Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324 DOI 10.1186/s13660-016-1266-y

RESEARCH

Open Access

Boundedness and compactness of a class of Hardy type operators Akbota M Abylayeva1 , Ryskul Oinarov1 and Lars-Erik Persson2,3* *

Correspondence: [email protected] Department of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå, 97187, Sweden 3 UiT, The Artic University of Norway, Tromsø, Norway Full list of author information is available at the end of the article 2

Abstract We establish characterizations of both boundedness and of compactness of a general class of fractional integral operators involving the Riemann-Liouville, Hadamard, and Erdelyi-Kober operators. In particular, these results imply new results in the theory of Hardy type inequalities. As applications both new and well-known results are pointed out. MSC: 26A33; 26D10; 47G10 Keywords: inequalities; Hardy type inequalities; fractional integral operator; Riemann-Liouville operator; Hadamard operator; Erdelyi-Kober operator; boundedness; compactness

1 Introduction Let I = (a, b),  ≤ a < b ≤ ∞. Let v and u be almost everywhere positive functions, which are locally integrable on the interval I. Let  < p < ∞ and p + p = . Denote by Lp,v ≡ Lp (v, I) the set of all functions f measurable b  on I such that f p,v := ( a |f (x)|p v(x) dx) p < ∞. Let W be a non-negative, strictly increasing and locally absolutely continuous function on I. Suppose that dWdx(x) = w(x), a.e. x ∈ I. We consider the Hardy type operator Tα,β defined by 

x

Tα,β f (x) := a

u(s)W β (s)f (s)w(s) ds , (W (x) – W (s))–α

x ∈ I.

(.)

When u ≡  and β =  the operator Tα,β is called the fractional integral operator of a function f with respect to a function W ([], p.). When u ≡  and W (x) = x the operator (.) becomes the Riemann-Liouville operator Iα defined by 

x

Iα f (x) := a

sβ f (s) ds . (x – s)–α

(.)

When u ≡  and W (x) ≡ ln ax , a > , this operator is the Hadamard operator Hα defined by 

x

Hα f (x) := a

(ln as )β f (s) ds . s(ln xs )–α

© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International 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.

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

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Moreover, when u ≡  and W (x) = xσ , σ > , we get the operator Eα,β of Erdelyi-Kober type ([], p.) defined by 

x

Eα,β f (x) := σ a

f (s)sσβ+σ – ds . (xσ – sσ )–α

There are a lot of works devoted to the mapping properties of the Riemann-Liouville operator Iα . Two-weighted estimates of the operator Iα of the order α >  in weighted Lebesgue spaces were first obtained in the papers [] and []. The singular case  < α <  was studied with different restrictions in [–] and some others. The most general results among them are given in [] and [] under the assumption that one of the weight functions is increasing or decreasing. In this work we investigate the problems of boundedness and compactness of the operator Tα,β defined by (.) from Lp,w to Lq,v when  < α < . When α >  the results follow from the results in []. The operator Tα,β was studied in [] and [] when u ≡ , β =  and u ≡ , β > – p , respectively. Due to the non-negativity and monotone increase of the function W the limit limx→a+ W (x) ≡ W (a) ≥  exists.  defined by We also consider the Hardy type operator Tα,β   Tα,β f (x) :=

x a

β

u(s)W (s)f (s)w(s) ds , (W (x) – W (s))–α

x ∈ I,

where W (x) = W (x) – W (a).   f (x) + W (a)Tα, f (x), Since we also suppose that β ≥ , for f ≥  we have Tα,β f (x) ≈ Tα,β where the equivalence constants do not depend on x and f . Therefore, without loss of generality, we can assume that W (a) = . For short writing we denote by K the norm of a linear operator K acting from one normalized space to another, since from the context we shall in each case clearly see which spaces the operator is acting between. The paper is organized as follows: In order not to disturb our discussions later on some auxiliary statements are given in Section . The main results concerning the boundedness of operator Tα,β , including the corresponding Hardy type inequalities, can be found in Section . The main results about the compactness are presented in Section . Moreover,  are given. Finally, Section  is in Section  some similar results for the dual operator Tα,β reserved for some applications (both new and well-known results). Conventions The indeterminate form  · ∞ is assumed to be zero. The relations A  B and A B, respectively, mean A ≤ cB and A ≥ cB, where a positive constant c can be dependent only on the parameters p, q, α and β. The relation A ≈ B is interpreted as A  B  A. The set of all integers is denoted by Z. Moreover, χ(c,a) (·) is the characteristic function of the interval (c, a) ⊂ I.

2 Auxiliary statements To prove the main results we shall need some auxiliary results from the standard literature on Hardy type inequalities (see [] and []).

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

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Together with the operator (.) we consider the Hardy type operator Hα,β defined by

Hα,β f (x) =





x

u(s)W β (s)f (s)w(s) ds.

W –α (x)

(.)

a

It is easy to see that for f ≥  we have Tα,β f (x) ≥ Hα,β f (x),

∀x ∈ I.

(.)

The problem of boundedness of operators of the form (.) in weighted Lebesgue spaces have been very well studied. The history and development of Hardy type inequalities with relevant references can be found in []. In view of [] the following statements are consequences of Theorem  of []. Lemma . Let  < p ≤ q < ∞ and let the operator Hα,β be defined by (.). Then the inequality 

b

q Hα,β f (x) v(x) dx

 q

a



b

≤C

p f (x) w(x) dx

 p (.)

a

holds if and only if 

z





up (s)W p β (s)w(s) ds

Aα,β = sup z∈I

  

b

p

a

W q(α–) (x)v(x) dx

 q

< ∞.

z

Moreover, C ≈ Aα,β . Lemma . Let  < q < p < ∞, p >  and let the operator Hα,β be defined by (.). Then the inequality (.) holds if and only if  b  Bα,β =

W a

p  p–q

b q(α–)

(x)v(x) dx

z

 ×

z

p

u (s)W

p β

 p(q–) p–q (s)w(s) ds

p

u (z)W

p β

 p–q pq (z)w(z) dz

< ∞.

a

Moreover, C ≈ Bα,β . Remark . In the case  < q < p < ∞, p >  it is well known and easy to prove that the value Bα,β is equivalent to the value  Bα,β =

 b 

W a

q  p–q

b q(α–)

(x)v(x) dx

z

 ×

z

p

u (s)W a

p β

 q(p–) p–q (s)w(s) ds

W

q(α–)

 p–q pq (z)v(z) dz

.

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

Page 4 of 18

3 Boundedness of the operator Tα ,β The main results in this section read as follows. Theorem . Let  < α < ,  < p ≤ q < ∞ and β ≥ . Let u be a non-increasing function on I. Then the operator Tα,β defined by (.) is bounded from Lp,w to Lq,v if and only if Aα,β < ∞. Moreover, Tα,β  ≈ Aα,β . Theorem . Let  < α < ,  < q < p < ∞, p > α and β ≥ . Let u be a non-increasing function on I. Then the operator Tα,β is bounded from Lp,w to Lq,v if and only if Bα,β < ∞. Moreover, Tα,β  ≈ Bα,β . These two theorems can be reformulated as the following new information in the theory of Hardy type inequalities. Theorem . Let  < α < , β ≥  and u be a non-increasing function on I. Then the inequality 

b

q

 q

Tα,β f (x) v(x) dx

a

 ≤C

b

p

 p

f (x) w(x) dx

(.)

a

holds if and only if (a) Aα,β < ∞ for the case  < p ≤ q < ∞, (b) Bα,β < ∞ for the case  < q < p < ∞, p > α . Moreover, for the best constant C in (.) it yields C ≈ Aα,β in case (a) and C ≈ Bα,β in case (b). Proof of Theorem . Necessity. Let the operator Tα,β be bounded from Lp,w to Lq,v . Then, in view of (.), the operator Hα,β is bounded from Lp,w to Lq,v , and Tα,β  ≥ Hα,β . Consequently, by Lemma . we have Aα,β < ∞ and Tα,β  Aα,β .

(.)

Sufficiency. Since the function W is continuous and strictly increasing on I and W (a) = , for any k ∈ Z we can define xk := sup{x : W (x) ≤ k , x ∈ I}. We obtain a sequence of points {xk }k>–∞ such that  < xk ≤ xk+ , ∀k ∈ Z, and if xk < b, then W (xk ) = k , k ≤ W (x) ≤  xk x k+ for xk ≤ x ≤ xk+ , xk– w(s) ds = k– , and if xk+ = b, then xkk+ w(s) ds ≤ k . These facts will be used below without reminders. We assume that Ik = [xk , xk+ ), k ∈ Z, Z = {k : k ∈ Z, Ik = ∅}. Then Z ⊆ Z and I = k∈Z Ik = k∈Z Ik . Since Ik = ∅, ∀k ∈ Z \ Z , and integrals over these intervals are equal to zero, in the sequel, without loss of generality, we can suppose that Z = Z . Let Aα,β < ∞. We need to prove that the inequality Tα,β f q,v  Aα,β f p,w ,

f ∈ Lp,w ,

holds, which means Tα,β   Aα,β and, together with (.), this gives Tα,β  ≈ Aα,β .

(.)

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324



Let f ≥ . Using the relation I =

k Ik ,



Page 5 of 18

we have

 u(s)W β (s)f (s)w(s) ds q dx (W (x) – W (s))–α a xk k  xk–  x 

 xk+ u(s)W β (s)f (s)w(s) ds q = v(x) + dx (W (x) – W (s))–α xk– xk a k   xk–

 xk+ u(s)W β (s)f (s)w(s) ds q  v(x) dx (W (x) – W (s))–α xk a k  x 

 xk+ u(s)W β (s)f (s)w(s) ds q v(x) dx := J + J . + –α xk– (W (x) – W (s)) xk



Tα,β f qq,v =

xk+

x

v(x)

(.)

k

We now estimate J and J separately. Using the monotonicity of W we find that J =





= 

u(s)W β (s)f (s)w(s) ds (W (x) – W (s))–α

a



xk+

xk–

q(–α)

a



q dx

u(s)W β (s)f (s)w(s) ds (W (xk ) – W (xk– ))–α

v(x) xk

k

xk–

v(x) xk

k

≤



xk+



xk+

q(–α) 



q dx q

xk–

β

u(s)W (s)f (s)w(s) ds dx k+ a  x q

 xk+ v(x)W q(α–) (x) u(s)W β (s)f (s)w(s) ds dx ≤ Hα,β f qq,v .  v(x)

xk

k

a

xk

k

Hence, by Lemma . we get q

J  Aα,β f qp,w .

(.)

Moreover, by using Hölder’s inequality and the fact that the function u is increasing, we obtain J =





≤



xk+

u(s)W β (s)f (s)w(s) ds (W (x) – W (s))–α  pq 

x p

xk– xk+

q

xk–

 pq

p

f (s)w(s) ds

dx 

x

f (s)w(s) ds

v(x)





up (s)W p β (s)w(s) ds (W (x) – W (s))p (–α)

 q p

dx

uq (xk– )

xk–

k

 ×

xk–

xk

k

x

v(x)

xk

k

≤



xk+



xk+

v(x) xk

a

x



W p β (s)w(s) ds (W (x) – W (s))p (–α)

 q p

(.)

dx.

A change of variables W (s) = W (x)t in the last integral, implies that  a

x





W p β+ (x) W p β (s)w(s) ds  (–α) = p (W (x) – W (s)) W p (–α) (x)









t p β ( – t)p (α–) dt. 

(.)

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

Since β ≥ , α > p , the Euler beta function from (.) and (.) it follows that J 



f (s)w(s) ds











t p β (–t)p (α–) dt converges. Consequently,

xk+

q

u (xk– ) xk



xk+

f p (s)w(s) ds

q p

q

uq (xk– )W p

q

(p β+)

dx W p v(x) W q(–α) (x)

(p β+)



xk+

(xk+ )

xk–

k

v(x)W q(α–) (x) dx

xk

(qβ+  )

p q

= 

 pq

p



xk–

k

≤

xk+

Page 6 of 18



xk+

 pq

p

f (s)w(s) ds xk–

k q

× uq (xk– )W p

(p β+)



xk+

(xk– )

v(x)W q(α–) (x) dx

xk





xk+

 pq

p

f (s)w(s) ds xk–

k



xk–

× uq (xk– )



W p β (s)w(s) ds

 q  p

a

≤



xk+



xk

×

xk

f p (s)w(s) ds

 pq

p

u (s)W

p β

 q  p

(s)w(s) ds

≤ Aα,β

b

v(x)W q(α–) (x) dx

xk

a



xk+

f p (s)w(s) ds

 pq

xk–

k



v(x)W q(α–) (x) dx

xk–

k

q

xk+

q

≤ Aα,β

  k

xk+

f p (s)w(s) ds

 pq

xk–

q Aα,β f qp,w .

(.) 

By combining (.), (.) and (.) we obtain (.). The proof is complete. Proof of Theorem . Necessity. Similarly to the proof of Theorem . and the estimate Tα,β  Bα,β ,

(.)

it follows from (.) and Lemma .. Sufficiency. Let Bα,β < ∞. If we show that Tα,β   Bα,β , then this fact and (.) imply that Tα,β  ≈ Bα,β . Next, we use relation (.). For the estimate J we have obtained J  q Hα,β f q,v . Hence, by Lemma . we obtain q

J  Bα,β f qp,w .

(.)

Moreover, from Theorem ., obvious estimates and Hölder’s inequality it follows that J 

 k

xk+

 pq

p

f (s)w(s) ds xk– q

× uq (xk– )W p

(p β+)



xk+

(xk+ ) xk

v(x)W q(α–) (x) dx

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

(qβ+  )  p β+ p q

= 



–

  q

xk+

Page 7 of 18

 pq

p

f (s)w(s) ds

p

xk–

k



q    × uq (xk– ) (p β+)(k–) – (p β+)(k–) p

xk+

v(x)W q(α–) (x) dx

xk





xk+

 pq

p

f (s)w(s) ds xk–

k



xk–

× uq (xk– )



W p β (s)w(s) ds

 q 

xk–





xk+

xk+

p

v(x)W q(α–) (x) dx

xk

f p (s)w(s) ds

 pq

xk–

k



xk–

×

p

u (s)W

p β

 q  p

(s)w(s) ds

xk–

xk+

v(x)W q(α–) (x) dx

xk

  p p we apply Hölder’s inequality with the conjugate exponents , q p–q q p p–q   xk+ p–q p p f p (s)w(s) ds  J f qp,w , ≤ J

(.)

xk–

k

where

J =



xk–





up (s)W p β (s)w(s) ds

  q(p–) p–q

xk–

k

xk+

v(x)W q(α–) (x) dx

p  p–q

.

xk

To estimate J we use the relation 

xk–





up (s)W p β (s)w(s) ds

 q(p–) p–q

xk–

 

xk–  t

p

u (s)W xk–

p β

 p(q–) p–q (s)w(s) ds





up (t)W p β (t)w(t) dt.

xk–

Then J 

xk–  t



xk–

k

p

× u (t)W





up (s)W p β (s)w(s) ds

 p(q–) p–q

xk– p β



xk+

(t)w(t) dt

v(x)W

q(α–)

p  p–q

(x) dx

xk

≤

xk–  t



u (s)W

xk–

k

p

 ×

p β

 p(q–) p–q (s)w(s) ds

a

b

v(x)W q(α–) (x) dx

p  p–q





up (t)W p β (t)w(t) dt

t qp p–q

≤ Bα,β .

(.)

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

Page 8 of 18

By substitution of (.) in (.) we obtain q

J  Bα,β f qp,w .

(.)

Now, by combining (.), (.) and (.) we obtain Tα,β f q,v  Bα,β f p,w . Consequently, Tα,β q,v  Bα,β . The proof is complete.



4 Compactness of the operator Tα ,β The main results in this section read as follows. Theorem . Let  < α < , α < p ≤ q < ∞ and β ≥ . Let u be a non-increasing function on I. Then the operator Tα,β is compact from Lp,w to Lq,v if and only if Aα,β < ∞ and lim Aα,β (z) = lim– Aα,β (z) = ,

z→a+

z→b

where 

z

Aα,β (z) =

p

u (s)W

p β

   (s)w(s) ds

W

a

 q

b

p

q(α–)

(x)v(x) dx

.

z

Theorem . Let  < α < , p > α and β ≥ . Let u be a non-increasing function on I. If b < ∞ and  < q < p < ∞ or b = ∞ and  < q < p < ∞, then the operator Tα,β is compact from Lp,w to Lq,v if and only if Bα,β < ∞. Proof of Theorem . Necessity. Let the operator Tα,β be compact from Lp,w to Lq,v . Then it is bounded and consequently, by Theorem ., we have Aα,β < ∞. First we need to show that limz→a+ Aα,β (z) = . Consider the family of functions {ft }t∈I , where 





t

ft (x) = χ(a,t) (x)up – (x)W (p –)β (x)





up (s)W p β (s)w(s) ds

– p

x ∈ I.

,

(.)

a

We note that 

b

ft (x) p w(x) dx

 p

 =

a

t

ft (x) p w(x) dx

 p

a



t





up (s)W p β (s)w(s) ds

=

– p

a

 ×

t

p

u (s)W

p β

 p (s)w(s) ds

= .

(.)

a

Next we show that the family of functions {ft }t∈I defined by (.) converges weakly to zero in Lp,w . Let g ∈ Lp ,w–p = (Lp,w )∗ . Then, by Hölder’s inequality and (.), we find that 

b



t

ft (x) p w(x) dx

ft (x)g(x) dx ≤

a



 p 

a t



g(s) p w–p (s) ds

= a

  p

t



g(s) p w–p (s) ds

  p

a

.

(.)

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

Page 9 of 18

Since g ∈ Lp ,w–p , the last integral in (.) converges to zero as t → a+ , which means weak convergence of the family of functions {ft } to zero as t → a+ . Therefore, from the compactness of the operator Tα,β from Lp,w to Lq,v it follows that lim Tα,β ft q,v = .

(.)

t→a+

Moreover,  Tα,β ft qq,v



b

=

x

u(s)W β (s)ft (s)w(s) ds (W (x) – W (s))–α

v(x) a

a

q dx

 u(s)W β (s)ft (s)w(s) ds q v(x) dx ≥ (W (x) – W (s))–α t a  t – pq  b v(x) dx p p β ≥ u (s)W (s)w(s) ds q(–α) (x) t W a  t q   × up (s)W p β (s)w(s) ds 



b

t

a

=

q Aα,β (t).

(.)

From (.) and (.) it follows that limt→a+ Aα,β (t) = . Now, we show that limt→b– Aα,β (t) = . From the compactness of the operator Tα,β from Lp,w to Lq,v compactness of the conjugate operator follows: ∗ g(s) = u(s)W p (s)w(s) Tα,β

 s

b

g(x) dx (W (x) – W (s))–α

from Lq ,v–q to Lp ,w–p . For t ∈ I we introduce the family {gt }t∈I of functions: 

b

W q(α–) (x)v(x) dx

gt (x) = χ[t,b) (x)

–  q

W (q–)(α–) (x)v(x).

(.)

t

The family {gt }t∈I of functions defined by (.) is correctly defined, since due to condition Aα,β < ∞ the involving integrals are finite. We show that for all t ∈ I the functions gt ∈ Lq ,v–q converge weakly to zero as t → b– . Indeed,  gt q ,v–q =



gt (x) q v–q (x) dx

b

  q

t



–  

b

=

W

q(α–)

(x)v(x) dx

t

 t



W (q–)(α–) (x)v(x) q v–q (x) dx

  q

t b

W q(α–) (x)v(x) dx

=

b

q

–  

b

q

W q(α–) (x)v(x) dx t

  q

= .

(.)

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By using (.) with f ∈ Lq,v = (Lq ,v–q )∗ we obtain 

b

 gs (x)f (x) dx ≤

a

b

 q

gt (x) q v– q (x) dx

   q

t

b

f (x) q v(x) dx

 q

t



b

f (x) q v(x) dx

≤ g 

t q ,v–q

 q



b

f (x) q v(x) dx

=

t

 q .

t

Since f ∈ Lq,v , the last integral tends to zero as t → b– , which gives the weak convergence ∗ : Lq ,v–q → Lp ,w–p it follows to zero of {gt }t∈I in Lq ,v–q as t → b– . By compactness of Tα,β that  ∗  lim– Tα,β gt p ,w–p = .

(.)

s→b

Furthermore, we note that  ∗  T gt   –p = α,β p ,w



b



u(s)W β (s)w(s) p

a



  p p gt (x) dx –p × w (s) ds –α (W (x) – W (s)) s p    t  b p gt (x) dx   ≥ up (s)W p β (s)w(s) ds –α a t (W (x) – W (s))  t    b –  p q p p β q(α–) ≥ u (s)W (s)w(s) ds W (x)v(x) dx b

a

 ×

b

W

(q–)(α–)

t

(x)v(x) dx W –α (x)

q

t

= Aα,β (t).

Hence, according to (.) we have lims→b– Aα,β (s) = . The proof of the necessity is complete. Sufficiency. For a < c < d < b we define Pc f := χ(a,c] f ,

Pcd f := χ(c,d] f ,

Qd f := χ(d,b) f .

Then f = Pc f + Pcd f + Qd f and since Pc Tα,β Pcd ≡ , Pc Tα,β Qd ≡ , Pcd Tα,β Qd ≡ , we have Tα,β f = Pcd Tα,β Pcd f + Pc Tα,β Pc f + Pcd Tα,β Pc f + Qd Tα,β f .

(.)

We show that the operator Pcd Tα,β Pcd is compact from Lp,w to Lq,v . Since Pcd Tα,β Pcd f (x) =  for x ∈ I \ (c, d), it is enough to show that the operator Pcd Tα,β Pcd is compact from Lp,w (c, d) to Lq,v (c, d). This, in turn, is equivalent to compactness of the operator 

d

K(x, s)f (s) ds

Tf (x) = c

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from Lp (c, d) to Lq (c, d) with the kernel 



u(s)W β (s)v q (x)χ(c,d) (x – s)w p (s) K(x, s) = . (W (x) – W (s))–α Let {xk }k∈Z be the sequence of points defined in the proof of Theorem .. There are points xi , xn+ , xi < xn+ such that xi ≤ c < xi+ , xn < d ≤ xn+ . We assume that the numbers c, d are chosen so that xi+ < xn . Similarly to obtaining estimates of J and J in Theorem ., we have  d  c

d



K(x, s) p ds

 q p

dx

c



q   up (s)W p β (s)w(s) ds p = v(x) dx p (–α) c c (W (x) – W (s))  xk–  x  p q  n  xk+

u (s)W p β (s)w(s) ds p ≤ v(x) + dx p (–α) xk– (W (x) – W (s)) a k=i xk 

d

x

q

≤ μ(n – i + )Aα,β < ∞, where the constant μ does not depend on i, n. Therefore, on the basis of the Kantarovich condition ([], p.), the operator T is compact from Lp (c, d) to Lq (c, d), which is equivalent to compactness of the operator Pcd Tα,β Pcd from Lp,w to Lq,v . From (.) it follows that Tα,β – Pcd Tα,β Pcd  ≤ Pc Tα,β Pc  + Pcd Tα,β Pc  + Qd Tα,β .

(.)

We will show that the right-hand side of (.) tends to zero at c → a and d → b. Then the operator Tα,β as the uniform limit of compact operators is compact from Lp,w to Lq,v . By using Theorem . we find that



v(x)

 q u(s)W β (s)f (s)w(s) ds

q dx (W (x) – W (s))–α a a  z   p p p β  sup u (s)W (s)w(s) ds 

Pc Tα,β Pc f q,v =

c

a
x

a

 ×

c

v(x)W q(α–) (x) dx

 q

f p,w

z

≤ sup Aα,β (z)f p,w . a
Consequently, Pc Tα,β Pc   supa
c→a+

c→a a
z→a

(.)

To estimate Pcd Tα,β Pc  we assume that vε (x) = v(x) for x ∈ (c, d] and vε (x) = εq v(x) for x ∈ (a, c], uε (s) = u(s) for s ∈ (a, c] and uε (s) = εu(s) for s ∈ (c, d], where  > ε > . Obviously,

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the function uε is non-increasing on I. Then, according to Theorem ., we obtain

 q u(s)W β (s)f (s)w(s) ds

q = dx (W (x) – W (s))–α c a

 x

 d  q

uε (s)W β (s)f (s)w(s) ds

q ≤ vε (x)

dx (W (x) – W (s))–α a a 

Pcd Tα,β Pc q,v

d



v(x)

c

 Aεα,β f p,w ,

(.)

where 

d

W q(α–) (x)vε (x) dx

Aεα,β = sup

a
 q 

z

z





upε (s)W p β (s)w(s) ds

a

  p

.

We estimate the expression Aεα,β from the above as follows:  Aεα,β

≤ sup



d

W

a
q(α–)

(x)v(x) dx + ε

W

c



q(α–)

(x)v(x) dx

z

z

×

 q

c

q





up (s)W p β (s)w(s) ds

  p

a

 + sup

W

c
 q

d q(α–)

(x)v(x) dx

z



c

×









z

up (s)W p β (s)w(s) ds + εp a

 ≤ sup



d

W q(α–) (x)v(x) dx

a
c



  p

c

z

   up (s)W p β (s)w(s) ds + εAα,β

a



 q 

d

+ sup c


up (s)W p β (s)w(s) ds

W

q(α–)

(x)v(x) dx

z

c

p

u (s)W

p β

  (s)w(s) ds

p

+ εAα,β

a

  ≤  Aα,β (c) + εAα,β .

(.)

Since the left side of (.) does not depend on ε > , substituting (.) in (.) and letting ε → , we get Pcd Tα,β Pc f   Aα,β (c)f p,w . Therefore Pcd Tα,β Pc   Aα,β (c) and we conclude that lim Pcd Tα,β Pc   lim+ Aα,β (c) = .

c→a+

c→a

Next, arguing as above we find that  Qd Tα,β f q,v =

b

d



v(x)

a

x

 q u(s)W β (s)f (s)w(s) ds

q dx (W (x) – W (s))–α

 sup Aα,β (z)f p,w . d
(.)

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Consequently, lim Qd Tα,β  ≤ lim– sup Aα,β (z) = lim– Aα,β (z) = .

d→b–

d→b d
(.)

z→b

From (.), (.) and (.) it follows that the right-hand side of (.) tends to zero as  c → a+ and d → b– . The proof is complete. Proof of Theorem . In the case b < ∞ and  < q < p < ∞ the statement of Theorem . follows from the Ando theorem and its generalizations []. Therefore, we only need to prove Theorem . in the case a = , b = ∞ and  < q < p < ∞. Necessity. Let the operator Tα,β be compact from Lp,w to Lq,v . Then the operator is bounded. Hence, by Theorem ., Bα,β < ∞. Sufficiency. Let Bα,β < ∞. Here Tα,β f = Pd Tα,β Pd f + Qd Tα,β f . Therefore Tα,β – Pd Tα,β Pd  ≤ Qd Tα,β .

(.)

Since d < ∞, the operator Pd Tα,β Pd is compact from Lp,w (, d) to Lq,v (, d), which is equivalent to its compactness from Lp,w to Lq,v . We show that the right-hand side of (.) tends to zero as d → ∞. Then the operator Tα,β is compact from Lp,w to Lq,v as the uniform limit of compact operators. Let  > ε > . To estimate Qd Tα,β f  we suppose that vε (x) = v(x) for x ∈ [d, ∞) and Bα,β (see Remark .), in view of vε (x) = εq v(x) for x ∈ (, d). Using the relations Bα,β ≈  Theorem ., we have 

∞

Qd Tα,β f  ≤ a



vε (x)

x a

 q u(s)W β (s)f (s)w(s) ds

q dx (W (x) – W (s))–α

 Bεα,β f p,w or Qd Tα,β    Bεα,β ,

(.)

where  Bεα,β =

∞  ∞



W a

q(α–)

q  p–q

(x)vε (x) dx

z



z

×

p

u (s)W

p β

 q(p–) p–q (s)w(s) ds

W

q(α–)

 p–q pq (z)vε (z) dz

.

a

Passing to the limit ε → + , from (.) it follows that ∞  ∞

 Qd Tα,β   d

W q(α–) (x)v(x) dx

q  p–q

z



z

×





up (s)W p β (s)w(s) ds a

 q(p–) p–q

W q(α–) (z)v(z) dz

 p–q pq .

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Hence, lim Qd Tα,β  = .

(.)

d→∞

Obviously, (.) implies that the right-hand side of (.) tends to zero as d → ∞. The proof is complete. 

5 Some dual results ∗ Here we consider the dual operator Kα,β defined by ∗ Kα,β g(s) =



b

s

u(s)W β (s)g(x)v(x) dx (W (x) – W (s))–α

(.)

and its mapping properties from Lp,v to Lq,w . We define A∗α,β (z) :=



z q

qβ

 q 

u (s)W (s)w(s) ds a

b

W

p (α–)

  (x)v(x) dx

p

,

z

A∗α,β = sup A∗α,β (z). z∈I

Our first main result here reads as follows.  Theorem . Let  < α < ,  < p ≤ q < –α and β ≥ . Let u be a non-increasing function ∗ on I. Then the operator Kα,β defined by (.) ∗  ≈ A∗α,β ; (i) is bounded from Lp,v to Lq,w if and only if A∗α,β < ∞ and moreover, Kα,β ∗ (ii) is compact from Lp,v to Lq,w if and only if Aα,β < ∞ and

lim A∗α,β (z) = lim– A∗α,β (z) = .

z→a+

z→b

∗ acting from Lp,v to Lq,w is conjugate to the operator Proof The operator Kα,β

 Kα,β f (x) = v(x) a

x

u(s)W β (s)f (s) ds (W (x) – W (s))–α

acting from Lq ,w–q to Lp ,v–p , which is equivalent to the action of the operator Tα,β ∗ from Lq ,w to Lp ,v . Consequently, the operator Kα,β is bounded and compact from Lp,v to Lq,w if and only if the operator Tα,β is, respectively, bounded and compact from Lq ,w ∗ to Lp ,v . Moreover, Kα,β  = Tα,β . Since, by the conditions of Theorem . we have    < q ≤ p < ∞, the statements (i) and (ii) in Theorem . follow directly from Theoα rem . and Theorem .. The proof is complete.  Similarly, in view of Theorem . we have the following.  Theorem . Let  < α < ,  < q < min{p, α– }, p >  and β ≥ . Let u be a non-increasing ∗ function on I. Then the operator Kα,β defined by (.) is bounded and compact from Lp,v to

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Lq,w if and only if B∗α,β < ∞, where

B∗α,β

 b 

b

=

W a

p (α–)

  q(p–) p–q (x)v(x) dx

q

qβ

u (s)W (s)w(s) ds

z

a

× u (s)W (s)w(s) ds q

q  p–q

z

qβ

 p–q pq .

Theorems . and . imply especially the following new information in the theory of Hardy type inequalities. Theorem . Let  < α < , β ≥  and u be a non-increasing function on I. Then 

b a

q ∗ Kα,β f (x) w(x) dx

 q

 ≤C

b

p f (x) v(x) dx

 p (.)

a

holds if and only if  (a) A∗α,β < ∞ for the case  < p ≤ q ≤ –α ,  ∗ (b) Bα,β < ∞ for the case  < q < min(p, α– ), p > . Moreover, for the best constant C in (.) it yields C ≈ A∗α,β in case (a) and C ≈ B∗α,β in case (b). Theorem . supplements the results of [].

6 Applications By applying our results in special cases we obtain both new and well-known results. Here we just consider the Riemann-Liouville, Erdelyi-Kober, and Hadamard operators mentioned in our introduction. We use the weight functions ρ and ω and consider these opα defined by erators on the forms  Iα ,  Eα,γ and H    Iα f (x) := ρ(x) Iα (f ω) (x),    Eα,γ f (x) := ρ(x) Eα,γ (f ω) (x),   α f (x) := ρ(x) Hα (f ω) (x), H where ρ and ω are almost everywhere positive functions locally summable on I with degrees q and p , respectively. The action of the operator Tα,β from Lp,v to Lq,w is equivalent to the action of the operator



 Tα,β f (x) = v q (x)

 a

x



u(s)W β (s)w p (s)f (s) ds (W (x) – W (s))–α 

from Lp to Lq . Therefore, in the case W (x) = x we have ρ(x) = v q (x), ω(x) = u(x)xβ and  Iα f (x) = ρ(x)

 a

x

ω(s)f (s) ds . (x – s)–α

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

If W (x) = xσ , σ > , then u(s)W β (s)w p (s) = u(s)s

σβ– σ –  p

= u(s)sσ γ +σ – , where γ = β – σσ– . p

 q

Consequently, ρ(x) = v (x), ω(s) = u(s) and  Eα,γ f (x) = ρ(x)



x

a

ω(s)sσ γ +σ – f (s) ds . (xσ – sσ )–α 



Now, we assume that a >  and W (x) = ln ax . Then u(s)W β (s)w p (s) = u(s)(ln as )β ( as ) p =









a p u(s)s p (ln as )β s . In this case ρ(x) = v q (x), ω(s) = u(s)s p (ln as )β and α f (x) = ρ(x) H



x

a

ω(s)f (s) ds . s(ln xs )–α

Below we present statements for boundedness and compactness of the operators  Iα , α from Lp to Lq . These statements are consequences of Theorems ., ., .,  Eα,γ and H and .. We define  Aα (z) :=

b

 q 

z

 

p

ω (s) ds

dx

z

p

,

b

ρ(x)xα– q dx

:= a

p   p–q

z

p

ω (s) ds

z

Aα := sup Aα (z), z∈I

a

 b  Bα

ρ(x)x

 α– q

 p(q–) p–q

 p–q pq

p

ω (z) dz

.

a

Corollary . Let  < α < , β ≥  and ω(s) = u(s)sβ . Let u be a non-increasing function on I. Then: Iα is bounded from Lp to Lq if and only if Aα < ∞ and, (i) for α < p ≤ q < ∞ the operator  moreover,  Iα  ≈ Aα ; it is compact from Lp to Lq if and only if Aα < ∞ and  limz→a+ Aα (z) = limz→b– Aα (z) = ; Iα is bounded (compact if b < ∞ or b = ∞ (ii) for  < q < p < ∞ and p > α the operator  and  ≤ q < p < ∞) from Lp to Lq if and only if Bα < ∞. Remark . Corollary . generalizes the results of Theorems  and ,  and  in [], where the case β =  was considered. Even in this case the results of Corollary . are different (and in a sense simpler to use) than those in [], because in [] the statements are given in terms of two expressions while here we only need one condition. We define  Aα,γ (z) :=

b

ρ(x)xσ (α–) q dx

 q 

z

z



ω(s)sσ γ +σ – p ds

  p

,

a

Aα,γ := sup Aα,γ (z), z∈I

 b  Bα,γ

:= a

b

ρ(x)xσ (α–) q dx

p  p–q

z

 × a

z



ω(s)sσ γ +σ – p ds

 p(p–) p–q



ω(z)zσ γ +σ – p dz

 p–q pq .

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Corollary . Let  < α < , σ > , β ≥  and γ = β – σσ– . Let ω be a non-increasing p function on I. Then: Eα,γ is bounded from Lp to Lq if and only if Aα,γ < ∞ (i) for α < p ≤ q < ∞ the operator   and, moreover,  Eα,γ  ≈ Aα,γ ; it is compact from Lp to Lq if and only if Aα,γ < ∞ and limz→a+ Aα,γ (z) = limz→b– Aα,γ (z) = ; Eα,γ is bounded (compact if b < ∞ or b = ∞ (ii) for  < q < p < ∞ and p > α the operator  and  ≤ q < p < ∞) from Lp to Lq if and only if Bα,γ < ∞. α we define To formulate statements corresponding to the operator H Aα (z) := Bα :=

 α– q  q  z  b  

p p

ρ(x) ln x

dx ω (s) ds ,

a z a

Aα := sup Aα (z), z∈I

p  α– q  p–q  b  b  p(q–)  p–q  z p–q pq

p p

ρ(x) ln x

dx ω (s) ds ω (z) dz .

a a z a 

Corollary . Let a > ,  < α < , β ≥  and ω(s) = u(s)s p (ln as )β . Let u be a nonincreasing function on I. Then: α is bounded from Lp to Lq if and only if A < ∞ (i) for α < p ≤ q < ∞ the operator H α α  ≈ A ; it is compact from Lp to Lq if and only if A < ∞ and and, moreover, H α α limz→a+ Aα (z) = limz→b– Aα (z) = ; α is bounded (compact if b < ∞ or b = ∞ (ii) for  < q < p < ∞ and p > α the operator H and  ≤ q < p < ∞) from Lp to Lq if and only if Bα < ∞. Finally, we consider the operator  Iα∗ g(s) = ρ(s)[Iα∗ (gω)](s), s ∈ I, acting from Lp to Lq , where Iα∗ is the Weyl operator Iα∗ g(s) =



b s

g(x) dx . (x – s)–α

∗ from Lp,v to Lq,w is equivalent to the action of the operator The action of the operator Kα,β



∗ α,β K g(s) = w q (s)u(s)W β (s)



b

s



v p (x)g(x) dx (W (x) – W (s))–α

from Lp to Lq . Therefore, when W (x) = x we have 

ω(x) = v p (x),

ρ(s) = u(s)sβ ,   Iα∗ g(s) = ρ(s)

b

s

ω(x)g(x) dx . (x – s)–α

We define A∗α (z) := B∗α :=



z

ρ q (s) ds a

b



ω(x)xα– p dx

  p

,

z

b



ω(x)xα– p dx

A∗α := sup A∗α (z), z∈I

z

 b  a

 q 

  q(p–) p–q a

z

ρ q (s) ds

q  p–q

ρ q (z) dz

 p–q pq .

Abylayeva et al. Journal of Inequalities and Applications (2016) 2016:324

Page 18 of 18

From Theorems . and . we have the following result. Corollary . Let  < α < , β ≥  and ρ(s) = u(s)sβ . Let u be a non-increasing function on I. Then:  the operator  Iα∗ is bounded from Lp to Lq if and only if A∗α < ∞ (i) for  < p ≤ q < –α ∗ and, moreover,  Iα  ≈ Aα ; it is compact from Lp to Lq if and only if A∗α < ∞ and limz→a+ A∗α (z) = limz→b– A∗α (z) = ;  )} < ∞ and p >  the operator  Iα∗ is bounded (compact) from Lp (ii) for  < q < {min(p, –α to Lq if and only if B∗α < ∞. Remark . From the results in Corollary .-. follow some corresponding Hardy type inequalities, which seem to be new even as they are special cases of our Theorems . and ..

Competing interests The authors declare that they have no competing interests. Authors’ contributions All the authors contributed equally and significantly in writing this paper. All the authors read and approved the final manuscript. Author details 1 Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, 2 Satpaev St., Astana, 010008, Kazakhstan. 2 Department of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå, 97187, Sweden. 3 UiT, The Artic University of Norway, Tromsø, Norway. Acknowledgements We thank both referees for very good remarks, which have helped us to improve the final version of this paper. Received: 2 September 2016 Accepted: 29 November 2016 References 1. Samko, SG, Kilbas, AA, Marichev, OI: Integrals and Derivatives of Fractional Order and Some of Their Applications. Science and Technology, Minsk (1987) (in Russian) 2. Stepanov, VD: Two-weighted estimates of Riemann-Liouville integrals. Proc. USSR Acad. Sci., Izv. AN SSSR Ser. Mat. 54(3), 645-655 (1990) (in Russian) 3. Stepanov, VD: On a weighted inequality of Hardy type for fractional Riemann-Liouville integrals. Sib. Math. J. 31(3), 186-197 (1990) (in Russian) 4. Andersen, KF, Sawyer, ET: Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators. Trans. Am. Math. Soc. 308, 547-558 (1988) 5. Farsani, SM: On the boundedness and compactness of the fractional Riemann-Liouville operators. Sib. Mat. Zh. 54(2), 468-479 (2013) (in Russian); translation in Sib. Math. J. 54(2), 368-378 (2013) 6. Lorente, M: A characterization of two weighted norm inequalities for one-sided operators of fractional type. Can. J. Math. 49, 1010-1033 (1997) 7. Meskhi, A: Solution of some weight problems for the Riemann-Liouville and Weyl operators. Georgian Math. J. 106, 727-733 (1989) 8. Prokhorov, DV: On the boundedness and compactness of a class of integral operators. J. Lond. Math. Soc. 61(2), 617-628 (2000) 9. Prokhorov, DV, Stepanov, VD: Weighted estimates for the Riemann-Liouville operators and applications. Proc. Math. Inst. RAS Steklov V.A. 248, 289-312 (2003) (in Russian) 10. Oinarov, R: Two-sided estimates of the norms for certain classes of integral operators. Proc. Math. Inst. RAS Steklov V.A. 204, 240-250 (1993) (in Russian) 11. Abylayeva, AM, Kaskirbaeva, D: Boundedness and compactness of fractional integral operator type Holmgren in weighted Lebesgue spaces. Eurasian Math. J. 2, 75-86 (2007) (in Russian) 12. Oinarov, R, Abylayeva, AM: Criteria for the boundedness of a class of fractional integral operators. Math. J. 4(2(12)), 5-14 (2004) (in Russian) 13. Kufner, A, Maligranda, L, Persson, LE: The Hardy Inequality. About Its History and Some Related Results. Vydavatelsky Servis Publishing House, Pilsen (2007) 14. Kufner, A, Persson, L-E: Weighted Inequalities of Hardy Type. World Scientific, Singapore (2003) 15. Sinnamon, G, Stepanov, VD: The weighted Hardy inequality: new proofs and the case p = 1. J. Lond. Math. Soc. 54, 89-101 (1996) 16. Kantarovich, LV, Akilov, GR: Functional Analysis. Nauka, Moscow (1977) (in Russian) 17. Krasnosel’skii, MA, Zabreiko, PP, Pustilnik, EN, Sobolewski, PE: Integral Operators in Spaces of Summable Functions. Nauka, Moscow (1966) (in Russian) 18. Abylayeva, AM: Boundedness, compactness for a class of fractional integral operators of Weyl type. Eurasian Math. J. 7(1), 9-27 (2016)