R E S E A R C H
Open Access
Boundedness and compactness of a class
of Hardy type operators
Akbota M Abylayeva
1, Ryskul Oinarov
1and Lars-Erik Persson
2,3**Correspondence: [email protected] 2Department of Engineering
Sciences and Mathematics, Luleå University of Technology, Luleå, 97187, Sweden
3UiT, The Artic University of Norway,
Tromsø, Norway
Full list of author information is available at the end of the article
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
LetI= (a,b), ≤a<b≤ ∞. Letvandube almost everywhere positive functions, which are locally integrable on the intervalI.
Let <p<∞andp+p = . Denote byLp,v≡Lp(v,I) the set of all functionsfmeasurable
onIsuch thatfp,v:= (ab|f(x)|pv(x)dx)p<∞.
LetWbe a non-negative, strictly increasing and locally absolutely continuous function onI. Suppose thatdWdx(x)=w(x), a.e.x∈I.
We consider the Hardy type operatorTα,βdefined by
Tα,βf(x) :=
x
a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α , x∈I. (.)
Whenu≡ and β= the operatorTα,β is called the fractional integral operator of
a functionf with respect to a functionW ([], p.). Whenu≡ and W(x) =x the operator (.) becomes the Riemann-Liouville operatorIαdefined by
Iαf(x) :=
x
a
sβf(s)ds
(x–s)–α. (.)
Whenu≡ andW(x)≡lnxa,a> , this operator is the Hadamard operatorHαdefined
by
Hαf(x) :=
x
a
(lns a)
βf(s)ds
s(lnxs)–α .
Moreover, whenu≡ andW(x) =xσ,σ> , we get the operatorE
α,βof Erdelyi-Kober
type ([], p.) defined by
Eα,βf(x) :=σ
x
a
f(s)sσβ+σ–ds
(xσ –sσ)–α .
There are a lot of works devoted to the mapping properties of the Riemann-Liouville operatorIα. Two-weighted estimates of the operatorIα of the orderα > 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 oper-atorTα,β defined by (.) fromLp,wtoLq,vwhen <α< . Whenα> the results follow
from the results in [].
The operatorTα,β was studied in [] and [] whenu≡,β= andu≡,β > –p,
respectively.
Due to the non-negativity and monotone increase of the function W the limit
limx→a+W(x)≡W(a)≥ exists.
We also consider the Hardy type operatorTα,βdefined by
Tα,βf(x) :=
x
a
u(s)Wβ(s)f(s)w(s)ds (W(x) –W(s))–α
, x∈I,
whereW(x) =W(x) –W(a).
Since we also suppose thatβ≥, forf≥ we haveTα,βf(x)≈Tα,βf(x) +W(a)Tα,f(x),
where the equivalence constants do not depend on xandf. Therefore, without loss of generality, we can assume thatW(a) = . For short writing we denote byKthe norm of a linear operatorKacting 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 operatorTα,β, including the corresponding Hardy type inequalities, can be found in
Section . The main results about the compactness are presented in Section . Moreover, in Section some similar results for the dual operatorT
α,βare given. Finally, Section is
reserved for some applications (both new and well-known results).
Conventions The indeterminate form · ∞is assumed to be zero. The relationsAB
andAB, respectively, meanA≤cBandA≥cB, where a positive constantccan be dependent only on the parameters p,q, α andβ. The relationA≈Bis interpreted as ABA. The set of all integers is denoted byZ. Moreover,χ(c,a)(·) is the characteristic
function of the interval (c,a)⊂I.
2 Auxiliary statements
Together with the operator (.) we consider the Hardy type operatorHα,βdefined by
Hα,βf(x) =
W–α(x)
x
a
u(s)Wβ(s)f(s)w(s)ds. (.)
It is easy to see that forf ≥ we have
Tα,βf(x)≥Hα,βf(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,βbe defined by(.).Then the
in-equality
b
a
Hα,βf(x)
q
v(x)dx
q ≤C
b
a
f(x)pw(x)dx
p
(.)
holds if and only if
Aα,β=sup
z∈I z
a
up(s)Wpβ(s)w(s)ds
p b
z
Wq(α–)(x)v(x)dx
q
<∞.
Moreover,C≈Aα,β.
Lemma . Let <q<p<∞,p> and let the operator Hα,βbe defined by(.).Then the
inequality(.)holds if and only if
Bα,β=
b
a b
z
Wq(α–)(x)v(x)dx p
p–q
× z
a
up(s)Wpβ(s)w(s)ds
p(q–)
p–q
up(z)Wpβ(z)w(z)dz
p–q pq
<∞.
Moreover,C≈Bα,β.
Remark . In the case <q<p<∞,p> it is well known and easy to prove that the
valueBα,βis equivalent to the value
Bα,β=
b
a b
z
Wq(α–)(x)v(x)dx
q p–q
× z a
up(s)Wpβ(s)w(s)ds
q(p–)
p–q
Wq(α–)(z)v(z)dz
p–q pq
3 Boundedness of the operatorTα,β
The main results in this section read as follows.
Theorem . Let <α< , <p≤q<∞andβ≥.Let u be a non-increasing function on I. Then the operator Tα,β defined by(.) is bounded from Lp,w to Lq,vif and only if
Aα,β<∞.Moreover,Tα,β ≈Aα,β.
Theorem . Let <α< , <q<p<∞,p> α andβ≥.Let u be a non-increasing function on I.Then the operator Tα,βis bounded from Lp,wto Lq,vif 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 in-equality
b
a
Tα,βf(x)
q
v(x)dx
q ≤C
b
a
f(x)pw(x)dx
p
(.)
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 operatorTα,βbe bounded fromLp,wtoLq,v. Then,
in view of (.), the operatorHα,βis bounded fromLp,wtoLq,v, andTα,β ≥ Hα,β.
Con-sequently, by Lemma . we haveAα,β<∞and
Tα,β Aα,β. (.)
Sufficiency. Since the functionWis continuous and strictly increasing onIandW(a) = , for anyk∈Z we can definexk:=sup{x:W(x)≤k,x∈I}. We obtain a sequence of
points{xk}k>–∞such that <xk≤xk+,∀k∈Z, and ifxk<b, thenW(xk) = k, k≤W(x)≤
k+forx
k≤x≤xk+, xk
xk–w(s)ds=
k–, and ifx
k+=b, then xk+
xk w(s)ds≤
k. These facts
will be used below without reminders. We assume thatIk= [xk,xk+),k∈Z,Z={k:k∈
Z,Ik=∅}. ThenZ⊆ZandI= k∈ZIk= k∈ZIk. SinceIk=∅,∀k∈Z\Z, and integrals over these intervals are equal to zero, in the sequel, without loss of generality, we can suppose thatZ=Z.
LetAα,β<∞. We need to prove that the inequality
Tα,βfq,vAα,βfp,w, f∈Lp,w, (.)
holds, which meansTα,β Aα,βand, together with (.), this gives
Letf ≥. Using the relationI= kIk, we have
Tα,βfqq,v =
k xk+ xk v(x) x a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q dx = k xk+ xk v(x) xk– a + x
xk–
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q dx k xk+ xk v(x) xk– a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q dx + k xk+ xk v(x) x xk–
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q
dx:=J+J. (.)
We now estimateJandJseparately. Using the monotonicity ofW we find that
J = k xk+ xk v(x) xk– a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q
dx
≤ k
xk+
xk
v(x)
xk–
a
u(s)Wβ(s)f(s)w(s)ds
(W(xk) –W(xk–))–α q
dx
= q(–α)
k xk+ xk v(x) k+
q(–α) xk–
a
u(s)Wβ(s)f(s)w(s)ds
q dx k xk+ xk
v(x)Wq(α–)(x)
x
a
u(s)Wβ(s)f(s)w(s)ds
q
dx≤ Hα,βfqq,v.
Hence, by Lemma . we get
JAqα,βfqp,w. (.)
Moreover, by using Hölder’s inequality and the fact that the function u is increasing, we obtain
J = k xk+ xk v(x) x xk–
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
q
dx
≤ k
xk+
xk
v(x)
x
xk–
fp(s)w(s)ds
q
p x
xk–
up(s)Wpβ(s)w(s)ds
(W(x) –W(s))p(–α) q p dx ≤ k xk+ xk–
fp(s)w(s)ds
q p
uq(xk–)
× xk+ xk v(x) x a
Wpβ(s)w(s)ds
(W(x) –W(s))p(–α) q
p
dx. (.)
A change of variablesW(s) =W(x)tin the last integral, implies that
x
a
Wpβ(s)w(s)ds
(W(x) –W(s))p(–α) =
Wpβ+(x)
Wp(–α)(x)
Sinceβ≥,α>p, the Euler beta functiontpβ(–t)p(α–)dtconverges. Consequently,
from (.) and (.) it follows that
J
k
xk+
xk–
fp(s)w(s)ds
q p
uq(xk–) xk+
xk
v(x)W
q
p(pβ+)dx
Wq(–α)(x)
≤ k
xk+
xk–
fp(s)w(s)ds
q p
uq(xk–)W q p(pβ+)(x
k+) xk+
xk
v(x)Wq(α–)(x)dx
= (qβ+
q
p)
k
xk+
xk–
fp(s)w(s)ds
q p
×uq(xk–)W q p(pβ+)(x
k–) xk+
xk
v(x)Wq(α–)(x)dx
k
xk+
xk–
fp(s)w(s)ds
q p
×uq(xk–) xk–
a
Wpβ(s)w(s)ds
q p xk+
xk
v(x)Wq(α–)(x)dx
≤ k
xk+
xk–
fp(s)w(s)ds
q p
× xk a
up(s)Wpβ(s)w(s)ds
q p b
xk
v(x)Wq(α–)(x)dx
≤Aqα,β
k
xk+
xk–
fp(s)w(s)ds
q p
≤Aqα,β
k xk+
xk–
fp(s)w(s)ds
q p
Aqα,βfqp,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. LetBα,β<∞. If we show thatTα,β Bα,β, then this fact and (.) imply
thatTα,β ≈Bα,β. Next, we use relation (.). For the estimateJwe have obtainedJ Hα,βfqq,v. Hence, by Lemma . we obtain
JBqα,βfqp,w. (.)
Moreover, from Theorem ., obvious estimates and Hölder’s inequality it follows that
J
k
xk+
xk–
fp(s)w(s)ds
q p
×uq(xk–)W q p(pβ+)(x
k+) xk+
xk
= (qβ+
q
p)pβ+– q
p
k
xk+
xk–
fp(s)w(s)ds
q p
×uq(xk–)
(pβ+)(k–)– (pβ+)(k–)pq xk+
xk
v(x)Wq(α–)(x)dx
k
xk+
xk–
fp(s)w(s)ds
q p
×uq(xk–) xk–
xk–
Wpβ(s)w(s)ds
q p xk+
xk
v(x)Wq(α–)(x)dx
k
xk+
xk–
fp(s)w(s)ds
q p
× xk– xk–
up(s)Wpβ(s)w(s)ds
q p xk+
xk
v(x)Wq(α–)(x)dx
we apply Hölder’s inequality with the conjugate exponents p q,
p p–q
≤J
p–q p
k xk+
xk–
fp(s)w(s)ds
q p
J
p–q p
fqp,w, (.)
where
J=
k xk–
xk–
up(s)Wpβ(s)w(s)ds
q(p–)
p–q xk+
xk
v(x)Wq(α–)(x)dx
p p–q
.
To estimateJwe use the relation
xk–
xk–
up(s)Wpβ(s)w(s)ds
q(p–)
p–q
xk–
xk–
t
xk–
up(s)Wpβ(s)w(s)ds
p(q–)
p–q
up(t)Wpβ(t)w(t)dt.
Then
J
k xk–
xk–
t
xk–
up(s)Wpβ(s)w(s)ds
p(q–)
p–q
×up(t)Wpβ(t)w(t)dt
xk+
xk
v(x)Wq(α–)(x)dx
p p–q
≤ k
xk–
xk–
t
a
up(s)Wpβ(s)w(s)ds
p(q–)
p–q
× b
t
v(x)Wq(α–)(x)dx
p p–q
up(t)Wpβ(t)w(t)dt
≤ B
qp p–q
By substitution of (.) in (.) we obtain
JBqα,βfqp,w. (.)
Now, by combining (.), (.) and (.) we obtain
Tα,βfq,vBα,βfp,w.
Consequently,Tα,βq,vBα,β. The proof is complete.
4 Compactness of the operatorTα,β
The main results in this section read as follows.
Theorem . Let <α< ,α <p≤q<∞andβ≥.Let u be a non-increasing function on I.Then the operator Tα,βis compact from Lp,wto Lq,vif and only if Aα,β<∞and
lim
z→a+Aα,β(z) =z→blim–Aα,β(z) = ,
where
Aα,β(z) =
z
a
up(s)Wpβ(s)w(s)ds
p b
z
Wq(α–)(x)v(x)dx
q
.
Theorem . Let <α< ,p>α andβ≥.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,wto Lq,vif and only if Bα,β<∞.
Proof of Theorem. Necessity. Let the operatorTα,βbe compact fromLp,wtoLq,v. Then
it is bounded and consequently, by Theorem ., we haveAα,β<∞. First we need to show
thatlimz→a+Aα,β(z) = . Consider the family of functions{ft}t∈I, where
ft(x) =χ(a,t)(x)up –
(x)W(p–)β(x)
t
a
up(s)Wpβ(s)w(s)ds
–
p
, x∈I. (.)
We note that
b
a
ft(x)pw(x)dx
p
=
t
a
ft(x)pw(x)dx
p
=
t
a
up(s)Wpβ(s)w(s)ds
–
p
× t
a
up(s)Wpβ(s)w(s)ds
p
= . (.)
Next we show that the family of functions{ft}t∈I defined by (.) converges weakly to zero inLp,w. Letg∈Lp,w–p= (Lp,w)∗. Then, by Hölder’s inequality and (.), we find that
b
a
ft(x)g(x)dx≤ t
a ft(x)
p
w(x)dx
p t
a
g(s)pw–p(s)ds
p
=
t
a
g(s)pw–p(s)ds
p
Sinceg∈Lp,w–p, the last integral in (.) converges to zero ast→a+, which means
weak convergence of the family of functions{ft}to zero ast→a+. Therefore, from the
compactness of the operatorTα,βfromLp,wtoLq,vit follows that
lim
t→a+Tα,βftq,v= . (.)
Moreover,
Tα,βftqq,v=
b
a
v(x)
x
a
u(s)Wβ(s)f
t(s)w(s)ds
(W(x) –W(s))–α
q
dx
≥ b t
v(x)
t
a
u(s)Wβ(s)f
t(s)w(s)ds
(W(x) –W(s))–α
q
dx
≥ b
t
v(x)dx Wq(–α)(x)
t
a
up(s)Wpβ(s)w(s)ds
–q p
× t
a
up(s)Wpβ(s)w(s)ds
q
=Aqα,β(t). (.)
From (.) and (.) it follows thatlimt→a+Aα,β(t) = . Now, we show thatlimt→b–Aα,β(t) = .
From the compactness of the operatorTα,βfromLp,wtoLq,vcompactness of the
conju-gate operator follows:
Tα∗,βg(s) =u(s)Wp(s)w(s)
b
s
g(x)dx (W(x) –W(s))–α
fromLq,v–q toLp,w–p.
Fort∈Iwe introduce the family{gt}t∈Iof functions:
gt(x) =χ[t,b)(x) b
t
Wq(α–)(x)v(x)dx
–
q
W(q–)(α–)(x)v(x). (.)
The family{gt}t∈Iof functions defined by (.) is correctly defined, since due to condition Aα,β<∞the involving integrals are finite. We show that for allt∈I the functionsgt ∈
Lq,v–q converge weakly to zero ast→b–.
Indeed,
gtq,v–q = b
t gt(x)
q
v–q(x)dx
q
=
b
t
Wq(α–)(x)v(x)dx
–
q b
t
W(q–)(α–)(x)v(x)qv–q(x)dx
q
=
b
t
Wq(α–)(x)v(x)dx –
q b
t
Wq(α–)(x)v(x)dx
q
By using (.) withf ∈Lq,v= (Lq,v–q)∗we obtain
b
a
gs(x)f(x)dx≤ b
t gt(x)
q
v–
q q(x)dx
q b
t
f(x)qv(x)dx
q
≤ gtq,v–q b
t
f(x)qv(x)dx
q
=
b
t
f(x)qv(x)dx
q
.
Sincef ∈Lq,v, the last integral tends to zero ast→b–, which gives the weak convergence
to zero of{gt}t∈IinLq,v–q ast→b–. By compactness ofTα∗,β:Lq,v–q→Lp,w–p it follows
that
lim s→b–T
∗
α,βgtp,w–p = . (.)
Furthermore, we note that
Tα∗,βgtp,w–p = b
a
u(s)Wβ(s)w(s)p
× b s
gt(x)dx
(W(x) –W(s))–α
p
w–p(s)ds
p
≥ t
a
up(s)Wpβ(s)w(s)
b
t
gt(x)dx
(W(x) –W(s))–α
p
ds
p
≥ t a
up(s)Wpβ(s)w(s)ds
p b
t
Wq(α–)(x)v(x)dx
–
q
× b
t
W(q–)(α–)(x)v(x)dx
W–α(x)
q
=Aα,β(t).
Hence, according to (.) we havelims→b–Aα,β(s) = . The proof of the necessity is com-plete.
Sufficiency. Fora<c<d<bwe define
Pcf :=χ(a,c]f, Pcdf:=χ(c,d]f, Qdf :=χ(d,b)f.
Then
f =Pcf+Pcdf+Qdf
and sincePcTα,βPcd≡,PcTα,βQd≡,PcdTα,βQd≡, we have
Tα,βf =PcdTα,βPcdf +PcTα,βPcf+PcdTα,βPcf +QdTα,βf. (.)
We show that the operatorPcdTα,βPcdis compact fromLp,wtoLq,v. SincePcdTα,βPcdf(x) =
for x∈I\(c,d), it is enough to show that the operatorPcdTα,βPcd is compact from
Lp,w(c,d) toLq,v(c,d). This, in turn, is equivalent to compactness of the operator
Tf(x) =
d
c
fromLp(c,d) toLq(c,d) with the kernel
K(x,s) =u(s)W
β(s)vq(x)χ
(c,d)(x–s)w
p(s)
(W(x) –W(s))–α .
Let{xk}k∈Zbe the sequence of points defined in the proof of Theorem .. There are points xi,xn+,xi<xn+such thatxi≤c<xi+,xn<d≤xn+. We assume that the numbersc,dare
chosen so thatxi+<xn. Similarly to obtaining estimates ofJ andJin Theorem ., we
have
d
c d
c
K(x,s)pds
q p
dx
=
d
c
v(x)
x
c
up(s)Wpβ(s)w(s)ds
(W(x) –W(s))p(–α) q
p
dx
≤ n
k=i xk+
xk
v(x)
xk–
a
+
x
xk–
up(s)Wpβ(s)w(s)ds
(W(x) –W(s))p(–α) q
p
dx
≤μ(n–i+ )Aqα,β<∞,
where the constantμdoes not depend oni,n.
Therefore, on the basis of the Kantarovich condition ([], p.), the operatorTis com-pact fromLp(c,d) toLq(c,d), which is equivalent to compactness of the operatorPcdTα,βPcd
fromLp,wtoLq,v.
From (.) it follows that
Tα,β–PcdTα,βPcd ≤ PcTα,βPc+PcdTα,βPc+QdTα,β. (.)
We will show that the right-hand side of (.) tends to zero atc→aandd→b. Then the operatorTα,βas the uniform limit of compact operators is compact fromLp,wtoLq,v.
By using Theorem . we find that
PcTα,βPcfq,v = c
a
v(x)
x
a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
qdx
q
sup a<z<c
z
a
up(s)Wpβ(s)w(s)ds
p
× c
z
v(x)Wq(α–)(x)dx
q fp,w
≤ sup a<z<c
Aα,β(z)fp,w.
Consequently,PcTα,βPc supa<z<cAα,β(z). Hence,
lim
c→a+PcTα,βPc c→alim+sup
a<z<c
Aα,β(z) = lim
z→a+Aα,β(z) = . (.)
To estimatePcdTα,βPcwe assume thatvε(x) =v(x) forx∈(c,d] andvε(x) =εqv(x) for
the functionuεis non-increasing onI. Then, according to Theorem ., we obtain
PcdTα,βPcq,v =
d
c
v(x)
c
a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
qdx
q
≤ d
a
vε(x)
x
a
uε(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
qdx
q
Aε
α,βfp,w, (.)
where
Aεα,β= sup
a<z<d d
z
Wq(α–)(x)vε(x)dx
q z
a
upε(s)Wp
β
(s)w(s)ds
p
.
We estimate the expressionAεα,βfrom the above as follows:
Aε
α,β≤ sup
a<z<c d
c
Wq(α–)(x)v(x)dx+εq c
z
Wq(α–)(x)v(x)dx
q
× z a
up(s)Wpβ(s)w(s)ds
p
+sup c<z<d
d
z
Wq(α–)(x)v(x)dx
q
× c a
up(s)Wpβ(s)w(s)ds+εp
z
c
up(s)Wpβ(s)w(s)ds
p
≤ sup a<z<c
d
c
Wq(α–)(x)v(x)dx
z
a
up(s)Wpβ(s)w(s)ds
+εAα,β
+sup c<z<d
d
z
Wq(α–)(x)v(x)dx
q c
a
up(s)Wpβ(s)w(s)ds
p
+εAα,β
≤Aα,β(c) +εAα,β
. (.)
Since the left side of (.) does not depend onε> , substituting (.) in (.) and lettingε→, we get
PcdTα,βPcf Aα,β(c)fp,w.
ThereforePcdTα,βPc Aα,β(c) and we conclude that
lim
c→a+PcdTα,βPc c→alim+Aα,β(c) = . (.)
Next, arguing as above we find that
QdTα,βfq,v =
b
d
v(x)
x
a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
qdx
q
sup d<z<b
Consequently,
lim
d→b–QdTα,β ≤d→blim– sup
d<z<b
Aα,β(z) = lim
z→b–Aα,β(z) = . (.)
From (.), (.) and (.) it follows that the right-hand side of (.) tends to zero as
c→a+andd→b–. The proof is complete.
Proof of Theorem. In the caseb<∞and <q<p<∞the statement of Theorem . follows from the Ando theorem and its generalizations []. Therefore, we only need to prove Theorem . in the casea= ,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. LetBα,β<∞. HereTα,βf =PdTα,βPdf +QdTα,βf. Therefore
Tα,β–PdTα,βPd ≤ QdTα,β. (.)
Sinced<∞, the operatorPdTα,βPdis compact fromLp,w(,d) toLq,v(,d), which is
equiv-alent to its compactness fromLp,wtoLq,v. We show that the right-hand side of (.) tends
to zero asd→ ∞. Then the operatorTα,βis compact fromLp,wtoLq,vas the uniform limit
of compact operators.
Let >ε> . To estimateQdTα,βfwe suppose thatvε(x) =v(x) forx∈[d,∞) and
vε(x) =εqv(x) forx∈(,d). Using the relationsBα,β≈Bα,β (see Remark .), in view of
Theorem ., we have
QdTα,βf ≤
∞
a
vε(x)
x
a
u(s)Wβ(s)f(s)w(s)ds
(W(x) –W(s))–α
qdx
q
Bεα,βfp,w
or
QdTα,β Bεα,β, (.)
where
Bεα,β=
∞
a
∞
z
Wq(α–)(x)vε(x)dx
q p–q
× z a
up(s)Wpβ(s)w(s)ds
q(p–)
p–q
Wq(α–)(z)vε(z)dz
p–q pq
.
Passing to the limitε→+, from (.) it follows that
QdTα,β
∞
d
∞
z
Wq(α–)(x)v(x)dx q
p–q
× z
a
up(s)Wpβ(s)w(s)ds
q(p–)
p–q
Wq(α–)(z)v(z)dz p–q
pq
Hence,
lim
d→∞QdTα,β= . (.)
Obviously, (.) implies that the right-hand side of (.) tends to zero asd→ ∞. The
proof is complete.
5 Some dual results
Here we consider the dual operatorKα∗,βdefined by
Kα∗,βg(s) =
b
s
u(s)Wβ(s)g(x)v(x)dx
(W(x) –W(s))–α (.)
and its mapping properties fromLp,vtoLq,w.
We define
A∗α,β(z) :=
z
a
uq(s)Wqβ(s)w(s)ds
q b
z
Wp(α–)(x)v(x)dx
p
,
A∗α,β=sup
z∈I A ∗
α,β(z).
Our first main result here reads as follows.
Theorem . Let <α< , <p≤q<–αandβ≥.Let u be a non-increasing function on I.Then the operator Kα∗,βdefined by(.)
(i) is bounded fromLp,vtoLq,wif and only ifA∗α,β<∞and moreover,Kα∗,β ≈A∗α,β; (ii) is compact fromLp,vtoLq,wif and only ifA∗α,β<∞and
lim z→a+A
∗
α,β(z) =z→blim–A
∗
α,β(z) = .
Proof The operatorKα∗,βacting fromLp,vtoLq,wis conjugate to the operator
Kα,βf(x) =v(x)
x
a
u(s)Wβ(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 operatorKα∗,β is bounded and compact from Lp,v
toLq,wif and only if the operatorTα,β is, respectively, bounded and compact fromLq,w
to Lp,v. Moreover, Kα∗,β=Tα,β. Since, by the conditions of Theorem . we have
α <q≤p<∞, the statements (i) and (ii) in Theorem . follow directly from
Theo-rem . and TheoTheo-rem .. The proof is complete.
Similarly, in view of Theorem . we have the following.
Lq,wif and only if B∗α,β<∞,where
B∗α,β=
b
a b
z
Wp(α–)(x)v(x)dx
q(p–)
p–q z
a
uq(s)Wqβ(s)w(s)ds
q p–q
×uq(s)Wqβ(s)w(s)ds
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
Kα∗,βf(x)qw(x)dx
q ≤C
b
a
f(x)pv(x)dx
p
(.)
holds if and only if
(a) A∗α,β<∞for the case <p≤q≤–α,
(b) B∗α,β<∞for the case <q<min(p,α–),p> .
Moreover,for the best constant C in(.)it yields C≈A∗α,βin 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 men-tioned in our introduction. We use the weight functionsρandωand consider these op-erators on the formsIα,Eα,γ andHαdefined by
Iαf(x) :=ρ(x)
Iα(fω)
(x),
Eα,γf(x) :=ρ(x)
Eα,γ(fω)
(x),
Hαf(x) :=ρ(x)
Hα(fω)
(x),
whereρandωare almost everywhere positive functions locally summable onIwith de-greesqandp, respectively.
The action of the operatorTα,βfromLp,vtoLq,wis equivalent to the action of the operator
Tα,βf(x) =v
q(x) x
a
u(s)Wβ(s)wp(s)f(s)ds
(W(x) –W(s))–α
fromLptoLq. Therefore, in the caseW(x) =xwe haveρ(x) =v
q(x),ω(x) =u(x)xβand
Iαf(x) =ρ(x)
x
a
IfW(x) =xσ,σ> , thenu(s)Wβ(s)wp(s) =u(s)sσβ–
σ–
p =u(s)sσ γ+σ–, whereγ =β–σ–
σp.
Consequently,ρ(x) =vq(x),ω(s) =u(s) and Eα,γf(x) =ρ(x)
x
a
ω(s)sσ γ+σ–f(s)ds
(xσ–sσ)–α .
Now, we assume thata> andW(x) =lnx
a. Thenu(s)W
β(s)wp(s) =u(s)(lns a)
β(a s)
p =
a
pu(s)sp(lns a)
β
s. In this caseρ(x) =v
q(x),ω(s) =u(s)sp(ln s a)
βand
Hαf(x) =ρ(x)
x
a
ω(s)f(s)ds s(lnxs)–α .
Below we present statements for boundedness and compactness of the operatorsIα,
Eα,γ andHαfromLptoLq. These statements are consequences of Theorems ., ., .,
and .. We define
Aα(z) :=
b
z
ρ(x)xα–qdx
q z
a
ωp(s)ds
p
, Aα:=sup
z∈I A
α(z),
Bα:=
b
a b
z
ρ(x)xα–q
dx
p p–q z
a
ωp(s)ds
p(q–)
p–q
ωp(z)dz
p–q pq
.
Corollary . Let <α< ,β≥andω(s) =u(s)sβ.Let u be a non-increasing function
on I.Then:
(i) for α <p≤q<∞the operatorIαis bounded fromLptoLqif and only ifAα<∞and, moreover,Iα ≈Aα;it is compact fromLptoLqif and only ifAα<∞and
limz→a+Aα(z) =limz→b–Aα(z) = ;
(ii) for <q<p<∞andp>α the operatorIαis bounded(compact ifb<∞orb=∞ and≤q<p<∞)fromLptoLqif and only ifBα<∞.
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
Aα,γ(z) :=
b
z
ρ(x)xσ(α–)qdx
q z
a
ω(s)sσ γ+σ–pds
p
,
Aα,γ:=sup
z∈I A
α,γ(z),
Bα,γ:=
b
a b
z
ρ(x)xσ(α–)qdx
p p–q
× z a
ω(s)sσ γ+σ–pds
p(p–)
p–q
ω(z)zσ γ+σ–pdz
p–q pq
Corollary . Let <α< ,σ > ,β≥ andγ =β– σσ–p. Letωbe a non-increasing function on I.Then:
(i) for α <p≤q<∞the operatorEα,γ is bounded fromLptoLqif and only ifAα,γ <∞ and,moreover,Eα,γ ≈Aα,γ;it is compact fromLptoLqif and only ifAα,γ <∞and
limz→a+Aα
,γ(z) =limz→b–Aα,γ(z) = ;
(ii) for <q<p<∞andp>α the operatorEα,γ is bounded(compact ifb<∞orb=∞ and≤q<p<∞)fromLptoLqif and only ifBα,γ <∞.
To formulate statements corresponding to the operatorHαwe define
Aα(z) :=
b
z ρ(x)
lnx
a
α– qdx
q z
a
ωp(s)ds
p
, Aα:=sup
z∈I A
α(z),
Bα:=
b
a b
z ρ(x)
lnx
a
α– qdx
p p–q z
a
ωp(s)ds
p(q–)
p–q
ωp(z)dz
p–q pq
.
Corollary . Let a> , <α< , β ≥ and ω(s) =u(s)sp(lns a)
β. Let u be a
non-increasing function on I.Then:
(i) for α <p≤q<∞the operatorHαis bounded fromLptoLqif and only ifAα<∞ and,moreover,Hα ≈Aα;it is compact fromLptoLqif and only ifAα<∞and
limz→a+Aα(z) =limz→b–Aα(z) = ;
(ii) for <q<p<∞andp>α the operatorHαis bounded(compact ifb<∞orb=∞ and≤q<p<∞)fromLptoLqif and only ifBα<∞.
Finally, we consider the operatorIα∗g(s) =ρ(s)[Iα∗(gω)](s),s∈I, acting from Lp to Lq,
whereIα∗is the Weyl operator
Iα∗g(s) =
b
s
g(x)dx (x–s)–α.
The action of the operatorKα∗,βfromLp,vtoLq,wis equivalent to the action of the operator
Kα∗,βg(s) =w
q(s)u(s)Wβ(s) b
s
v
p(x)g(x)dx
(W(x) –W(s))–α
fromLptoLq. Therefore, whenW(x) =xwe have
ρ(s) =u(s)sβ, ω(x) =vp(x), Iα∗g(s) =ρ(s)
b
s
ω(x)g(x)dx (x–s)–α .
We define
A∗α(z) :=
z
a
ρq(s)ds
q b
z
ω(x)xα–pdx
p
, A∗α:=sup
z∈IA ∗
α(z),
B∗α:=
b
a b
z
ω(x)xα–p
dx
q(p–)
p–q z
a
ρq(s)ds
q p–q
ρq(z)dz
p–q pq
From Theorems . and . we have the following result.
Corollary . Let <α< ,β≥andρ(s) =u(s)sβ.Let u be a non-increasing function
on I.Then:
(i) for <p≤q<–α the operatorI∗αis bounded fromLptoLqif and only ifA∗α<∞ and,moreover,Iα ≈A∗α;it is compact fromLptoLqif and only ifA∗α<∞and
limz→a+A∗α(z) =limz→b–A∗α(z) = ;
(ii) for <q<{min(p,–α)}<∞andp> the operatorI∗αis bounded(compact)fromLp
toLqif and only ifB∗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
1Department of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, 2 Satpaev St., Astana, 010008,
Kazakhstan. 2Department of Engineering Sciences and Mathematics, Luleå University of Technology, Luleå, 97187,
Sweden.3UiT, 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
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