chi _{(a,c)}(x)=\frac {c-a}{c-x}$ and $c-x\geq 2|I|$, from which it follows that $a-x=c-x-|I|\geq \frac {c-x}{2}$. Hence, we have that by (4.2)

\begin{align*} M^{\sharp,+}f(x) & \lesssim M^+f(x)\leq M^+g(x)+M^+(\chi_{(a,c)})(x)\\ & \leq\frac{\int_{a}^{c}g(y)\,{\rm d}y}{a-x}+M^+(\chi_{(a,c)})(x)\\ & \lesssim\frac{c-a}{c-x}\frac{1}{c-a}\int_{a}^{c}g(y)\,{\rm d}y+M^+(\chi_{(a,c)})(x)\\ & \lesssim\frac{c-a}{c-x}+M^+(\chi_{(a,c)})(x)\lesssim M^+(\chi_{(a,c)})(x). \end{align*}

Gathering the estimates above we have that

\[ M^{\sharp,+}f(x)\lesssim M^+(\chi_{(a,c)})(x)\quad x\in\mathbb{R}. \]

Taking into account the preceding estimate and (4.4), we have that

\begin{align*} w(E) & =\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}\int_{E}w(x)\,{\rm d}x\\ & =\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\int_{E}|f(x)|^{p}w(x)\,{\rm d}x\\ & \leq\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\int_{\mathbb{R}}(M^+f(x))^{p}w(x)\,{\rm d}x\\ & \lesssim\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\int_{\mathbb{R}}|M^{\sharp,+}f(x)|^{p}w(x)\,{\rm d}x\\ & \lesssim\frac{1}{\left[1+\log^+\left(\frac{|(b,c)|}{|E|}\right)\right]^{p}}\int_{\mathbb{R}}(M^+\chi_{(a,c)})^{p}w(x)\,{\rm d}x \end{align*}

and we are done.

As we mentioned above, we are left with settling (4.2) and (4.3).

To establish (4.2), note that

\[ \frac{1}{|(a,c)|}\int_{(a,c)}\log^+\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right)=\frac{1}{|(a,c)|}\int_{(a,c)\cap F}\log\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right) \]

where $F=\left \{ \frac {|(b,c)|}{|E|}M^{-}(\chi _{E})\geq 1\right \} .$ If $|(a,c)\cap F|=0$ then there is nothing to show. Hence we may assume that $|(a,c)\cap F|\not =0$. Taking that into account we argue as follows. Let $\delta \in (0,1)$. Then

\begin{align*} & \frac{1}{|(a,c)|}\int_{(a,c)\cap F}\log\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right)\\ & =\frac{1}{\delta}\frac{|(a,c)\cap F|}{|(a,c)|}\frac{1}{|(a,c)\cap F|}\int_{(a,c)\cap F}\log\left(\left(\frac{|(b,c)|}{|E|}\right)^{\delta}M^{-}(\chi_{E})^{\delta}\right)\\ & \leq\frac{1}{\delta}\frac{|(a,c)\cap F|}{|(a,c)|}\log\left(\frac{1}{|(a,c)\cap F|}\int_{(a,c)\cap F}\left(\frac{|(b,c)|}{|E|}\right)^{\delta}M^{-}(\chi_{E})^{\delta}\right)\\ & =\frac{|(a,c)\cap F|}{|(a,c)|}\log\left(\frac{|(b,c)|}{|E|}\left(\frac{1}{|(a,c)\cap F|}\int_{(a,c)\cap F}M^{-}(\chi_{E})^{\delta}\right)^{\frac{1}{\delta}}\right)\\ & \leq\frac{|(a,c)\cap F|}{|(a,c)|}\log\left(\frac{|(b,c)|}{|E|}c_{\delta}\frac{|E|}{|(a,c)\cap F|}\right)\\ & =\frac{|(a,c)\cap F|}{|(a,c)|}\log\left(\frac{|(b,c)|}{|(a,c)\cap F|}\right)\\ & \leq\frac{|(a,c)\cap F|}{|(a,c)|}\log\left(\frac{|(a,c)|}{|(a,c)\cap F|}\right)\leq1 \end{align*}

where in the third line we used Jensen's inequality, in the fifth Kolmogorov's inequality and in the last one that $\log (t)\leq t$ for every $t\geq 1$.

Now we focus on (4.3). First we observe that

\[ \|f\|_{BMO^+}\leq\|g\|_{BMO^+}+\|\chi_{I}\|_{BMO^+}\leq\|g\|_{BMO^+}+3 \]

so it suffices to provide a bound for $\|g\|_{BMO^+}$. It is not hard to check that

\[ \|\max\left\{ h_{1},h_{2}\right\} \|_{BMO^+}\lesssim\max\left\{ \|h_{1}\|_{BMO^+},\|h_{2}\|_{BMO^+}\right\} . \]

Hence

\begin{align*} \|g\|_{BMO^+}& =\left\Vert \max\left\{ \log\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right),0\right\} \right\Vert _{BMO^+}\\ & \lesssim\left\Vert \log\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right)\right\Vert _{BMO^+} \end{align*}

Now we observe that

\begin{align*} \left\Vert \log\left(\frac{|(b,c)|}{|E|}M^{-}(\chi_{E})\right)\right\Vert _{BMO^+} & =\left\Vert \log\left(\frac{|(b,c)|}{|E|}\right)+\log\left(M^{-}(\chi_{E})\right)\right\Vert _{BMO^+}\\ & =\left\Vert \log\left(M^{-}(\chi_{E})\right)\right\Vert _{BMO^+}\\ & =2\left\Vert \log\left(M^{-}(\chi_{E})^{\frac{1}{2}}\right)\right\Vert _{BMO^+} \end{align*}

and since for every $\delta \in (0,1)$ we have that $M^{-}(\chi _{E})(\cdot )^{\delta }\in A_{1}^+$ with constant depending just on $\delta$, then using results in [Reference Martín-Reyes and de la Torre27, § 2],

\[ \left\Vert \log\left(M^{-}(\chi_{E})^{\frac{1}{2}}\right)\right\Vert _{BMO^+}\lesssim1. \]

Combining the estimates above yields (4.3). This ends the proof.

We devote the remainder of the section to settle lemma 4.3. First we will need the following lemma.

Lemma 4.4 Let $1< p<\infty$. Assume that $w$ is a weight such that

\[ \int_{E}w\lesssim\frac{1}{\left(1+\log^+\left(\frac{c-b}{|E|}\right)\right)^{p}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w \]

for every $a< b< c$ with $c-b< b-a$ and where $E$ is any measurable set contained in $(a,b)$. Then, for every family $\{I_{k}\}_{k=1}^{n}$ of disjoint subintervals of an interval $I$, if we denote

\[ \Delta(x)=\sum_{k=1}^{n}M^+(\chi_{I_{k}})^{p}(x), \]

then

(4.5)\begin{equation} \int_{\mathbb{R}}\Delta w\lesssim\frac{1}{\left(1+\log\left(\frac{|I|}{\sum_{k=1}^{n}|I_{k}|}\right)\right)^{p-1}}\int_{\mathbb{R}}\left(M^+\chi_{I}\right)^{p}w.\end{equation}

Proof. Let $I=(a,c)$ be an interval and $\{I_{k}\}$ a family of disjoint subintervals of $I$. First we note that

\[ M^+(\chi_{I_{k}})(x)=0 \]

for each $x>c$. Then we have that

\[ \int_{\mathbb{R}}\Delta w=\int_{-\infty}^{c}\Delta w=\int_{a'}^{c}\Delta w+\int_{-\infty}^{a'}\Delta w. \]

where $a'=a-|I|$. First we deal with the second term. Observe that if $d_{k}$ is the right endpoint of $I_{k}$, then

\begin{align*} \int_{-\infty}^{a'}\Delta w & =\int_{-\infty}^{a'}\sum_{k=1}^{n}\left(M^+\chi_{I_{k}}\right)^{p}(x)w(x)\,{\rm d}x\leq\int_{-\infty}^{a'}\sum_{k=1}^{n}\left(\frac{|I_{k}|}{d_{k}-x}\right)^{p}w(x)\,{\rm d}x\\ & \leq\int_{-\infty}^{a'}\left(\sum_{k=1}^{n}\frac{|I_{k}|}{d_{k}-x}\right)^{p}w(x)\,{\rm d}x\lesssim\int_{-\infty}^{a'}\left(\sum_{k=1}^{n}\frac{|I_{k}|}{c-x}\right)^{p}w(x)\,{\rm d}x\\ & \leq\int_{-\infty}^{a}\left(\frac{|I|}{c-x}\right)^{p}w(x)\,{\rm d}x\lesssim\int_{-\infty}^{c}(M^+\chi_{I})^{p}w(x)\,{\rm d}x. \end{align*}

Now we deal with the first term. Let $j$ be the least integer such that

\[ \log\left(\frac{\sum_{k=1}^{n}|I_{k}|}{|I|}\right)\leq j \]

and $J$ the least integer such that for some $D\in (0,1)$ to be chosen later,

\[ \log\left(\frac{1}{D}\log\left(\frac{e|I|}{\sum_{k=1}^{n}|I_{k}|}\right)\right)\leq J. \]

Note that since $j\leq 0$ and $J>0$ we have that $j< J$. Let

\begin{align*} Q & =\{\Delta(x)\leq e^{j}\}\\ S & =\{e^{j}<\Delta(x)\leq e^{J}\}\\ T & =\{\Delta(x)>e^{J}\} \end{align*}

For $Q$ we have that

\begin{align*} \int_{(a',c)\cap Q}\Delta(x)w(x)\,{\rm d}x & \leq\int_{(a',c)\cap Q}e^{j}w(x)\,{\rm d}x\\ & =e\int_{(a',c)\cap Q}\,{\rm e}^{j-1}w(x)\,{\rm d}x\\ & \leq e\int_{(a',c)\cap Q}\,{\rm e}^{\log\left(\frac{\sum_{k=1}^{n}|I_{k}|}{|I|}\right)}w(x)\,{\rm d}x\\ & \leq e\frac{\sum_{k=1}^{n}|I_{k}|}{|I|}\int_{(a',c)}w(x)\,{\rm d}x \end{align*}

and the right-hand side is bounded by the right-hand side of (4.5). We continue with $S$.

\begin{align*} \int_{(a',c)\cap S}\Delta(x)w(x)\,{\rm d}x & \leq\sum_{k=j}^{J-1}\,{\rm e}^{k+1}\int_{\{\Delta(x)>e^{k}\}\cap(a',c)}w(x)\,{\rm d}x\\ & =\sum_{k=j}^{J-1}e^{k+1}w(E_{e^{k}}) \end{align*}

where

\[ E_{\lambda}=\{\Delta(x)>\lambda\}\cap(a',c). \]

To continue with the argument, we borrow ideas from [Reference Riveros and de la Torre33, p. 406]. We begin noting that there exists $B>1$ and that we can choose the $D\in (0,1)$ above in such a way that

\[ |E_{\lambda}|\leq B\,{\rm e}^{{-}D\lambda}|(a',c)|. \]

At this point we use our hypothesis on the weight $w$. For that purpose we split $(a',c)$ as follows. Let $x_{0}=a'$ and let us define recursively $x_{i}-x_{i-1}=c-x_{i}$. Associated to the collection of intervals $(x_{i},x_{i+1})$ we consider the sets

\[ E_{\lambda}^{i}=E_{\lambda}\cap(x_{i},x_{i+1}). \]

Observe that for each $i$ we may assume that the elements that we consider in the sum $\Delta (x)$ are contained in $(x_{i},c)$ (the remaining terms are zero). Hence we have that

\[ |E_{\lambda}^{i}|\leq B\,{\rm e}^{{-}D\lambda}|(x_{i},c)|=4B\,{\rm e}^{{-}D\lambda}(x_{i+2}-x_{i+1}). \]

Now we use the hypothesis for $x_{i}$, $x_{i+1}$, $x_{i+2}$ and we have that

\begin{align*} w(E_{\lambda}^{i}) & \lesssim\frac{1}{\left(1+\log^+\left(\frac{x_{i+2}-x_{i+1}}{|E_{\lambda}^{i}|}\right)\right)^{p}}\int_{-\infty}^{\infty}(M^+\chi_{(x_{i},x_{i+2})})^{p}w\\ & \lesssim\frac{1}{\left(1+\log^+\left(\frac{1}{4B\,{\rm e}^{{-}D\lambda}}\right)\right)^{p}}\int_{-\infty}^{\infty}(M^+\chi_{(x_{i},x_{i+2})})^{p}w. \end{align*}

Summing in $i$ we have that it follows from the definition of the partition $x_{i}$, which leads to a geometric series, that

\[ \sum_{i}\int_{-\infty}^{\infty}\left(M^+(\chi_{(x_{i},x_{i+2})})\right)^{p}w\leq C\int_{-\infty}^{\infty}\left(M^+\chi_{(a',c)}\right)^{p}w. \]

Hence,

\begin{align*} w(E_{\lambda}) & \lesssim\frac{1}{\left(1+\log^+\left(\frac{1}{4Be^{{-}D\lambda}}\right)\right)^{p}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a',c)}\right)^{p}w\\ & \lesssim\frac{1}{\left(1+\log^+\left(\frac{1}{4B\,{\rm e}^{{-}D\lambda}}\right)\right)^{p}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w \end{align*}

and we have that

\begin{align*} \int_{(a',c)\cap S}\Delta(x)w(x)\,{\rm d}x & \leq\sum_{k=j}^{J}e^{k+1}w(E_{e^{k}})\\ & \lesssim\sum_{k=j}^{J}\frac{1}{\left(1+\log^+\left(\frac{e^{De^{k}}}{4B}\right)\right)^{p}}e^{k+1}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w \end{align*}

and it suffices to estimate the sum. We proceed as follows.

\begin{align*} & \sum_{k=j}^{J}\frac{1}{\left(1+\log^+\left(\frac{e^{De^{k}}}{4B}\right)\right)^{p}}e^{k+1}\\ & \leq\sum_{k=j}^{\left\lfloor \log\left(\frac{1}{D}\log(4B)\right)\right\rfloor }e^{k+1}+\sum_{k=\left\lfloor \log\left(\frac{1}{D}\log(4B)\right)\right\rfloor +1}^{J}\frac{1}{\left(1+De^{k}-\log(4B)\right)^{p}}e^{k+1}\\ & \lesssim e^{j}\frac{(1-e^{\log\left(\frac{1}{D}\log(4B)\right)})}{1-e}+\sum_{k=\left\lfloor \log\left(\frac{1}{D}\log(4B)\right)\right\rfloor +1}^{J}\frac{1}{\left(De^{k}+1-De^{\log\left(\frac{1}{D}\log(4B)\right)}\right)^{p}}e^{k+1}\\ & \lesssim e^{j}\frac{\frac{1}{D}\log(4B)-1}{e-1}+\sum_{k=\left\lfloor \log\left(\frac{1}{D}\log(4B)\right)\right\rfloor +1}^{J}\frac{e}{c^{p}D^{p}e^{k(p-1)}}\\ & \lesssim e^{j-1}\frac{1}{D}\log(4B)+\kappa e^{{-}J(p-1)}\\ & \lesssim\frac{\sum_{k=1}^{n}|I_{k}|}{|I|}+\kappa\frac{1}{\left[\frac{1}{D}\log\left(\frac{e|I|}{\sum_{k=1}^{n}|I_{k}|}\right)\right]^{p-1}} \end{align*}

and again this term is bounded by the right-hand side of (4.5).

Finally, for $T$ we have that

\begin{align*} \int_{T\cap(a',c)}\Delta(x)w(x)\,{\rm d}x & \leq\sum_{i=J}^{\infty}e^{i+1}\int_{\{\Delta(x)>e^{i}\}\cap(a',c)}w(x)\,{\rm d}x\\ & \lesssim\sum_{i=J}^{\infty}e^{i+1}\frac{1}{\left(1+\log^+\left(\frac{1}{4Be^{{-}De^{i}}}\right)\right)^{p}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a',c)}\right)^{p}w\\ & \lesssim\sum_{i=J}^{\infty}\frac{1}{e^{i(p-1)}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w\\ & \lesssim\frac{1}{e^{J(p-1)}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w\\ & \lesssim\frac{1}{\left[\log\left(\frac{e|I|}{\sum_{k=1}^{n}|I_{k}|}\right)\right]^{p-1}}\int_{-\infty}^{\infty}\left(M^+\chi_{(a,c)}\right)^{p}w. \end{align*}

Armed with the preceding lemma we can finally settle lemma 4.3.

Proof of lemma 4.3 Let $I=(a,c)$ be an interval. Let $\delta >0$ such that if $\sum |I_{k}|\leq 2\delta |I|$ then

(4.6)\begin{equation} \int_{\mathbb{R}}\Delta w\leq\frac{1}{2}\int_{\mathbb{R}}(M^+\chi_{I})^{p}w.\end{equation}

Now assume that $a< b< c$ where $c-b< b-a$ and let $E\subset (a,b)$ be a measurable set. Let $n$ be the least integer such that $\delta ^{n}|(a,b)|<|E|.$ Now we let $E_{j}= \{x\,:M^+(\chi _{E})(x)>\delta ^{j}\}$ for $1\leq j\leq n$. Let $J_{i}^{j}$ be the component intervals of $E_{j}$ and $\Delta _{j}(x)=\sum _{i}M^+(\chi _{J_{i}^{j}})(x){}^{p}$. We claim that for $2\leq j\leq n$

\[ \int_{\mathbb{R}}\Delta_{j-1}(x)w(x)\,{\rm d}x\leq\frac{1}{2}\int_{\mathbb{R}}\Delta_{j}(x)w(x)\,{\rm d}x. \]

Assume by now that the claim holds. Note that since $ {\chi _{E}(x)\leq \Delta _{1}(x)}$ then

\[ w(E)\leq\int_{\mathbb{R}}\Delta_{1}(x)w(x)\,{\rm d}x \]

and iterating the preceding inequality,

\[ \int_{\mathbb{R}}\Delta_{1}(x)w(x)\,{\rm d}x\leq\frac{1}{2^{n-1}}\int_{\mathbb{R}}\Delta_{n}(x)w(x)\,{\rm d}x \]

and consequently

\[ w(E)\leq2\frac{1}{2^{n}}\int_{\mathbb{R}}\Delta_{n}(x)w(x)\,{\rm d}x. \]

Note that by the definition of $n$, and taking into account that $b-c< b-a$,

\begin{align*} \delta^{n}|(a,b)|<|E| & \iff\frac{1}{2^{n(-\log_{2}\delta)}}<\frac{|E|}{|(a,b)|}\\ & \iff\frac{1}{2^{n}}<\left(\frac{|E|}{|(a,b)|}\right)^{\frac{1}{-\log_{2}\delta}}\leq\left(\frac{|E|}{|(b,c)|}\right)^{\frac{1}{-\log_{2}\delta}} \end{align*}

so if we can show that

(4.7)\begin{equation} \int_{\mathbb{R}}\Delta_{n}(x)w(x)\,{\rm d}x\lesssim\int_{\mathbb{R}}(M^+\chi_{(a,c)}(x))^{p}w(x)\,{\rm d}x\end{equation}

then

\[ w(E)\lesssim2\left(\frac{|E|}{|(b,c)|}\right)^{\frac{1}{-\log_{2}\delta}}\int_{\mathbb{R}}(M^+\chi_{(a,c)})^{p}w \]

and we would be done.

Let us settle (4.7). Observe that since

\[ \frac{|E\cap(a,b)|}{|(a,b)|}=\frac{|E|}{|(a,b)|}>\delta^{n}, \]

we have that $a\in J_{i}^{n}$ for some component $J_{i}^{n}$ of $E_{n}$. Let us call $J_{i}^{n}=(a',b')$. Observe that since $a\in J_{i}^{n}$ then $a'< a$. Observe that $M^+\chi _{E}(a')=\delta ^{n}$ and since $(a',b')$ is a component, for some $\varepsilon >0$, we have that if $x\in [a'-\varepsilon,a')$, then $M^+\chi _{E}(x)\leq \delta ^{n}.$ Note that since $E\subset (a,b)$ and $a>a'$ this yields that for every $x< a'-\varepsilon$

\[ M^+\chi_{E}(x)\leq M^+\chi_{E}(a'-\varepsilon)\leq\delta^{n}. \]

This yields that all the connected components are contained in the interval $(a',b''),$ for some $b''\leq b$, since $M^+\chi _{E}(x)=0$ for every $x>b$. Now, that by the weak type $(1,1)$ of $M^+$, combined with the definition of $n$,

\begin{align*} \sum_{i}|J_{i}^{n}| & =\left|\left\{ x\in\mathbb{R}\,:\,M^+\chi_{E}(x)>\delta^{n}\right\} \right|\leq\frac{1}{\delta^{n}}|E|\\ & =\frac{1}{\delta}\frac{1}{\delta^{n-1}}\frac{|E|}{|(a,b)|}|(a,b)|\leq\frac{1}{\delta}|(a,b)|. \end{align*}

Observe that since $(a',b')$ is some component $J_{i}^{n}$ in $E_{n}$ then

\[ a-a'\leq b'-a'\leq\sum_{i}|J_{i}^{n}|\leq\frac{1}{\delta}|(a,b)|. \]

Consequently

\begin{align*} b-a' & =b-a+a-a'\\ & \leq b-a+\frac{1}{\delta}(b-a)=\left(1+\frac{1}{\delta}\right)(b-a). \end{align*}

Since all the intervals $J_{i}^{n}$ are contained in $(a',b)$, by (4.5) we have that

\begin{align*} \int_{\mathbb{R}}\Delta w & \lesssim\frac{1}{\left(1+\log\left(\frac{|(a',b)|}{\sum_{i}|J_{i}^{n}|}\right)\right)^{p-1}}\int_{\mathbb{R}}M^+(\chi_{(a',b)})^{p}w.\\ & \leq\int_{\mathbb{R}}(M^+\chi_{(a,b)})^{p}w\leq\int_{\mathbb{R}}(M^+\chi_{(a,c)})^{p}w \end{align*}

We end the proof of (4.7) just noting that

\[ M^+(\chi_{(a'',b)})\lesssim M^+(\chi_{(a,c)}) \]

since taking into account that $|(a'',b)|\leq \frac {1}{\delta }(\frac {1}{\delta }+1)|(a,b)|$ yields that $M^+(\chi _{(a'',b)})\simeq M^+(\chi _{(a,b)})$.

Since as we have just shown (4.7) holds, we are left with settling the claim. We argue as follows. Let $H$ be a component of $E_{j}$. Note that, since $H$ is a component

(4.8)\begin{equation} \frac{|H\cap E|}{|H|}=\delta^{j}.\end{equation}

Our next step is to show that

(4.9)\begin{equation} H\cap E_{j-1}=\{M^+\chi_{H\cap E}>\delta^{j-1}\}.\end{equation}

First, we observe that the components of $E_{j-1}$ are contained in the components of $E_{j}$. Hence

\[ H\cap E_{j-1}=\bigcup I_{k}^{j-1} \]

where the $I_{k}^{j-1}$ are the components of $E_{j-1}$ contained in $H$. Then we have that if $x\in H\cap E_{j-1}$ then $x\in I_{k}^{j-1}=(\alpha,\beta )$ for some $k$. Since $x\in E_{j-1}$, then $M^+(\chi _{E})(x)>\delta ^{j-1}$ and

\[ \frac{1}{|(x,\beta)|}\int_{x}^{\beta}\chi_{E}>\delta^{j-1}. \]

Observe that $(x,\beta )\subset I_{k}^{j-1}\subset H$, so

\[ \frac{1}{|(x,\beta)|}\int_{x}^{\beta}\chi_{E\cap H}>\delta^{j-1}. \]

Consequently

\[ M^+\chi_{H\cap E}(x)>\delta^{j-1} \]

and this yields $\{M^+\chi _{H\cap E}>\delta ^{j-1}\}\supset H\cap E_{j-1}$.

Now we prove the converse inclusion. Observe that if $M^+\chi _{H\cap E}(x)>\delta ^{j-1}$ then

\[ M^+\chi_{E}(x)\geq M^+\chi_{H\cap E}(x)>\delta^{j-1} \]

and consequently $x\in E_{j-1}$. Now we have to see that $x\in H$. Let us call $H=(d,e)$. Observe that $M^+\chi _{H\cap E}(x)=0$ for every $x>e$, and hence $x\not \in \{M^+\chi _{H\cap E}>\delta ^{j-1}\}$. On the other hand, observe that since $H$ is a component of $E_{j}$ there exist some $\varepsilon <0$ such that if $x\in [d-\varepsilon,d]$ then $M^+\chi _{E}(x)\leq \delta ^{j}$. Relying upon this fact, note that if $x< d$ then we have that if $x\in [d-\varepsilon,d]$, then $M^+\chi _{H\cap E}(x)\leq M^+\chi _{E}(x)\leq \delta ^{j}$ and consequently $x\not \in \{M^+\chi _{H\cap E}>\delta ^{j-1}\}$ and if $x< d-\varepsilon$ then, $M^+\chi _{H\cap E}(x)\leq M^+\chi _{H\cap E}(d-\varepsilon )\leq \delta ^{j}$ and also $x\not \in \{M^+\chi _{H\cap E}>\delta ^{j-1}\}$. Hence

\[ \{M^+\chi_{H\cap E}>\delta^{j-1}\}\subset H \]

and we are done.

The weak type $(1,1)$ of $M^+$ combined with (4.9) yields

\[ |H\cap E_{j-1}|\leq\delta^{1-j}|E\cap H| \]

and combining this with (4.8) we have that

\[ |H\cap E_{j-1}|\leq\delta|H|. \]

If we denote

\[ \Delta_{\mathcal{H}}(x)=\sum_{I_{k}\in\mathcal{H}}(M^+\chi_{I_{k}}(x))^{p} \]

where $\mathcal {H}$ is the set of component intervals in $H\cap E_{j-1}$, we have that

\[ \sum_{I_{k}\in\mathcal{H}}|I_{k}|\leq|H\cap E_{j-1}|\leq\delta|H|. \]

By the definition of $\delta$ and (4.6) we have that

\[ \int_{-\infty}^{\infty}\Delta_{\mathcal{H}}(x)w(x)\,{\rm d}x\leq\frac{1}{2}\int_{-\infty}^{\infty}M^+(\chi_{H})^{p}w(x)\,{\rm d}x. \]

Adding those inequalities for all the components $H$ of $E_{j}$ gives

\[ \int_{\mathbb{R}}\Delta_{j-1}(x)w(x)\leq\frac{1}{2}\int_{\mathbb{R}}\Delta_{j}(x)w(x)\,{\rm d}x \]

for $2\leq j\leq n$ as we wanted to show. This ends the proof of the theorem.