Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator

R E S E A R C H Open Access

Some new Hardy-type inequalities for Riemann-Liouville fractional q-integral operator

Lars-Erik Persson^1,2*and Serikbol Shaimardan³

*Correspondence: larserik@ltu.se

1Luleå University of Technology, Luleå, 971 87, Sweden

2Narvik University College, P.O. Box 385, Narvik, 8505, Norway Full list of author information is available at the end of the article

Abstract

We consider theq-analog of the Riemann-Liouville fractionalq-integral operator of ordern∈N. Some new Hardy-type inequalities for this operator are proved and discussed.

MSC: Primary 26D10; 26D15; secondary 33D05; 39A13

Keywords: inequalities; Hardy-type inequalities; Riemann-Liouville operator; integral operator;q-calculus;q-integral

1 Introduction

In  FH Jackson definedq-derivative and definiteq-integral [] (see also []). It was the starting point ofq-analysis. Today the interest in the subject has exploded. Theq-analysis has numerous applications in various fields of mathematics,e.g., dynamical systems, num- ber theory, combinatorics, special functions, fractals and also for scientific problems in some applied areas such as computer science, quantum mechanics and quantum physics (see,e.g., [–]). For further development and recent results inq-analysis, we refer to the books [, ] and [] and the references given therein. The first results concerning integral inequalities inq-analysis were proved in  by Gauchman []. Later on some further q-analogs of the classical inequalities have been proved (see [–]). Moreover, in 

Maligrandaet al.[] derived aq-analog of the classical Hardy inequality. Further devel- opment of Hardy’s original inequality from  (see [] and []) has been enormous.

Some of the most important results and applications have been presented and discussed in the books [, ] and []. Hence, it seems to be a huge new research area to investigate which of these so-called Hardy-type inequalities have theirq-analogs.

The aim of this paper is to obtain some q-analogs of Hardy-type inequalities for the Riemann-Liouville fractional integral operator of ordern∈Nand to find necessary and sufficient conditions of the validity of these inequalities for all non-negative real functions (see Theorems . and .).

The paper is organized as follows. In order not to disturb our discussions later on, some preliminaries are presented in Section . The main results can be found in Section , while the detailed proofs are given in Section .

©2015 Persson and Shaimardan. This article is distributed under the terms of the Creative Commons Attribution 4.0 Interna- tional License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

2 Preliminaries

First we recall some definitions and notations inq-analysis from the recent books [, ]

and [].

Letq∈(, ). Then aq-real number [α]qis defined by [α]q:= –q^α

 –q , α∈R, wherelim_q→^–q_–q^α =α.

Theq-analog of the binomial coefficients is defined by

[n]_q! :=

 ifn= , []_q×[]_q× · · · ×[n]_q ifn∈N,

n k

q

:= [n]q! [n–k]q![k]q!. We introduce theq-analog of a polynomial in the following way:

(x–a)ⁿ_q:=

 ifn= ,

(x–a)(x–qa)· · ·(x–q^n–a) ifn∈N, () (x–a)^n+m_q = (x–a)^m_q

x–q^man

q, n,m= , , , . . . . ()

Theq-gamma function_qis defined by _q(n+ ) := [n]q!, n∈N.

Forf : [,b)→R,  <b≤ ∞, we define theq-derivative as follows:

Dqf(x) :=f(x) –f(qx)

( –q)x , x∈[,b).

Clearly, if the functionf(x) is differentiable at a pointx∈(, ), thenlim_q→Dqf(x) =f(x).

Let  <a≤b<∞. The definiteq-integral (also called theq-Jackson integral) of a func- tionf(x) is defined by the formulas

_a



f(x)dqx:= ( –q)a ∞

k=

q^kf q^ka

. ()

Moreover, the improperq-integral of a functionf(x) is defined by _∞



f(x)dqx:= ( –q) ∞ k=–∞

q^kf q^k

, ()

provided that the series on the right-hand sides of () and () converge absolutely.

Suppose thatf(x) andg(x) are two functions which are defined on (,∞). Then _∞



f(x)Dq

g(x)

= ∞

j=

f q^j

g q^j

–g q^j+

. ()

Letbe a subset of (,∞) andX(t) denote the characteristic function of. For allz:

 <z<∞, we have that _∞



X(,z](t)f(t)dqt= ( –q)

qⁱ≤z

qⁱf qⁱ

()

and _∞



X[z,∞)(t)f(t)dqt= ( –q)

qⁱ≥z

qⁱf qⁱ

. ()

Al-Salam (see [] and also []) introduced the fractional q-integral of the Riemann- Liouville operatorIq,nof ordern∈Nby

Iq,nf(x) :=  _q(n)

_x



Kn–(x,s)f(s)dqs, whereK_n–(x,s) = (x–qs)^n–_q .

Next we will present a lemma (Lemma .) concerning discrete Hardy-type inequalities which are proved in []. In this paper all authors studied inequalities of the form

∞ j=

u^r_j (S_nf)_jr

^_r

≤C ∞

i=

v^p_if_i^{p }_p

, ∀f ≥ ()

for then-multiple discrete Hardy operator with weights of the form (Snf)j=

∞ k=j

ω,k k

k=

ω,k k

k=

ω,k· · ·

kn–

kn–=

ωn–,kn–

kn–

i=

fi= ∞

i=j

An–(i,j)fi,

where u={ui}^∞_i=, v={vi}^∞_i=, ω_i={ω_i,k}^∞_k= are positive sequences of real numbers (i.e., weight sequences). She also studied inequality () for the operatorS^∗_ndefined by

S_n^∗f

i:=

i

j=

fiAn–,(i,j),

which is the conjugate to the operatorSn, whereAn–,(i,j)≡ forn=  and An–,(i,j) =

i

kn–=j

ω_{n–,kn–}

i

kn–=kn–

ω_{n–,kn–}· · · i

k=k

ω_,k

forn≥.

We consider the following Hardy-type inequalities:

∞ j=–∞

u^r_j (Snf)j

r

^_r

≤C ∞ i=–∞

v^p_if_i^p _p^

()

and ∞ i=–∞

u^r_i S_n^∗f

i

r

^_r

≤C^∗ ∞ i=–∞

v^p_if_i^p _p^

. ()

In the sequel, for anyp> , the conjugate numberpis defined byp:=p/(p– ), and the considered functions are assumed to be non-negative. Moreover, the symbolMK means that there existsα>  such thatM≤αK, whereαis a constant which may depend only on parameters such asp,q,r. Similarly, the caseKM. IfMKM, then we writeM≈K.

Lemma .

(i) Let <p≤r<∞andn≥.Then inequality()holds if and only if A(n) =max_{≤m≤n–}A_m(n) <∞,where

A_m(n) =sup

k∈Z

∞ j=k

A^p_m, (j,k)v^–p_j ^

p k

i=–∞

A^r_{n–,m+}(k,i)u^r_i ^_r

, n∈N.

Moreover,A(n)≈C,whereCis the best constant in().

(ii) Let <p≤r<∞andn≥.Then inequality()holds if and only if A^∗(n) =max_{≤m≤n–}A^∗_m(n) <∞,where

A^∗_m(n) =sup

k∈Z

∞ i=k

A^r_m,(i,k)u^r_i ^_{r k}

j=–∞

A^p_{n–,m+} (k,j)v^–p_j ^

p

, n∈N.

Moreover,A^∗(n)≈C,whereCis the best constant in().

We also need the corresponding result for the case  <r<p<∞, which was proved in [].

Lemma .

(i) Let <r<p<∞andn≥.Then inequality()holds if and only if B(n) =max_{≤m≤n–}B_m(n) <∞,where

B_m(n) = _∞

i=–∞

∞ j=i

A^p_m, (j,i)v^–p_j

^p(r–)_{p–r i}

k=–∞

A^r_{n–,m+}(i,k)u^r_{k p–r}^p

× ⁺ ∞

j=i

A^p_m, (j,i)v^–p_j ^p–r

pr

, ⁺E_i,j=E_i,j–E_i,j+,n∈N.

Moreover,B(n)≈C,whereCis the best constant in().

(ii) Let <r<p<∞andn≥.Then inequality()holds if and only if B^∗(n) =max_{≤m≤n–}B^∗_m(n) <∞,where

B^∗_m(n) = _∞

i=–∞

∞ j=i

A^r_m,(j,i)u^r_j

_p–r^r _i

k=–∞

A^p_{n–,m+} (i,k)v^–p_k ^r(p–)_p–r

× ⁺ ∞

j=i

A^r_m,(j,i)u^r_j ^p–r

pr

, ⁺Ei,j=Ei,j–Ei,j+,∀n∈N.

Moreover,B^∗(n)≈C^∗,whereC^∗is the best constant in().

Let (a⁽ⁿ⁾_i,j) be a matrix whose elements are non-negative and non-increasing in the second index for alli,j:∞>i≥j> –∞, and the entries of the matrixa⁽ⁿ⁾_i,j satisfy the following (so- called discrete Oinarov condition):

a⁽ⁿ⁾_i,j ≈ n

γ=

a^(γ)_i,kd_k,j^n,γ, γ = , , . . . ,n– ,n∈N ()

for all∞>i≥k≥j> –∞.

Remark . Note that the matrices (d^γ_k,j^,m), γ = , , . . . ,m, m≥, are arbitrary non- negative matrices which satisfy () (see []).

Moreover, in [] necessary and sufficient conditions for inequalities () and () were proved for matrix operators with a matrix (a⁽ⁿ⁾_i,j) which satisfies (). For our purposes we need such characterization on the following form.

Lemma .

(i) Let <p≤r<∞and the entries of the matrix(a⁽ⁿ⁾_i,j)satisfy condition().Then inequality()for the operator(A^–f)_j:=_∞

i=ja⁽ⁿ⁾_i,jf_i,j∈Z,holds if and only if at least one of the conditionsB⁺<∞orB^–<∞holds,where

B^–=sup

k∈Z

∞ i=k

v^–p_i k j=–∞

a⁽ⁿ⁾_i,jr

u^r_j ^p_r^

p

,

B⁺=sup

k∈Z

k j=–∞

u^r_j ∞

i=k

a⁽ⁿ⁾_i,jp

v^–p_i ^r

p^_r .

Moreover,B⁺≈B^–≈C,whereCis the best constant in().

(ii) Let <p≤r<∞.Let the entries of the matrix(a⁽ⁿ⁾_i,j)satisfy condition().Then inequality()for the operator(A⁺f)i:=_i

j=–∞a⁽ⁿ⁾_i,jfj,i∈Z,holds if and only if at least one of the conditionsA⁺<∞orA^–<∞holds,where

A^–=sup

k∈Z

∞ i=k

u^r_i k j=–∞

a⁽ⁿ⁾_i,jp

v^–p_j ^r

p^_r ,

A⁺=sup

k∈Z

k j=–∞

v^–p_j ∞

i=k

a⁽ⁿ⁾_i,jr

u^r_i ^p_rp^

.

Moreover,A⁺≈A^–≈C,whereCis the best constant in().

3 The main results

Let  <r,p≤ ∞. Then theq-analog of the two-weighted inequality for the operatorIq,n

of the form _∞



u^r(x)

Iq,nf(x)r

d_qx ^_r

≤C _∞



v^p(x)f^p(x)d_qx ^_p

()

has several applications in various fields of science. In the classical analysis two-weighted estimates for the Riemann-Liouville fractional operator were derived by Stepanov for the case with parameters greater than one (see [, ]).

We consider the operatorIq,nof the following form:

Iq,nf(x) =  _q(n)

_∞



X(,x](s)K_n–(x,s)f(s)d_qs,

which is defined for allx> . Although it does not coincide with the operatorIq,n(they coincide at the pointsx=q^k,k∈Z), we have the equality

_∞



u^r(x)

Iq,nf(x)r

dqx= _∞



u^r(x)

Iq,nf(x)r

dqx.

Therefore, inequality () can be rewritten as _∞



u^r(x)

Iq,nf(x)r

dqx ^_r

≤C _∞



v^p(x)f^p(x)dqx _p^

. ()

Its conjugate operatorI^∗_q,ncan be defined by

I_q,n^∗ f(s) :=  _q(n)

_∞



X[s,∞)(x)K_n–(x,s)f(x)d_qx,

with the same kernel. The dual inequality of inequality () reads as follows:

_∞



u^r(x)

I_q,n^∗ f(x)r

dqx ^_r

≤C^∗ _∞



v^p(x)f^p(x)dqx _p^

, ()

whereCandC^∗ are positive constants independent off andu(·),v(·) are positive real- valued functions on (,∞),i.e., weight functions. In what follows we investigate inequal- ities () and ().

Let N= N∪ {}. Then, for ≤m≤n– ,m,n∈N, we use the following notations:

Q^n–_m = _∞



_∞



X(,z](s)K_m^p(z,s)v^–p(s)d_qs ^p(r–)_p–r

× _∞



X[z,∞)(x)K_n–m–^r (x,z)u^r(x)d_qx _p–r^p

×D_{q ∞}



X(,z](s)K_m^p(z,s)v^–p(s)d_qs ^p–r

pr,

Q^n–_m = _∞



_∞



X(,z](s)K_m^r(z,s)u^r(s)dqs _p–r^r

× _∞



X[z,∞)(x)K_n–m–^–p (z,x)v^–p(x)d_qx ^r(p–)_p–r

×D_{q ∞}



X(,z](s)K_m^r(z,s)u^r(x)d_qs ^p–r

pr,

H_m^n–=sup

z>

_∞



X[z,∞)(x)K_n–m–^r (x,z)u^r(x)dqx

^_{r ∞}



X(,z](s)K_m^p(z,s)v^–p(s)dqs _p^

, H^n–_m =sup

z>

_∞



X(,z](x)K_m^r(z,x)u^r(x)d_qx

^_{r ∞}



X[z,∞)(s)K_n–m–^p (s,z)v^–p(s)d_qs ^

p

,

A⁺(z) = ∞



X[z,∞)(x)u^r(x) ∞



X(,z](t)K_n–^p (x,t)v^–p(t)dqt ^r

p

dqx ^_r

,

A^–(z) = _∞



X(,z](t)v^–p(t) _∞



X[z,∞)(x)K_n–^r (x,t)u^r(x)dqx ^p_r

dqt ^

p

.

A⁺(z) = _∞



X[z,∞)(t)v^–p(t) _∞



X(,z](x)K_n–^r (t,x)u^r(x)d_qx ^p_r

d_qt _p^

, A^–(z) =

_∞



X(,z](x)u^r(x) _∞



X[z,∞)(t)K_n–^p (t,x)v^–p(t)dqt _p^r

dqx ^_r

, Hn–= max

≤k≤n–H_k^n–, Hn–= max

≤k≤n–H^n–_k , A⁺_q=sup

z>

A⁺(z), A^–_q=sup

z>

A^–(z), A⁺_q=sup

z>A⁺(z), A^–_q=sup

z>A^–(z), Qn–= max

≤k≤n–Q^n–_k and Qn–= max

≤k≤n–Q^n–_k . Our main results read as follows.

Theorem .

(i) Let <r<p<∞.Then inequality()holds if and only ifQn–<∞.Moreover, Qn–≈C,whereCis the best constant in().

(ii) Let <p≤r<∞.Then inequality()holds if and only if at least one of the conditionsH_n–<∞orA⁺_q<∞orA^–_q<∞holds.Moreover,H_n–≈A⁺_q≈A^–_q≈C, whereCis the best constant in().

Theorem .

(i) Let <r<p<∞.Then inequality()holds if and only ifQn–<∞.Moreover, Qn–≈C^∗,whereC^∗is the best constant in().

(ii) Let <p≤r<∞.Then inequality()holds if and only if at least one of the conditionsHn–<∞orA⁺_q<∞,orA^–_q<∞holds.Moreover,Hn–≈A⁺_q≈A^–_q≈C, whereCis the best constant in().

For the proofs of these results, we need the following lemmata of independent interest.

Lemma . Let x,t,s:  <s≤t≤x<∞.Then

≤m≤n–max K_n–m–(x,t)K_m(t,s)≤K_n–(x,s) ≤ n–

m=

n–  m

q

K_n–m–(x,t)K_m(t,s) ()

for m: ≤m≤n– ,n,m– ∈Nand where K_n–(x,s) = (x–qs)^n–_q . Lemma . Let f and g be non-negative functions on(,∞),α,β∈Rand

I(z) :=

_∞



X(,z](t)f(t)d_qt

α _∞



X[z,∞)(x)g(x)d_qx β

.

Then

sup

z>

I(z) = ( –q)^α+βsup

k∈Z

∞ j=k

q^jf

q^jα k i=–∞

qⁱg qⁱβ

, ()

where at least one ofαandβis non-negative.

This result was proved in [], but for the readers’ convenience we will include in Sec- tion  a proof which is slightly simpler than that in the Russian version given in [].

Lemma . Letα,β∈R⁺,K(·,·)be a non-negative function and I⁺(z) :=

∞



X[z,∞)(x)g(x) ∞



X(,z](t)K(x,t)f(t)dqt α

dqx β

, I^–(z) :=

∞



X(,z](t)f(t) ∞



X[z,∞)(x)K(x,t)g(x)dqx α

dqt β

. Then

sup

z>

I⁺(z) =sup

k∈Z ( –q) k j=–∞

q^jg q^j

( –q) ∞

i=k

qⁱK q^j,qⁱ

f qⁱαβ

()

and

sup

z>

I^–(z) =sup

k∈Z ( –q) ∞

j=k

q^jf q^j

( –q) k j=–∞

q^jK q^j,qⁱ

g q^jαβ

. ()

Lemma . Let Q^n–_m ,Q^n–_m <∞for <m≤n– .Then

Q^n–_m = _∞

i=–∞

( –q) ∞

t=i

q^tK_m^p qⁱ,q^t

v^–p

q^tp(r–)_p–r

× ( –q) i j=–∞

q^jK_n–m–^r q^j,qⁱ

u^r q^j_p–r^p

× ⁺ ∞

n=i

( –q)qⁿK_m^r qⁱ,qⁿ

v^–p

q^np–r

pr

and Q^n–_m =

_∞

i=–∞

( –q) ∞

t=i

q^tK_m^r qⁱ,q^t

u^r q^t_p–r^r

× ( –q) i j=–∞

q^jK_n–m–^p q^j,qⁱ

v^–p

q^jr(p–)_p–r

× ⁺ ∞

n=i

( –q)qⁿK_m^r qⁱ,qⁿ

u^r

q^np–r

pr

,

where⁺En,i=En,i–En,i+.

4 Proofs

Proof of Lemma. Let  <s≤t≤x<∞. First we prove the lower estimate. By using () we find that

Kn–m–(x,t)Km(t,s) = (x–qt)^n–m–_q (t–qs)^m_q

≤(x–qs)^n–m–_q (x–qs)^m_q

≤(x–qs)^n–m–_q

x–q^n–msm q

= (x–qs)^n–_q =K_n–(x,s)

for  <s≤t≤x<∞and ≤m≤n– ,m– ,n∈N. Hence,

≤m≤n–max Kn–m–(x,t)Km(t,s)≤Kn–(x,s), and the lower estimate in () is proved.

According to () we get thatK(x,t)K(t,s) =K(x,s)≡ forn= . Moreover, we have that

K(x,s) = (x–qs)_q< (x–qt)_q+ (t–qs)_q

= 

m=

 m

q

K–m–(x,t)Km(t,s)

forn= .

This means that the inequality

Kn–(x,s) <

n–

m=

n– 

m

q

Kn–m–(x,t)Km(t,s) ()

holds forn= . Our aim is now to use induction, and we assume that () holds forn=l– , l≥, and we will prove that it then holds also forn=l.

We use our induction assumption, make some calculations and obvious estimates and find that

Kl–(x,s) =Kl–(x,s)

x–q^l–s

<

l–

m=

l– 

m

q

Kl–m–(x,t)Km(t,s)

x–q^l–s

<

l–

m=

l– 

m

q

Kl–m–(x,t)Km(t,s)

x–q^l–m–t+q^l–m–t–q^l–s

= l–

m=

l–  m

q

K_l–m–(x,t)K_m(t,s)

x–q^l–m–t

+ l–

m=

l–  m

q

Kl–m–(x,t)Km(t,s)q^l–m–

t–q^m+s

= l– 



q

Kl–(x,t)K(t,s) + l–

m=

l–  m

q

Kl–m–(x,t)Km(t,s)

+ l–

m=

q^l–m–

l–  m– 

q

K_l–m–(x,t)K_m(t,s) + l– 

l– 

q

K_(x,t)K_l–(t,s)

= l– 



q

Kl–(x,t)K(t,s)

+ l–

m=

q^l–m–

l–  m– 

q

+ l– 

m

q

K_l–m–(x,t)K_m(t,s)

+

l–  l– 

q

K(x,t)Kl(t,s).

Since, for anym≥ (q^l–m–l–  m– 

q+l–  m

q=l–  m

q), we get that Kl–(x,s) <

l–

m=

l– 

m

q

Kl–m–(x,t)Km(t,s).

Hence, () holds also withn=lwhich, by the induction axiom, means that also the

upper estimate in () is proved. The proof is complete.

Proof of Lemma. From () and () it follows that

I(z) = ( –q)^α+β

q^j≤z

q^jf

q^jα

qⁱ≥z

qⁱg q^iβ

.

Ifz=q^k, then, fork∈Z, I(z) = ( –q)^α+β

∞ j=k

q^jf

q^jα k i=–∞

qⁱg qⁱβ

.

Ifq^k<z<q^k–, then, fork∈Z, I(z) = ( –q)^α+β

∞ j=k

q^jf

q^jα k–

i=–∞

qⁱg qⁱβ

.

Hence, fork∈Zandβ> , we find that

sup

q^k≤z<q^k–

I(z) = ( –q)^α+β ∞

j=k

q^jf

q^jα k i=–∞

qⁱg qⁱβ

.

Therefore sup

z>

I(z) =sup

k∈Z sup

q^k≤z<q^k–

I(z)

= ( –q)^α+βsup

k∈Z

∞ j=k

q^jf

q^jα k i=–∞

qⁱg qⁱβ

. ()

We have proved that () holds whereverβ> .

Next we assume thatα> . Letq^k+<z<q^k,k∈Z. Then we get that

I(z) = ( –q)^α+βsup

k∈Z

∞ j=k+

q^jf

q^jα k i=–∞

qⁱg qⁱβ

,

and analogously as above we find that

sup

q^k+<z≤q^k

I(z) = ( –q)^α+β ∞

j=k

q^jf

q^jα k i=–∞

qⁱg qⁱβ

,

and () holds also for the caseα> . The proof is complete.

Proof of Lemma. Letz=q^k,k∈Z. By using () and () we have that

I⁺(z) = ( –q) k j=–∞

q^jg q^j

( –q) ∞

i=k

qⁱK q^j,qⁱ

f

qⁱαβ

.

For the casesq^k+<z<q^k,k∈Zandq^k<z<q^k–,k∈Z, we find that

I⁺(z) = ( –q) k j=–∞

q^jg q^j

( –q) ∞ i=k+

qⁱK q^j,qⁱ

f

qⁱαβ

and

I⁺(z) = ( –q) k–

j=–∞

q^jg q^j

( –q) ∞

i=k

qⁱK q^j,qⁱ

f

qⁱαβ

,

respectively.

Hence, we conclude that

sup

q^k+<z<q^k–

I⁺(z) = ( –q) k j=–∞

q^jg q^j

( –q) ∞

i=k

qⁱK q^j,qⁱ

f qⁱαβ

.

Sincesup_z>I⁺(z) =sup_k∈Zsup_qk+<z<q^k–I⁺(z), we find that () holds. The identity () can be proved in a similar way as (). The proof is complete.

Proof of Lemma. Without loss of generality we may assume thatQ^n–_m <∞. By using (), () and () we can deduce that

Q^n–_m = _∞

i=–∞

_∞



X(,qⁱ](s)K_m^p qⁱ,s

v^–p(s)d_qs ^p(r–)_p–r

× ∞



X[qⁱ,∞)(x)K_n–m–^r x,qⁱ

u^r(x)dqx _p–r^p

× _∞



X_(,qⁱ_](s)K_m^p qⁱ,s

v^–p(s)d_qs

– _∞



X(,q^i+](s)K_m^p q^i+,s

v^–p(s)dqs ^p–r_pr

= _∞

i=–∞

( –q) ∞

t=i

q^tK_m^p qⁱ,q^t

v^–p

q^tp(r–)_p–r

× ( –q) i j=–∞

q^jK_n–m–^r q^j,qⁱ

u^r q^j_p–r^p

× ⁺ ∞

n=i

( –q)qⁿK_m^r qⁱ,qⁿ

v^–p

q^np–r

pr

,

and the first equality in Lemma . is proved.

The second inequality can be proved in a similar way, so we leave out the details. The

proof is complete.

Proof of Theorem. By using formulas () and () we find that inequality () can be rewritten as

∞ j=–∞

( –q)^r+q^ju^r q^{j ∞}

i=j

qⁱf qⁱ

Kn–

q^j,qⁱr^_r

≤C ∞ i=–∞

( –q)qⁱf^p qⁱ

v^p q^i_p

. ()

Let

u^r_j = ( –q)^r+q^ju^r q^j

, fi=qⁱf qⁱ

, v^p_i = ( –q)q^i(–p)v^p

qⁱ

, W⁽ⁿ⁾(i,j) =Kn–

q^j,qⁱ .

()

Then we get that inequality () can be rewritten as the discrete weighted Hardy-type inequality (see,e.g., [])

∞ j=–∞

u^r_j ∞

i=j

W⁽ⁿ⁾(i,j)fi

r^_r

≤C ∞ i=–∞

v^p_ia^p_i ^_p

. ()

Hence, inequality () is equivalent to inequality (), where (W⁽ⁿ⁾(i,j)) is the non- negative triangular matrix which has entriesW⁽ⁿ⁾(i,j)≥ forj≤iandW⁽ⁿ⁾(i,j)≡ for j>iand is non-decreasing in the first index for alli≥j> –∞.

First we will prove that, forn∈N, ( –q)^n–

i

kn–=j

[n– ]qq^kn–

i

kn–=kn–

[n– ]qq^kn–· · · i

k=k

[]qq^k=W⁽ⁿ⁾(i,j). ()

We will use induction and first we note thatW^()(i,j) = (q^j–qⁱ)^_q≡ forn= . Ifn= , then

( –q) i

k=j

q^k= i

k=j

[]_q

q^k–q^k+

=

q^j–q^i+

q=W^()(i,j).