R E S E A R C H Open Access
Boundedness and compactness of a class of Hardy type operators
Akbota M Abylayeva1, Ryskul Oinarov1and Lars-Erik Persson2,3*
*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:= (b
a |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
(lnas)βf(s)ds s(lnxs)–α .
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Moreover, whenu≡ andW(x) =xσ,σ> , we get the operatorEα,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,β defined by
Tα,β f(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,β f(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 operatorTα,β are 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
To prove the main results we shall need some auxiliary results from the standard literature on Hardy type inequalities (see [] and []).
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)p
w(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–qp
× z
a
up(s)Wpβ(s)w(s)ds p(q–)p–q
up(z)Wpβ(z)w(z)dz p–qpq
<∞.
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 p–qq
× z a
up(s)Wpβ(s)w(s)ds q(p–)p–q
Wq(α–)(z)v(z)dz p–qpq
.
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)p
w(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+forxk≤x≤xk+,xk
xk–w(s)ds= k–, and ifxk+=b, thenxk+
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 Tα,β ≈Aα,β.
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 qp x
xk–
up(s)Wpβ(s)w(s)ds (W(x) –W(s))p(–α)
pq dx
≤
k
xk+
xk–
fp(s)w(s)ds qp
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)
tpβ( –t)p(α–)dt. (.)
Sinceβ≥,α>p, the Euler beta function
tpβ(–t)p(α–)dtconverges. Consequently, from (.) and (.) it follows that
J
k
xk+
xk–
fp(s)w(s)ds qp
uq(xk–) xk+
xk
v(x)W
pq(pβ+)
dx Wq(–α)(x)
≤
k
xk+
xk–
fp(s)w(s)ds qp
uq(xk–)W
pq(pβ+)
(xk+) xk+
xk
v(x)Wq(α–)(x)dx
= (qβ+
q
p)
k
xk+
xk–
fp(s)w(s)ds qp
×uq(xk–)W
q p(pβ+)
(xk–) xk+
xk
v(x)Wq(α–)(x)dx
k
xk+
xk–
fp(s)w(s)ds qp
×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 qp
× xk a
up(s)Wpβ(s)w(s)ds pq b
xk
v(x)Wq(α–)(x)dx
≤Aqα,β
k
xk+
xk–
fp(s)w(s)ds qp
≤Aqα,β
k
xk+
xk–
fp(s)w(s)ds qp
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 qp
×uq(xk–)W
q p(pβ+)
(xk+) xk+
xk
v(x)Wq(α–)(x)dx
= (qβ+
q p)
pβ+– q
p
k
xk+
xk–
fp(s)w(s)ds qp
×uq(xk–)
(pβ+)(k–)– (pβ+)(k–)q
p xk+
xk
v(x)Wq(α–)(x)dx
k
xk+
xk–
fp(s)w(s)ds qp
×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 qp
× xk–
xk–
up(s)Wpβ(s)w(s)ds
pq 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 qp
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–qp
.
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–qp
≤
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–qp
up(t)Wpβ(t)w(t)dt
≤ B
qp p–q
α,β. (.)
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
z→alim+Aα,β(z) = lim
z→b–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)pw(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
t→alim+Tα,βftq,v= . (.)
Moreover, Tα,βftqq,v=
b
a
v(x) x
a
u(s)Wβ(s)ft(s)w(s)ds (W(x) –W(s))–α
q
dx
≥ b t
v(x) t
a
u(s)Wβ(s)ft(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 –qp
× 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,β∗ g(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)qv–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)qv–
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
s→blim–Tα,β∗ gt
p,w–p = . (.)
Furthermore, we note that Tα,β∗ gt
p,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
K(x,s)f(s)ds
fromLp(c,d) toLq(c,d) with the kernel
K(x,s) =u(s)Wβ(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,
c→alim+PcTα,βPc lim
c→a+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 x∈(a,c],uε(s) =u(s) fors∈(a,c] anduε(s) =εu(s) fors∈(c,d], where >ε> . Obviously,
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
Aεα,βfp,w, (.)
where
Aεα,β= sup
a<z<d
d z
Wq(α–)(x)vε(x)dx q z
a
upε(s)Wpβ(s)w(s)ds
p
. We estimate the expressionAεα,βfrom the above as follows:
Aεα,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
c→alim+PcdTα,βPc lim
c→a+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
Aα,β(z)fp,w.
