Boundedness of weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Ces\`{a}ro function spaces

TL;DR

This paper characterizes the boundedness of weighted Hardy-type operators involving suprema from weighted Lebesgue spaces to weighted Cesàro spaces, including applications to fractional maximal functions.

Contribution

It provides new characterizations of the boundedness of iterated Hardy-type operators involving suprema between weighted Lebesgue and Cesàro spaces, extending previous results.

Findings

Characterization of boundedness of $T_{u,b}$ and $T_{u,b}^*$ operators.

Boundedness criteria for the supremal operator $R_u$ on monotone functions.

Norm calculation of the fractional maximal function $M_{eta}$ between specific function spaces.

Abstract

In this paper the boundedness of the weighted iterated Hardy-type operators $T_{u,b}$ and $T_{u,b}^*$ involving suprema from weighted Lebesgue space $L_p(v)$ into weighted Ces\`{a}ro function spaces ${\operatorname{Ces}}_{q}(w,a)$ are characterized. These results allow us to obtain the characterization of the boundedness of the supremal operator $R_u$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. For the convenience of the reader, we formulate the statement on the boundedness of the weighted Hardy operator $P_{u,b }$ from $L^p(v)$ into ${\operatorname{Ces}}_{q}(w,a)$ on the cone of monotone non-increasing functions. Under additional condition on $u$ and $b$, we are able to characterize the boundedness of weighted iterated Hardy-type operator $T_{u,b}$ involving suprema from $L^p(v)$ into ${\operatorname{Ces}}_q(w,a)$ on the cone of monotone non-increasing functions. At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function $M_{\gamma}$ from $\Lambda^p(v)$ into $\Gamma^q(w)$.

Equations

$$(Pf)(x):=\frac{1}{x}{\int }_0^{x}f(t)dt\text{mbox}and(Qf)(x):={\int }_x^{\infty }\frac{f(t)}{t}dt,(x>0),$$

$$\operatorname{Ces}(p):=\bigg{\{}f:\,\|f\|_{\operatorname{Ces}(p)}:=\bigg{(}\int_{0}^{\infty}\bigg{(}\frac{1}{x}\int_{0}^{x}|f(t)|\,dt\bigg{)}^{p}\,dx\bigg{)}^{\frac{1}{p}}<\infty\bigg{\}},$$

$$\operatorname{Cop}(p):=\bigg{\{}f:\,\|f\|_{\operatorname{Cop}(p)}:=\bigg{(}\int_{0}^{\infty}\bigg{(}\int_{x}^{\infty}\frac{|f(t)|}{t}\,dt\bigg{)}^{p}\,dx\bigg{)}^{\frac{1}{p}}<\infty\bigg{\}},$$

$$V(x):={\int }_0^{x}v(t)dt.$$

$$\|f\|_{p,w,I}:=\left\{\begin{array}[]{cl}\left(\int_{I}|f(x)|^{p}w(x)\,dx\right)^{1/p}&\qquad\mbox{if}\qquad p<\infty\\ \operatornamewithlimits{ess\,sup}_{I}|f(x)|w(x)&\qquad\mbox{if}\qquad p=\infty.\end{array}\right.$$

$$L^p(w,I)=\{f\in \mathfrak{M}(I):\Vert f{\Vert }_{p,w,I}<\infty \}$$

$${\mathrm{Ces}}_p(u,v):$$

$${\mathrm{Cop}}_p(u,v):$$

$$(T_{u,b}g)(t)$$

$$(T_{u,b}^{\ast }g)(t)$$

$$(T_{u,b}h)(t)=(T_{\bar{u},b}h)(t),$$

$$\bar{u}(t):=B(t)\underset{t\le \tau }{\sup }\frac{u(\tau )}{B(\tau )},t\in (0,\infty ).$$

$$\underset{0<t<\infty }{\sup }\frac{u(t)}{B(t)}{\int }_0^t\frac{b(\tau )}{u(\tau )}d\tau <\infty .$$

$$(T_{u,b}f)(t)\approx (R_{u}f)(t)+(P_{\bar{u},b}f)(t),t\in (0,\infty ),$$

$$(R_{u}h)(t):$$

$$(P_{u,b}h)(t):$$

$$\Vert R_{u}f{\Vert }_{L^q(w)}\le C\Vert f{\Vert }_{L^p(v)},f\in {\mathfrak{M}}^{+,\downarrow }(0,\infty )$$

$$\Vert P_{u,b}(f){\Vert }_{q,w,(0,\infty )}\le c\Vert f{\Vert }_{p,v,(0,\infty )},f\in {\mathfrak{M}}^{+,\downarrow }(0,\infty )$$

$$\Vert T_{u,b}f{\Vert }_{q,w,(0,\infty )}\le C\Vert f{\Vert }_{p,v,(0,\infty )},f\in {\mathfrak{M}}^{+,\downarrow }(0,\infty )$$

$$\displaystyle A_{1}:=\int_{0}^{\infty}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)[f(t)-\lim_{t\rightarrow+\infty}f(t)]\,dt<\infty\quad\Longleftrightarrow\quad A_{2}:=\int_{(0,\infty)}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}\,d[-f(t)]<\infty.$$

$$A_{1}=\int_{0}^{\infty}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)f(t)\,dt<\infty.$$

$$\int_{0}^{x}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)f(t)\,dt\rightarrow 0,\quad\mbox{as}\quad x\rightarrow 0+.$$

$$\displaystyle\int_{0}^{x}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)f(t)\,dt\geq f(x)\,\int_{0}^{x}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)\,dt\approx f(x)\,\bigg{(}\int_{0}^{x}g\bigg{)}^{\alpha+1},\quad x>0,$$

$$f(x)\,\bigg{(}\int_{0}^{x}g\bigg{)}^{\alpha+1}\rightarrow 0,\quad\mbox{as}\quad x\rightarrow 0+.$$

$$A_2$$

$$\displaystyle=\lim_{t\rightarrow 0+}f(t)\,\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}-\lim_{t\rightarrow+\infty}f(t)\,\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}+(\alpha+1)\int_{0}^{\infty}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)f(t)\,dt$$

$$\displaystyle\leq(\alpha+1)\int_{0}^{\infty}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha}g(t)f(t)\,dt=(\alpha+1)A_{1}.$$

$$A_2\lesssim A_1.$$

$$A_{2}:=\int_{(0,\infty)}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}\,d[-f(t)]<\infty.$$

$$\int_{[x,\infty)}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}\,d[-f(t)]\rightarrow 0,\quad\mbox{as}\quad x\rightarrow+\infty.$$

$$\displaystyle\int_{[x,\infty)}\bigg{(}\int_{0}^{t}g\bigg{)}^{\alpha+1}\,d[-f(t)]$$

$$\displaystyle=\bigg{(}\int_{0}^{x}g\bigg{)}^{\alpha+1}[f(x)-\lim_{x\rightarrow+\infty}f(x)]=f(x)\,\bigg{(}\int_{0}^{x}g\bigg{)}^{\alpha+1},\quad x>0,$$

$$f(x)\,\bigg{(}\int_{0}^{x}g\bigg{)}^{\alpha+1}\rightarrow 0,\quad\mbox{as}\quad x\rightarrow+\infty.$$

Full text

1. Introduction

Many Banach spaces which play an important role in functional analysis and its applications are obtained in a special way: the norms of these spaces are generated by positive sublinear operators and by $L_{p}$-norms.

In connection with Hardy and Copson operators

[TABLE]

the classical Cesàro function space

[TABLE]

and the classical Copson function space

[TABLE]

where $1<p\leq\infty$, with the usual modifications if $p=\infty$, are of interest.

The classical Cesàro function spaces $\operatorname{Ces}(p)$ have been introduced in 1970 by Shiue [shiue]. These spaces have been defined analogously to the Cesàro sequence spaces that appeared two years earlier in [prog] when the Dutch Mathematical Society posted a problem to find a representation of their dual spaces. In 1971 Leibowitz proved that $\operatorname{ces}_{1}=\{0\}$ and for $1<q<p\leq\infty$, $\ell_{p}$ and $\operatorname{ces}_{q}$ sequence spaces are proper subspaces of $\operatorname{ces}_{p}$ [Leibowitz]. The problem posted [prog] was resolved by Jagers [jagers] in 1974 who gave an explicit isometric description of the dual of Cesàro sequence space. In [syzhanglee], Sy, Zhang and Lee gave a description of dual spaces of $\operatorname{Ces}(p)$ spaces based on Jagers’ result. In 1996 different, isomorphic description due to Bennett appeared in [bennett1996]. In [bennett1996, Theorem 21.1] Bennett observes that the classical Cesàro function space and the classical Copson function space coincide for $p>1$. He also derives estimates for the norms of the corresponding inclusion operators. The same result, with different estimates, is due to Boas [boas1970], who in fact obtained the integral analogue of the Askey-Boas Theorem [boas1967, Lemma 6.18] and [askeyboas]. These results generalized in [grosse] using the blocking technique. In [astasmal2009] they investigated dual spaces for $\operatorname{Ces}(p)$ for $1<p<\infty$. Their description can be viewed as being analogous to one given for sequence spaces in [bennett1996]. For a long time, Cesàro function spaces have not attracted a lot of attention contrary to their sequence counterparts. In fact there is quite rich literature concerning different topics studied in Cesàro sequence spaces as for instance in [CuiPluc, cuihud1999, cuihud2001, chencuihudsims, cuihudli]. However, recently in a series of papers, Astashkin and Maligranda started to study the structure of Cesàro function spaces. Among others, in [astasmal2009] they investigated dual spaces for $\operatorname{Ces}(p)$ for $1<p<\infty$. Their description can be viewed as being analogous to one given for sequence spaces in [bennett1996] (For more detailed information about history of classical Cesàro spaces see recent survey paper [asmalsurvey]).

Throughout the paper we assume that $I:=(a,b)\subseteq(0,\infty)$. By ${\mathfrak{M}}(I)$ we denote the set of all measurable functions on $I$. The symbol ${\mathfrak{M}}^{+}(I)$ stands for the collection of all $f\in{\mathfrak{M}}(I)$ which are non-negative on $I$, while ${\mathfrak{M}}^{+,\downarrow}(I)$ is used to denote the subset of those functions which are non-increasing on $I$, respectively. A weight is a function $v\in{\mathfrak{M}}^{+}(0,\infty)$ such that $0<V(x)<\infty$ for all $x\in(0,\infty)$, where

[TABLE]

The family of all weight functions (also called just weights) on $(0,\infty)$ is given by ${\mathcal{W}}(0,\infty)$.

For $p\in(0,\infty]$ and $w\in{\mathfrak{M}}^{+}(I)$, we define the functional $\|\cdot\|_{p,w,I}$ on ${\mathfrak{M}}(I)$ by

[TABLE]

If, in addition, $w\in{\mathcal{W}}(I)$, then the weighted Lebesgue space $L^{p}(w,I)$ is given by

[TABLE]

and it is equipped with the quasi-norm $\|\cdot\|_{p,w,I}$.

When $I=(0,\infty)$, we write $L^{p}(w)$ instead of $L^{p}(w,(0,\infty))$.

We adopt the following usual conventions.

Convention 1.1**.**

We adopt the following conventions:

• •

Throughout the paper we put $0\cdot\infty=0$, $\infty/\infty=0$ and $0/0=0$.

• •

If $p\in[1,+\infty]$, we define $p^{\prime}$ by $1/p+1/p^{\prime}=1$.

• •

If $0<q<p<\infty$, we define $r$ by $1/r=1/q-1/p$.

• •

If $I=(a,b)\subseteq{\mathbb{R}}$ and $g$ is monotone function on $I$, then by $g(a)$ and $g(b)$ we mean the limits $\lim_{x\rightarrow a+}g(x)$ and $\lim_{x\rightarrow b-}g(x)$, respectively.

Throughout the paper, we always denote by $c$ and $C$ a positive constant, which is independent of main parameters but it may vary from line to line. However a constant with subscript or superscript such as $c_{1}$ does not change in different occurrences. By $a\lesssim b$, ($b\gtrsim a$) we mean that $a\leq\lambda b$, where $\lambda>0$ depends on inessential parameters. If $a\lesssim b$ and $b\lesssim a$, we write $a\approx b$ and say that $a$ and $b$ are equivalent.

Unless a special remark is made, the differential element $dx$ is omitted when the integrals under consideration are the Lebesgue integrals.

The weighted Cesàro and Copson function spaces are defined as follows:

Definition 1.2**.**

Let $0<p\leq\infty$, $u\in{\mathfrak{M}}^{+}(0,\infty)$ and $v\in{\mathcal{W}}(0,\infty)$. The weighted Cesàro and Copson spaces are defined by

[TABLE]

respectively.

When $v\equiv 1$ on $(0,\infty)$, we simply write $\operatorname{Ces}_{p}(u)$ and $\operatorname{Cop}_{p}(u)$ instead of $\operatorname{Ces}_{p}(u,v)$ and $\operatorname{Cop}_{p}(u,v)$, respectively.

Recall that $\operatorname{Ces}_{p}(u,v)$ and $\operatorname{Cop}_{p}(u,v)$ are contained in the scale of weighted Cesàro and Copson function spaces $\operatorname{Ces}_{p,q}(u,v)$ and $\operatorname{Cop}_{p,q}(u,v)$ defined in [gmu_2017]. Obviously, $\operatorname{Ces}(p)=\operatorname{Ces}_{p}(x^{-1})$ and $\operatorname{Cop}(p)=\operatorname{Cop}_{p}(x^{-1})$. In [kamkub], Kamińska and Kubiak computed the dual norm of the Cesàro function space $\operatorname{Ces}_{p}(u)$, generated by $1<p<\infty$ and an arbitrary positive weight $u$. A description presented in [kamkub] resembles the approach of Jagers [jagers] for sequence spaces.

Let $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$, $b\in{\mathcal{W}}(0,\infty)$ and $B(t):=\int_{0}^{t}b(s)\,ds$. Assume that $b$ is a weight such that $b(t)>0$ for a.e. $t\in(0,\infty)$. The weighted iterated Hardy-type operators involving suprema $T_{u,b}$ and $T_{u,b}^{*}$ are defined at $g\in\mathfrak{M}^{+}(0,\infty)$ by

[TABLE]

Such operators have been found indispensable in the search for optimal pairs of rearrangement-invariant norms for which a Sobolev-type inequality holds (cf. [kerp]). They constitute a very useful tool for characterization of the associate norm of an operator-induced norm, which naturally appears as an optimal domain norm in a Sobolev embedding (cf. [pick2000], [pick2002]). Supremum operators are also very useful in limiting interpolation theory as can be seen from their appearance for example in [evop], [dok], [cwikpys], [pys]. Recall that $T_{u,b}$ successfully controls non-increasing rearrangements of wide range of maximal functions (see, for instance, [musbil] and references therein).

It was shown in [gop] that for every $h\in{\mathfrak{M}}^{+}(0,\infty)$ and $t\in(0,\infty)$

[TABLE]

where

[TABLE]

Moreover, if the condition

[TABLE]

holds, then for all $f\in{\mathfrak{M}}^{+,\downarrow}(0,\infty)$,

[TABLE]

where the supremal operator $R_{u}$ and the weighted Hardy operator $P_{u,b}$ are defined for $h\in{\mathfrak{M}}^{+}(0,\infty)$ and $t\in(0,\infty)$ by

[TABLE]

respectively.

Recall that the boundedness of $R_{u}$ from $L^{p}(v)$ into $L^{q}(w)$ on the cone of monotone non-increasing functions, that is, the validity of the inequality

[TABLE]

was completely characterized in [gop] in the case $0<p\leq q<\infty$. In the case $0<q<p<\infty$, [gop] provides solution when $u$ is equivalent to a non-decreasing function on $(0,\infty)$. The complete solution of inequality (1.3) using a certain reduction method was presented in [GogMusISI]. Another solution of (1.3) was obtained in [krep].

Note that inequality

[TABLE]

was considered by many authors and there exist several characterizations of this inequality (see, papers [cpss, bengros, gjop, cgmp2008, GogStep, GogMusIHI]).

The complete characterizations of inequality

[TABLE]

for $0<q\leq\infty$, $0<p\leq\infty$ were given in [GogMusISI] and [musbil]. Inequality (1.5) was characterized in [gop, Theorem 3.5] under condition (1.1). Note that the case when $0<p\leq 1<q<\infty$ was not considered in [gop]. It is also worth to mention that in the case when $1<p<\infty$, $0<q<p<\infty$, $q\neq 1$ [gop, Theorem 3.5] contains only discrete condition. In [gogpick2007] the new reduction theorem was obtained when $0<p\leq 1$, and this technique allowed to characterize inequality (1.5) when $b\equiv 1$, and in the case when $0<q<p\leq 1$, [gogpick2007] contains only discrete condition. Using the results in [PS_Proc_2013, PS_Dokl_2013, PS_Dokl_2014, P_Dokl_2015], another characterization of (1.5) was obtained in [StepSham] and [Sham].

In this paper we investigate the boundedness of $T_{u,b}$ and $T_{u,b}^{*}$ from the weighted Lebesgue spaces $L_{p}(v)$ into the weighted Cesàro spaces $\operatorname{Ces}_{q}(w,a)$, when $1<p,\,q<\infty$ (see, Theorems 3.1 and 3.3). These results allow us to obtain the characterization of the boundedness of $R_{u}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions (see, Theorem 4.1). For the convenience of the reader, we formulate the statement on the boundedness of $P_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions (see, Theorem 5.1). In view of (1.2), we are able to characterize the boundedness of $T_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions (see, Theorem 6.1). At the end of the paper, as an application of obtained results, we calculate the norm of the fractional maximal function $M_{\gamma}$ from $\Lambda^{p}(v)$ into $\Gamma^{q}(w)$.

The paper is organized as follows. We start with formulations of ”an integration by parts” formula in Section 2. The boundedness results for $T_{u,b}$ and $T_{u,b}^{*}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ are presented in Section 3. The characterizations of the boundedness of $R_{u}$, $P_{u,b}$ and $T_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions are given in Sections 4, 5 and 6, respectively. Finally, the obtained in previous sections results are applied to calculate the norm of the operator $M_{\gamma}:\Lambda_{p}(v)\rightarrow\Gamma_{q}(w)$ in Section 7.

2. ”An integration by parts” formula

We recall the following ”an integration by parts” formula. For the convenience of the reader we give the proof here (cf. [step_1993, Lemma, p. 176]).

Theorem 2.1**.**

Let $\alpha>0$. Let $g$ be a non-negative function on $(0,\infty)$ such that $0<\int_{0}^{t}g<\infty$, $t>0$ and let $f$ be a non-negative non-increasing right-continuous function on $(0,\infty)$. Then

[TABLE]

Moreover, $A_{1}\approx A_{2}$.

Proof.

Assume at first that $\lim_{t\rightarrow+\infty}f(t)=0$. Let

[TABLE]

Then

[TABLE]

Since

[TABLE]

we have that

[TABLE]

Integrating by parts, we get that

[TABLE]

Thus

[TABLE]

Now assume that

[TABLE]

Then

[TABLE]

Since

[TABLE]

we obtain that

[TABLE]

Thus, integrating by parts, we get that

[TABLE]

Hence

[TABLE]

We have shown that if $\lim_{x\rightarrow+\infty}f(x)=0$, then

[TABLE]

and

[TABLE]

Now assume that $\lim_{x\rightarrow+\infty}f(x)>0$. Then, applying previous statement to the function $f(x)-\lim_{x\rightarrow+\infty}f(x)$, we arrive at

[TABLE]

The proof is completed. ∎

Remark 2.2**.**

Note that if $f\in{\mathfrak{M}}^{+,\downarrow}(0,\infty)$ is such that $\lim_{x\rightarrow+\infty}f(x)>0$, then

[TABLE]

Indeed: for each $x\in(0,\infty)$

[TABLE]

holds. Thus

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

Corollary 2.3**.**

Let $\alpha>0$. Let $g$ be a non-negative function on $(0,\infty)$ such that $0<\int_{0}^{t}g<\infty$, $t>0$ and let $f$ be a non-negative non-increasing right-continuous function on $(0,\infty)$. Then

[TABLE]

Proof.

If $\lim_{x\rightarrow+\infty}f(x)=0$, then the statement follows by Theorem 2.1. If $\lim_{x\rightarrow+\infty}f(x)>0$, then by Remark 2.2, we know that

[TABLE]

Therefore, by Theorem 2.1, we get that

[TABLE]

The proof is completed. ∎

Theorem 2.4**.**

Let $\alpha>0$. Let $g$ be a non-negative function on $(0,\infty)$ such that $0<\int_{t}^{\infty}g<\infty$, $t>0$ and let $f$ be a non-negative non-decreasing left-continuous function on $(0,\infty)$. Then

[TABLE]

Moreover, $B_{1}\approx B_{2}$.

Proof.

Assume at first that $f(0+)=0$. Let

[TABLE]

Then

[TABLE]

Since

[TABLE]

we have that

[TABLE]

Hence, integrating by parts, we get that

[TABLE]

Now assume that

[TABLE]

Then

[TABLE]

Since

[TABLE]

we obtain that

[TABLE]

Thus, integrating by parts, we get that

[TABLE]

We have shown that if $f(0+)=0$, then

[TABLE]

and

[TABLE]

Now assume that $f(0+)>0$. Then, applying previous statement to the function $f(x)-f(0+)$, we arrive at

[TABLE]

The proof is completed. ∎

Remark 2.5**.**

Note that if $f$ is a non-negative non-decreasing function on $(0,\infty)$ such that $f(0+)>0$, then

[TABLE]

Indeed: for each $x\in(0,\infty)$

[TABLE]

holds. Thus

[TABLE]

Hence

[TABLE]

Therefore

[TABLE]

Corollary 2.6**.**

Let $\alpha>0$. Let $g$ be a non-negative function on $(0,\infty)$ such that $0<\int_{t}^{\infty}g<\infty$, $t>0$ and let $f$ be a non-negative non-decreasing left-continuous function on $(0,\infty)$. Then

[TABLE]

Proof.

If $f(0+)=0$, then the statement follows by Theorem 2.4. If $f(0+)>0$, then by Remark 2.5, we know that

[TABLE]

Therefore, by Theorem 2.4, we get that

[TABLE]

The proof is completed. ∎

3. The boundedness of $T_{u,b}$ and $T_{u,b}^{*}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$

In this section we give solutions of the following two inequalities

[TABLE]

and

[TABLE]

where $1<p\leq q<\infty$ and $a,\,u,\,v,\,w\in{\mathcal{W}}(0,\infty)$. Using the duality argument, we reduce the problem to the boundedness for the dual of integral Volterra operator with a kernel satisfying Oinarov’s condition and weighted Stieltjes operator.

Note that the characterization of inequalities

[TABLE]

and

[TABLE]

can be reduced to the solutions of (3.1) and (3.2).

Recall that, if $F$ is a non-negative non-decreasing function on $(0,\infty)$, then

[TABLE]

likewise, when $F$ is a non-negative non-increasing function on $(0,\infty)$, then

[TABLE]

(see, for instance, [gp2, p. 85]).

We need the following notations:

[TABLE]

Theorem 3.1**.**

Let $1<p,\,q<\infty$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$. Moreover, assume that

[TABLE]

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

Assume that $1<p\leq q<\infty$. By duality, using Fubini’s Theorem, and interchanging the suprema, we get that

[TABLE]

Applying [gop, Theorems 4.4], on using (3.5), we arrive at

[TABLE]

where

[TABLE]

Integrating by parts (applying Corollary 2.6), on using Fubini’s Theorem, we arrive at

[TABLE]

Similarly, integrating by parts (applying Corollary 2.3), on using Fubini’s Theorem, we get at

[TABLE]

(i) Let $p\leq q$. By [Oinar, Theorem 1.1], we obtain that

[TABLE]

By [mazya, Theorem 1, p. 40 and Theorem 3, p. 44], respectively, we have that

[TABLE]

and

[TABLE]

By duality, we have that

[TABLE]

Thus, we get that

[TABLE]

Combining (3.7) and (3.8), we arrive at

[TABLE]

(ii) Let now $q<p$. By [Oinar, Theorem 1.2], we obtain that

[TABLE]

By [mazya, Theorem 2, p. 48], we have that

[TABLE]

and

[TABLE]

Consequently, we arrive at

[TABLE]

Combining (3.9) and (3.10), we arrive at

[TABLE]

The proof is completed. ∎

Theorem 3.2**.**

Let $1<p,\,q<\infty$ and $b\in{\mathcal{W}}(0,\infty)$ be such that $b(t)>0$ for a.e. $t\in(0,\infty)$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$. Moreover, assume that

[TABLE]

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

The statement follows by Theorem 3.1 at once if we note that

[TABLE]

∎

Theorem 3.3**.**

Let $1<p,\,q<\infty$ and $b\in{\mathcal{W}}(0,\infty)$ be such that $b(t)>0$ for a.e. $t\in(0,\infty)$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$. Moreover, assume that

[TABLE]

Denote by

[TABLE]

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

By [GogMusIHI, Corollary 3.5], we have that

[TABLE]

(i) Let $p\leq q$. By Theorem 3.1, (i), we get that

[TABLE]

(ii) Let $q<p$. By Theorem 3.1, (ii), we obtain that

[TABLE]

The proof is completed. ∎

4. The boundedness of $R_{u}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions

In this section we characterize the boundedness of $R_{u}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions.

Theorem 4.1**.**

Let $1<p,\,q<\infty$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$.

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

By [GogStep, Theorem 3.2] (cf. [GogMusIHI, Theorem 2.3]), we get that

[TABLE]

By Theorem 3.1, we have that

(i) if $p\leq q$, then

[TABLE]

(ii) if $q<p$, then

[TABLE]

∎

5. The boundedness of $P_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions

In this section we characterize the boundedness of weighted Hardy operator $P_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions.

Theorem 5.1**.**

Let $1<p,\,q<\infty$ and $b\in{\mathcal{W}}(0,\infty)$ be such that $b(t)>0$ for a.e. $t\in(0,\infty)$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$.

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

By [GogStep, Theorem 3.1], using Fubini’s Theorem, we get that

[TABLE]

(i) Let $p\leq q$. Using the characterizations of weighted Hardy-type inequalities (see, for instance, [ok, Section 1]), by [Oinar, Theorem 1.1], we obtain that

[TABLE]

(ii) Let now $q<p$. Using the characterizations of weighted Hardy-type inequalities (see, for instance, [ok, Section 1]), by [Oinar, Theorem 1.2], we obtain that

[TABLE]

The proof is completed. ∎

6. The boundedness of $T_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions

In this section we combine the results from previous two sections to present the characterization of the boundedness of $T_{u,b}$ from $L^{p}(v)$ into $\operatorname{Ces}_{q}(w,a)$ on the cone of monotone non-increasing functions.

Theorem 6.1**.**

Let $1<p,\,q<\infty$ and $b\in{\mathcal{W}}(0,\infty)$ be such that $b(t)>0$ for a.e. $t\in(0,\infty)$. Assume that $u\in{\mathcal{W}}(0,\infty)\cap C(0,\infty)$ and $a,\,v,\,w\in{\mathcal{W}}(0,\infty)$. Moreover, assume that condition (1.1) holds.

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

By (1.2), we have that

[TABLE]

It remains to apply Theorems 4.1 and 5.1. ∎

7. The boundedness of $M_{\gamma}$ from $\Lambda^{p}(v)$ into $\Gamma^{q}(w)$

Suppose that $f$ is a measurable a.e. finite function on ${\mathbb{R}}^{n}$. Then its non-increasing rearrangement $f^{*}$ is given by

[TABLE]

and let $f^{**}$ denotes the Hardy-Littlewood maximal function of $f^{*}$, i.e.

[TABLE]

Quite many familiar function spaces can be defined by using the non-increasing rearrangement of a function. One of the most important classes of such spaces are the so-called classical Lorentz spaces.

Let $p\in(0,\infty)$ and $w\in{\mathcal{W}}(0,\infty)$. Then the classical Lorentz spaces $\Lambda^{p}(w)$ and $\Gamma^{p}(w)$ consist of all measurable functions $f$ on $\mathbb{R}^{n}$ for which $\|f\|_{\Lambda^{p}(w)}:=\|f^{*}\|_{p,w,(0,\infty)}<\infty$ and $\|f\|_{\Gamma^{p}(w)}:=\|f^{**}\|_{p,w,(0,\infty)}<\infty$, respectively. For more information about the Lorentz $\Lambda$ and $\Gamma$ spaces see e.g. [cpss] and the references therein.

The fractional maximal operator, $M_{\gamma}$, $\gamma\in(0,n)$, is defined at a locally integrable function $f$ on $\mathbb{R}^{n}$ by

[TABLE]

It was shown in [ckop, Theorem 1.1] that

[TABLE]

for every locally integrable function $f$ on $\mathbb{R}^{n}$ and $t\in(0,\infty)$, where $\tilde{f}(x):=f^{*}(\omega_{n}|x|^{n})$ and $\omega_{n}$ is the volume of the unit ball in $\mathbb{R}^{n}$.

The characterization of the boundedness of $M_{\gamma}$ between classical Lorentz spaces $\Lambda^{p}(v)$ and $\Lambda^{q}(w)$ was obtained in [ckop] for the particular case when $1<p\leq q<\infty$ and in [o, Theorem 2.10] in the case of more general operators and for extended range of $p$ and $q$ (For the characteriation of the boundedness of more general fractional maximal functions between $\Lambda^{p}(v)$ and $\Lambda^{q}(w)$, see [musbil], and the references therein).

As an application of obtained results, we calculate the norm of the fractional maximal function $M_{\gamma}$ from $\Lambda^{p}(v)$ into $\Gamma^{q}(w)$.

Theorem 7.1**.**

Let $1<p,\,q<\infty$ and $0<\gamma<n$. Assume that $v,\,w\in{\mathcal{W}}(0,\infty)$.

(i)* If $p\leq q$, then*

[TABLE]

(ii)* If $q<p$, then*

[TABLE]

Proof.

From inequalities (7.1), we have that

[TABLE]

with $u(\tau)=\tau^{\gamma/n}$ and $b\equiv 1$. Note that

[TABLE]

in this case. So, it remains to apply Theorem 6.1. ∎