# Discrete Rubio de Francia extrapolation theorem via factorization of weights
and iterated algorithms

S. H. Saker1, A. I. Saied2 and R. P. Agarwal3 1Department of Mathematics,
Faculty of Science, Mansoura University, Mansoura-Egypt, e-mail:
<EMAIL_ADDRESS>
2Department of Mathematics, Faculty of Science, Benha university, Benha-
Egypt, e-mail<EMAIL_ADDRESS>
3Department of Mathematics, Texas A & M University- Kingsvilie,
Texas, 78363, USA.
E-mail<EMAIL_ADDRESS>
###### Abstract.

In this paper, we prove a discrete Rubio de Francia extrapolation theorem via
factorization of discrete Muckenhoupt weights and discrete iterated Rubio de
Francia algorithm and its duality. Precisely, we will prove that if for some
$p_{0}>1$ and $w\in\mathcal{A}_{p_{0}}$ the $\ell^{p_{0}}(w)$ inequality
$\left\|\mathcal{T}f\right\|_{\ell^{p_{0}}(w)}\leq
C\left\|f\right\|_{\ell^{p_{0}}(w)}$ holds for pairs of the form
($\mathcal{T}f,f)$, then for all $p$ and $w\in\mathcal{A}_{p}$ the
$\ell^{p}(w)$ inequality $\left\|\mathcal{T}f\right\|_{\ell^{p}(w)}\leq
C\left\|f\right\|_{\ell^{p}(w)}$ also holds for such pairs, where
$\mathcal{T}$ is a given operator. The new approach, among other things,
allows us to obtain explicit estimates of the constants and do not assume that
the operator $\mathcal{T}$ is linear or even sublinear. The theorem will be
proved for a fixed exponent $1<p<\infty$ and the case when $0<p<1$ is an open
problem.

2010 Mathematics Subject Classification: 44A55, 26D15, 40A05, 39A70, 46B45.

Key words and phrases: Extrapolation theorem, Muckenhoupt classes, Bounded
operator and Rubio de Francia algorithm.

##  1\. Introduction

The study of boundedness of operators is one of the main subjects in modern
harmonic analysis. The ideas related to the boundedness of operators and the
weighted inequalities appeared with the birth of singular integrals but a
better understanding of the subject was only in the 1970s. In 1972 Muckenhoupt
[18] introduced the $A_{p}-$weights (now often referred to as Muckenhoupt
weights): for a given exponent $p>1$ a weight $w$ is said to belongs to
$A_{p}$ if there exists a real constant $\mathcal{C}>1$ such that,

(1.1)
$\left(\frac{1}{\left|I\right|}\int_{I}w(x)dx\right)\left(\frac{1}{\left|I\right|}\int_{I}w^{-\frac{1}{p-1}}(x)dx\right)^{p-1}\leq\mathcal{C},$

holds for every interval $I\subseteq I_{0}$. The infimum of all such
$\mathcal{C}$ is denoted by $[w]_{A_{p}}$ and is called the
$A_{p}\emph{-}$norm. The full characterization of the weights $w,$ for which
the Hardy-Littlewood maximal operator

(1.2) $\mathcal{M}f(x):=\sup_{x\in I}\frac{1}{\left|I\right|}\int_{I}f(y)dy,$

is bounded on $L_{w}^{p}(\mathbb{R}^{+})$ by means of the so-called
$A_{p}$_-condition_ was achieved by Muckenhoupt and the weighted norm
inequality

(1.3) $\int_{0}^{\infty}w(t)\left(\mathcal{M}f(t)\right)^{p}dt\leq
C\int_{0}^{\infty}w(t)f^{p}(t)dt.$

has been proved for a nonnegative function $f\in L_{w}^{p}(\mathbb{R}^{+})$
and the constant $C$ depends of the norm $[w]_{A_{p}}$ of the weight $w.$ The
weight $w$ is in the Muckenhoupt class $A_{1}$ if there exists a constant
$\mathcal{C}>1$ such that

(1.4) $\mathcal{M}w(x)\leq\mathcal{C}w(x),\text{ \ for all
}x\in\mathbb{R}^{+},$

where the operator $f\rightarrow\mathcal{M}f$ is the Hardy-Littlewood maximal
operator. The infimum of all such $\mathcal{C}$ is denoted by $[w]_{A_{1}}$
and is called the $A_{1}\emph{-}$norm.

Muckenhoupt’s result became a landmark in the theory of weighted norm
inequalities because most of the previously known results for classical
operators had been obtained for special classes of weights (like power
weights) and has been extended to cover several operators like Hardy’s
operator, Hilbert’s operator, Calderón-Zygmund’s singular operator, fractional
integral operator. For more details about the properties of Muckenhoupt
weights and their applications, we refer the reader to the papers [14, 19, 1,
2] and the references cited therein. By Hölder’s inequality it is easy to
prove that if $w$ satisfies the $A_{p}$ condition (1.1) then $w$ satisfies the
Muckenhoupt condition

(1.5)
$\left(\frac{1}{\left|I\right|}\int_{I}w(x)dx\right)\left(\frac{1}{\left|I\right|}\int_{I}w^{-\frac{1}{p_{1}-1}}(x)dx\right)^{p_{1}-1}\leq\mathcal{C},\text{
for }p_{1}>p.$

The main problem that has been solved in [18] by Muckenhoupt is the (backward
propagation) self-improving property of the Muckenhoupt class of weights which
sates that: If $w\in A_{p}(\mathcal{C)}$ then there exists
$\epsilon=\epsilon(p,\mathcal{C})>0$ such that

(1.6)
$\left(\frac{1}{\left|I\right|}\int_{I}w(t)dt\right)\left(\frac{1}{\left|I\right|}\int_{I}w^{-\frac{1}{q-1}}(t)dt\right)^{q-1}\leq\mathcal{C}_{1},$

for all $q=p-\epsilon$, where the constant
$\mathcal{C}_{1}=\mathcal{C}_{1}(p,\mathcal{C}).$ In other words, Muckenhoupt
result for self-improving property shows that if $w\in A_{q}(\mathcal{C})$
then there exists $\epsilon>0$ such that $w\in
A_{q-\epsilon}(\mathcal{C}_{1}),$ and then

(1.7) $A_{q}(\mathcal{C})\subset A_{q-\epsilon}(\mathcal{C}_{1}).$

In 1982 Rubio de Francia [20] discovered a very different perspective theorem
on the theory of weighted norm inequalities. This theorem called the
extrapolation theorem which studies the boundedness of operators in different
spaces via the Muckenhoupt weights. In particular, it has been proved that for
a given sublinear operator $\mathcal{T},$ for some $p_{0}$ such that $1\leq
p_{0}<\infty,$ and every $w\in A_{p_{0}},$ there exists a constant
$\mathcal{C}>0$ depends on $[w]_{A_{p_{0}}}$ such that

$\int_{0}^{\infty}\left|\mathcal{T}f(x)\right|^{p_{0}}w(x)dx\leq\mathcal{C(}[w]_{A_{p_{0}}})\int_{0}^{\infty}\left|f(x)\right|^{p_{0}}w(x)dx,$

then for every $1<p<\infty$ and every $w\in A_{p},$ there exists a constant
$\mathcal{C}_{1}$ depends on $[w]_{A_{p}}$ such that

$\int_{0}^{\infty}\left|\mathcal{T}f(x)\right|^{p}w(x)dx\leq\mathcal{C}_{1}([w]_{A_{p}})\int_{0}^{\infty}\left|f(x)\right|^{p}w(x)dx.$

The original proof of Rubio de Francia extrapolation theorem was quite complex
and depends on a connection between vector-valued estimates and a weighted
norm inequalities. A more direct proof that depend only on weighted norm
inequalities was given by Garciá-Cuerva [13]. However this approach requires
two complicated lemmas on the structure of $A_{p}-$ weights.

In [10] the authors proved the extrapolation theorem with sharp bounds with a
different method depends on the applications of Rubio de Francia algorithm. In
[12] Duoandikoetxea proved the extrapolation theorem for Muckenhoupt weights
with explicit bounds and avoided the iteration algorithm when $p_{0}>1.$ The
proof has two steps: first prove the desired inequality holds for $1<p<p_{0}$
and $w\in A_{1}$ and then use this to prove the full result. The proof also
requires the very deep self-improving property of $A_{p}$ weights that: if
$w\in A_{p}$ then there exists $\epsilon>0$ such that $w\in A_{p-\epsilon}.$

In [9] Carro and Lorente proved a new version of Rubio de Francia
extrapolation theorem in the setting of $B_{p}-$weights, instead of the
$A_{p}-$weights, for pairs of positive decreasing functions defined on
$\mathbb{R}^{+}$. A weight $w$ is said to be belong to the class $B_{p}(C)$
for $0<p<\infty\ $if it satisfies the condition

(1.8)
$\int_{t}^{\infty}\frac{w(x)}{x^{p}}dx\leq\frac{C}{t^{p}}\int_{0}^{t}w(x)dx\text{,
\ for all }t>0.$

The smallest constant $C>0$ satisfying (1.8) is called the $B_{p}-$constant of
the class and is denoted by $\left[B_{p}(w)\right]$. This class has been
introduced by Ariňo and Muckenhoupt [3] in connection with the boundedness of
the Hardy operator $\mathcal{H}f(t)=(1/t)\int_{0}^{t}f(x)dx$, for $t>0$ on the
space $L_{w}^{p}[\mathbb{R}^{+}).$ The proof also requires the very deep self-
improving property of $B_{p}$ weights that: if $w\in B_{p}$ then there exists
$\epsilon>0$ such that $w\in B_{p-\epsilon}.$

The theory of Rubio de Francia extrapolation has found a host of applications
in harmonic analysis. One of the important applications is due to
Duoandikoetxea and Rubio de Francia [11]. They have proved the boundedness of
the rough singular integrals on $L^{p}(w)$ with kernels via the Muckenhoupt
weights $w\in A_{p}$. The key to the proof was the extrapolation theorem,
since this reduced the problem to proving that the singular operator is
bounded on $L^{2}(w)$,$\ w\in A_{2}.$

In the last years, there has been renewed interest in the area of discrete
harmonic analysis and then it becomes an active field of research. For
example, the study of discrete $\ell_{p}(w)$ analogues for $L^{p}(w)$ results
has been considered by some authors, see for example [5, 6, 7, 8, 17, 23, 24,
25, 28] and the references cited therein. In [4] the authors mentioned that
the study of discrete norm inequalities is not an easy task and more difficult
to analyze than its integral counterparts and discovered that the discrete
conditions do not correspond, in any natural way, with those that are obtained
by discretization the results of functions but the reverse is true. This means
that what goes for sums goes, with the obvious modifications, for integrals
which in fact proved the first part of basic principle of Hardy, Littlewood
and Polya [15, p. 11].

Indeed the proofs for series translate immediately and become much simpler,
when applied to integrals. For illustration, we present the following examples
to show the big difference of discrete and continuous inequalities. In [16]
the author proved the inequality

(1.9)
$\int_{a}^{b}\frac{\lambda(t)}{\Lambda^{1-\alpha}(t)}\left(\mathcal{M}v\left(t\right)\right)^{\beta}dt\leq\left(\frac{\beta}{\beta-\alpha}\right)^{\beta}\int_{a}^{b}\frac{\lambda(t)}{\Lambda^{1-\alpha}(t)}v^{\beta}\left(t\right)dt,$

where

$\mathcal{M}v(t):=\frac{1}{\Lambda(t)}\int_{a}^{t}\lambda(s)v\left(s\right)ds\text{,
and }\Lambda(t)=\int_{a}^{t}\lambda(s)ds.$

The discrete analogy of this inequality is given by (translation from integral
to series directly): Let $v$ be a positive sequence and $\lambda$ be a
nonnegative sequence and $\Lambda[1,n]=\sum\limits_{k=1}^{n}\lambda(k)$ for
all $n\in\mathbb{Z}_{+}$. Suppose that $0<\alpha<1$, $\beta<0$ or $\beta\geq
1$ and $\mathcal{M}v$ is defined by

$\mathcal{M}v(k):=\frac{1}{\Lambda[1,k]}\sum\limits_{s=1}^{k}\lambda(s)v\left(s\right).$

Then

(1.10)
$\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}\left(\mathcal{M}v\left(k\right)\right)^{\beta}\leq\left(\frac{\beta}{\beta-\alpha}\right)^{\beta}\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}v^{\beta}\left(k\right).$

In the unweighted case $\lambda(k)=1$ and $\beta-\alpha=1$, the inequality
(1.10) becomes

(1.11)
$\sum\limits_{k=1}^{N}\frac{1}{k^{2}}\left(\sum\limits_{s=1}^{k}v\left(s\right)\right)^{\beta}\leq\beta^{\beta}\sum\limits_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}\left(k\right).$

Numerical (counter) example: Set in (1.11) $N=4$, $\beta=1.2$, $\alpha=0.2$,
$v(1)=100$, $v(2)=v(3)=v)4)=1$, we see that the right side is given by

$\sum\limits_{k=1}^{N}\frac{1}{k^{2}}\left(\sum\limits_{s=1}^{k}v\left(s\right)\right)^{\beta}=359.587$

and the left hand side is given by

$\beta^{\beta}\sum\limits_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}\left(k\right)=314.263$

For the case $\beta<0$ and $\beta-\alpha=-1,$ then (1.10) becomes (unweighted
case):

(1.12)
$\sum\limits_{k=1}^{N}\left(\sum\limits_{s=1}^{k}v\left(s\right)\right)^{\beta}\leq(-\beta)^{\beta}\sum\limits_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}\left(k\right).$

By choosing $\beta=-0.9$, $\alpha=0.1$, $N=5,$ $v(1)=1,$ $v(2)=5$, $v(3)=9$,
$v(4)=13,$ $v(5)=17.$ Numerical gives the right hand side

$\sum\limits_{k=1}^{N}\left(\sum\limits_{s=1}^{k}v\left(s\right)\right)^{\beta}=1.36913\text{,
}$

and the left hand side

$(-\beta)^{\beta}\sum\limits_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}\left(k\right)=1.3406.$

This shows that (1.11) and (1.12) are not valid and a constant $\zeta>1$
should be exists in the right hand side. This proves that the discrete
conditions do not correspond, in any natural way, with those that are obtained
by discretization the results of functions. So the inequality (1.10) should be

$\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}\left(\mathcal{M}v\left(k\right)\right)^{\beta}\leq\zeta\left(\frac{\beta}{\beta-\alpha}\right)^{\beta}\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}v^{\beta}\left(k\right),$

where $\Lambda[1,k]\leq\zeta\Lambda[1,k-1]$ for some $\zeta>1$ and all $k$.
Another choice is to put the inequality in the form

(1.13)
$\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}\left(\mathcal{M}v\left(k-1\right)\right)^{\beta}\leq\left(\frac{\beta}{\beta-\alpha}\right)^{\beta}\sum\limits_{k=1}^{N}\frac{\lambda(k)}{\Lambda^{1-\alpha}[1,k]}v^{\beta}\left(k\right).$

For the unweighted case ($\lambda(s)=1$ for every $s$) and $\beta-\alpha=1$,
inequality (1.13) becomes:

(1.14)
$\sum_{k=1}^{N}\frac{1}{k^{1-\alpha}}\left(\frac{1}{k-1}\sum_{s=1}^{k-1}v(s)\right)^{\beta}\leq\beta^{\beta}\sum_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}(k).$

The numerical gives us that: For $N=4,$ $\beta=1.2,$ $\alpha=0.2,$
$v_{1}=100,v_{2}=1,$ $v_{3}=1,$ $v_{4}=1$, then the left-side of (1.14) gives:

$\frac{1}{1^{0.8}}(0)^{1.2}+\frac{1}{2^{0.8}}\left(v_{1}\right)^{1.2}+\frac{1}{3^{0.8}}\left(\frac{v_{1}+v_{2}}{2}\right)^{1.2}+\frac{1}{4^{0.8}}\left(\frac{v_{1}+v_{2}+v_{3}}{3}\right)^{1.2}=212.922,$

and the right-side gives:

$1.2^{1.2}\ast(v_{1}^{1.2}+\frac{v_{2}^{1.2}}{2^{0.8}}+\frac{v_{3}^{1.2}}{3^{0.8}}+\frac{v_{4}^{1.2}}{4^{0.8}})=314.263.$

This shows that (1.13) holds. For the unweighted case and $\beta<0,$
$\beta-\alpha=-1$, inequality (1.13) becomes:

(1.15)
$\sum_{k=1}^{N}\frac{1}{k^{1-\alpha}}\left(\frac{1}{k-1}\sum_{s=1}^{k-1}v(s)\right)^{\beta}\leq\left(-\beta\right)^{\beta}\sum_{k=1}^{N}\frac{1}{k^{1-\alpha}}v^{\beta}(k).$

For $N=5,\beta=-0.9,$ $\alpha=0.1,$ $v_{1}=1,$ $v_{2}=5,$ $v_{3}=9,$
$v_{4}=13,$ $v_{5}=17$, the left-side of (1.15) gives:

$\frac{1}{1^{0.9}}(0)^{0.9}+\frac{1}{2^{0.9}}\left(v_{1}\right)^{-0.9}+\frac{1}{3^{0.9}}\left(\frac{v_{1}+v_{2}}{2}\right)^{-0.9}$
$+\frac{1}{4^{0.9}}\left(\frac{v_{1}+v_{2}+v_{3}}{3}\right)^{-0.9}+\frac{1}{5^{0.9}}\left(\frac{v_{1}+v_{2}+v_{3}+v_{4}}{4}\right)^{-0.9}=0.7743$

and the right-side gives:

$(0.9)^{-0.9}\ast\left(v_{1}^{-0.9}+\frac{v_{2}^{-0.9}}{2^{0.9}}+\frac{v_{3}^{-0.9}}{3^{0.9}}+\frac{v_{4}^{-0.9}}{4^{0.9}}+\frac{v_{5}^{-0.9}}{5^{0.9}}\right)=1.3516.$

This shows that (1.13) holds and explains our aim to prove the discrete
results and do not translate the continuous results directly to the discrete
versions and also gives an answer to the questions from the referees to our
discrete papers.

In the following, for completeness, we present some of the related results of
the discrete Muckenhoupt $\mathcal{A}_{p}$ weights and discrete extrapolation
theorem of discrete operators to show the motivation of this paper. Throughout
the paper, we assume that $1<p<\infty$ and assume that $J\mathbb{\subset
Z}_{+}=\\{1,2,3,...\\},$ where $J$ is of the form $J=\\{1,2,\ldots,n\\}$. A
discrete weight $w$ is a sequence $\left\\{w(n)\right\\}_{n=1}^{\infty}$ of
nonnegative real numbers. Most often $w$ will appear in the role of the weight
in $\ell_{p}(w)-$estimates, i.e. we shall consider the norm

$\left\|f\right\|_{\ell_{p}(w)}:=\left(\sum_{k=1}^{\infty}w(k)\left|f(k)\right|^{p}\right)^{\frac{1}{p}}<\infty.$

A discrete nonnegative weight $w$ defined on $\mathbb{Z}_{+}=\\{1,2,\ldots\\}$
belongs to the discrete Muckenhoupt class $\mathcal{A}_{1}(C)$ for $p>1$ and
$C>1$ if the inequality

(1.16)
$\left(\frac{1}{n}\sum\limits_{k=1}^{n}{w(k)}\right){\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}}\leq
C,$

holds for every $n>1.$ For a given exponent $p>1$ we define the
$\mathcal{A}_{p}(A)-$norm of the discrete weight $w$ by the following quantity

$\left[w\right]_{\mathcal{A}_{p}}:=\sup_{n\geq
1}\frac{1}{n}\sum\limits_{k=1}^{n}w(n)\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{1}{1-p}}(n)\right)^{p-1}.$

The boundedness of discrete Hardy-Littlewood maximal operator

(1.17)
$\mathcal{M}f(n):=\sup_{n>1}\frac{1}{n}\sum\limits_{k=1}^{n}f(k)\text{,}$

where $f(n)$ is nonnegative sequence has been characterized in [26] in terms
of the Muckenhoupt weights $\mathcal{A}_{p}$. They proved that $\mathcal{M}f$
is bounded in $\ell_{p}(w)$ iff $w\in\mathcal{A}_{p}$ and

$\left\|\mathcal{M}f\right\|_{\ell^{p}(w)}\leq
C\left\|f\right\|_{\ell^{p}(w)},$

where the constant $C$ depends on the norm $[w]_{\mathcal{A}_{p}}.$ The
discrete weight $w$ belongs to the discrete Muckenhoupt class
$\mathcal{A}_{1}$ if there exists a constant $C>1$ such that

$\mathcal{M}w(n)\leq Cw(n),\text{ \ \ \ }n\in\mathbb{Z}_{+},$

where $\mathcal{M}$ is the Hardy-Littlewood maximal operator. The infimum of
all such $C$ is denoted by $[w]_{\mathcal{A}_{1}}$ and is called the
$\mathcal{A}_{1}\emph{-}$norm. A discrete weight $w$ is said to be belong to
the discrete Muckenhoupt class $\mathcal{A}_{\infty}$ if there exists a
constant $C>1$ such that

$\left(\frac{1}{n}\sum\limits_{k=1}^{n}w\right)\left(\exp\frac{1}{n}\sum\limits_{k=1}^{n}\log\frac{1}{w}\right)\leq
C.$

The infimum of all such $C$ is denoted by $[w]_{\mathcal{A}_{\infty}}$ and is
called the $\mathcal{A}_{\infty}\emph{-}$norm. The prototypical
$\mathcal{A}_{p}-$weights are the power weights. The authors in [29] (see also
[27]) investigated the following estimates for power low discrete weights. If
$p>1$ and $-1<\lambda<p-1$, then $n^{\lambda}\in A_{p}$ and its norm is given
by $\left[n^{\lambda}\right]_{\mathcal{A}_{p}}\simeq\Phi(p,\lambda)$, where

(1.18)
$\Phi(p,\lambda)=\frac{1}{(1+\lambda)}\left(\frac{p-1}{p-\lambda-1}\right)^{p-1}$

Taking limit as $p\rightarrow 1,$ we get the following: If $-1<\lambda<0$,
then $n^{\lambda}\in\mathcal{A}_{1}$ and the norm
$\left[n^{\lambda}\right]_{\mathcal{A}_{1}}\simeq\frac{1}{(1+\lambda)}$.

In [21] Saker and Agarwal proved the discrete Rubio de Francia extrapolation
theorem via the discrete Muckenhoupt weights. Precisely they proved that if
$\varphi$ is an increasing function on $(0,\infty)$ and $\mathcal{T}$ is a
sublinear operator defined on $\mathbb{Z}_{+}$ such that

$\sum_{k=1}^{\infty}\left|\mathcal{T}f(k)\right|^{p_{0}}w(k)\leq\varphi\left(\left[w\right]_{\mathcal{A}_{p_{0}}}\right)\sum_{k=1}^{\infty}\left|f(k)\right|^{p_{0}}w(k),$

for some $p_{0},$ $1\leq p_{0}<\infty,$ and every weight
$w\in\mathcal{A}_{p_{0}},$ then

$\sum_{k=1}^{\infty}\left|\mathcal{T}f(k)\right|^{p}w(k)\leq\varphi\left(\left[w\right]_{\mathcal{A}_{p}}\right)\sum_{k=1}^{\infty}\left|f(k)\right|^{p}w(k),$

for every $p$ with $1<p<\infty$ and every $w\in\mathcal{A}_{p}.$ The proof
depends on the weighted norm inequality

(1.19)
$\left\|\mathcal{M}^{w}f\right\|_{\ell_{w}^{r}\left(\mathbb{Z}_{+}\right)}\leq
C\left\|f\right\|_{\ell_{w}^{r}\left(\mathbb{Z}_{+}\right)},$

for $r>1$ and $f$ be a nonnegative series defined on $\mathbb{Z}_{+}$, where
$\mathcal{M}^{w}f$ is the weighted Hardy-Littlewood maximal operator

(1.20) $\left[\mathcal{M}^{w}\left(f\right)\right](k)=\sup_{k\in
J}\frac{1}{\Lambda(J)}\sum_{s\in J}w(s)f(s),\text{ for }k\in\mathbb{Z}_{+},$

where $\Lambda(J)=\sum_{s\in J}w(s)$. However this approach requires two
complicated lemmas on the structure of $\mathcal{A}_{p}-$ weights and the
self-improving property of the discrete $\mathcal{A}_{p}$ Muckenhoupt weights:
if $v\in\mathcal{A}_{q}(C)$ then there exists an $\epsilon>0$ and a constant
$C_{1}=C_{1}(p,$ $C)$ such that $v\in\mathcal{A}_{q-\epsilon}(C_{1})$.

In [22] Saker and Agarwal proved the discrete Rubio de Francia extrapolation
theorem via discrete $\mathcal{B}_{p}$ weights by employing the self-improving
property of the discrete $\mathcal{B}_{p}$ weights that: if
$v\in\mathcal{B}_{q}(C)$ then there exists an $\epsilon>0$ and a constant
$C_{1}=C_{1}(q,$ $C)$ such that $v\in\mathcal{B}_{q-\epsilon}(C_{1})$. A
nonnegative discrete sequence $w$ defined on $\mathbb{Z}_{+}$ is said to be
belong to the discrete class $\mathcal{B}_{p}(B)$ for $p>0$ and $B>0$ if $w$
satisfies the condition

(1.21)
$\sum_{k=n}^{\infty}\frac{w(k)}{k^{p}}\leq\frac{B}{n^{p}}\sum_{k=1}^{n}w(n),\text{
\ for all }n\in\mathbb{Z}_{+}.$

In [4] Bennett and Grosse-Erdmann proved that the discrete Hardy operator

$\mathcal{H}f(n)=\frac{1}{n}\sum_{k=1}^{n}f(k),$

is bounded on $\ell_{p}(w)$ for all decreasing sequence $f$ if and only if
$w\in\mathcal{B}_{p}(B).$

In this paper, we give a new proof of the discrete extrapolation theorem via
Muckenhoupt weights $\mathcal{A}_{p}$ with explicit constants. The proof is
different from the proof in [21] and simpler and more direct, since it yields
the desired inequality directly without cases of intermediate steps and uses
the iteration algorithm

$\mathcal{N}\left(h(n)\right):=\sum_{s=0}^{\infty}\frac{\mathcal{M}^{s}h(n)}{2^{s}\left\|\mathcal{M}\right\|_{\ell_{p}(w)}^{s}},$

where for $s>0,$ $\mathcal{M}^{s}h=\mathcal{M\circ}...$ $\mathcal{\circ M}h$
denotes $s$ iterations of the Hardy-Littlewood maximal operator and
$\mathcal{M}^{0}h=h,$ and the dual iteration algorithm

(1.22)
$\mathcal{N}^{\prime}h(n):=\sum_{s=0}^{\infty}\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}},$

with

(1.23)
$\mathcal{M}^{\prime}h(n)=\frac{\mathcal{M}\left(wh\right)(n)}{w(n)},\text{ \
}n\in\mathbb{Z}_{+}\text{,}$

where $\mathcal{M}h$ is the Hardy-Littlewood maximal operator and
$\left(\mathcal{M}^{\prime}\right)^{s}h=\mathcal{M}^{\prime}\mathcal{\circ}...$
$\mathcal{\circ M}^{\prime}h$ denotes $s$ iterations of the operator
$\mathcal{M}^{\prime},$ and $\left(\mathcal{M}^{\prime}\right)^{0}h=h.$ In
addition, we will use some of the properties of $\mathcal{A}_{p}$ weights that
are presented in the following lemmas for the sake of completeness.

######  Lemma 1.1.

[28] Assume that $w$ is a nonnegative weight and $1<p,$ $q<\infty\ $ are
positive real numbers. Then the following properties hold.

$(i).$ If $w\in\mathcal{A}_{p},$ then $w^{\alpha}\in\mathcal{A}_{p}$ for
$0\leq\alpha\leq 1$ with
$\left[w^{\alpha}\right]_{\mathcal{A}_{p}}\leq\left(\left[w\right]_{\mathcal{A}_{p}}\right)^{\alpha}$.

$(ii).$ $w\in\mathcal{A}_{p}$ if and only if
$w^{1-p^{\prime}}\in\mathcal{A}_{p^{\prime}},$ $p^{\prime}=p/\left(p-1\right)$
and

(1.24) $\
\left[w^{1-p^{\prime}}\right]_{\mathcal{A}_{p^{\prime}}}=\left[w\right]_{\mathcal{A}_{p}}^{p^{\prime}-1}.$

######  Lemma 1.2.

[28] Let $w,$ $w_{1},$ $w_{2}$ be discrete weights. Then the following
properties hold.

$1)$ $w\in\mathcal{A}_{p},$ $1<p<\infty$ if and only if there exist $w_{1},$
$w_{2}\in\mathcal{A}_{1}$ such that $w=w_{1}w_{2}^{1-p}.$

$2)$ If $w_{1},$ $w_{2}\in\mathcal{A}_{p},$ then
$w_{1}^{\alpha}w_{2}^{1-\alpha}\in\mathcal{A}_{p},$ $0<\alpha\leq 1$ with a
constant

$\left[w_{1}^{\alpha}w_{2}^{1-\alpha}\right]_{\mathcal{A}_{p}}=\left[w_{1}\right]_{\mathcal{A}_{p}}^{\alpha}\left[w_{2}\right]_{\mathcal{A}_{p}}^{1-\alpha}.$

$3).$ $\mathcal{A}_{p}\subset\mathcal{A}_{q}$ for all $1<p\leq q<\infty\ $and
$\left[w\right]_{\mathcal{A}_{q}}\leq\left[w\right]_{\mathcal{A}_{p}}.$

The paper is organized as follows: In Section $2,$ we present the basic
properties of the Rubio de Francia iterated algorithm proved in [21]. Next, we
prove the properties of the Dual Rubio de Francia iterated algorithm by using
the result proved in [21]. In Section $3,$ we prove some weighted norm
inequalities of pairs of sequences $(f,g)$ by employing the Rubio de Francia
iterated algorithms and the reverse factorization properties of Muckenhoupt
weights $\mathcal{A}_{p}$. We consider two cases: the first case when
$1<p<p_{0}<\infty$ and the second case when $1<p_{0}<p<\infty,$ where $p_{0}$
is a fixed number, $1<p_{0}<\infty.$ The results will be extended to the
extrapolation theorem via $\mathcal{A}_{\infty}$ weights. The proof of the
extrapolation theorem with exact constants will be obtained by combining the
two cases and replacing the pair $(f,g)$ by $(\mathcal{T}f,f).$ The advantage
of this approach is that a number of different results become special cases of
the extrapolation theorem with exact constants. The case when $0<p<1$ will be
considered in future and also left to the interested readers.

The features of the new proof of the discrete Rubio de Francia extrapolation
are the combining of some interesting results: norm inequalities for the
Hardy-Littlewood maximal operator $\mathcal{M}$, duality
$\mathcal{M}^{{}^{\prime}}$, and the reverse factorization of
$\mathcal{A}_{p}-$weights. In the proof we also do not assume that the
operator $\mathcal{T}$ is linear or sublinear as posed in [21]. More
precisely, we are talking about the properties:

$1).$ $\mathcal{M}$ is sublinear, positive and bounded in $\ell_{p}(w)$ if
$w\in\mathcal{A}_{p}$ ([26])

$2).$ $\mathcal{M}^{{}^{\prime}}$ is sublinear, positive and bounded in
$\ell_{p^{{}^{\prime}}}(w)$ if $w\in\mathcal{A}_{p}$ (will be proved later).

$3).$ If $w_{1},$ $w_{2}\in\mathcal{A}_{1}$
then$w_{1}w_{2}^{1-p}\in\mathcal{A}_{p}$ (see Lemma 1.2).

$4).$ $w\in\mathcal{A}_{p}$ if and only if
$w^{1-p^{\prime}}\in\mathcal{A}_{p^{\prime}},$ $p^{\prime}=p/\left(p-1\right)$
(see Lemma 1.1)

##  2\. Preliminaries and Basic Lemmas

In [26] the authors proved that the Hardy-Littlewood maximal operator
$\mathcal{M}$ is bounded in $\ell_{p}(w)$ if $w$ is a Muckenhoupt weight. In
[21] the authors used this result and proved the following results which will
play crucial roles in proving the main results in Section 3.

######  Theorem 2.1.

Assume that $0<\gamma\leq 1$ and $w\in\mathcal{A}_{p}$ for $1<p<\infty.$ For a
nonnegative sequence $g,$ we define

$Gg:=\left(\frac{\mathcal{M}\left(g^{1/\gamma}w\right)}{w}\right)^{\gamma}.$

Then the operator $G$ is bounded in $\ell_{p^{\prime}/\gamma}(w),$
$p^{\prime}=p/\left(p-1\right)$ and there exists a constant $C>0$ such that

$\left\|Gg\right\|_{\ell_{p^{\prime}/\gamma}(w)}\leq
C\left\|g\right\|_{\ell_{p^{\prime}/\gamma}(w)}.$

######  Lemma 2.1.

(Discrete Rubio de Francia algorithm) Fix $1<p<\infty$ and assume that
$w\in\mathcal{A}_{p}.$ For any nonnegative sequence $h\in\ell_{p}(w)$, we
define

$\mathcal{N}h(n):=\sum_{s=0}^{\infty}\frac{\mathcal{M}^{s}h(n)}{2^{s}\left\|\mathcal{M}\right\|_{\ell_{p}(w)}^{s}},$

where for $s>0,$ $\mathcal{M}^{s}h=\mathcal{M\circ}...$ $\mathcal{\circ M}h$
denotes $s$ iterations of the Hardy-Littlewood maximal operator and
$\mathcal{M}^{0}h=h.$ Then

$(i)$ $h(n)\leq\mathcal{N}h(n),$

$(ii)$ $\left\|\mathcal{N}h\right\|_{\ell_{p}(w)}\leq
2\left\|h\right\|_{\ell_{p}(w)},$

$(iii)$ $\mathcal{N}h\in\mathcal{A}_{1}$ and
$\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\leq
2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}.$

Now, we define the dual Discrete Rubio de Francia algorithm and prove the
properties of it.

######  Lemma 2.2.

Assume that $w\in\mathcal{A}_{p},$ for $1<p<\infty.$ For a nonnegative
sequence $h,$ we define a general Hardy-Littlewood maximal operator by

$\mathcal{M}^{\prime}h:=\frac{\mathcal{M}\left(wh\right)}{w}.$

Then the operator $\mathcal{M}^{\prime}$ is bounded in $\ell_{p^{\prime}}(w),$
$p^{\prime}=p/\left(p-1\right)$ and there exists a constant $C>0$ such that

(2.1) $\left\|\mathcal{M}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}\leq
C\left\|h\right\|_{\ell_{p^{\prime}}(w)}.$

###### Proof.

Applying Theorem 2.1 with $g=h$ and $\gamma=1,$ we see that the operator
$\mathcal{M}^{\prime}$ which is defined by

$\mathcal{M}^{\prime}h:=\frac{\mathcal{M}\left(wh\right)}{w},$

is bounded in $\ell_{{}^{p^{\prime}}}(w)$ and satisfies that

$\left\|\mathcal{M}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}\leq
C\left\|h\right\|_{\ell_{p^{\prime}}(w)},$

which is the desired inequality (2.1). The proof is complete.

######  Lemma 2.3.

(Discrete Dual Rubio de Francia algorithm) Fix $1<p<\infty$ and
$w\in\mathcal{A}_{p}.$ For any nonnegative sequence
$h\in\ell_{p^{\prime}}(w),$ $p^{\prime}=p/(p-1),$ we define

(2.2)
$\mathcal{N}^{\prime}h(n):=\sum_{s=0}^{\infty}\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}},$

with

(2.3)
$\mathcal{M}^{\prime}h(n)=\frac{\mathcal{M}\left(wh\right)(n)}{w(n)},\text{ \
}n\in\mathbb{Z}_{+}\text{,}$

where $\mathcal{M}h$ is the Hardy-Littlewood maximal operator and
$\left(\mathcal{M}^{\prime}\right)^{s}h=\mathcal{M}^{\prime}\mathcal{\circ}...$
$\mathcal{\circ M}^{\prime}h$ denotes $s$ iterations of the operator
$\mathcal{M}^{\prime},$ and $\left(\mathcal{M}^{\prime}\right)^{0}h=h.$ Then

$\,(a).$ $h(n)\leq\mathcal{N}^{\prime}h(n),$

$(b).$ $\left\|\mathcal{N}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}\leq
2\left\|h\right\|_{\ell_{p^{\prime}}(w)},$

$(c).$ $\left(\mathcal{N}^{\prime}h\right)w\in\mathcal{A}_{1}$ and
$\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}\leq
2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}.$

###### Proof.

From (2.2), we have (where $\left(\mathcal{M}^{\prime}\right)^{0}h=h$) that

$\mathcal{N}^{\prime}h(n)=h(n)+\sum_{s=1}^{\infty}\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}},$

and then

$h(n)\leq\mathcal{N}^{\prime}h(n),$

which proves the property $(a).$ Applying Minkowski’s inequality, on the term
$\left\|\mathcal{N}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)},$ we see that

(2.4)
$\displaystyle\left\|\mathcal{N}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}$
$\displaystyle=$
$\displaystyle\left(\sum_{n=1}^{\infty}w(n)\left(\mathcal{N}^{\prime}h(n)\right)^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\left(\sum_{n=1}^{\infty}w(n)\left(\sum_{s=0}^{\infty}\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\right)^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}\frac{1}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\left(\sum_{n=1}^{\infty}w(n)\left(\left(\mathcal{M}^{\prime}\right)^{s}h(n)\right)^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\sum_{s=0}^{\infty}\frac{1}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\left\|\left(\mathcal{M}^{\prime}\right)^{s}h\right\|_{\ell_{p^{\prime}}(w)}.$

From Lemma 2.2, since the operator $\mathcal{M}^{\prime}$ is bounded, the
inequality (2.4) becomes

$\left\|\mathcal{N}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}\leq\sum_{s=0}^{\infty}\frac{1}{2^{s}}\left\|h\right\|_{\ell_{p^{\prime}}(w)}=2\left\|h\right\|_{\ell_{p^{\prime}}(w)},$

which proves the property $(b).$ Since the operator
$h\rightarrow\mathcal{M}^{\prime}h$ is sublinear, then

$\displaystyle\mathcal{M}^{\prime}\left(\mathcal{N}^{\prime}h(n)\right)$
$\displaystyle=$
$\displaystyle\mathcal{M}^{\prime}\left(\sum_{s=0}^{\infty}\left(\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\right)\right)$
$\displaystyle\leq$
$\displaystyle\sum_{s=0}^{\infty}\mathcal{M}^{\prime}\left(\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\right)=\sum_{s=0}^{\infty}\frac{1}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\left(\mathcal{M}^{\prime}\right)^{s+1}h(n)$
$\displaystyle\leq$ $\displaystyle
2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}\sum_{s=0}^{\infty}\frac{\left(\mathcal{M}^{\prime}\right)^{s}h(n)}{2^{s}\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}^{s}}\leq
2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}\mathcal{N}^{\prime}h(n),$

and then by using (2.3), we have that

$\mathcal{M}\left(w\mathcal{N}^{\prime}h\right)(n)\leq
2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}\left(w\mathcal{N}^{\prime}h\right)(n).$

This proves that $w\mathcal{N}^{\prime}h\in\mathcal{A}_{1}$ with the norm

$\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}\leq
2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)},$

which proves the property $(c).$ The proof is complete.

######  Lemma 2.4.

Assume that $w,$ $h$ are positive weights and $1<p<p_{0}<\infty.$ If
$w\in\mathcal{A}_{p},$ then
$w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\in\mathcal{A}_{p_{0}}$ and

$\left[w\mathcal{N}h^{-\left(p_{0}-p\right)}\right]_{\mathcal{A}_{p_{0}}}\leq\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}},$

where the operator $\mathcal{N}$ is defined as in Lemma 2.1.

###### Proof.

From the property $(iii)$ of Lemma 2.1, we see that
$\mathcal{N}h\in\mathcal{A}_{1},$ and then

$\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\leq\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\mathcal{N}h(k),$

and thus we have for $p_{0}>p$ that

(2.5)
$\displaystyle\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\mathcal{N}h^{-\left(p_{0}-p\right)}(k)$
$\displaystyle\leq$
$\displaystyle\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{-\left(p_{0}-p\right)}\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}$
$\displaystyle=$
$\displaystyle\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{-\left(p_{0}-p\right)}.$

Applying Hölder’s inequality on the term

$\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left[w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right]^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1},$

with $\gamma=\left(p_{0}-1\right)/\left(p-1\right)>1$ and
$\nu=\left(p_{0}-1\right)/\left(p_{0}-p\right)$ ($1/\gamma+1/\nu=1$), we get
that

(2.6)
$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left[w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right]^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{n}\right)^{p_{0}-1}\left(\sum\limits_{k=1}^{n}w^{\frac{-1}{p_{0}-1}}(k)\left(\mathcal{N}h\right)^{\frac{p_{0}-p}{p_{0}-1}}(k)\right)^{p_{0}-1}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{n}\right)^{p_{0}-1}\left(\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{p_{0}-p}\left(\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}$
$\displaystyle=$
$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{p_{0}-p}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}.$

From (2.5) and (2.6), since $w\in\mathcal{A}_{p},$ $1<p<\infty,$ we have that

$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left[w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right]^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle\leq$
$\displaystyle\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{-\left(p_{0}-p\right)}$
$\displaystyle\times\left(\frac{1}{n}\sum\limits_{k=1}^{n}\mathcal{N}h(k)\right)^{p_{0}-p}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}$
$\displaystyle=$
$\displaystyle\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}.$

Taking supremum, we have that
$w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\in\mathcal{A}_{p_{0}}$
with the constant

$\displaystyle\left[w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\right]_{\mathcal{A}_{p_{0}}}$
$\displaystyle=$
$\displaystyle\sup_{n}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right)$
$\displaystyle\times\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left[w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right]^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle\leq$
$\displaystyle\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}},$

which is the desired inequality. The proof is complete.

######  Lemma 2.5.

Assume that $w,$ $h$ are positive sequences on $\mathbb{Z}_{+}\ $and
$1<p_{0}<p<\infty.$ If $w\in\mathcal{A}_{p},$ then
$w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\in\mathcal{A}_{p_{0}}$
and

$\left[w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\right]_{\mathcal{A}_{p_{0}}}\leq\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}},$

where the operator $\mathcal{N}^{\prime}$ is defined as in Lemma 2.3.

###### Proof.

Since $p>p_{0}>1,$ $p_{0}-1>0,$ $p-p_{0}>0,$ we have that

(2.7)
$\frac{p-1}{p-p_{0}}=\frac{\left(p-p_{0}\right)+\left(p_{0}-1\right)}{p-p_{0}}=1+\frac{p_{0}-1}{p-p_{0}}>1.$

Then by applying the weighted Hölder inequality

(2.8)
$\sum\limits_{k=1}^{n}w(k)f(k)g(k)\leq\left[\sum\limits_{k=1}^{n}w(k)f^{\gamma}(k)\right]^{\frac{1}{\gamma}}\left[\sum\limits_{k=1}^{n}w(k)g^{\nu}(k)\right]^{\frac{1}{\nu}},$

on the term

$\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k),$

with indices $\gamma=\left(p-1\right)/\left(p-p_{0}\right)>1$ and
$\nu=\left(p-1\right)/\left(p_{0}-1\right)$ (where $1/\gamma+1/\nu=1$), we see
that

$\displaystyle\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)=\sum\limits_{k=1}^{n}w(k).1.\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)$
$\displaystyle\leq$
$\displaystyle\left(\sum\limits_{k=1}^{n}w(k)\right)^{\frac{p_{0}-1}{p-1}}\left(\sum\limits_{k=1}^{n}w(k)\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p-1}},$

and then

(2.9)
$\displaystyle\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\right)^{\frac{p_{0}-1}{p-1}}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p-1}}.$

Also, we note that

$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle=\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p_{0}-1}}(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}(k)\right)^{p_{0}-1}$
$\displaystyle=\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}w^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}w^{\frac{p-p_{0}}{\left(p_{0}-1\right)\left(p-1\right)}}(k)w^{\frac{-1}{p_{0}-1}}(k)\right)^{p_{0}-1}$
(2.10)
$\displaystyle=\left(\frac{1}{n}\right)^{p_{0}-1}\left(\sum\limits_{k=1}^{n}\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}w^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}(k)w^{\frac{-1}{p-1}}(k)\right)^{p_{0}-1}.$

From the property $(c)$ in Lemma 2.3, we observe that
$\left(\mathcal{N}^{\prime}h\right)w\in\mathcal{A}_{1},$ and then

(2.11)
$\frac{1}{n}\sum\limits_{k=1}^{n}\left(\mathcal{N}^{\prime}h\right)(k)w(k)\leq\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}\left(\mathcal{N}^{\prime}h\right)(k)w(k).$

Substituting (2.11) into (2.10), we have for $p>p_{0}>1,$
$\left(p-p_{0}\right)/\left[\left(p_{0}-1\right)\left(p-1\right)\right]>0,$
that

$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle\leq\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(\mathcal{N}^{\prime}h\right)(k)w(k)\right)^{\frac{-\left(p-p_{0}\right)}{\left(p_{0}-1\right)\left(p-1\right)}}\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{\left(p_{0}-1\right)\left(p-1\right)}}w^{\frac{-1}{p-1}}(k)\right)^{p_{0}-1}$
(2.12)
$\displaystyle=\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(\mathcal{N}^{\prime}h\right)(k)w(k)\right)^{\frac{-\left(p-p_{0}\right)}{p-1}}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p_{0}-1}.$

From (2.9) and (2.12), we observe that

$\displaystyle\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}$
$\displaystyle\leq$
$\displaystyle\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}w^{\frac{-1}{p-1}}(k)\right)^{p-1}\right]^{\frac{p_{0}-1}{p-1}},$

and then we have for $p>p_{0}>1$ and $w\in\mathcal{A}_{p},$ that

$\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}.$

Taking the supremum of the two sides of the last inequality, we see that
$w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\in\mathcal{A}_{p_{0}}$
with the constant

$\displaystyle\left[w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\right]_{\mathcal{A}_{p_{0}}}=\sup_{n\in\mathbb{Z}_{+}}\left(\frac{1}{n}\sum\limits_{k=1}^{n}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)$
$\displaystyle\times\left(\frac{1}{n}\sum\limits_{k=1}^{n}\left(w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{-1}{p_{0}-1}}\right)^{p_{0}-1}\leq\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}},$

which is the desired inequality. The proof is complete.

##  3\. Main results

In this section, we prove the main extrapolation theorem via the Muckenhoupt
weights with sharp constants. We assume that $f,$ $g$ are nonnegative
sequences belong to and $w_{0},$ $w$ are nonnegative weights.

######  Theorem 3.1.

Assume that for some $p_{0},$ $1\leq p_{0}<\infty,$ there exists a positive
increasing function $\mathcal{\varphi}_{p_{0}}$ on $[1,\infty)$ such that for
every $w_{0}\in\mathcal{A}_{p_{0}},$

(3.1)
$\left(\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}\leq\mathcal{\varphi}_{p_{0}}\left([w_{0}]_{\mathcal{A}_{p_{0}}}\right)\left(\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Then for all $p,$ $1<p<p_{0}<\infty$ and for all $w\in\mathcal{A}_{p},$

(3.2)
$\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}\leq\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}},$

where

(3.3)
$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right).$

###### Proof.

Let $\varepsilon>0$ and define

(3.4)
$h(k):=\varepsilon\frac{f(k)}{\left\|f\right\|_{\ell_{p}(w)}}+\frac{g(k)}{\left\|g\right\|_{\ell_{p}(w)}},\text{
for \ }k\in\mathbb{Z}_{+}.$

Applying the weighted Minkowski inequality, we have

$\displaystyle\left\|h\right\|_{\ell_{p}(w)}$
$\displaystyle=\left(\sum_{k=1}^{\infty}w(k)h^{p}(k)\right)^{\frac{1}{p}}$
$\displaystyle=\left(\sum_{k=1}^{\infty}w(k)\left(\varepsilon\frac{f(k)}{\left\|f\right\|_{\ell_{p}(w)}}+\frac{g(k)}{\left\|g\right\|_{\ell_{p}(w)}}\right)^{p}\right)^{\frac{1}{p}}$
$\displaystyle\leq\frac{\varepsilon}{\left\|f\right\|_{\ell_{p}(w)}}\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}+\frac{1}{\left\|g\right\|_{\ell_{p}(w)}}\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}}$
(3.5) $\displaystyle=\varepsilon+1.$

Since

(3.6) $\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{\infty}f^{p}(k)\mathcal{N}h^{\frac{-\left(p_{0}-p\right)p}{p_{0}}}(k)\mathcal{N}h^{\frac{\left(p_{0}-p\right)p}{p_{0}}}(k)w(k)\right)^{\frac{1}{p}},$

we have by applying the weighted Hölder inequality on the right hand side of
(3.6) with $\gamma=p_{0}/p>1$ and $\nu=p_{0}/\left(p_{0}-p\right)$, that

(3.7)
$\left\|f\right\|_{\ell_{p}(w)}\leq\left(\sum_{k=1}^{\infty}w(k)f^{p_{0}}(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right)^{\frac{1}{p_{0}}}\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{p}(k)\right)^{\frac{p_{0}-p}{pp_{0}}}.$

By applying the property $(ii)$ of Lemma 2.1, we have for $p_{0}>p,$ that

$\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{p}(k)\right)^{\frac{p_{0}-p}{pp_{0}}}=\left\|\mathcal{N}h\right\|_{\ell_{p}(w)}^{\frac{p_{0}-p}{p_{0}}}\leq\left(2\left\|h\right\|_{\ell_{p}(w)}\right)^{\frac{p_{0}-p}{p_{0}}},$

and then by using (3.5), we obtain

(3.8)
$\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{p}(k)\right)^{\frac{p_{0}-p}{pp_{0}}}\leq\left[2\left(\varepsilon+1\right)\right]^{\frac{p_{0}-p}{p_{0}}}.$

Substituting (3.8) into (3.7), we have that

(3.9)
$\left\|f\right\|_{\ell_{p}(w)}\leq\left[2\left(\varepsilon+1\right)\right]^{\frac{p_{0}-p}{p_{0}}}\left(\sum_{k=1}^{\infty}w(k)f^{p_{0}}(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)\right)^{\frac{1}{p_{0}}}.$

From Lemma 2.4, we have that
$w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\in\mathcal{A}_{p_{0}}$
with the constant

(3.10)
$\left[w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\right]_{\mathcal{A}_{p_{0}}}\leq\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}.$

Since
$w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}\in\mathcal{A}_{p_{0}},$ we
can apply (3.1) with
$w_{0}=w\mathcal{N}h^{-\left(p_{0}-p\right)}\in\mathcal{A}_{p_{0}}$ and then

(3.11)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left([w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}]_{\mathcal{A}_{p_{0}}}\right)\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Since the function $\mathcal{\varphi}_{p_{0}}$ is increasing, we have from
(3.10) that

$\mathcal{\varphi}_{p_{0}}\left([w\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}]_{\mathcal{A}_{p_{0}}}\right)\leq\mathcal{\varphi}_{p_{0}}\left(\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right),$

and then the inequality (3.11) becomes

(3.12)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

From the property $(iii)$ in Lemma 2.1, we observe that

$\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\leq
2\left\|\mathcal{M}\right\|_{\ell_{p}(w)},$

and then we have for $p_{0}>p$ and the increasing function
$\mathcal{\varphi}_{p_{0}}$ that

(3.13)
$\mathcal{\varphi}_{p_{0}}\left(\left(\left[\mathcal{N}h\right]_{\mathcal{A}_{1}}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\leq\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right).$

Substituting (3.13) into (3.12), we obtain

(3.14)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Substituting (3.14) into (3.9), we get

(3.15) $\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle\leq$
$\displaystyle\left[2\left(\varepsilon+1\right)\right]_{p_{0}}^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}h\right)^{-\left(p_{0}-p\right)}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

From (3.4), where $\varepsilon>0$ and the property $(i)$ in Lemma 2.1, we
observe that

(3.16) $\frac{g(k)}{\left\|g\right\|_{\ell_{p}(w)}}\leq
h(k)\leq\mathcal{N}h(k).$

Substituting (3.16) into (3.15), we have for $p_{0}>p,$ that

$\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle\leq$
$\displaystyle\left[2\left(\varepsilon+1\right)\right]_{p_{0}}^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)$
$\displaystyle\times\left\|g\right\|_{\ell_{p}(w)}^{\frac{p_{0}-p}{p_{0}}}\left(\sum_{k=1}^{\infty}w(k)\left[g(k)\right]^{-\left(p_{0}-p\right)}g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle=$
$\displaystyle\left[2\left(\varepsilon+1\right)\right]^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)$
$\displaystyle\times\left\|g\right\|_{\ell_{p}(w)}^{\frac{p_{0}-p}{p_{0}}}\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle=$
$\displaystyle\left[2\left(\varepsilon+1\right)\right]^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\left\|g\right\|_{\ell_{p}(w)},$

i.e.

$\displaystyle\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}$
$\displaystyle\leq$
$\displaystyle\left[2\left(\varepsilon+1\right)\right]^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}}.$

Taking the limit when $\varepsilon\rightarrow 0,$ we have that

$\displaystyle\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle
2^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}},$

which satisfies (3.2). The proof is complete.

######  Theorem 3.2.

Assume that for some $p_{0},$ $1\leq p_{0}<\infty,$ there exists a positive
increasing function $\mathcal{\varphi}_{p_{0}}$ on $[1,\infty)$ such that for
every $w_{0}\in\mathcal{A}_{p_{0}},$

(3.17)
$\left(\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}\leq\mathcal{\varphi}_{p_{0}}\left([w_{0}]_{\mathcal{A}_{p_{0}}}\right)\left(\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Then for all $p,$ $1<p_{0}<p<\infty$ and for all $w\in\mathcal{A}_{p},$

(3.18)
$\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}\leq\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}},$

where

(3.19)
$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right),$

with $p^{\prime}=p/\left(p-1\right).$

###### Proof.

Since $f\in\ell_{p}(w),$ we may assume without loss of generality that there
exists a nonnegative sequence $h\in\ell_{p^{\prime}}(w)$,
$\left\|h\right\|_{\ell_{p^{\prime}}(w)}=1,$ such that

(3.20) $\left\|f\right\|_{\ell_{p}(w)}=\sum_{k=1}^{\infty}f(k)h(k)w(k).$

By applying the property $(a)$ in Lemma 2.3 on the right hand side of (3.20),
we have for $p>p_{0}>1$ that

(3.21) $\displaystyle\sum_{k=1}^{\infty}f(k)h(k)w(k)$ $\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}f(k)h^{\frac{p-p_{0}}{p_{0}\left(p-1\right)}}(k)h^{\frac{p\left(p_{0}-1\right)}{p_{0}\left(p-1\right)}}(k)w(k)$
$\displaystyle\leq$
$\displaystyle\sum_{k=1}^{\infty}f(k)\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p_{0}\left(p-1\right)}}h^{\frac{p\left(p_{0}-1\right)}{p_{0}\left(p-1\right)}}(k)w(k).$

Applying the weighted Hölder inequality on the term

$\sum_{k=1}^{\infty}f(k)\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p_{0}\left(p-1\right)}}h^{\frac{p\left(p_{0}-1\right)}{p_{0}\left(p-1\right)}}(k)w(k),$

with indices $p_{0}>1$ and $p_{0}=p_{0}/\left(p_{0}-1\right),$ we see that

$\displaystyle\sum_{k=1}^{\infty}f(k)\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p_{0}\left(p-1\right)}}h^{\frac{p\left(p_{0}-1\right)}{p_{0}\left(p-1\right)}}(k)w(k)$
$\displaystyle\leq$
$\displaystyle\left(\sum_{k=1}^{\infty}f^{p_{0}}(k)\left(\mathcal{N}^{\prime}h(k)\right)^{\frac{p-p_{0}}{p-1}}w(k)\right)^{\frac{1}{p_{0}}}\left(\sum_{k=1}^{\infty}h^{\frac{p}{p-1}}(k)w(k)\right)^{\frac{p_{0}-1}{p_{0}}},$

and then since $\left\|h\right\|_{\ell_{p^{\prime}}(w)}=1$, we that

(3.22)
$\sum_{k=1}^{\infty}f(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p_{0}\left(p-1\right)}}(k)h^{\frac{p\left(p_{0}-1\right)}{p_{0}\left(p-1\right)}}(k)w(k)\leq\left(\sum_{k=1}^{\infty}f^{p_{0}}(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)w(k)\right)^{\frac{1}{p_{0}}}.$

Substituting (3.22) into (3.21), we observe that

$\sum_{k=1}^{\infty}f(k)h(k)w(k)\leq\left(\sum_{k=1}^{\infty}f^{p_{0}}(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)w(k)\right)^{\frac{1}{p_{0}}},$

and then we have from (3.20) that

(3.23)
$\left\|f\right\|_{\ell_{p}(w)}\leq\left(\sum_{k=1}^{\infty}f^{p_{0}}(k)w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)\right)^{\frac{1}{p_{0}}}.$

From Lemma 2.5, we have that
$w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\in\mathcal{A}_{p_{0}}$
with the constant

(3.24)
$\left[w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\right]_{\mathcal{A}_{p_{0}}}\leq\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}},$

Since
$w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\in\mathcal{A}_{p_{0}},$
we can apply (3.17) with
$w_{0}=w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\in\mathcal{A}_{p_{0}}$
to get

(3.25)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left([w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}]_{\mathcal{A}_{p_{0}}}\right)\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Since $\mathcal{\varphi}_{p_{0}}$ is an increasing function, we have from
(3.24) that

$\mathcal{\varphi}_{p_{0}}\left(\left[w\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}\right]_{\mathcal{A}_{p_{0}}}\right)\leq\mathcal{\varphi}_{p_{0}}\left(\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right),$

and then the inequality (3.25) becomes

(3.26)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Again by applying the property $(c)$ in Lemma 2.3 , we have for $p>p_{0}>1$
that

$\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\leq\left(2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}.$

Since $\mathcal{\varphi}_{p_{0}}$ is an increasing function, we get

(3.27)
$\mathcal{\varphi}_{p_{0}}\left(\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)\leq\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right).$

Since
$\mathcal{M}^{\prime}f(k)=\left(1/w(k)\right)\mathcal{M}\left(fw\right)(k),$
we have that

(3.28)
$\displaystyle\left\|\mathcal{M}^{\prime}\right\|_{\ell_{p^{\prime}}(w)}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{M}^{\prime}f(k)\right)^{p^{\prime}}\right)^{\frac{1}{p^{\prime}}}$
$\displaystyle=$
$\displaystyle\left(\sum_{k=1}^{\infty}w^{1-p^{\prime}}(k)\mathcal{M}^{p^{\prime}}\left(fw\right)(k)\right)^{\frac{1}{p^{\prime}}}=\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}.$

Substituting (3.28) into (3.27), we get

$\mathcal{\varphi}_{p_{0}}\left(\left[\left(\mathcal{N}^{\prime}h\right)w\right]_{\mathcal{A}_{1}}^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)\leq\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right),$

and then the inequality (3.26) becomes

(3.29)
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)f^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}$
$\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

From (3.23) and (3.29), we have that

(3.30) $\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell^{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k)\right)^{\frac{1}{p_{0}}}.$

Applying the weighted Hölder inequality on the term

$\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k),$

with $\gamma=p/p_{0}>1$ and $\nu=p/\left(p-p_{0}\right),$ we see that

(3.31)
$\displaystyle\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p-p_{0}}{p-1}}(k)g^{p_{0}}(k)$
$\displaystyle\leq$
$\displaystyle\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p}{p-1}}(k)\right)^{\frac{p-p_{0}}{p}}\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{p_{0}}{p}}.$

Substituting (3.31) into (3.30), we obtain

(3.32) $\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle\leq$
$\displaystyle\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p}{p-1}}(k)\right)^{\frac{p-p_{0}}{pp_{0}}}\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}}.$

From the property $(b)$ in Lemma 2.3, we have for $p>p_{0}>1$ that

$\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p}{p-1}}(k)\right)^{\frac{p-p_{0}}{pp_{0}}}=\left\|\mathcal{N}^{\prime}h\right\|_{\ell_{p^{\prime}}(w)}^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\leq\left(2\left\|h\right\|_{\ell_{p^{\prime}}(w)}\right)^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}.$

Since $\left\|h\right\|_{\ell_{p^{\prime}}(w)}=1$, we have that

(3.33)
$\left(\sum_{k=1}^{\infty}w(k)\left(\mathcal{N}^{\prime}h\right)^{\frac{p}{p-1}}(k)\right)^{\frac{p-p_{0}}{pp_{0}}}\leq
2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}.$

Substituting (3.33) into (3.32), we obtain

$\displaystyle\left\|f\right\|_{\ell_{p}(w)}$ $\displaystyle\leq$
$\displaystyle
2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell^{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}},$

i.e.

$\displaystyle\left(\sum_{k=1}^{\infty}w(k)f^{p}(k)\right)^{\frac{1}{p}}$
$\displaystyle\leq$ $\displaystyle
2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell^{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right)$
$\displaystyle\times\left(\sum_{k=1}^{\infty}w(k)g^{p}(k)\right)^{\frac{1}{p}},$

which satisfies (3.18). The proof is complete.

By combining Theorem 3.1 and Theorem 3.2, we get the following theorem.

######  Theorem 3.3.

Assume that for some $p_{0},$ $1\leq p_{0}<\infty,$ there exists a positive
increasing function $\mathcal{\varphi}_{p_{0}}$ on $[1,\infty)$ such that for
every $w_{0}\in\mathcal{A}_{p_{0}},$

(3.34)
$\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\leq\mathcal{\varphi}_{p_{0}}\left([w_{0}]_{\mathcal{A}_{p_{0}}}\right)\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k).$

Then for all $p,$ $1\leq p<\infty$ and for all $w\in\mathcal{A}_{p},$

$\sum_{k=1}^{\infty}w(k)f^{p}(k)\leq\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)\sum_{k=1}^{\infty}w(k)g^{p}(k).$

If $p<p_{0},$ then

$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right),$

and if $p>p_{0},$ then

$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{(w^{1-p^{\prime}})}^{p^{\prime}}}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right),$

with $p^{\prime}=p/\left(p-1\right).$

Now, we are ready to state and prove the main theorem.

######  Theorem 3.4.

Assume that $\mathcal{T}$ be a given operator defined on $\mathbb{Z}_{+}$ and
there exists a positive an increasing function $\mathcal{\varphi}_{p_{0}}$ on
$[1,\infty)$ such that for every $w_{0}\in\mathcal{A}_{p_{0}}$

$\sum_{k=1}^{\infty}\left|\mathcal{T}f(k)\right|^{p_{0}}w_{0}(k)\leq\varphi_{p_{0}}\left(\left[w\right]_{\mathcal{A}_{p_{0}}}\right)\sum_{k=1}^{\infty}\left|f(k)\right|^{p_{0}}w_{0}(k),$

for some $p_{0},$ $1\leq p_{0}<\infty.$ Then

$\sum_{k=1}^{\infty}\left|\mathcal{T}f(k)\right|^{p_{0}}w(k)\leq\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)\sum_{k=1}^{\infty}\left|f(k)\right|^{p_{0}}w(k),$

for every $w\in\mathcal{A}_{p}$ such that $1\leq p<\infty$.

If $p<p_{0},$ then

$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p_{0}-p}{p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p}(w)}\right)^{p_{0}-p}\left[w\right]_{\mathcal{A}_{p}}\right).$

If $p>p_{0},$ then

$\mathcal{\varphi}_{p}\left(p_{0},p,[w]_{\mathcal{A}_{p}}\right)=2^{\frac{p-p_{0}}{\left(p-1\right)p_{0}}}\mathcal{\varphi}_{p_{0}}\left(\left(2\left\|\mathcal{M}\right\|_{\ell_{p^{\prime}}(w^{1-p^{\prime}})}\right)^{\frac{p-p_{0}}{p-1}}\left[w\right]_{\mathcal{A}_{p}}^{\frac{p_{0}-1}{p-1}}\right),$

where $p^{\prime}=p/\left(p-1\right).$

###### Proof.

The proof is obtained from Theorem 3.3 by considering the pair
$(\mathcal{T}f,f)$ instead of $(f,g).$ The proof is complete.

In the following, we prove two corollaries of Theorem 3.3. These results
further illustrate the value of the extrapolation of families of pairs of
functions. To prove these results we need the following inclusion properties
of Muckenhoupt classes that has been proved in [28].

######  Theorem 3.5.

Let $w$ be a positive weight, $p,$ $q$ be nonnegative real numbers. Then the
following inclusion relations hold:

$(1)$. $\mathcal{A}_{1}\subset\mathcal{A}_{p}\subset\mathcal{A}_{\infty}$, for
all $1<p<\infty,$

$(2)$. $\mathcal{A}_{p}\subset\mathcal{A}_{q}$ for all $1<p\leq q,$

$(3).$ $\mathcal{A}_{\infty}=\bigcup\limits_{1<p}\mathcal{A}_{p}$ with
$\left[w\right]_{\mathcal{A}_{\infty}}=\lim_{p\rightarrow\infty}\left[w\right]_{\mathcal{A}_{p}}$
and $\mathcal{A}_{1}\subset\bigcap{}_{p>1}\mathcal{A}_{p}.$

Our first theorem yields $\ell_{p}$ inequalities with weights in
$\mathcal{A}_{p/r}$, $p\geq r>0.$ Hereafter, by $\Omega$ we mean a family of
pairs $(f\,,g)$ of nonnegative sequences.

######  Corollary 3.1.

Suppose that for some $r>0$, and some $p_{0}$, $r\leq p_{0}<\infty,$ and for
every $w_{0}\in\mathcal{A}_{p_{0}/r},$

(3.35) $\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\leq
C\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)\text{.}$

Then for all $p,$ $r\leq p<\infty$ and all $w\in\mathcal{A}_{p/r},$

(3.36) $\sum_{k=1}^{\infty}w(k)f^{p}(k)\leq C\sum_{k=1}^{\infty}w(k)g^{p}(k).$

###### Proof.

Define a new family $\Omega_{r}$ consisting of the pairs $(f^{r},g^{r})$,
where $(f,g)\in\Omega.$ Then for $w_{0}\in\mathcal{A}_{p_{0}/r}$, by (3.35)

$\displaystyle\sum_{k=1}^{\infty}w_{0}(k)\left(f^{r}(k)\right)^{p_{0}/r}$
$\displaystyle=$ $\displaystyle\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\leq
C\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)$ $\displaystyle=$ $\displaystyle
C\sum_{k=1}^{\infty}w_{0}(k)\left(g^{r}(k)\right)^{p_{0}/r}.$

Therefore, we can apply Theorem 3.3 with this as our initial hypothesis and
conclude that for all $q>1$ and all $w\in\mathcal{A}_{q},$

$\sum_{k=1}^{\infty}w(k)\left(f^{r}(k)\right)^{q}\leq
C\sum_{k=1}^{\infty}w(k)\left(g^{r}(k)\right)^{q}\text{.}$

By putting $q=p/r$ for some $p>r$ and this is equivalent to (3.36). The proof
is complete.

######  Corollary 3.2.

Suppose that for some $p_{0}$, $0<p_{0}<\infty,$ and for every
$w_{0}\in\mathcal{A}_{\infty},$

(3.37) $\sum_{k=1}^{\infty}w_{0}(k)f^{p_{0}}(k)\leq
C\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)\text{.}$

Then for all $p,$ $0<p<\infty$ and all $w\in\mathcal{A}_{\infty},$

(3.38) $\sum_{k=1}^{\infty}w(k)f^{p}(k)\leq C\sum_{k=1}^{\infty}w(k)g^{p}(k).$

###### Proof.

Fix $r,$ $1<r<\infty$ and define a new family $\Omega_{r}$ consisting of the
pairs $(f^{p_{0}/r},g^{p_{0}/r})$, with $(f,g)\in\Omega.$ Then by inequality
(3.37), for for every $w_{0}\in\mathcal{A}_{r}\subset\mathcal{A}_{\infty}$ and
$(f^{p_{0}/r},g^{p_{0}/r})\in\Omega_{r}$

$\displaystyle\sum_{k=1}^{\infty}w_{0}(k)\left(f^{p_{0}/r}(k)\right)^{r}$
$\displaystyle=$
$\displaystyle\sum_{k=1}^{\infty}w_{0}(k)\left(f^{p_{0}}(k)\right)\leq
C\sum_{k=1}^{\infty}w_{0}(k)g^{p_{0}}(k)$ $\displaystyle=$ $\displaystyle
C\sum_{k=1}^{\infty}w_{0}(k)\left(g^{p_{0}/r}(k)\right)^{r}.$

This gives the initial estimate (3.34) for $\Omega_{r}$ and with $r$ in place
of $p_{0}.$ Therefore by Theorem 3.3, we have that for every $s,$ $1<s<\infty$
and $w\in\mathcal{A}_{s}$

(3.39) $\sum_{k=1}^{\infty}w(k)\left(f^{p_{0}/r}(k)\right)^{s}\leq
C\sum_{k=1}^{\infty}w(k)\left(g^{p_{0}/r}(k)\right)^{s}.$

Fix $p>1$ and $w\in\mathcal{A}_{\infty}.$ By Theorem 3.5, since
$\mathcal{A}_{p}$ are nested, we may assume without loss of generality that
$w\in\mathcal{A}_{s}\subset\mathcal{A}_{\infty}$ for $s>p/p_{0}.$ Therefore,
we can choose $r>1$ such that $\left(p_{0}/r\right)s=p$ and then the
inequality (3.39) yields the inequality (3.38). The proof is complete.

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