Some new weighted weak-type iterated and bilinear modified Hardy inequalities

Abstract

We characterize the good weights for some weighted weak-type iterated and bilinear modified Hardy inequalities to hold.

1 Introduction and results

The initial problem in the theory of weighted Hardy inequalities was the one of characterizing the positive functions w, v, the weights, such that

$$\begin{aligned} \left( \int \limits _a^b\left( \int \limits _a^xf\right) ^qw(x) \text {d}x\right) ^{\frac{1}{q}}\le C\left( \int \limits _a^bf^pv\right) ^{\frac{1}{p}} \end{aligned}$$

(1.1)

holds for all positive measurable function f with a positive constant C independent of f, which means that the Hardy operator $Tf(x)=\int _a^xf$ is bounded from $L^p(v)$ to $L^q(w)$.

This problem was solved by Talenti [31], Muckenhoupt [23] and Bradley [4] in the case $p\le q$, by Mazja [22] when $1\le q<p$, Sinnamon [27, 28] for $0<q<1<p$ and Sinnamon and Stepanov [29] for $0<q<1=p$. Their results are the following ones.

Theorem A

([4, 22, 23, 29, 31]) Let $1<q<\infty$, $1\le p<\infty$ and let w, v be positive measurable functions on (a, b), where $-\infty \le a<b\le \infty$. Then there exists a positive constant C such that inequality (1.1) holds for all nonnegative functions f if and only if

• (i) in the case $p\le q$,

  $$\begin{aligned} B_1\equiv \sup _{s\in (a,b)}\left( \int \limits _s^bw\right) ^{\frac{1}{q}}\Vert \chi _{(a,s)}v^{-\frac{1}{p}}\Vert _{p'}<\infty , \end{aligned}$$

  and the best constant C in inequality (1.1) verifies $B_1\le C\le K(q,p)B_1$, where $K(q,p)=\left( 1+\frac{q}{p'}\right) ^{\frac{1}{q}}\left( 1+\frac{p'}{q}\right) ^{\frac{1}{p'}}$ if $p>1$ and $K(q,1)=1$;

• (ii) in the case $q<p$,

  $$\begin{aligned} B_2\equiv \left( \int \limits _a^b\left( \int \limits _t^bw\right) ^{\frac{r}{q}}\Vert \chi _{(a,t)}v^{-\frac{1}{p}}\Vert _{p'}^{\frac{rp'}{q'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{r}}<\infty , \end{aligned}$$

  where $\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$, and the best constant C in inequality (1.1) verifies $q\left( \frac{p'}{r}\right) ^{\frac{1}{q'}}B_2\le C\le q^{\frac{1}{q}}(p')^{\frac{1}{q'}}B_2$.

Weighted weak-type inequalities for T were also studied. By a weighted weak-type (p, q) inequality for T we mean the boundedness of T from $L^p(v)$ to $L^{q,\infty }(w)$, where

$$\begin{aligned} L^{q,\infty }(w)=\left\{ f: \Vert f\Vert _{q,\infty ;w}=\sup _{\lambda>0}\lambda \left( \int \limits _{\{ x: \vert f(x)\vert >\lambda \}}w\right) ^{\frac{1}{q}}<\infty \right\} . \end{aligned}$$

Really, weighted weak-type inequalities have been studied for the modified Hardy operators $T_{\beta }f(x)=\beta (x)\int _a^xf$. This kind of inequalities are technically more difficult than the strong-type ones. In fact, the problem of characterizing the boundedness of $T_{\beta }$ from $L^p(v)$ to $L^{q,\infty }(w)$ in the case $q<p$ is not completely solved yet.

The first results on weighted weak-type inequalities for modified Hardy operators are due to Andersen and Muckenhoupt [2], who worked with $\beta (x)=x^{\alpha }$, $\alpha \in {\mathbb {R}}$, on $(0,\infty )$. The weighted weak-type inequalities with more general functions $\beta$ were characterized in [6, 20, 21]. The following two theorems contain such characterizations.

Theorem B

([6, 21]) Let $1\le p\le q<\infty$ and $\beta$, v and w be positive measurable functions on (a, b), where $-\infty \le a<b\le \infty$. Then there exists a positive constant C such that inequality

$$\begin{aligned} \left\| \beta (x)\left( \int \limits _a^xf\right) \right\| _{q,\infty ;w}\le C\Vert f\Vert _{p, v} \end{aligned}$$

(1.2)

holds for all nonnegative functions f if and only if

$$\begin{aligned} B_3\equiv \sup _{a<s<b}\Vert \beta \chi _{(s,b)}\Vert _{q,\infty ;w}\Vert \chi _{(a,s)}v^{-\frac{1}{p}}\Vert _{p'}<\infty , \end{aligned}$$

(1.3)

and the best constant C in inequality (1.2) verifies $B_3\le C\le 4B_3$.

Theorem C

([20]) Let $0<q<p<\infty$ with $p\ge 1$ and $\beta$, v and w be positive measurable functions on (a, b), where $-\infty \le a<b\le \infty$ and $\beta$ is a monotone function. Then there exists a positive constant C such that inequality (1.2) holds for all nonnegative functions f if and only if the function $\Psi$ defined on (a, b) by

$$\begin{aligned} \Psi (x)=\sup _{b>c>x}\left[ \left( \inf _{y\in (x,c)}\beta (y)\right) \left( \int \limits _x^cw\right) ^{\frac{1}{p}}\right] \Vert \chi _{(a,x)}v^{-\frac{1}{p}}\Vert _{p'} \end{aligned}$$

belongs to $L^{r,\infty }(w)$, where $\frac{1}{r}=\frac{1}{q}-\frac{1}{p}$. In this case, the best constant C in inequality (1.2) verifies $2^{-\frac{1}{p}}\Vert \Psi \Vert _{r,\infty ;w}\le C\le (1+4^p)^{\frac{1}{q}}\Vert \Psi \Vert _{r,\infty ;w}$.

It is worth noting that weighted weak-type inequalities for modified linear or sublinear operators are included in the topic of weighted mixed weak-type inequalities, which goes back to the work of Andersen and Muckenhoupt [2] and have been studied by several authors (see [5, 16,17,18,19, 21, 26]).

Two new kinds of Hardy inequalities are the weighted iterated and bilinear Hardy inequalities. On one hand, weighted iterated Hardy inequalities are of the form

$$\begin{aligned} \left\| \left( \int \limits _a^x\left( \int \limits _a^tf\right) ^ru(t) \text {d}t\right) ^{\frac{1}{r}}\right\| _{q,w}\le C\Vert f\Vert _{p, v} \end{aligned}$$

(1.4)

or

$$\begin{aligned} \left\| \left( \int \limits _a^x\left( \int \limits _t^xf\right) ^ru(t) \text {d}t\right) ^{\frac{1}{r}}\right\| _{q,w}\le C\Vert f\Vert _{p, v}, \end{aligned}$$

(1.5)

and have been studied by many authors [3, 8,9,10,11, 24, 25, 30].

On the other hand, weighted strong-type bilinear Hardy inequalities

$$\begin{aligned} \left\| \left( \int \limits _a^xf\right) \left( \int \limits _a^xg\right) \right\| _{q,w}\le C\Vert f\Vert _{p_1, v_1}\Vert g\Vert _{p_2, v_2} \end{aligned}$$

(1.6)

were characterized in [1] and some of their generalizations and variants have also been studied later (see, for instance, [12,13,14, 30]).

Recently, the authors have characterized in [7] the weights $w, v_1, v_2$ for which the weighted weak-type bilinear modified Hardy inequality

$$\begin{aligned} \left\| \beta (x)\left( \int \limits _a^xf\right) \left( \int \limits _a^xg\right) \right\| _{q,\infty ;w}\le C\Vert f\Vert _{p_1, v_1}\Vert g\Vert _{p_2, v_2} \end{aligned}$$

(1.7)

holds in the cases $0<q<\infty$, $1\le p_1, p_2<\infty$, $q<p_1$, $q<p_2$ and $\frac{1}{q}\le \frac{1}{p_1}+\frac{1}{p_2}$. In this paper, we will complete the characterization of inequality (1.7) solving the problem for the case $\frac{1}{q}>\frac{1}{p_1}+\frac{1}{p_2}$.

As we showed in [7], inequality (1.7) is equivalent to two weighted weak-type iterated modified Hardy inequalities of the form

$$\begin{aligned} \left\| \alpha (x)\left\| u(t)\chi _{(a,x)}(t)\int \limits _a^t f\right\| _r\right\| _{q,\infty ;w}\le C \Vert f \Vert _{p,v}, \end{aligned}$$

(1.8)

where $q<p$. Therefore, we will solve the problem of the characterization of (1.8) in the case $q<p$ and then we will get immediately the characterization of (1.7). It is worth noting that the good weights for (1.8) to hold in the case $p\le q$ were characterized by the authors in [7].

In order to state the results for the iterated inequality (1.8), we define two functions $\Phi , \Psi$ on (a, b) by

$$\begin{aligned} \Phi (x)=\sup _{a<e<c<x<d<b} \left( \inf _{t\in (c,d)} \alpha (t) \Vert \chi _{(e,t)}u\Vert _r\right) \left( \int \limits _c^d w \right) ^{\frac{1}{p}}\Vert \chi _{(a,e)}v^{-\frac{1}{p}}\Vert _{p'} \end{aligned}$$

and

$$\begin{aligned} \begin{array}{rl} \displaystyle \Psi (x)=\displaystyle \sup _{a<c<x<d<b} &{}\displaystyle \left( \inf _{t\in (c,d)} \alpha (t) \right) \left( \int \limits _c^d w \right) ^{\frac{1}{p}}\\ &{}\displaystyle \times \left( \int \limits _a^c\left( \int \limits _t^cu^r\right) ^{\frac{\theta }{r}}\left( \int \limits _a^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'} (t)\text {d}t\right) ^{\frac{1}{\theta }}, \end{array} \end{aligned}$$

where $\frac{1}{\theta }=\frac{1}{r}-\frac{1}{p}$.

The results are the following ones.

Theorem 1

Let p, q, r with $0<q<p$, $1\le p<\infty$ and $p\le r\le \infty$. Let $\alpha$ be a positive function in (a, b) such that

$$\begin{aligned} \inf _{t\in (\rho , \nu )}\alpha (t)>0 \end{aligned}$$

(1.9)

for all $\rho , \nu$ with $a<\rho<\nu < b$. Let us suppose that for all $e\in (a,b)$ and all measurable sets $\Omega \subset (e,b)$, the function $\alpha (t) \Vert \chi _{(e,t)}u\Vert _r$ verifies

$$\begin{aligned} \inf _{t\in \Omega } \{\alpha (t) \Vert \chi _{(e,t)}u\Vert _r \}= \inf _{t\in (\rho _1,\rho _2)} \{\alpha (t) \Vert \chi _{(e,t)}u\Vert _r \}, \end{aligned}$$

(1.10)

where $\rho _1=\inf \Omega$ and $\rho _2=\sup \Omega$. Then, (1.8) holds for all nonnegative functions f if and only if $\Phi \in L^{\eta , \infty }(w)$, where $\frac{1}{\eta }=\frac{1}{q} - \frac{1}{p}$. Moreover, the best constant C in inequality (1.8) verifies

$$\begin{aligned} 2^{\frac{-1}{p}}\Vert \Phi \Vert _{\eta ,\infty ; w}\le C\le (\Vert \Phi \Vert _{\eta ,\infty ; w}^{\eta }+2^p4^{(1+\frac{1}{r})p}+2^p4^{\frac{p}{r}}K(r,p)^p)^{\frac{1}{q}} \end{aligned}$$

if $r<\infty$ and

$$\begin{aligned} 2^{\frac{-1}{p}}\Vert \Phi \Vert _{\eta ,\infty ; w}\le C\le (2^{\eta }\Vert \Phi \Vert _{\eta ,\infty ;w}^{\eta }+8^p+2^p)^{\frac{1}{q}} \end{aligned}$$

if $r=\infty$.

Theorem 2

Let p, q, r with $0<q<p$ and $1<r<p<\infty$. Let $\alpha$ be a positive monotone function in (a, b) and let us suppose that (1.10) holds. Then, the weighted iterated weak-type modified Hardy inequality (1.8) holds for all nonnegative functions f if and only if $\Phi ,\Psi \in L^{\eta , \infty }(w)$, where $\frac{1}{\eta }=\frac{1}{q} - \frac{1}{p}$. Moreover, the best constant C in (1.8) verifies

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \max \{2^{\frac{-1}{p}}\Vert \Phi \Vert _{\eta ,\infty ; w}, 2^{\frac{-1}{p}}r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\Vert \Psi \Vert _{\eta ,\infty ;w}\}\le C\\ &{}\displaystyle \le (\Vert \Phi \Vert _{\eta ,\infty , w}^{\eta }+\Vert \Psi \Vert _{\eta ,\infty ; w}^{\eta }+2^p4^{(1+\frac{1}{r})p}+2^p4^{\frac{p}{r}}C_{r,p}^p)^{\frac{1}{q}}, \end{array} \end{aligned}$$

where $C_{r,p}=r^{\frac{1}{r}}(p')^{\frac{1}{r'}}$.

Observe that condition (1.10) holds if the function $\alpha (t) \Vert \chi _{(e,t)}u\Vert _r$ is monotone or increases in an interval $(e,x_0)$ and decreases in $(x_0,b)$. In the same way, condition (1.9) holds, for instance, if $\alpha$ is a positive monotone function.

As consequences of Theorems 1 and 2 we get the results for the weighted weak-type bilinear modified Hardy inequalities. In order to state them, we define the next functions on (a, b):

$$\begin{aligned} \alpha _i(x)=\sup _{c>x}\left( \inf _{(x,c)}\beta \right) \left( \int \limits _x^cw\right) ^{\frac{1}{p_i}}, \quad i=1,2, \end{aligned}$$

$$\begin{aligned} \begin{array}{rl} \Phi _1(x)=\displaystyle \sup _{a<e<c<x<d<b}&{}\displaystyle \left( \inf _{t\in (c,d)} \alpha _1 (t) \Vert \chi _{(e,t)}v_1^{\frac{-1}{p_1}}\Vert _{p_1'}\right) \\ &{}\displaystyle \times \left( \int \limits _c^d w \right) ^{\frac{1}{p_2}}\Vert \chi _{(a,e)}v_2^{-\frac{1}{p_2}}\Vert _{p_2'}, \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rl} \Phi _2(x)=\displaystyle \sup _{a<e<c<x<d<b} &{}\displaystyle \left( \inf _{t\in (c,d)} \alpha _2 (t) \Vert \chi _{(e,t)}v_2^{\frac{-1}{p_2}}\Vert _{p_2'}\right) \\ &{}\displaystyle \times \left( \int \limits _c^d w \right) ^{\frac{1}{p_1}}\Vert \chi _{(a,e)}v_1^{-\frac{1}{p_1}}\Vert _{p_1'}, \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rl} \displaystyle \Psi _1(x)=\sup _{a<c<x<d<b}&{}\displaystyle \left( \inf _{t\in (c,d)} \alpha _1 (t) \right) \left( \int \limits _c^d w \right) ^{\frac{1}{p_2}}\\ &{}\displaystyle \times \left( \int \limits _a^c\left( \int \limits _t^cv_1^{1-p_1'}\right) ^{\frac{\theta }{p_1'}} \left( \int \limits _a^tv_2^{1-p_2'}\right) ^{\frac{\theta }{p_1}}v_2^{1-p_2'}(t)\text {d}t\right) ^{\frac{1}{\theta }}, \end{array} \end{aligned}$$

$$\begin{aligned} \begin{array}{rl} \displaystyle \Psi _2(x)=\sup _{a<c<x<d<b}&{}\displaystyle \left( \inf _{t\in (c,d)} \alpha _2 (t) \right) \left( \int \limits _c^d w \right) ^{\frac{1}{p_1}}\\ &{}\displaystyle \times \left( \int \limits _a^c\left( \int \limits _t^cv_2^{1-p_2'}\right) ^{\frac{\theta }{p_2'}} \left( \int \limits _a^tv_1^{1-p_1'}\right) ^{\frac{\theta }{p_2}}v_1^{1-p_1'}(t)\text {d}t\right) ^{\frac{1}{\theta }}, \end{array} \end{aligned}$$

where $\frac{1}{\theta }=\frac{1}{p_2'}-\frac{1}{p_1}=\frac{1}{p_1'}-\frac{1}{p_2}=1-\frac{1}{p_2}-\frac{1}{p_1}$.

The theorems read as follows.

Theorem 3

Let $0<q<p_1,p_2<\infty$, $p_1,p_2\ge 1$, $\frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{q}$ and $p_2\le p_1'$. Let $w, v_1, v_2, \beta$ be positive measurable functions on (a, b) with $\beta$ monotone. Assume that the functions $\alpha _i$ verify (1.9) and that for all $e\in (a,b)$ and all measurable sets $\Omega \subset (e,b)$, the functions $\alpha _i(t) \Vert \chi _{(e,t)}v_i^{\frac{-1}{p_i}}\Vert _{p_i'}$, i=1,2, verify (1.10). Then the weighted weak-type bilinear modified Hardy inequality (1.7) holds if and only if $\Phi _1, \Phi _2\in L^{\eta ,\infty }(w)$, where $\frac{1}{\eta }=\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}$.

Theorem 4

Let $0<q<p_1,p_2<\infty$, $p_1,p_2\ge 1$, $\frac{1}{p_1}+\frac{1}{p_2}<\frac{1}{q}$ and $p_2> p_1'$. Let $w, v_1, v_2, \beta$ be positive measurable functions on (a, b) with $\beta$ monotone. Assume that the functions $\alpha _i$ are monotone and that for all $e\in (a,b)$ and all measurable sets $\Omega \subset (e,b)$, the functions $\alpha _i(t) \Vert \chi _{(e,t)}v_i^{\frac{-1}{p_i}}\Vert _{p_i'}$, i=1,2, verify (1.10). Then the weighted weak-type bilinear modified Hardy inequality (1.7) holds if and only if $\Phi _1, \Phi _2, \Psi _1, \Psi _2\in L^{\eta ,\infty }(w)$, where $\frac{1}{\eta }=\frac{1}{q}-\frac{1}{p_1}-\frac{1}{p_2}$.

The proofs of Theorems 1 and 2 are included in Sects. 2 and 3, respectively, while Sect. 4 contains the proofs of Theorems 2 and 4.

2 Proof of Theorem 1

First of all, let us prove the necessity of the condition. Assume that the weak-type inequality (1.8) holds and let us see that $\Phi \in L^{\eta ,\infty }(w)$. Let $\lambda >0$ and $S_\lambda =\{x\in (a,b):\Phi (x)>\lambda \}$. We will prove that

$$\begin{aligned} \lambda \left( \int \limits _{S_\lambda }w\right) ^{\frac{1}{\eta }} \le 2^{\frac{1}{p}}C, \end{aligned}$$

where C is the constant in (1.8). Let K be a compact subset of $S_\lambda$. For all $z\in K$, $z\in S_\lambda$ and then $\Phi (z)>\lambda$. This implies the existence of $c_z, d_z, e_z$ with $e_z<c_z<d_z$ such that $z\in (c_z,d_z)$ and

$$\begin{aligned} \inf _{t\in (c_z,d_z)}\left[ \alpha (t)\Vert \chi _{(e_z,t)}u\Vert _r\right] \left( \int \limits _{c_z}^{d_z}w\right) ^{\frac{1}{p}}\left( \int \limits _a^{e_z}v^{1-p'}\right) ^{\frac{1}{p'}}>\lambda . \end{aligned}$$

(2.1)

Then, $K\subset \bigcup _{z\in K} (c_z, d_z)$. Since K is compact, there are $(c_{z_1},d_{z_1}), (c_{z_2},d_{z_2})\dots$ $(c_{z_N}, d_{z_N})$ such that $K\subset \bigcup _{j=1}^N (c_{z_j},d_{z_j})$. We can also suppose that

$$\begin{aligned} \sum _{j=1}^N \chi _{(c_{z_j},d_{z_j})}\le 2 \chi _{\cup _{j=1}^N (c_{z_j}, d_{z_j})}. \end{aligned}$$

Let, for all $j\in \{1, 2, \ldots , N\}$ and $x\in (a,b)$,

$$\begin{aligned} f_j(x)=\left( \inf _{y\in (c_{z_j},d_{z_j})}\left[ \alpha (y)\Vert \chi _{(e_{z_j},y)}u\Vert _r\right] \left( \int \limits _a^{e_{z_j}}v^{1-p'}\right) \right) ^{-p}v(x)^{-p'}\chi _{(a,e_{z_j})}(x) \end{aligned}$$

and

$$\begin{aligned} f = \left( \sum _{j=1}^N f_j\right) ^{\frac{1}{p}}. \end{aligned}$$

Let $\varepsilon$ with $0<\varepsilon <1$. Let us see that

$$\begin{aligned} \bigcup _{j=1}^N (c_{z_j},d_{z_j}) \subset \left\{ x \in (a,b):\alpha (x)\left\| \chi _{(a,x)}(t)u(t)\int \limits _a^t f\right\| _r > \varepsilon \right\} . \end{aligned}$$

(2.2)

Indeed, if $z\in (c_{z_j},d_{z_j})$, then

$$\begin{aligned} \begin{array}{rl} \displaystyle \alpha (z)\left\| \chi _{(a,z)}(t)u(t)\int \limits _a^t f \right\| _r&{}\displaystyle \ge \alpha (z)\left\| \chi _{(e_{z_j},z)}(t)u(t)\int \limits _a^t f \right\| _r\\ {} &{}\\ &{}\displaystyle \ge \alpha (z) \Vert \chi _{(e_{z_j}, z)}(t) u(t)\Vert _r \int \limits _a^{e_{z_j}}f\\ {} &{}\\ &{}\displaystyle \ge \alpha (z) \Vert \chi _{(e_{z_j}, z)}(t) u(t)\Vert _r \int \limits _a^{e_{z_j}}f_j^{\frac{1}{p}}\\ {} &{}\\ &{}\displaystyle =\frac{\alpha (z)\Vert \chi _{(e_{z_j}, z)}(t)u(t)\Vert _r \displaystyle \int \limits _a^{e_{z_j}}v^{-\frac{p'}{p}}}{\displaystyle \inf _{y\in (c_{z_j},d_{z_j})}[\alpha (y)\Vert \chi _{(e_{z_j}, y)} u \Vert _r] \displaystyle \int \limits _a^{e_{z_j}}v^{1-p'}}\\ {} &{}\\ &{}\displaystyle \ge 1 > \varepsilon . \end{array} \end{aligned}$$

This proves (2.2). Applying the weak-type inequality,

$$\begin{aligned} \displaystyle \int \limits _{\cup _{j=1}^N (c_{z_j},d_{z_j})} w \le \displaystyle \int \limits _{\left\{ x \in (a,b):\alpha (x)\left\| \chi _{(a,x)}(t)u(t)\int \limits _a^t f\right\| _r > \varepsilon \right\} }w \displaystyle \le \frac{C^q}{\varepsilon ^q}\Vert f \Vert _{p,v}^q. \end{aligned}$$

Since the last inequality holds for all $\varepsilon$ with $0<\varepsilon <1$, letting $\varepsilon \rightarrow 1^-$ we get

$$\begin{aligned} \int \limits _{\cup _{j=1}^N (c_{z_j},d_{z_j})} w\le C^q \Vert f \Vert _{p,v}^q. \end{aligned}$$

Let $\gamma _j= \inf _{y\in (c_{z_j},d_{z_j})} [\alpha (y)\Vert \chi _{(e_{z_j},y)}u\Vert _r]$. Then the inequality above and (2.1) yield

$$\begin{aligned} \begin{array}{rl}&{} \displaystyle \int \limits _{\cup _{j=1}^N (c_{z_j},d_{z_j})} w\le C^q \displaystyle \left( \int \limits _a^b \left( \sum _{j=1}^N f_j(x)\right) v(x)\text{ d }x\right) ^{\frac{q}{p}} \\ {} &{}{}=\displaystyle C^q \left( \int \limits _a^b \sum _{j=1}^N \frac{v^{-p'}(x)\chi _{(a,e_{z_j})}(x)}{\gamma _j^p\left( \displaystyle \int _a^{e_{z_j}}v^{1-p'}\right) ^p}v(x)\text{ d }x\right) ^{\frac{q}{p}}\\ {} &{}{}\\ {} &{}{}=\displaystyle C^q \left( \sum _{j=1}^N \frac{1}{\gamma _j^p\left( \displaystyle \int _a^{e_{z_j}}v^{1-p'}\right) ^p}\int \limits _a^{e_{z_j}}v^{1-p'}\right) ^{\frac{q}{p}}\\ {} &{}{}=\displaystyle C^q \left( \sum _{j=1}^N \frac{1}{\gamma _j^p\left( \displaystyle \int _a^{e_{z_j}}v^{1-p'}\right) ^{p-1}}\right) ^{\frac{q}{p}}\le \displaystyle C^q\left( \sum _{j=1}^N \frac{1}{\lambda ^p} \int \limits _{c_{z_j}}^{d_{z_j}} w \right) ^{\frac{q}{p}}\\ {} &{}{}\\ {} &{}{}=\displaystyle \frac{C^q}{\lambda ^q} \left( \sum _{j=1}^N \int \limits _{c_{z_j}}^{d_{z_j}}w\right) ^{\frac{q}{p}}\le \frac{2^{\frac{q}{p}} C^q}{\lambda ^q}\left( \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \right) ^{\frac{q}{p}}. \end{array} \end{aligned}$$

Then, we have

$$\begin{aligned} \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \le \frac{2^{\frac{q}{p}} C^q}{\lambda ^q}\left( \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \right) ^{\frac{q}{p}}, \end{aligned}$$

i.e.,

$$\begin{aligned} \lambda \left( \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \right) ^{\frac{1}{\eta }}\le 2^{\frac{1}{p}}C. \end{aligned}$$

The last inequality implies

$$\begin{aligned} \lambda \left( \int \limits _K w \right) ^{\frac{1}{\eta }}\le 2^{\frac{1}{p}}C. \end{aligned}$$

Since the inequality above holds for all compact $K\subset S_{\lambda }$, the regularity of the measure $w(x)\text {d}x$ gives

$$\begin{aligned} \lambda \left( \int \limits _{S_{\lambda }}w \right) ^{\frac{1}{\eta }} \le 2^{\frac{1}{p}}C, \end{aligned}$$

what proves that $\Vert \Phi \Vert _{\eta , \infty ; w}\le 2^{\frac{1}{p}}C$, as we wished to show.

Now, let us prove the sufficiency of the condition. Let f be a positive function such that $\int _a^b f^p v=1$. Let $\lambda >0$ and $O_{\lambda }=\{x \in (a,b): \alpha (x)\Vert \chi _{(a,x)}(t)u(t)\int _a^t f \Vert _r>\lambda \}$. Then,

$$\begin{aligned} \int \limits _{O_\lambda } w = \int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)>\lambda ^{\frac{q}{\eta }}\}}w+\int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}\}}w=I+II. \end{aligned}$$

The estimation of I is as follows:

$$\begin{aligned} \lambda ^q \int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)>\lambda ^{\frac{q}{\eta }}\}}w \le \sup _{z>0} z^\eta \int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)>z\}}w=\Vert \Phi \Vert _{\eta , \infty ; w}^\eta . \end{aligned}$$

Now, we will estimate II. Assume first that $r<\infty$. Let us suppose that $\int _a^b f <\infty$ and $\int _a^b \left( \int _a^t f \right) ^ru^r(t)\text {d}t < \infty$, too. Let $\{ x_k\}$ be the sequence defined by $x_0=b$ and

$$\begin{aligned} \int \limits _a^{x_{k+1}}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t=\int \limits _{x_{k+1}}^{x_k}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t. \end{aligned}$$

The sequence $\{ x_k\}$ decreases to a and verifies

$$\begin{aligned} \int \limits _a^{x_{k}}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t=4\int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t \end{aligned}$$

(2.3)

for all k. Let $E_k=O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}\}\cap (x_{k+1},x_k)$. If $x\in E_k$, we have

$$\begin{aligned} \begin{array}{rl} \lambda &{} <\displaystyle \alpha (x)\left( \int \limits _a^x\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}}\\ &{}\le \displaystyle \alpha (x)\left( \int \limits _a^{x_k}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}}\\ &{} =\displaystyle 4^{\frac{1}{r}}\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _a^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}}\\ &{}\le \displaystyle 4^{\frac{1}{r}}\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}} \\ &{} +\displaystyle 4^{\frac{1}{r}}\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\int \limits _a^{x_{k+2}}f. \end{array} \end{aligned}$$

(2.4)

It is clear that, for each k, $E_k= E_{k,1}\cup E_{k,2}$, where

$$\begin{aligned} E_{k,1}=\left\{ x\in E_k: \alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}}>\frac{\lambda }{2\cdot 4^{\frac{1}{r}}} \right\} \end{aligned}$$

and

$$\begin{aligned} E_{k,2}=\left\{ x\in E_k: \alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\int \limits _a^{x_{k+2}}f>\frac{\lambda }{2\cdot 4^{\frac{1}{r}}} \right\} . \end{aligned}$$

Since $p\le r$, by Theorem A (i), we have that

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}} \\ &{}\displaystyle \le K(r,p)\sup _{x_{k+2}<\gamma <x_{k+1}}\left( \int \limits _{\gamma }^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(x_{k+2},\gamma )}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{array} \end{aligned}$$

(2.5)

Let us see that the supremum in (2.5) is finite. Let $\gamma \in (x_{k+2},x_{k+1})$. As $\Phi \in L^{\eta , \infty }(w)$, $\Phi$ is finite almost everywhere. Let $\rho , \nu$ with $x_{k+1}<\rho<\nu <b$ and let $t\in (\rho ,\nu )$ such that $\Phi (t)<\infty$. Then,

$$\begin{aligned} 1+\Phi (t)> \left( \inf _{x\in (\rho ,\nu )}\alpha (x)\left( \int \limits _\gamma ^x u^r\right) ^{\frac{1}{r}}\right) \left( \int \limits _\rho ^\nu w\right) ^{\frac{1}{p}}\Vert \chi _{(a,\gamma )}v^{-\frac{1}{p}}\Vert _{p'}. \end{aligned}$$

Thus, there is ${\tilde{x}}\in (\rho , \nu )$, which depends on $\gamma$, such that

$$\begin{aligned} \begin{array}{rl} \displaystyle 1+\Phi (t)&{}\displaystyle > \alpha ({\tilde{x}})\left( \int \limits _\gamma ^{{\tilde{x}}} u^r\right) ^{\frac{1}{r}}\left( \int \limits _\rho ^\nu w\right) ^{\frac{1}{p}}\Vert \chi _{(a,\gamma )}v^{-\frac{1}{p}}\Vert _{p'}\\ &{}\\ &{}\displaystyle \ge \alpha ({\tilde{x}})\left( \int \limits _\gamma ^{x_{k+1}} u^r\right) ^{\frac{1}{r}}\left( \int \limits _\rho ^\nu w\right) ^{\frac{1}{p}}\Vert \chi _{(x_{k+2},\gamma )}v^{-\frac{1}{p}}\Vert _{p'}. \end{array} \end{aligned}$$

Then, applying (1.9), we get

$$\begin{aligned} \left( \int \limits _{\gamma }^{x_{k+1}}u^r\right) ^{\frac{1}{r}} \Vert \chi _{(x_{k+2},\gamma )}v^{-\frac{1}{p}}\Vert _{p'}< \displaystyle \frac{ 1 +\Phi (t)}{\left( \displaystyle \inf _{(\rho ,\nu )}\alpha \right) \left( \displaystyle \int _\rho ^\nu w \right) ^{\frac{1}{p}} }<\infty . \end{aligned}$$

Therefore, the supremum in (2.5) is finite. Then, for all $x\in E_{k,1}$ we have

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \frac{\lambda }{2\cdot 4^{\frac{1}{r}}K(r,p)}\\ &{}<\displaystyle \alpha (x)\sup _{x_{k+2}<\gamma <x_{k+1}}\left( \int \limits _{\gamma }^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(x_{k+2},\gamma )}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{array} \end{aligned}$$

Let $\varepsilon >1$. For every k, there is $\gamma _k \in (x_{k+2},x_{k+1})$ such that

$$\begin{aligned} \begin{array}{rl} \displaystyle \sup _{x_{k+2}<\gamma<x_{k+1}}&{}\displaystyle \left( \int \limits _{\gamma }^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(x_{k+2},\gamma )}v^{-\frac{1}{p}}\Vert _{p'}\\ &{}<\displaystyle \varepsilon \left( \int \limits _{\gamma _k}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(x_{k+2},\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}. \end{array} \end{aligned}$$

Therefore, for all $x\in E_{k,1}$ the following inequality holds:

$$\begin{aligned} \frac{\lambda }{2\cdot 4^{\frac{1}{r}}K(r,p)}<\varepsilon \alpha (x)\left( \int \limits _{\gamma _k}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(x_{k+2},\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{aligned}$$

Since $x_{k+1}<x$, we get

$$\begin{aligned} \frac{\lambda }{2\cdot 4^{\frac{1}{r}}K(r,p)}<\varepsilon \alpha (x)\left( \int \limits _{\gamma _k}^{x}u^r\right) ^{\frac{1}{r}}\Vert \chi _{(a,\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{aligned}$$

The last inequality holds for all $x\in E_{k,1}$. Then,

$$\begin{aligned} \frac{\lambda }{2\cdot 4^{\frac{1}{r}}K(r,p)}\le \varepsilon \inf _{x\in E_{k,1}}\left( \alpha (x)\left( \int \limits _{\gamma _k}^{x}u^r\right) ^{\frac{1}{r}}\right) \Vert \chi _{(a,\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{aligned}$$

Now, if we multiply both sides of this inequality by $\displaystyle \left( \int _{E_{k,1}} w \right) ^{\frac{1}{p}}$ and apply condition (1.10), we have

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \frac{\lambda }{2\cdot 4^{\frac{1}{r}}K(r,p)\varepsilon }\left( \int \limits _{E_{k,1}} w \right) ^{\frac{1}{p}}\displaystyle \le \inf _{x\in (\rho _k^1, \rho _k^2)}\left\{ \alpha (x)\left( \int \limits _{\gamma _k}^{x}u^r\right) ^{\frac{1}{r}}\right\} \\ &{}\\ &{}\displaystyle \times \left( \int \limits _{\rho _k^1}^{\rho _k^2}w\right) ^{\frac{1}{p}}\Vert \chi _{(a,\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}\\ &{}\\ &{}\displaystyle \le \lambda ^{\frac{q}{\eta }}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}, \end{array} \end{aligned}$$

where $\rho _k^1=\inf E_{k,1}$, $\rho _k^2=\sup E_{k,1}$ and the last inequality holds since

$$\begin{aligned} \inf _{x\in (\rho _k^1, \rho _k^2)}\left\{ \alpha (x)\left( \int \limits _{\gamma _k}^{x}u^r\right) ^{\frac{1}{r}}\right\} \left( \int \limits _{\rho _k^1}^{\rho _k^2}w\right) ^{\frac{1}{p}}\Vert \chi _{(a,\gamma _k)}v^{-\frac{1}{p}}\Vert _{p'}\le \Phi (t)\le \lambda ^{\frac{q}{\eta }} \end{aligned}$$

for all $t\in E_{k,1}$. Thus,

$$\begin{aligned} \lambda \left( \int \limits _{E_{k,1}}w\right) ^{\frac{1}{p}}\le \varepsilon 2 \cdot 4^{\frac{1}{r}} K(r,p) \lambda ^{\frac{q}{\eta }}\left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{aligned}$$

Since this inequality holds for every $\varepsilon >1$, letting $\varepsilon \rightarrow 1^+$ and raising to p we get

$$\begin{aligned} \int \limits _{E_{k,1}}w\le 2^p 4^{\frac{p}{r}}K(r,p)^p \lambda ^{\frac{qp}{\eta }-p} \int \limits _{x_{k+2}}^{x_{k+1}}f^p v. \end{aligned}$$

Now, summing up in k, we have

$$\begin{aligned} \int \limits _{\cup _k E_{k,1}} w \le \frac{2^p\cdot 4^{\frac{p}{r}}K(r,p)^p}{\lambda ^q}\int \limits _a^b f^pv=\frac{2^p\cdot 4^{\frac{p}{r}}K(r,p)^p}{\lambda ^q}, \end{aligned}$$

what finishes the estimation of $\int _{\cup _k E_{k,1}}w$. In order to estimate $\int _{\cup _kE_{k,2}}w$, we will use the technique, due to Lai [15], that we have already used in [7]. Let us define the sequence $\{ y_m'\}$ as $y_0'=b$ and $\int _a^{y_{m+1}'}f=\int _{y'_{m+1}}^{y_m'}f$. Let $\{ y_n\}$ be the subsequence of $\{ y_m'\}$ defined by $y_0=y_0'$ and by deleting $y'_{m+1}$ if $[y'_{m+1}, y'_m)\cap \{ x_k\}=\emptyset$. In this way, if $y'_{m+1}=y_{n+1}\le x_{k+2}<y_n$, then $x_{k+2}\le y'_m$ and $y_{n+2}\le y'_{m+2}$, which yields

$$\begin{aligned} \int \limits _a^{x_{k+2}}f\le \int \limits _a^{y'_m}f=4\int _{y'_{m+2}}^{y'_{m+1}}f\le 4\int \limits _{y_{n+2}}^{y_{n+1}}f. \end{aligned}$$

(2.6)

Let $E_2^n=\bigcup _{\{k: y_{n+1}\le x_{k+2}<y_n\}}E_{k,2}$. If $x\in E_2^n$, there exists k with $y_{n+1}\le x_{k+2}<y_n$ such that $x\in E_{k,2}$ and then, by (2.6),

$$\begin{aligned} \frac{\lambda }{2\cdot 4^{\frac{1}{r}}}<\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\int \limits _a^{x_{k+2}}f\le 4 \alpha (x)\left( \int \limits _{y_{n+1}}^{x}u^r\right) ^{\frac{1}{r}}\int \limits _{y_{n+2}}^{y_{n+1}}f. \end{aligned}$$

(2.7)

Since (2.7) holds for all $x\in E_2^n$, we have

$$\begin{aligned} \frac{\lambda }{2\cdot 4^{1+\frac{1}{r}}}\le \inf _{x\in E_2^n}\left[ \alpha (x)\left( \int \limits _{y_{n+1}}^{x}u^r\right) ^{\frac{1}{r}}\right] \int \limits _{y_{n+2}}^{y_{n+1}}f. \end{aligned}$$

(2.8)

Multiplying both sides of (2.8) by $\left( \int _{E_2^n}w\right) ^{\frac{1}{p}}$, applying Holder’s inequality and (1.10), we get

$$\begin{aligned} \begin{array}{rl} \displaystyle \frac{\lambda }{2\cdot 4^{1+\frac{1}{r}}}\left( \int \limits _{E_2^n}w\right) ^{\frac{1}{p}} &{}\le \displaystyle \inf _{x\in (\rho _1^n,\rho _2^n)}\left[ \alpha (x)\left( \int \limits _{y_{n+1}}^{x}u^r\right) ^{\frac{1}{r}}\right] \\ &{}\\ &{}\displaystyle \times \left( \int \limits _{\rho _1^n}^{\rho _2^n}w\right) ^{\frac{1}{p}}\Vert \chi _{(a,y_{n+1})}v^{-{\frac{1}{p}}} \Vert _{p'}\left( \int \limits _{y_{n+2}}^{y_{n+1}}f^pv\right) ^{\frac{1}{p}}\\ &{}\\ &{}\displaystyle \le \lambda ^{\frac{q}{\eta }}\left( \int \limits _{y_{n+2}}^{y_{n+1}} f^pv\right) ^{\frac{1}{p}}, \end{array} \end{aligned}$$

(2.9)

where we have used that

$$\begin{aligned} \inf _{x\in (\rho _1^n,\rho _2^n)}\left[ \alpha (x)\left( \int \limits _{y_{n+1}}^{x}u^r\right) ^{\frac{1}{r}}\right] \left( \int \limits _{\rho _1^n}^{\rho _2^n}w\right) ^{\frac{1}{p}}\Vert \chi _{(a,y_{n+1})}v^{-{\frac{1}{p}}}\Vert _{p'}\le \Phi (t) \end{aligned}$$

for all $t\in E_2^n$.

Then, raising to the p in (2.9) and summing, we get

$$\begin{aligned} \begin{array}{rl} \displaystyle \int \limits _{\cup _kE_{k,2}}w&{}\displaystyle =\sum _{k=0}^{\infty }\int \limits _{E_{k,2}}w=\sum _{n=0}^{\infty }\sum _{\{k: y_{n+1}\le x_{k+2}<y_n\}}\int \limits _{E_{k,2}}w=\sum _{n=0}^{\infty }\int \limits _{E_2^n}w\\ &{}\\ &{}\displaystyle \le \frac{2^p\cdot 4^{(1+\frac{1}{r})p}}{\lambda ^q}\left( \int \limits _a^bf^pv\right) =\frac{2^p\cdot 4^{(1+\frac{1}{r})p}}{\lambda ^q}, \end{array} \end{aligned}$$

(2.10)

which implies

$$\begin{aligned} II\le \int \limits _{\cup _k E_{k,1}}w+\int \limits _{\cup _k E_{k,2}}w\le \frac{1}{\lambda ^q} (2^p\cdot 4^{(1+\frac{1}{r})p}+2^p\cdot 4^{\frac{p}{r}}K(r,p)^p). \end{aligned}$$

This finishes the proof of the sufficiency in the case $r<\infty$. Now, we will deal with the case $r=\infty$. Let us consider two sequences $\{a_n\}$ and $\{b_n\}$, with $\{ a_n\}$ decreasing to a and $\{ b_n\}$ increasing to b. Then, $\Phi \in L^{\eta , \infty }(w)$ implies that $\Phi _n \in L^{\eta , \infty }(w, (a_n,b_n))$, where

$$\begin{aligned} \begin{array}{rl} \Phi _n(x)=\displaystyle \sup _{a_n<e<c<x<d<b_n}&{}\displaystyle \left( \inf _{t\in (c,d)} \left( \alpha (t) \Vert \chi _{(e,t)}u\Vert _{\infty } \right) \right) \\ &{}\displaystyle \times \left( \int \limits _c^d w \right) ^{\frac{1}{p}} \Vert \chi _{(a_n,e)}v^{-\frac{1}{p}}\Vert _{p'}. \end{array} \end{aligned}$$

For fixed n, there is $r_0>p$ such that $(b_n-a_n)^{\frac{1}{r}} \le 2$ for all $r\ge r_0$. Then, if $r\ge r_0$ and $a_n<e<x<b_n$, we have

$$\begin{aligned} \Vert \chi _{(e,x)}u \Vert _r =\left( \int \limits _e^x \vert u \vert ^r\right) ^{\frac{1}{r}}\le \Vert \chi _{(e,x)}u \Vert _\infty (b_n-a_n)^{\frac{1}{r}} \le 2 \Vert \chi _{(e,x)}u \Vert _\infty . \end{aligned}$$

Therefore, if we define $\Phi _{n,r}(x)$ as

$$\begin{aligned} \begin{array}{rl} \Phi _{n,r}(x)=\displaystyle \sup _{a_n<e<c<x<d<b_n}&{}\displaystyle \left( \inf _{t\in (c,d)} \left( \alpha (t) \Vert \chi _{(e,t)}u\Vert _{r} \right) \right) \\ &{}\displaystyle \times \left( \int \limits _c^d w \right) ^{\frac{1}{p}} \Vert \chi _{(a_n,e)}v^{-\frac{1}{p}}\Vert _{p'}, \end{array} \end{aligned}$$

we have that $\Phi _{n,r}\in L^{\eta , \infty }(w, (a_n,b_n))$ for all $r\ge r_0$ and their norms are bounded by $2\Vert \Phi \Vert _{\eta ,\infty ,w}$. Now, applying the Theorem in the case which we have already proved, we have that the weak-type inequality

$$\begin{aligned} \left\| \chi _{(a_n,b_n)}(x)\alpha (x)\left\| \chi _{(a_n,x)}(t) u(t)\int \limits _{a_n}^t f \right\| _r \right\| _{q,\infty ; w}\le C_{r,p,q}\Vert \chi _{(a_n,b_n)}f \Vert _{p,v} \end{aligned}$$

(2.11)

holds, where $C_{r,p,q}=(2^{\eta }\Vert \Phi \Vert _{\eta ,\infty ,w}^{\eta }+2^p4^{\frac{p}{r}}(4^p+K(r,p)^p))^{\frac{1}{q}}$. Since

$$\begin{aligned} \left\| \chi _{(a_n,x)}(t)u(t)\int \limits _{a_n}^tf\right\| _{\infty }=\lim _{r\rightarrow \infty }\left\| \chi _{(a_n,x)}(t)u(t)\int \limits _{a_n}^tf\right\| _r \end{aligned}$$

for every x, by Fatou’s lemma we have

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \left\| \chi _{(a_n,b_n)}(x) \alpha (x)\left\| \chi _{(a_n,x)}(t)u(t)\int \limits _{a_n}^tf\right\| _{\infty }\right\| _{q,\infty ;w}\\ &{}\displaystyle \le \lim \inf _{r\rightarrow \infty } \left\| \chi _{(a_n,b_n)}(x)\alpha (x)\left\| \chi _{(a_n,x)}(t)u(t)\int _{a_n}^tf\right\| _r\right\| _{q,\infty ;w}. \end{array} \end{aligned}$$

(2.12)

Now, from (2.11) and (2.12) we get

$$\begin{aligned} \left\| \chi _{(a_n,b_n)}(x)\alpha (x)\left\| \chi _{(a_n,x)}(t)u(t)\int \limits _{a_n}^tf\right\| _{\infty }\right\| _{q,\infty ;w}\le C_{p,q} \Vert \chi _{(a_n,b_n)} f\Vert _{p, v}, \end{aligned}$$

(2.13)

where $C_{p,q}=(2^{\eta }\Vert \Phi \Vert _{\eta ,\infty ,w}^{\eta }+8^p+2^p)^{\frac{1}{q}}$. Finally, since (2.13) holds for all n with a constant independent of n, letting n tend to infinity and applying the monotone convergence theorem, we get (1.8) in the case $r=\infty$.

3 Proof of Theorem 2

The necessity of condition $\Phi \in L^{\eta ,\infty }(w)$ follows as in the proof of Theorem 1. Therefore, the best constant C in (1.8) verifies $C\ge 2^{\frac{-1}{p}}\Vert \Phi \Vert _{\eta ,\infty , w}$. Let us prove now that (1.8) implies $\Psi \in L^{\eta , \infty }(w)$. Let $\lambda >0$ and $S_{\lambda }=\{ x\in (a,b): \Psi (x)>\lambda \}$. Let K be a compact subset of $S_{\lambda }$. If $z\in K$, there exist $c_z, d_z$ with $c_z<z<d_z$ such that

$$\begin{aligned} \left( \inf _{(c_z,d_z)}\alpha \right) \left( \int \limits _{c_z}^{d_z}w\right) ^{\frac{1}{p}}\left( \int \limits _a^{c_z}\left( \int \limits _t^{c_z}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _a^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }}>\lambda . \end{aligned}$$

(3.1)

Since K is compact, there exist $z_1, z_2,\ldots ,z_N\in K$ such that $K\subset \displaystyle \bigcup _{j=1}^N(c_{z_j},d_{z_j})$ and

$$\begin{aligned} {} \sum \limits _{j=1}^N \chi _{(c_{z_j},d_{z_j})}\le 2 \chi _{\cup _{j=1}^N (c_{z_j}, d_{z_j})}. \end{aligned}$$

(3.2)

Let, for each $j\in \{ 1,2,\ldots ,N\}$,

$$\begin{aligned} \begin{array}{rl} f_j(x)&{} =\displaystyle \left( \inf _{(c_{z_j},d_{z_j})}\alpha \right) ^{-p}\chi _{(a,c_{z_j})}(x) \left( \int \limits _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}}\left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{-p'}(x)\\ &{}\\ &{}\times \displaystyle \left( \int \limits _a^{c_{z_j}}\left( \int _t^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _a^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{-\frac{p}{r}} \end{array} \end{aligned}$$

and

$$\begin{aligned} f = \left( \sum _{j=1}^N f_j\right) ^{\frac{1}{p}}. \end{aligned}$$

If $z\in (c_{z_j},d_{z_j})$ and $\gamma _j=\inf _{(c_{z_j},d_{z_j})}\alpha$, we have

$$\begin{aligned} \displaystyle \alpha (z)\left( \int \limits _a^z\left( \int \limits _a^t f\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}\ge \alpha (z)\left( \int \limits _a^{c_{z_j}}\left( \int \limits _a^t f_j^{\frac{1}{p}}\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}\nonumber \\ \begin{array}{rl} &{}\displaystyle =\frac{\alpha (z)}{\gamma _j\left( \displaystyle \int _a^{c_{z_j}} \left( \int _t^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}}\left( \int _a^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{r}}}\\ {} &{}\\ &{}\displaystyle \times \left( \int \limits _a^{c_{z_j}} \left( \int \limits _a^t\left( \int \limits _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{r'p}}v^{1-p'}(x)\text {d}x\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}. \end{array} \end{aligned}$$

(3.3)

If $h(x)=\displaystyle \left( \int _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int _a^xv^{1-p'}\right) ^{\frac{\theta }{r'p}}v^{1-p'}(x)$, the last factor in (3.3) can be written as follows

$$\begin{aligned} \displaystyle \left( \int \limits _a^{c_{z_j}}\left( \int \limits _a^th(x)\text {d}x\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}{} & {} =\displaystyle \left( \int \limits _a^{c_{z_j}}\left( \int \limits _a^t\left[ \left( \int \limits _a^s h\right) ^r\right] '(s)\text {d}s\right) u^r(t)\text {d}t\right) ^{\frac{1}{r}}\nonumber \\{} & {} =r^{\frac{1}{r}}\left( \int \limits _a^{c_{z_j}}\left( \int \limits _a^t\left( \int \limits _a^s h\right) ^{r-1}h(s)\text {d}s\right) u^r(t)\text {d}t\right) ^{\frac{1}{r}}\nonumber \\{} & {} \displaystyle =r^{\frac{1}{r}}\left( \int \limits _a^{c_{z_j}}\left( \int \limits _s^{c_{z_j}}u^r(t)\text {d}t\right) \left( \int \limits _a^s h\right) ^{r-1}h(s)\text {d}s\right) ^{\frac{1}{r}}.\nonumber \\ \end{aligned}$$

(3.4)

Let us estimate now $\displaystyle \int _a^sh$:

$$\begin{aligned} \begin{array}{rl} \displaystyle \int \limits _a^sh(x)\text {d}x &{}=\displaystyle \int \limits _a^s\left( \int _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{r'p}}v^{1-p'}(x)\text {d}x\\ &{}\\ &{}\displaystyle \ge \left( \int \limits _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}}\int \limits _a^s \left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{r'p}}v^{1-p'}(x)\text {d}x\\ &{}\\ &{}=\displaystyle \frac{rp'}{\theta }\left( \int \limits _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{rp'}}. \end{array} \end{aligned}$$

Taking this estimate to (3.4), we get

$$\begin{aligned}{} & {} \displaystyle \left( \int \limits _a^{c_{z_j}}\left( \int \limits _a^th(x)\text {d}x\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}\displaystyle \ge r^{\frac{1}{r}}\left( \frac{rp'}{\theta }\right) ^{\frac{1}{r'}}\\{} & {} \quad \displaystyle \times \left( \int \limits _a^{c_{z_j}}\left( \int \limits _s^{c_{z_j}}u^r\right) \left( \left( \int \limits _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{rp}} \left( \int \limits _a^xv^{1-p'}\right) ^{\frac{\theta }{rp'}}\right) ^{r-1}h(s)\text {d}s\right) ^{\frac{1}{r}}\\{} & {} \quad \displaystyle =r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\left( \int \limits _a^{c_{z_j}}\left( \int \limits _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _a^sv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(s)\text {d}s\right) ^{\frac{1}{r}}. \end{aligned}$$

Going back to (3.3) we get that for all $z\in (c_{z_j}, d_{z_j})$,

$$\begin{aligned} \alpha (z)\left( \int \limits _a^z\left( \int \limits _a^t f\right) ^ru^r(t)\text {d}t\right) ^{\frac{1}{r}}\ge r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\frac{\alpha (z)}{\gamma _j}\ge r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}. \end{aligned}$$

Therefore, by (1.8), we have

$$\begin{aligned} \int \limits _{{\cup _{j=1}^N (c_{{z_{j}}},d_{{z_{j}}})}} w\le \frac{C^q}{\left( r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\right) ^{q}} \Vert f \Vert _{p,v}^q. \end{aligned}$$

(3.5)

Let us estimate $\Vert f \Vert _{p,v}^q$. By definition of f, (3.1) and (3.2),

$$\begin{aligned} \Vert f \Vert _{p,v}^q =\left[ \displaystyle \int _a^b\sum _{j=1}^N\frac{\displaystyle \left( \int _x^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}}\left( \int _a^xv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(x) \chi _{(a,c_{z_j})}(x)}{\displaystyle \gamma _j^p\left( \int _a^{c_{z_j}} \left( \int _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}}\left( \int _a^sv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(s)\text {d}s\right) ^{\frac{p}{r}}}\text {d}x\right] ^{\frac{q}{p}}\nonumber \\ \begin{array}{rl} &{}\displaystyle =\left[ \sum _{j=1}^N\gamma _j^{-p}\left( \int \limits _a^{c_{z_j}} \left( \int \limits _s^{c_{z_j}}u^r\right) ^{\frac{\theta }{r}}\left( \int \limits _a^sv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(s)\text {d}s\right) ^{-\frac{p}{\theta }}\right] ^{\frac{q}{p}}\\ &{}\\ &{}\displaystyle \le \left( \sum _{j=1}^N\frac{1}{\lambda ^p}\int \limits _{c_{z_j}}^{d_{z_j}}w\right) ^{\frac{q}{p}}\le \frac{2^{\frac{q}{p}}}{\lambda ^q}\left( \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \right) ^{\frac{q}{p}}. \end{array} \end{aligned}$$

(3.6)

Finally, from (3.5) and (3.6) we get

$$\begin{aligned} \lambda ^q\left( \int \limits _{\cup _{j=1}^N(c_{z_j},d_{z_j})} w \right) ^{\frac{q}{\eta }}\le \frac{2^{\frac{q}{p}}C^q}{\left( r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\right) ^q}, \end{aligned}$$

which implies

$$\begin{aligned} \lambda \left( \int \limits _Kw\right) ^{\frac{1}{\eta }}\le \frac{2^{\frac{1}{p}}C}{r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}}. \end{aligned}$$

Since the inequality above holds for all compact set $K\subset S_{\lambda }$, we have that $\Psi \in L^{\eta ,\infty }(w)$ and $C\ge 2^{\frac{-1}{p}}r\left( \frac{p'}{\theta }\right) ^{\frac{1}{r'}}\Vert \Psi \Vert _{\eta ,\infty ;w}$.

Let us prove now the sufficiency. Let f be a nonnegative function with $f\in L^1$ and $\int _a^bf^pv=1$. Let $\lambda >0$ and

$$\begin{aligned} O_{\lambda }=\left\{ x \in (a,b): \alpha (x)\Vert \chi _{(a,x)}(t)u(t)\int \limits _a^t f \Vert _r>\lambda \right\} . \end{aligned}$$

Then, as in the proof of Theorem 1,

$$\begin{aligned} \int \limits _{O_\lambda } w = \int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)>\lambda ^{\frac{q}{\eta }}\}}w+\int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}\}}w=I+II. \end{aligned}$$

The estimation of I can be done as in the case $p\le r$. For the estimation of II, we work as follows:

$$\begin{aligned} \begin{array}{rl} II&{}=\displaystyle \int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}, \Psi (x)>\lambda ^{\frac{q}{\eta }}\}}w+\int \limits _{O_\lambda \cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}, \Psi (x)\le \lambda ^{\frac{q}{\eta }}\}}w\\ &{}\\ &{}=III+IV. \end{array} \end{aligned}$$

Firstly,

$$\begin{aligned} III\le \int \limits _{O_\lambda \cap \{ x\in (a,b): \Psi (x)>\lambda ^{\frac{q}{\eta }}\}}w \end{aligned}$$

and then

$$\begin{aligned} \displaystyle \int \limits _{O_\lambda \cap \{ x\in (a,b):\Psi (x)>\lambda ^{\frac{q}{\eta }}\}}w\le \frac{\Vert \Psi \Vert _{\eta , \infty ; w}^\eta }{\lambda ^q}=\frac{\Vert \Psi \Vert _{\eta , \infty ; w}^\eta }{\lambda ^q}\Vert f \Vert _{p,v}^q. \end{aligned}$$

Now, we will work on IV. Let $\{ x_k\}$ be the sequence defined as in the proof of Theorem 1 and

$$\begin{aligned} E_k=O_{\lambda }\cap (x_{k+1}, x_k)\cap \{ x\in (a,b):\Phi (x)\le \lambda ^{\frac{q}{\eta }}, \Psi (x)\le \lambda ^{\frac{q}{\eta }}\}. \end{aligned}$$

If $x\in E_k$,

$$\begin{aligned} \lambda <4^{\frac{1}{r}}\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}} +4^{\frac{1}{r}}\alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\int _a^{x_{k+2}}f. \end{aligned}$$

It is clear that, for each k, $E_k= E_{k,1}\cup E_{k,2}$, where

$$\begin{aligned} E_{k,1}=\left\{ x\in E_k: \alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}}>\frac{\lambda }{2\cdot 4^{\frac{1}{r}}} \right\} \end{aligned}$$

and

$$\begin{aligned} E_{k,2}=\left\{ x\in E_k: \alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}u^r\right) ^{\frac{1}{r}}\int \limits _a^{x_{k+2}}f>\frac{\lambda }{2\cdot 4^{\frac{1}{r}}} \right\} . \end{aligned}$$

Since $r<p$, by Theorem A (ii) we have

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _{x_{k+2}}^tf\right) ^ru^r(t) \text {d}t\right) ^{\frac{1}{r}} \\ &{}\displaystyle \le C_{r,p}\left( \int \limits _{x_{k+2}}^{x_{k+1}} \left( \int \limits _t^{x_{k+1}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{x_{k+2}}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }}\\ &{}\displaystyle \times \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}, \end{array} \end{aligned}$$

(3.7)

where $C_{r,p}=r^{\frac{1}{r}}(p')^{\frac{1}{r'}}$. The first integral in the right-hand side of (3.7) is finite due to the monotonicity of $\alpha$ and the proof of this fact follows the pattern of the one in Theorem 1.

If $x\in E_{k,1}$,

$$\begin{aligned} \begin{array}{rl} \lambda &{}<\displaystyle 2\cdot 4^{\frac{1}{r}} C_{r,p} \alpha (x)\left( \int \limits _{x_{k+2}}^{x_{k+1}}\left( \int \limits _t^{x_{k+1}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{x_{k+2}}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }}\\ &{}\displaystyle \times \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}, \end{array} \end{aligned}$$

which implies, due to the monotonicity of $\alpha$,

$$\begin{aligned} \begin{array}{rl} &{}\lambda \le \displaystyle 2\cdot 4^{\frac{1}{r}} C_{r,p}\left( \int \limits _{x_{k+2}}^{x_{k+1}} \left( \int \limits _t^{x_{k+1}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{x_{k+2}}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }}\\ &{}\displaystyle \times \left( \inf _{(\rho _k^1, \rho _k^2)} \alpha \right) \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}. \end{array} \end{aligned}$$

If we multiply both terms of the last inequality by $\displaystyle \left( \int _{E_{k,1}} w \right) ^{\frac{1}{p}}$, we get

$$\begin{aligned} \begin{array}{rl} &{}\displaystyle \left( \int \limits _{E_{k,1}} w \right) ^{\frac{1}{p}}\displaystyle \le \frac{2\cdot 4^{\frac{1}{r}} C_{r,p}}{\lambda }\left( \inf _{(\rho _k^1, \rho _k^2)} \alpha \right) \left( \int \limits _{\rho _k^1}^{\rho _k^2}w\right) ^{\frac{1}{p}}\\ &{}\\ &{}\displaystyle \times \left( \int _{x_{k+2}}^{x_{k+1}}\left( \int \limits _t^{x_{k+1}}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{x_{k+2}}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }} \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}\\ &{}\\ &{}\displaystyle \le \frac{2\cdot 4^{\frac{1}{r}} C_{r,p}}{\lambda }\left( \inf _{(\rho _k^1, \rho _k^2)} \alpha \right) \left( \int \limits _{\rho _k^1}^{\rho _k^2}w\right) ^{\frac{1}{p}} \end{array} \\ \begin{array}{rl} &{}\displaystyle \times \left( \int \limits _a^{\rho _k^1}\left( \int \limits _t^{\rho _k^1}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{a}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }} \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}\\ &{}\\ &{}\displaystyle \le \frac{2\cdot 4^{\frac{1}{r}} C_{r,p}}{\lambda }\lambda ^{\frac{q}{\eta }} \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) ^{\frac{1}{p}}, \end{array} \end{aligned}$$

where the last inequality holds since

$$\begin{aligned} \displaystyle \left( \inf \limits _{(\rho _k^1, \rho _k^2)} \alpha \right) \left( \int \limits _{\rho _k^1}^{\rho _k^2}w\right) ^{\frac{1}{p}}\left( \int \limits _a^{\rho _k^1}\left( \int \limits _t^{\rho _k^1}u^r\right) ^{\frac{\theta }{r}} \left( \int \limits _{a}^tv^{1-p'}\right) ^{\frac{\theta }{r'}}v^{1-p'}(t)\text {d}t\right) ^{\frac{1}{\theta }} \end{aligned}$$

$\le \Psi (t)$ for all $t\in E_{k,1}$. Raising to p, we have that

$$\begin{aligned} \int \limits _{E_{k,1}}w \le \frac{2^p\cdot 4^{\frac{p}{r}} C_{r,p}^p}{\lambda ^{p-\frac{pq}{\eta }}} \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) =\frac{2^p\cdot 4^{\frac{p}{r}} C_{r,p}^p}{\lambda ^q} \left( \int \limits _{x_{k+2}}^{x_{k+1}}f^pv\right) . \end{aligned}$$

Now, summing up in k,

$$\begin{aligned} \int \limits _{\cup _k E_{k,1}}w \le \frac{2^p\cdot 4^{\frac{p}{r}} C_{r,p}^p}{\lambda ^q} \left( \int \limits _a^b f^pv\right) =\frac{2^p\cdot 4^{\frac{p}{r}} C_{r,p}^p}{\lambda ^q}. \end{aligned}$$

The estimation of $\int _{\cup _k E_{k,2}} w$ is the same as the one in Theorem 1, because the relationship between r and p is not taken into account. Therefore, the proof is complete.

4 Proofs of Theorems 3 and 4

Working as in ([7], proof of Theorem 3), we have that (1.8) is equivalent to the two weighted weak-type bilinear inequalities

$$\begin{aligned} \left\| \beta (x)\int \limits _a^xf(t)\left( \int \limits _a^tg\right) \text {d}t\right\| _{q,\infty ;w}\le C\Vert f\Vert _{p_1, v_1}\Vert g\Vert _{p_2, v_2} \end{aligned}$$

(4.1)

and

$$\begin{aligned} \left\| \beta (x)\int \limits _a^xg(t)\left( \int \limits _a^tf\right) \text {d}t\right\| _{q,\infty ;w}\le C\Vert f\Vert _{p_1, v_1}\Vert g\Vert _{p_2, v_2}. \end{aligned}$$

(4.2)

Inequality (4.1) is equivalent to

$$\begin{aligned} \left\| \beta (x)\int \limits _a^xh\right\| _{q,\infty ;w}\le C\Vert h\Vert _{p_1, \tilde{v}_1^g}, \end{aligned}$$

(4.3)

where $\tilde{v}_1^g(x)=v_1(x)\left( \int _a^x\frac{g}{\Vert g\Vert _{p_2,v_2}}\right) ^{-p_1}$ and the constant C does not depend on g.

Since $q<p_1$ and $\beta$ is a monotone function, by Theorem C inequality (4.3) holds if and only if there exists $C>0$ such that

$$\begin{aligned} \Vert \Psi _g\Vert _{r_1,\infty ;w}\le C \end{aligned}$$

(4.4)

for all g, where $\frac{1}{r_1}=\frac{1}{q}-\frac{1}{p_1}$ and

$$\begin{aligned} \begin{array}{rl} \displaystyle \Psi _g(x)&{}=\displaystyle \sup _{c>x}\left( (\inf _{y\in (x,c)}\beta (y)) \left( \int \limits _x^cw\right) ^{\frac{1}{p_1}}\right) \Vert \chi _{(a,x)}(\tilde{v}_1^g)^{-\frac{1}{p_1}}\Vert _{p_1'}\\ &{}\\ &{}=\displaystyle \alpha _1(x)\Vert \chi _{(a,x)}(\tilde{v}_1^g)^{-\frac{1}{p_1}}\Vert _{p_1'}. \end{array} \end{aligned}$$

Then (4.4) can be written as

$$\begin{aligned} \left\| \alpha _1(x)\left\| \chi _{(a,x)}(t)\left( v_1^{-\frac{1}{p_1}}(t)\int \limits _a^tg\right) \right\| _{p_1'}\right\| _{r_1,\infty ;w}\le C\Vert g\Vert _{p_2,v_2}. \end{aligned}$$

(4.5)

Therefore, inequality (4.1) holds if and only if inequality (4.5) holds. Since $p_2> r_1$, by Theorems 1 and 2, (4.5) holds if and only $\Phi _1\in L^{\eta ,\infty }(w)$ in the case $p_2\le p_1'$ and $\Phi _1, \Psi _1\in L^{\eta ,\infty }(w)$ in the case $p_1'<p_2$.

In the same way, we see that (4.2) holds if and only if $\Phi _2\in L^{\eta ,\infty }(w)$ in the case $p_2\le p_1'$ and $\Phi _2, \Psi _2\in L^{\eta ,\infty }(w)$ in the case $p_1'<p_2$.