1 Introduction

Let f be a locally integrable function on \(\mathbb {R}^n\), the Hardy–Littlewood maximal function is defined as

$$\begin{aligned} Mf(x) = \sup _{x\in Q} \frac{1}{|Q|} \int _{Q} |f(y)|dy, \end{aligned}$$

where the supremum is taken over all cubes Q containing x in \(\mathbb {R}^n\), with sides parallel to the axis. In 1972, Muckenhoupt [18] showed that Hardy–Littlewood maximal function was of weak type (p, p) with respect to a pair of weights (u, v) if and only if \((u,v) \in A_p, 1\le p<\infty \). Since then many scholars have tried to extend Muckenhoupt’s result to various function spaces. In particular, let \(\varphi \) be a Young function, L. Pick [21] generalized Muckenhoupt’s result to the following weak and extra–weak type inequalities in Orlicz classes:

  1. (1)

    The weak type inequality:

    $$\begin{aligned} \varphi (\lambda ) v(\{x: Mf(x) > \lambda \}) \le C\int _{\mathbb {R}} \varphi (c|f(x)|)u(x)dx, \end{aligned}$$

    where the constants C and c are independent of f and \(\lambda >0\).

  2. (2)

    The extra–weak type inequality:

    $$\begin{aligned} v(\{x: Mf(x) > \lambda \}) \le C\int _{\mathbb {R}} \varphi \left( \frac{C|f(x)|}{\lambda } \right) u(x)dx, \end{aligned}$$

    where the constant C is independent of f and \(\lambda >0\). Besides, since weighted weak and extra–weak type inequalities for maximal functions are an important part of weighted theory, some related work in Orlicz classes was extensively studied(see [1, 3, 4, 6, 8, 9, 22]).

The one-sided Hardy–Littlewood maximal function \(M^+f\) is defined as

$$\begin{aligned} M^+ f(x) = \sup _{h>0} \frac{1}{h} \int _x^{x+h} |f(y)|dy, \end{aligned}$$

where \(f: \mathbb {R} \rightarrow \mathbb {R}\) is a measurable function. In 1986, E. Sawyer [26] began to study the one-sided Hardy–Littlewood maximal function and characterized the weak weighted inequalities for the one-sided Hardy–Littlewood maximal function on \(\mathbb {R}\). Since these results played an important role in ergodic theory, the research on weighted inequalities for the one-sided maximal function has aroused the interest of many scholars (see [11, 12, 14,15,16,17, 20, 23,24,25, 27]et al). For instance, P. Ortega Salvador and L. Pick [20] characterized two-weight weak and extra-weak type inequalities for the one-sided Hardy–Littlewood maximal function on \(\mathbb {R}\). Since the study of the one-sided Hardy–Littlewood maximal function in higher dimensions is very complex and difficult, in the last decade, only a few scholars studied the case of higher dimensions [2, 5, 7, 13, 19]. A fundamental result in higher dimensions case is attributed to S. Ombrosi [19], where the author introduced a dyadic one–sided maximal function on \(\mathbb {R}^n\) and characterized the pair of weights (w, v) such that the operator associated with this maximal function applied \(L^p(v)\) into weak–\(L^p(w)\) for \(1 \le p< \infty \). Further, L. Forzani et al [5] showed a characterization of the pairs of weights (w, v) such that the operator on \(\mathbb {R}^2\) is of weak type (p, p). In particular, A. Ghosh and P. Mohanty [7] ingeniously extended the ideas developed in [5, 19, 21] and successfully generalized extra–weak and weak type inequalities for the one–sided maximal function from \(\mathbb {R}\) to \(\mathbb {R}^2\), which motivated the study for the one-sided Hardy–Littlewood maximal function in higher dimensions. Inspired by the above work, in this paper, we obtain a necessary and sufficient condition for a three-weight weak type one-sided Hardy–Littlewood maximal inequality on \(\mathbb {R}^2\) of the form

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: M^+f(x)> \lambda \}) \le C\int _{\mathbb {R}^2}\varphi _2 \left( C\frac{|f(x)|\varrho (x)}{\gamma (\lambda )} \right) \sigma (x)dx. \end{aligned}$$
(1.1)

In this process, we find that inequalities (3.1) and (3.2) (see Sect. 3) can be unified in the above inequality (1.1) (see Remark 4.1), which is a novel and interesting result.

The remainder of this paper is organized as follows: In Sect. 2, as preliminaries we recapitulate some basic notions and lemmas. In Sect. 3, we review extra–weak and weak type characterizations for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). Finally, in Sect. 4, we introduce the definition of a three–weight weak type inequality for the one-sided Hardy–Littlewood maximal function and establish its characterization theorem on \(\mathbb {R}^2\).

2 Preliminaries

Throughout this paper, let \(\Phi \) be the set of all functions \(\varphi \), where \(\varphi : (0,\infty ] \rightarrow (0,\infty ]\) are even, continuous and increasing functions such that \(\varphi (t) \rightarrow 0\) as \(t \rightarrow 0^+\), \(\varphi (t) \rightarrow \infty \) as \(t \rightarrow \infty \).

A function \(\omega \in \Phi \) is called a Young function if it is convex and satisfies \(\lim _{t \rightarrow 0^+}\frac{\omega (t)}{t} = \lim _{t \rightarrow \infty } \frac{t}{\omega (t)} =0\). We say \(\varphi \) is a quasi–convex function if for any \(t \in [0,\infty ) \) and \(c >0\), the inequality \(\omega (t) \le \varphi (t) \le c\omega (ct)\) is true, where \(\omega \) is a Young function. \(\varphi \) is quasi–convex on \([0,\infty )\) if and only if there exists a constant \(C >0\) such that the inequality

$$\begin{aligned} \varphi (tx_1+(1-t)x_2) \le C (t\varphi (Cx_1) + (1-t)\varphi (Cx_2)) \end{aligned}$$

holds for all \(x_1,x_2 \in [0,+\infty )\) and \(t \in (0,1)\), where C is independent of \(x_1,x_2,t\)(see [10]).

For a quasi–convex function \(\varphi \), its complementary function \({\tilde{\varphi }}\) is defined by

$$\begin{aligned} {\tilde{\varphi }}(t) = \sup _{s \ge 0} (st - \varphi (s)). \end{aligned}$$

We denote \(R_{\varphi }(t) = \frac{\varphi (t)}{t}\) and \(S_{\varphi }(t)=\frac{{\tilde{\varphi }}(t)}{t}\). The subadditivity of the supremum readily implies that \({\tilde{\varphi }}\) is a Young function, and from the definition of the complementary function \({\tilde{\varphi }}\) we obtain the Young inequality

$$\begin{aligned} st \le \varphi (s) + {\tilde{\varphi }}(t),~~~~s,t\ge 0. \end{aligned}$$
(2.1)

Lemma 2.1

[26] For a quasi–convex function \(\varphi \), we have

$$\begin{aligned} \varepsilon \varphi (t) \le \varphi (C\varepsilon t), ~t>0,~ \varepsilon >1 \end{aligned}$$
(2.2)

and

$$\begin{aligned} \varphi (\gamma t) \le \gamma \varphi (Ct),~ t>0,~ \gamma <1, \end{aligned}$$
(2.3)

where the constants C do not depend on \(t, ~\varepsilon ,\) and \(\gamma \).

Lemma 2.2

[26] Let \(\varphi \in \Phi \) be a quasi–convex function, then there is a constant \(\delta > 0\) such that for an arbitrary \(t >0\), we have

$$\begin{aligned} {\tilde{\varphi }}\left( \delta \frac{\varphi (t)}{t}\right) \le \varphi (t) \le {\tilde{\varphi }}\left( 2\frac{\varphi (t)}{t}\right) \end{aligned}$$

and

$$\begin{aligned} \varphi \left( \delta \frac{{\tilde{\varphi }}(t)}{t}\right) \le {\tilde{\varphi }}(t) \le \varphi \left( 2\frac{{\tilde{\varphi }}(t)}{t}\right) . \end{aligned}$$
(2.4)

Throughout this paper, we use C and \(C_i\) to denote positive constants and may denote different constants at different occurrences.

For any interval \(I = [a,b), I^+:= [b,2b-a)\) and for any cube \(Q = I_1 \times \cdots \times I_n\) in \(\mathbb {R}^n\), \(Q^+:= I_1^+ \times \cdots \times I_n^+\). Cubes with side–length of the form \(2^k, k \in \mathbb {Z}\), are called cubes of dyadic size.

we will show the definition of the one-sided Hardy–Littlewood maximal function \(M^+f\) on \(\mathbb {R}^2\).

Definition 2.1

[7] For a locally integrable function f on \(\mathbb {R}^2\), the one-sided Hardy–Littlewood maximal function \(M^+f\) is defined by

$$\begin{aligned} M^+f(x) = \sup _{h>0} \frac{1}{h^2} \int _{Q_{x,h}} |f(y)|dy, \end{aligned}$$

where \(x=(x_1,x_2) \in \mathbb {R}^2\), h is a positive real number and \(Q_{x,h} = [x_1, x_1+h) \times [x_2, x_2 +h).\)

Corresponding to any cube \(Q:= [a-h,a) \times [b-h,b), Q^+:= [a,a+h) \times [b,b+h)\), \({\tilde{Q}}:= [a-h,a+\frac{h}{2}) \times [b-\frac{3\,h}{2},b)\) and \(Q^2:= [a-\frac{h}{2},a) \times [b-h,b-\frac{h}{2})\), respectively. See. Fig. 1.

Fig. 1
figure 1

Cubes  \(Q,~ Q^+, ~{\tilde{Q}}\) and \(Q^{+2}\)

Full size image

An almost everywhere positive local integrable function \(\omega : X \rightarrow \mathbb {R}\) is called a weight function. Given a weight function \(\omega \) and a measurable set E, we use the notation

$$\begin{aligned} \omega (E) = \int _{E} \omega (x) dx \end{aligned}$$

to denote the \(\omega \)–measure of the E. We also denote that \(g_Q:= \frac{1}{|Q|}\int _{Q} g(x) dx\), where g is a locally integrable function and Q is a cube.

3 Extra–Weak and Weak Type Inequalities for the One–Sided Maximal Function on \(\mathbb {R}^2\)

In [7], A. Ghosh and P. Mohanty showed extra–weak and weak type characterizations for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). Now we recall some relevant definitions and results for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\).

Definition 3.1

[7] Let (u, v) be a pair of weights. If there are constants \(C_1 >0\) and \(\varepsilon >0\) such that

$$\begin{aligned} \sup _{Q}\int _{Q^+} {\tilde{\varphi }}\left( \frac{\varepsilon v_Q}{u(x)} \right) \frac{u(x)}{v(Q)}dx \le C_1 <\infty , \end{aligned}$$

we say (u, v) satisfies \(A^+_{ew}(\varphi )\) condition (or \((u,v) \in A^+_{ew}(\varphi ))\).

Definition 3.2

[7] Let (u, v) be a pair of weights. If there is \(\varepsilon >0\) such that

$$\begin{aligned} \sup _{\alpha >0} \sup _{Q} \frac{\alpha v(Q)}{|Q|} R_{\varphi }\left( \frac{\varepsilon }{|Q|}\int _{Q^+} S_{\varphi } \left( \frac{1}{\alpha u(x)}\right) dx \right) <\infty , \end{aligned}$$

we say (u, v) satisfies \(A^+_{weak}(\varphi )\) condition (or \((u,v) \in A^+_{weak}(\varphi ))\).

Fig. 2
figure 2

Subcubes

Full size image

In [5], Forzani et al showed that square \(Q_{x,h} = [x_1, x_1+h) \times [x_2,x_2+h)\) can be divided into four squares (see, Fig. 2):

$$\begin{aligned} Q_{x,h}= Q_{x,\frac{h}{2}}~ \cup ~Q_{x,h}^1 ~\cup ~ Q_{x,h}^2 ~\cup ~ Q_{x,h}^3, \end{aligned}$$

where

$$\begin{aligned} Q_{x,\frac{h}{2}}= & {} \left[ x_1, x_1+\frac{h}{2}\right) \times \left[ x_2, x_2+\frac{h}{2}\right) , \\ Q_{x,h}^1= & {} \left[ x_1+\frac{h}{2}, x_1+h\right) \times \left[ x_2+\frac{h}{2}, x_2+h\right) ,\\ Q_{x,h}^2= & {} \left[ x_1+\frac{h}{2}, x_1+h\right) \times \left[ x_2, x_2+\frac{h}{2}\right) ,\\ Q_{x,h}^3= & {} \left[ x_1, x_1+\frac{h}{2}\right) \times \left[ x_2+\frac{h}{2}, x_2+h\right) . \end{aligned}$$

They defined

$$\begin{aligned} M^{+1}f(x)= & {} \sup _{h>0} \frac{1}{|Q_{x,h}^1|}\int _{Q_{x,h}^1}|f(y)|dy,\\ M^{+2}f(x)= & {} \sup _{h>0} \frac{1}{|Q_{x,h}^2|}\int _{Q_{x,h}^2}|f(y)|dy,\\ M^{+3}f(x)= & {} \sup _{h>0} \frac{1}{|Q_{x,h}^3|}\int _{Q_{x,h}^3}|f(y)|dy. \end{aligned}$$

Then they proved that \(M^+\) is essentially equivalent to the sum of the maximal operators \(M^{+i}, i=1,2,3.\) i.e the following Proposition 3.1:

Proposition 3.1

[5] The following inequality holds for every measurable function:

$$\begin{aligned} \frac{1}{12}(M^{+1}f(x) + M^{+2}f(x) +M^{+3}f(x))\le & {} M^{+}f(x)\\\le & {} \frac{1}{3}(M^{+1}f(x) + M^{+2}f(x) +M^{+3}f(x)). \end{aligned}$$

Like in [5, 7], for the convenience of the proof of our main result, we also shall use the maximal operator \({\mathcal {M}}^{+}\) defined by

$$\begin{aligned} {\mathcal {M}}^{+}f(x) = \sup _{k \in \mathbb {Z}}\frac{1}{|Q_{x,2^k}|}\int _{Q_{x,2^k}}|f(y)|dy, \end{aligned}$$

that is, we only take cubes \(Q_{x,h}\) of dyadic size. This operator is essentially equivalent to \(M^+\). At the same time, we also consider operators \({\mathcal {M}}^{+i},i=1,2,3,\) defined by

$$\begin{aligned} {\mathcal {M}}^{+i}f(x) = \sup _{k \in \mathbb {Z}}\frac{1}{|Q_{x,2^k}^i|}\int _{Q_{x,2^k}^i}|f(y)|dy. \end{aligned}$$

Similarly, \({\mathcal {M}}^+\) is essentially equivalent to the sum of the maximal operators \({\mathcal {M}}^{+i}, i=1,2,3.\) So we also have the Proposition 3.2:

Proposition 3.2

[5] The following inequality holds for every measurable function:

$$\begin{aligned} \frac{1}{12}({\mathcal {M}}^{+1}f(x)+ & {} {\mathcal {M}}^{+2}f(x) +{\mathcal {M}}^{+3}f(x)) \le {\mathcal {M}}^{+}f(x) \\\le & {} \frac{1}{3}({\mathcal {M}}^{+1}f(x) + {\mathcal {M}}^{+2}f(x) +{\mathcal {M}}^{+3}f(x)). \end{aligned}$$

So in the proof of the main result we can just use the maximal operator \({\mathcal {M}}^{+}\).

In [7], A. Ghosh and P. Mohanty made use of the maximal operator \({\mathcal {M}}^{+}\) establishing Theorems 3.1 and 3.2.

Theorem 3.1

[7] Let \(\varphi \) be a Young function and (u, v) be a pair of weights on \(\mathbb {R}^2\). Then there exist constants \(C_2,C_3 >0\) such that

$$\begin{aligned} v(\{x \in \mathbb {R}^2: M^+ f(x) > \lambda \}) \le C_2 \int _{\mathbb {R}^2} \varphi \left( \frac{C_3|f(x)|}{\lambda } \right) u(x)dx, \end{aligned}$$
(3.1)

holds for all \(\lambda >0\) and \(f \in L_{\varphi }(u)\) if and only if \((u,v) \in A^+_{ew}(\varphi )\).

Theorem 3.2

[7] Let \(\varphi \) be a Young function and (u, v) be a pair of weights on \(\mathbb {R}^2\). Then there exists a constant \(C_4>0\) such that for each \(\lambda >0\) and every locally integrable function f the inequality

$$\begin{aligned} \varphi (\lambda )v(\{x \in \mathbb {R}^2: M^+ f(x) > \lambda \}) \le C_4 \int _{\mathbb {R}^2} \varphi (C_4|f(x)|)u(x)dx, \end{aligned}$$
(3.2)

holds if and only if \((u,v) \in A^+_{weak}(\varphi )\).

4 Main Result

This section is devoted to proving a three–weight weak type characterization for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). We start by defining appropriate weights.

Definition 4.1

Let \(\omega ,\varrho ,\sigma \) be three weight functions. we say \((\omega ,\varrho ,\sigma )\) satisfies \(A^+_{weak}(\varphi _2)\) condition (or \((\omega ,\varrho ,\sigma )\in A^+_{weak}(\varphi _2))\) if there are constants \(C_5>0, \varepsilon >0\) such that

$$\begin{aligned} \sup _Q\sup _{\lambda >0} \frac{1}{\varphi _1(\lambda )\omega (Q)}\int _{Q^+} {\tilde{\varphi }}_2 \left( \varepsilon \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{ |Q| \sigma (x)\varrho (x)} \right) \sigma (x)dx \le C_5 < \infty , \end{aligned}$$

where \(\varphi _1\) and \(\gamma \) are nondecreasing functions defined on \([0,\infty )\), \(\varphi _2\) is a quasi-convex function.

To state and prove the main result of this section, we need the following Lemma 4.1.

Lemma 4.1

[7] Let \(\lambda >0\) and K be any compact set in \(E^2_{\lambda }\) (see(4.6)). Then there exist a finite family of cubes \(\{Q_i\}_{i \in \Gamma }\) and sets \(\{P_i\}_{i \in \Gamma }\) such that \(P_i\subset {\tilde{Q}}_i^+\) with

  1. 1.

    \(\omega (K) \le 2 \sum _{i \in \Gamma } \omega ({\tilde{Q}}_i)\), where \(\omega \) is a weight function;

  2. 2.

    \(\frac{1}{|{\tilde{Q}}_i|} \int _{P_i} |f(x)|dx > \frac{\lambda }{8}\) with \(\sum _{i \in \Gamma } \chi _{P_i} \le k\), where the constant k is independent of everything.

The main result of this paper is stated as follows:

Theorem 4.1

Let \(\varphi _1\) and \(\gamma \) be nondecreasing functions defined on \([0,\infty )\), \(\varphi _2\) be a quasi-convex function and \(\omega ,\varrho ,\sigma \) be three weight functions on \(\mathbb {R}^2\). Then there exists a constant \(C_6>0\) such that for all \(\lambda >0\) and every locally integrable function f the inequality

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: M^+f(x)> \lambda \}) \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx \end{aligned}$$
(4.1)

holds if and only if \((\omega ,\varrho ,\sigma )\in A^+_{weak}(\varphi _2)\).

Proof

Sufficient: Since \(M^+\) and \({\mathcal {M}}^{+}\) are essentially equivalent, it is enough to prove (4.1) for the operator \({\mathcal {M}}^{+}\). That is, we are going to prove the following inequality:

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: {\mathcal {M}}^+f(x)> \lambda \}) \le C_6 \int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx. \end{aligned}$$
(4.2)

Observe that (4.2) follows from the inequality

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: \lambda < {\mathcal {M}}^+f(x)\le 2\lambda \}) \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx.\nonumber \\ \end{aligned}$$
(4.3)

In fact, if (4.3) holds, we have

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: {\mathcal {M}}^+f(x)> \lambda \})\le & {} \sum _{k \ge 0}\varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: 2^k \lambda< {\mathcal {M}}^+f(x)\le 2^{k+1}\lambda \} ) \\= & {} \sum _{k \ge 0}\varphi _1(\lambda )\omega \left( \left\{ x \in \mathbb {R}^2: \lambda < {\mathcal {M}}^+\frac{f(x)}{2^k}\le 2\lambda \right\} \right) \\\le & {} C_6\sum _{k \ge 0}\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{2^k\gamma (\lambda )}\right) \sigma (x)dx\\\le & {} C_6\left( \sum _{k \ge 0}\frac{1}{2^k}\right) \int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx. \end{aligned}$$

So if (4.3) holds, we obtain that (4.2) is true.

In order for (4.3) to hold, by Proposition 3.2, we only have to prove that

$$\begin{aligned}{} & {} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: \lambda < {\mathcal {M}}^{+i}f(x), {\mathcal {M}}^{+}f(x)\le 2\lambda \}) \nonumber \\{} & {} \quad \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx, ~~for~~ i=1,2,3. \end{aligned}$$
(4.4)

To avoid difficulty arising from very small cubes we work with the following truncated maximal function, for any \(\xi >0\), we denote

$$\begin{aligned} {\mathcal {M}}^{+i}_\xi f(x) = \sup _{k \in \mathbb {Z}~:~h=2^k>\xi }\frac{1}{|Q^i_{x,2^k}|} \int _{Q^i_{x,2^k}}|f(y)|dy, ~~for~~i=1,2,3. \end{aligned}$$

Since \({\mathcal {M}}^{+i}_\xi f \uparrow {\mathcal {M}}^{+i} f\) as \(\xi \downarrow 0^+\), by monotone convergence theorem (4.5) implies (4.4).

$$\begin{aligned}{} & {} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: \lambda < {\mathcal {M}}^{+i}_\xi f(x), {\mathcal {M}}^{+}f(x)\le 2\lambda \})\nonumber \\{} & {} \quad \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx, ~~for~~ i=1,2,3. \end{aligned}$$
(4.5)

So for any \(\lambda >0\), consider

$$\begin{aligned} E^i_{\lambda } = \{x \in \mathbb {R}^2: \lambda < {\mathcal {M}}^{+i}_\xi f(x),{\mathcal {M}}^{+}f(x) \le 2\lambda \},~~ for ~~i=1,2,3. \end{aligned}$$
(4.6)

We need only to prove the following inequality

$$\begin{aligned} \varphi _1(\lambda )\omega (E^i_{\lambda }) \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx,~~~~~~~~for ~~i=1,2,3. \end{aligned}$$

Here, we only prove the case for \(i=2\), as the other cases are similar. We notice that the weighted measure \(\omega (x)dx\) is finite on compact sets since \(\omega \) is locally integrable. Therefore it is enough to prove the following inequality

$$\begin{aligned} \varphi _1(\lambda )\omega (K) \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx, \end{aligned}$$
(4.7)

for all compact set \(K \subset E^2_{\lambda }\).

In fact, let K be any compact set in \(E^2_{\lambda }\). By applying Lemma 4.1, there exist a finite index set \(\Gamma \), cubes \(\{{\tilde{Q}}_{x_i}: i \in \Gamma \}\) and sets \(F_{x_i} \subset {\tilde{Q}}_{x_i}^+\) such that

$$\begin{aligned} \omega (K) \le 2\sum _{i \in \Gamma }\omega ({\tilde{Q}}_{x_i}),~~ \frac{1}{|{\tilde{Q}}_{x_i}|}\int _{F_{x_i}}|f(x)|dx > \frac{\lambda }{8}, ~~and ~~\sum _{i \in \Gamma }\chi _{F_{x_i}} \le k. \end{aligned}$$
(4.8)

According to (4.8), (2.1) and Definition 4.1, we have

$$\begin{aligned} \varphi _1(\lambda )\omega ({\tilde{Q}}_{x_i})< & {} \varphi _1(\lambda )\omega ({\tilde{Q}}_{x_i})\frac{8}{\lambda |{\tilde{Q}}_{x_i}|}\int _{F_{x_i}}|f(x)|dx \\= & {} 8\int _{F_{x_i}}\frac{|f(x)|\varrho (x)}{\gamma (\lambda )} \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda } \frac{\omega ({\tilde{Q}}_{x_i})}{|{\tilde{Q}}_{x_i}|} \frac{1}{\varrho (x)\sigma (x)} \sigma (x)dx\\\le & {} \frac{1}{2C_5} \int _{F_{x_i}}\varphi _2\left( \frac{16C_5}{\varepsilon }\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx \\{} & {} +\frac{1}{2C_5} \int _{F_{x_i}}{\tilde{\varphi }}_2\left( \varepsilon \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda } \frac{\omega ({\tilde{Q}}_{x_i})}{|{\tilde{Q}}_{x_i}|} \frac{1}{\varrho (x)\sigma (x)}\right) \sigma (x)dx\\\le & {} \frac{1}{2C_5} \int _{F_{x_i}}\varphi _2\left( \frac{16C_5}{\varepsilon }\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx +\frac{1}{2}\varphi _1(\lambda )\omega ({\tilde{Q}}_{x_i}). \end{aligned}$$

This implies

$$\begin{aligned} \varphi _1(\lambda )\omega ({\tilde{Q}}_{x_i}) \le \frac{1}{C_5} \int _{F_{x_i}}\varphi _2\left( \frac{16C_5}{\varepsilon }\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx. \end{aligned}$$

Then we have

$$\begin{aligned} \varphi _1(\lambda )\omega ({\tilde{Q}}_{x_i}) \le C_6 \int _{F_{x_i}}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx, \end{aligned}$$

where \(C_6= \max \left\{ \frac{1}{C_5},\frac{16C_5}{\varepsilon }\right\} \).

Now summing over all \(i \in \Gamma \) and using \(\sum _{i \in \Gamma }\chi _{F_{x_i}} \le k\), we obtain the inequality (4.7):

$$\begin{aligned} \varphi _1(\lambda )\omega (K) \le C_6\int _{\mathbb {R}^2}\varphi _2\left( C_6\frac{|f(x)|\varrho (x)}{\gamma (\lambda )}\right) \sigma (x)dx. \end{aligned}$$

Necessary: Let Q be a square and f be a nonnegative and locally integrable function supported on \(Q^+\). For \(k \in \mathbb {N}\) and \(\lambda > 0\), put \(E_k = \{x \in Q^+: \sigma (x)\varrho (x) > \frac{1}{k}\}\) and

$$\begin{aligned} g(x) = \left( \frac{\varphi _1(\lambda )}{\lambda }\frac{\omega (Q)}{ |Q| \sigma (x)\varrho (x)} \right) ^{-1}{\tilde{\varphi }}_2\left( \varepsilon \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{|Q|\sigma (x)\varrho (x)}\right) \chi _{E_k}(x), \end{aligned}$$

where \(\chi _{E_k}(x)\) denotes the characteristic function of the set \(E_k\) and \(\varepsilon \) will be specified later. Then

$$\begin{aligned} I= & {} \int _{E_k} {\tilde{\varphi }}_2\left( \varepsilon \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{|Q|\sigma (x)\varrho (x)}\right) \sigma (x)dx\\= & {} \frac{\varphi _1(\lambda )}{\lambda } \frac{\omega (Q)}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx. \end{aligned}$$

If \(\frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx < \lambda \), then we have

$$\begin{aligned} I \le \varphi _1(\lambda )\omega (Q); \end{aligned}$$

if \(\frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx > \lambda \), put \(f(x) = \lambda \left( \frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx\right) ^{-1}\frac{g(x)}{\varrho (x)}\), then by (4.1) and (2.2) of Lemma 2.1, we have

$$\begin{aligned} I\le & {} \frac{\varphi _1(\lambda )}{\lambda }\frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}\omega (\{x \in \mathbb {R}^2: M^+ f(x) > \lambda \})dx \\\le & {} \frac{1}{\lambda }\frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx \cdot C_6 \int _{\mathbb {R}^2}\varphi _2\left( C_6\left( \frac{1}{|Q|}\int _{Q^+}\frac{g(x)}{\varrho (x)}dx\right) ^{-1}\frac{g(x)\lambda }{\gamma (\lambda )}\right) \sigma (x)dx \\\le & {} C_6 \int _{\mathbb {R}^2}\varphi _2\left( CC_6\frac{g(x)}{\gamma (\lambda )}\right) \sigma (x)dx, \end{aligned}$$

where C is the constant in (2.2).

Consequently,

$$\begin{aligned} I \le \varphi _1(\lambda )\omega (Q) + C_6 \int _{\mathbb {R}^2}\varphi _2\left( CC_6\frac{g(x)}{\gamma (\lambda )}\right) \sigma (x)dx. \end{aligned}$$

Then choose \(\varepsilon \) so small that \(\frac{C ^2 C_6 \varepsilon }{\delta } <1\), where the constant \(\delta \) is from (2.4). By (2.3) of Lemma 2.1, (2.4) of Lemma 2.2 and the definition of g we obtain, from the inequality, we have

$$\begin{aligned} I \le \varphi _1(\lambda )\omega (Q) + \frac{C^2 C_6^2 \varepsilon }{\delta }I. \end{aligned}$$
(4.9)

Next we will show that I is finite for a sufficiently small \(\varepsilon \). If \(\lim _{t \rightarrow \infty }\frac{\varphi _2(t)}{t} = \infty \), then \({\tilde{\varphi }}_2\) is finite everywhere and thus

$$\begin{aligned} I \le {\tilde{\varphi }}_2\left( \varepsilon k \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{ |Q|}\right) \sigma (Q^+) < \infty , \end{aligned}$$

since \(\sigma \) and \(\omega \) are locally integrable.

If \(\frac{\varphi _2(t)}{t}\) is bounded, let now \(\varphi _2(t) \le At, A >0\). Then (4.1) implies

$$\begin{aligned} \gamma (\lambda )\varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: M^+f(x)> \lambda \}) \le C_6\int _{\mathbb {R}^2}|f(x)|\varrho (x)\sigma (x)dx. \end{aligned}$$

Now take \(f(x)=\lambda \frac{|Q|}{|E|}\chi _E(x)\), where E is a measurable subset of Q, then

$$\begin{aligned} \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{|Q|} \le \frac{C_6}{|E|} \int _E \sigma (x)\varrho (x)dx, \end{aligned}$$

which yields the estimate

$$\begin{aligned} \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{|Q|\sigma (x)\varrho (x)} \le C_6, \end{aligned}$$

almost everywhere on Q, where \(C_6\) is independent of Q and \(\lambda \).

Therefore we conclude that

$$\begin{aligned} I \le {\tilde{\varphi }}_2(\varepsilon C_6)\sigma (Q^+). \end{aligned}$$

So we can choose \(\varepsilon \) so small that \({\tilde{\varphi }}_2(\varepsilon C_6) < \infty \), we see that I is finite.

Further, if \(\frac{C^2 C_6^2 \varepsilon }{\delta } <1\), it follows from (4.9) that

$$\begin{aligned} \int _{E_k} {\tilde{\varphi }}_2\left( \varepsilon \frac{\varphi _1(\lambda )\gamma (\lambda )}{\lambda }\frac{\omega (Q)}{|Q| \sigma (x)\varrho (x)}\right) \sigma (x)dx \le \frac{1}{1-\frac{C^2 C_6^2 \varepsilon }{\delta }}\varphi _1(\lambda )\omega (Q). \end{aligned}$$

Now let \(k \rightarrow \infty \), we have \((\omega ,\varrho ,\sigma )\in A^+_{weak}(\varphi _2)\). \(\square \)

Remark 4.1

We all know that A. Ghosh and P. Mohanty have shown extra-weak and weak type characterizations for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). Now, in (4.1), if we put \(\varphi _1(\lambda ) = 1,\omega = v, \varphi _2 = \varphi , \gamma (\lambda ) =\lambda , \varrho (x)=1, \sigma = u,\) then (4.1) will become (3.1), i.e the extra-weak type characterization for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). Besides, in (4.1), if we put \(\varphi _1 = \varphi _2 = \varphi , \omega = v, \varrho (x)=1, \gamma (\lambda ) =1, \sigma = u,\) then (4.1) will become (3.2), i.e the weak type characterization for the one–sided maximal function \(M^+f\) on \(\mathbb {R}^2\). That is, (3.1) and (3.2) can be unified in (4.1), which is an interesting result.