R E S E A R C H
Open Access
Optimal couples of rearrangement invariant
spaces for the Riesz potential on the bounded
domain
Shin Min Kang
1, Arif Rafiq
2, Waqas Nazir
2, Irshaad Ahmad
3, Faisal Ali
4and Young Chel Kwun
5**Correspondence:
yckwun@dau.ac.kr
5Department of Mathematics,
Dong-A University, Pusan, 614-714, Korea
Full list of author information is available at the end of the article
Abstract
We prove continuity of the Riesz potential operator in optimal couples of
rearrangement invariant function spaces defined inRnwith the Lebesgue measure.
MSC: 46E30; 46E35
Keywords: Riesz potential operator; rearrangement invariant function spaces; real interpolation
1 Introduction
LetMbe the space of all locally integrable functionsf on⊂Rnwith the Lebesgue measure, finite almost everywhere, and letM+be the space of all non-negative locally integrable functions on (,∞) with respect to the Lebesgue measure, finite almost every-where. We shall also need the following two subclasses ofM+. The subclassMconsists of those elementsgofM+for which there exists anm> such thattmg(t) is increasing. The subclassMconsists of those elementsgofM+which are decreasing.
The Riesz potential operatorRs
, <s<n,n≥ is defined formally by
Rsf(x) =
f(y)|x–y|s–ndy, f ∈M+; ||= . (.)
We shall consider rearrangement invariant quasi-Banach spacesE, continuously embed-ded inL(Rn) +L∞(Rn), such that the quasi-normf
EinEis generated by a quasi-norm ρE, defined onM+with values in [,∞], in the sense thatf
E=ρE(f∗). In this way equiv-alent quasi-normsρEgive the same spaceE. We suppose thatEis nontrivial. Heref∗is the decreasing rearrangement off, given by
f∗(t) =infλ> :μf(λ)≤t
, t> ,
whereμf is the distribution function off, defined by
μf(λ) =x∈Rn:f(x)>λn,
| · |ndenoting the Lebesguen-measure. Note thatf∗(t) = , ift> .
There is an equivalent quasi-normρpthat satisfies the triangle inequalityρpp(g+g)≤ ρpp(g) +ρpp(g) for somep∈(, ) that depends only on the spaceE(see []).
We say that the normρEisK-monotone (cf.[], p., and also [], p.) if
t
g∗(s)ds≤
t
g∗(s)ds implies ρE
g∗≤ρE
g∗, g,g∈M+. (.)
ThenρEis monotone,i.e.,g≤gimpliesρE(g)≤ρE(g).
We use the notationsaa oraa for non-negative functions or functionals to mean that the quotienta/a is bounded; also,a≈ameans thataa andaa. We say thatais equivalent toaifa≈a.
We say that the normρEsatisfies the Minkovski inequality if for the equivalent quasi-normρp,
ρpp gj
ρpp(gj), gj∈M+. (.)
For example, ifEis a rearrangement invariant Banach function space as in [], then by the Luxemburg representation theoremfE=ρE(f∗) for some normρEsatisfying (.) and (.). More general example is given by the Riesz-Fischer monotone spaces as in [], p..
Recall the definition of the lower and upper Boyd indicesαEandβE. Let
hE(u) =sup
ρE(gu∗)
ρE(g∗):g∈M +
, gu(t) :=g(t/u)
be the dilation function generated byρE. Then
αE:=sup <t<
loghE(t)
logt and βE:=<inft<∞
loghE(t)
logt .
If ρE is monotone, then the function hE is submultiplicative, increasing, hE() = ,
hE(u)hE(/u)≥, hence ≤αE≤βE. IfρEisK-monotone, then by interpolation (anal-ogously to [], p.), we see thathE(s)≤max(,s). Hence in this case we have alsoβE≤. Using the Minkovski inequality for the equivalent quasi-normρp and monotonicity of
f∗, we see that
ρE
f∗≈ρE
f∗∗ ifβE< , (.)
where f∗∗(t) = ttf∗(s)ds. The main goal of this paper is to prove continuity of the Riesz potential operatorRs
:E→Gin optimal couples of rearrangement invariant
func-tion spacesE andG, wherefG :=ρG(f∗). It is convenient to introduce the following classes of quasi-norms, where the optimality ofRs:E→Gis investigated. LetNdstand for all domain quasi-normsρE, which are monotone, rearrangement invariant, satisfying Minkowski’s inequality,ρE(χ(,)) <∞and
LetNt consist of all target quasi-normsρG that are monotone, satisfy Minkowski’s in-equality,ρG(χ(,)) <∞,ρG(χ(,∞)ts/n–) <∞and
G→ ∞t–s/nRn, (.)
whereχ(a,b) is the characteristic function of the interval (a,b), <a<b≤ ∞. Note that technically it is more convenient not to require that the target quasi-norm ρG is re-arrangement invariant. Of course, the target space Gis rearrangement invariant, since
fG=ρG(f∗). Finally, letN:=Nd×Nt.
Definition .(Admissible couple) We say that the couple (ρE,ρG)∈N is admissible for the Riesz potential if the following estimate is valid:
ρG
Rsf ∗∗
ρE
f∗. (.)
Moreover,ρE(E) is called domain quasi-norm (domain space), andρG(G) is called a target quasi-norm (target space).
For example, by Theorem . below (the sufficient part), the coupleE= q(ts/nw)(),
G= q(v), ≤q≤ ∞, is admissible ifβ
E< andvis related towby the Muckenhoupt condition []:
t
v(s)qds/s
/q ∞
t
w(s)–rds/s
/r
, /q+ /r= . (.)
Definition .(Optimal target quasi-norm) Given the domain quasi-normρE∈Nd, the optimal target quasi-norm, denoted byρG(E), is the strongest target quasi-norm,i.e.,
ρG
g∗ρG(E)
g∗, g∈M+, (.)
for any target quasi-normρG∈Ntsuch that the coupleρE,ρGis admissible.
Definition .(Optimal domain quasi-norm) Given the target quasi-normρG∈Nt, the optimal domain quasi-norm, denoted byρE(G), is the weakest domain quasi-norm,i.e.,
ρE(G)
g∗ρE
g∗, g∈M+, (.)
for any domain quasi-normρE∈Ndsuch that the coupleρE,ρGis admissible.
Definition .(Optimal couple) The admissible coupleρE,ρG is said to be optimal if ρE=ρE(G)andρG=ρG(E).
The optimal quasi-norms are uniquely determined up to equivalence, while the corre-sponding optimal quasi-Banach spaces are unique.
2 Admissible couples
Theorem .(General caseβE≤) The couple(ρE,ρG)∈N is admissible if and only if ρG(χ(,)Sg)ρE(g), g∈M+or g∈M, (.)
where
Sg(t) :=
⎧ ⎨ ⎩
ts/n–tg(u)du+tus/ng(u)du/u, <t< , <s<n,n≥,
ts/n–
g(u)du, t> , <s<n,n≥.
(.)
Proof First we prove
Rsf
∗∗
Sf∗. (.)
We are going to use real interpolation for quasi-Banach spaces. First we recall some basic definitions. Let (A,A) be a couple of two quasi-Banach spaces (see [, ]) and let
K(t,f) =K(t,f;A,A) = inf f=f+f
fA+tfA
, f∈A+A
be the K-functional of Peetre (see []). By definition, the K-interpolation space A=
(A,A)has a quasi-norm fA=K(t,f),
whereis a quasi-normed function space with a monotone quasi-norm on (,∞) with the Lebesgue measure and such thatmin{,t} ∈. Then (see [])
A∩A→A→A+A,
where byX→Y we mean thatXis continuously embedded inY. Ifg= (
∞
t–
θq×
gq(t)dt/t)/q, <θ< , <q≤ ∞, we write (A
,A)θ,qinstead of (A,A)(see []).
Using the Hardy-Littlewood inequalityRn|f(x)g(x)|dx≤
∞
f∗(t)g∗(t)dt, we get the well-known mapping property
Rs:
ts/n()→L∞Rn
and by the Minkovski inequality for the normf∗∗we get
Rs:L()→ ∞t–s/nRn. Hence
t–s/nRsf ∗∗
(t)Kt–s/n,f;L(), ts/n(), therefore (see [], Section .)
t–s/nRsf ∗∗
(t)
⎧ ⎨ ⎩ t
f∗(u)du+t–s/n
t us/nf∗(u)du/u, <t< ,
implies
Rsf ∗∗
(t)Sf∗(t).
It is clear that (.) follows from (.) and (.).
Now we prove that (.) implies (.). To this end we choose the test function in the form
f(x) =g(c|x|n),g∈M+, so thatf∗(t) =g∗(t) for some positive constantc(cf.[]). Then
Rsf(x) =
|y|<|x|g
c|y|n|x–y|s–ndy+
|y|>|x|g
c|y|n|x–y|s–ndy,
whence
Rsf(x)|x|s–n c|x|n
g(u)du+
||=
c|x|n u
s/n–g(u)duχ (,)(Sg)
c|x|n.
Note thatχ(,)Sg≈χ(,)QTg+χ(,)
g(u)du, where
Qg:=
t
g(u)du/u, t< ,
and
Tg(t) :=
⎧ ⎨ ⎩
ts/n–t
g(u)du, <t< , <s<n,n≥,
ts/n–
g(u)du, t> , <s<n,n≥, henceχ(,)Sgis decreasing, therefore
Rsf ∗
(t)χ(,)Sg(t). (.)
Thus, if (.) is given, then (.) implies (.).
In the sublimiting caseβE< we can simplify the condition (.), replacingS byT. Here
Tg(t) :=
⎧ ⎨ ⎩
ts/n–
t us/ng(u)du/u, <t< , <s<n,n≥, , t> .
(.)
Theorem .(Sublimiting caseβE< ) The couple(ρE,ρG)∈N is admissible if and only
if
ρG(χ(,)Tg)ρE(g), g∈M, (.)
where we recall that
Proof LetρE,ρGbe an admissible couple, then ρG(χ(,)Sg)ρE(g).
SinceρG(χ(,)Tg)ρG(χ(,)Sg), it follows thatρG(χ(,)Tg)ρE(g),g∈M. Now we need to prove sufficiency of (.). We have
χ(,)Sg∗≈χ(,)Tg∗∗+χ(,)g∗∗(), so
ρG
χ(,)Sg∗
ρG
χ(,)Tg∗∗
+ρG(χ(,))g∗∗() implies
ρG
χ(,)Sg∗
ρE
g∗.
In the subcritical caseαE>s/nwe have another simplification of (.).
Theorem .(CaseαE>s/n) The couple(ρE,ρG)∈N is admissible if and only if ρG
χ(,)Tg
ρE(g), g∈M:=
g∈M+,g is decreasing, (.)
where
Tg(t) :=
⎧ ⎨ ⎩
ts/n–t
g(u)du, <t< , <s<n,n≥,
ts/n–
g(u)du, t> , <s<n,n≥.
Proof Let (ρE,ρG)∈N be admissible, then ρG(χ(,)Sg)ρE(g), g∈M. As
ρG
χ(,)Tg
ρG(χ(,)Sg),
we have
ρG
χ(,)Tg
ρE(g).
For the reverse, it is enough to check that (.) implies (.) forg∈M, or ρG(χ(,)Tg)ρE(g), g∈M.
As
χ(,)Tgχ(,)T
t–s/nχ(,)Tg
so
ρG(χ(,)Tg)ρE
t–s/nχ(,)Tg
≈ρE
t–s/nQ
ts/ngρE(g).
Here we use
ρE
Q
t–s/ngρ E
t–s/ng, g∈M
,αE>s/n,t< .
2.1 Optimal quasi-norms
Here we give a characterization of the optimal domain and optimal target quasi-norms. We can define an optimal target quasi-norm by using Theorem ..
Definition . (Construction of the optimal target quasi-norm) For a given domain quasi-normρE∈Ndwe set
ρGE(χ(,)g) :=inf
ρE(h) :χ(,)g≤χ(,)Sh,h∈M+
, g∈M+. (.)
Then
ρG(E)(g) :=ρGE(χ(,)g) +sup
t>
t–s/ng.
Theorem . LetρE∈Ndbe a given domain quasi-norm.ThenρG(E)∈Nt,the coupleρE, ρG(E)is admissible and the target quasi-norm is optimal.By definition,
G(E) :=f ∈M: lim
t→∞f
∗(t) = ,ρ
G(E)
f∗<∞. (.)
Proof To see thatρG(E)is a quasi-norm, we first prove (.), for that we first prove
sup
<t<
t–s/ng∗ρ GE
g∗, g∈M+. (.)
Takeg∈M+and consider an arbitraryh∈M+such that, fort< ,g∗≤S
h. By the Hardy inequalityg∗S(h∗). Then,
t–s/ng∗≤Kt–s/n,h;L(), ts/n(). Hence
sup
<t<
t–s/ng∗≤K,h;L(), ts/n()ρE(h).
Taking the infimum over allhsuch thatg∗≤Sh, we get (.). HenceGE→ ∞(t–s/n)(, ), alsoρG(χ(,∞)g) =supt>t–s/ng. And these two together give (.).ρ
G(E) is indeed a quasi-norm onM+. Sinceχ
(,)(Rsf)∗χ(,)Sf∗, which givesρGE(χ(,)(R
s
f)∗)ρE(f∗). Also
sup
t>
t–s/nRsf
∗
sup
t>
t–s/nSf∗=
f∗(u)duρE
HenceρE,ρG(E)is admissible couple. Now we are going to prove thatρG(E)is optimal. For this purpose, suppose that the couple (ρE,ρG)∈N is admissible. Then by Theorem .,
ρG(χ(,)Sg)ρE(g), g∈M
+.
Therefore ifχ(,)g∗≤χ(,)Sh,h∈M+, then ρG
χ(,)g∗
≤ρG(χ(,)Sh)ρE(h),
so taking the infimum on the right-hand side, we get
ρG
χ(,)g∗
ρGE
χ(,)g∗
,
henceρG(g∗)ρG(E)(g∗).
In the sublimiting caseβE< we can simplify the optimal target quasi-norm.
Theorem . IfρE∈Ndbe a given domain quasi-norm.Then for g∈M+, ρGE
χ(,)g∗
≈ρχ(,)g∗
,
ρ(χ(,)g) :=inf
ρE(h) :χ(,)g≤χ(,)Th,h∈M
,
(.)
i.e.,
ρG(E)(g)≈ρ(χ(,)g) +sup t>
t–s/ng.
Proof Ifχ(,)g∗≤χ(,)Th,h∈M, thenχ(,)g∗≤χ(,)Sh, therefore ρGE
χ(,)g∗
≤ρE(h)
and taking the infimum, we get
ρGE
χ(,)g∗
≤ρχ(,)g∗
.
Now for the reverse, letχ(,)g∗≤χ(,)Sh,h∈M+. Then
χ(,)g∗χ(,)S
h∗≈χ(,)T
h∗∗+χ(,)f∗∗(),
so
χ(,)g∗–χ(,)f∗∗()χ(,)T
h∗∗, which gives, sinceh∗∗∈M,
ρχ(,)g∗–χ(,)f∗∗()
ρE
h∗∗≈ρE
and this implies
ρχ(,)g∗
ρE(h) +f∗∗(),
which gives
ρχ(,)g∗
ρE(h).
Taking the infimum, we getρ(χ(,)g∗)ρGE(χ(,)g∗), henceρ(χ(,)g∗)≈ρGE(χ(,)g∗).
A simplification of the optimal target quasi-norm is possible also in the subcritical case αE>s/n.
Theorem . LetρE∈Ndbe a given domain quasi-norm.Then for g∈M+, ρGE
χ(,)g∗
≈ρ
χ(,)g∗
,
ρ(χ(,)g) :=inf
ρE(h) :χ(,)g≤Th,h∈M
,
(.)
i.e.,
ρG(E)(g)≈ρ(χ(,)g) +sup t>
t–s/ng.
Proof Ifχ(,)g∗≤χ(,)Th,h∈M, then χ(,)g∗≤χ(,)Sh.
Therefore
ρGE
χ(,)g∗
≤ρE(h),
and taking the infimum, we get
ρGE
χ(,)g∗
≤ρ
χ(,)g∗
.
For the reverse, letχ(,)g∗≤χ(,)Sh. Thenχ(,)g∗χ(,)T(h∗) +χ(,)T(h∗). As χ(,)Tgχ(,)T
t–s/nχ(,)Tg
,
we get
χ(,)g∗χ(,)T
h∗+t–s/nχ(,)T
h∗, whence
ρ
χ(,)g∗
ρE
t–s/nχ(,)T
h∗+ρE(h)
≈ρE
t–s/nQ
where we use
ρE
Q
t–s/ngρE
t–s/ng, g∈M,αE>s/n,t< . Therefore, taking the infimum we arrive at
ρ
g∗ρGE
g∗.
We can construct an optimal domain quasi-normρE(G)by Theorem . as follows.
Definition .(Construction of an optimal domain norm) For a given target quasi-normρG∈Nt, we construct an optimal domain quasi-normρE(G)by
ρE(G)(g) :=ρG
χ(,)Sg∗
, g∈M+. (.)
Theorem . IfρG∈Ntis a given target quasi-norm,then the domain quasi-normρE(G)
is optimal.Moreover,ifβG< –s/n,then the coupleρE(G),ρGis optimal.
Proof Sinceχ(,)Sg∗≈χ(,)Tg∗∗+χ(,)g∗∗(), so ρE(G)(g)≈ρG
χ(,)Tg∗∗+χ(,)g∗∗()
,
it follows thatρE(G)is a quasi-norm. To prove the property (.), we notice that
ρE(G)
f∗=ρG
χ(,)Sf∗
≥ρG(χ(,))
Sf∗()
f∗(t)dt≈ fL().
The coupleρE(G),ρG is admissible sinceρE(G)(g) =ρG(χ(,)Sg∗)≥ρG(χ(,)Sg). More-over,ρE(G)is optimal, since for any admissible couple (ρE,ρG)∈N we haveρG(χ(,)Sh)
ρE(h),h∈M+. Therefore,
ρE(G)
g∗≤ρE
g∗.
To check that ifβG< –s/n, the coupleρE(G),ρGis optimal, we need only to prove that ρG is an optimal target quasi-norm,i.e.,ρ(g∗)ρG(g∗), whereρ=ρG(E(G))is defined by (.), sinceβE(G)< . We haveχ(,)g∗∗(t) –χ(,)g∗∗() =χ(,)Th, whereh(t) =t–s/n[g∗∗(t) –
g∗(t)]∈M,t< , therefore, ρGE(G)
χ(,)g∗∗(t) –χ(,)g∗∗()
≤ρE(G)(h) =ρG
χ(,)Sh∗
implies
ρGE(G)
χ(,)g∗∗(t)
ρG
χ(,)Sh∗
+g∗∗(), since
so
ρGE(G)
χ(,)g∗∗(t)
ρG
χ(,)ts/nh∗∗
+ρG
χ(,)Th∗∗
+g∗∗(). Now we define
Pg(t) :=
t
t
g(u)du, t< .
Fort< , sinceh∗ Qh, we have h∗∗=Ph∗ QPh, therefore Th∗∗TQ(Ph)
T(Ph). AlsoT(Ph)≈Th+ts/nPhandPh≤h∗∗. Therefore, ρGE(G)
χ(,)g∗
ρG(χ(,)Th) +ρG
χ(,)ts/nh∗∗
+g∗∗() ρG
χ(,)g∗∗
+ρG
χ(,)ts/nh∗∗
+g∗∗().
Fort< , sinceh(t)≤t–s/ng∗∗(t) we haveh∗(t)≤t–s/ng∗∗, therefore usingβ
G< –s/n, Minkowski’s inequality, and monotonicity ofρG, we have
ρG
χ(,)ts/nh∗∗
ρG
χ(,)g∗∗
.
Thus
ρGE(G)
χ(,)g∗
ρG
χ(,)g∗∗
≈ρG
χ(,)g∗
,
henceρ(g∗)ρG(g∗).
Example . IfG=Cconsists of all bounded continuous functions such thatf∗(∞) = and ρG(g) =g∗() =g∗∗(), thenαG =βG = and ρE(G)(g)≈
t
s/ng∗∗dt/t, i.e., E=
(ts/n)() and the coupleE,Gis optimal.
Example . LetG= ∞(v) withβG< –s/nand let
ρE(g) =supv(t)
t
us/ng∗∗(u)du/u.
Then, the coupleE,Gis optimal andβE< . In particular, this is true ifvis slowly varying since thenαG=βG= andαE=βE=s/n< .
2.2 Subcritical case
Here we suppose thats/n<αE.
Theorem .(Sublimiting caseβE< ) For a given domain quasi-normρE∈Ndwith ρE(χ(,)(t)t–s/n) <∞,we have
ρGE
χ(,)g∗
≈ρE
t–s/ng∗≈ρE
t–s/ng∗∗, (.)
i.e., ρG(E)
g∗≈ρGE
χ(,)g∗
+sup
t>
t–s/ng.
Proof Ifχ(,)g∗≤χ(,)Th,h∈M, then fort< ,t–s/ng∗≤h∗∗, whence ρE
t–s/ng∗ρE
h∗∗≈ρE
h∗≈ρE(h). Taking the infimum, we get
ρE
t–s/ng∗ρGE
χ(,)g∗
.
For the reverse, we notice that χ(,)T(t–s/ng∗) χ(,)g∗ =g∗, hence ρGE(χ(,)g∗)
ρE(t–s/ng∗).
It remains to prove that the domain quasi-normρEis also optimal. LetρE,ρG(E)be an
admissible couple inN. Then
ρE
g∗ρG(E)
χ(,)Sg∗
=ρGE
χ(,)Sg∗
+sup
t>
t–s/nχ(,)Sg∗
≈ρE
t–s/nχ(,)Sg∗
+
ρE
t–s/nχ(,)Tg∗
ρE
χ(,)g∗∗
≈ρE
χ(,)g∗
≈ρE
g∗.
Now we give an example.
Example . Let
E= qtαw
()∩ rtβw
(), s/n<α<β< , <q,r≤ ∞,
wherewandware slowly varying. Then we haveαE=α,βE=β. Now by applying the previous theorem, we get
G(E) = qtα–s/nw
∩ r
tβ–s/nw
,
and the couple (E,G(E)) is optimal.
In the limiting caseβE= , the formula for the optimal target quasi-norm is more com-plicated.
Theorem .(Limiting case) Let
ρE(g) :=ρH
χ(,)g∗∗
, ρG(g) :=ρH
t–sup
<u<t
u–s/ng(u)
,
whereρH is a monotone quasi-norm withαH=βH= ,ρH(χ(,)) <∞,ρH(χ(,∞)t–) <∞
and let
E:=f ∈M:tf∗∗(t)→as t→andρE
f∗<∞,
G:=
f ∈M: sup
<u<t
u–s/nf∗∗(u)→as t→andρG
f∗<∞
Define
ρG(g) :=ρG(χ(,)g) +sup
t>
t–s/ng.
Then the coupleρE,ρGis optimal.
Proof Note that
E→L().
Indeed,ρE(f∗) =ρH(χ(,)g∗∗)f∗∗() =
f∗(u)du. Hence the above embedding follows. Consequently,ρE∈Nd. On the other hand,
ρG
f∗≥ρH
χ(,∞)t–sup <u<t
u–s/nf∗(u)
≥ sup
<u<
u–s/nf∗(u)ρH
χ(,∞)t–
.
Hence G → ∞(t–s/n)(, ). This together with ρG(χ(,∞)) =supt>t–s/ng gives G→
∞(t–s/n). Then from the conditions onG
it follows thatρG∈Nt. Also,αE=βE= and αG=βG= –s/n. On the other hand, if <u< , then
u–s/nRsf ∗∗
(u)
u
f∗(v)dv+u–s/n
u
vs/n–f∗(v)dv.
For everyε> , we can find aδ> , such thatvf∗∗(v) <εfor all <v<δ. Then for <t< ,
sup
<u<t
u–s/nRsf ∗∗
(u)
t
f∗(v)dv+ε+t–s/n
δ
vs/n–f∗(v)dv. (.)
Now it is easy to check thatlimt→sup<u<tu–s/n(Rsf)∗∗= iff ∈E.
To prove thatRs:E→Gwe need to check that the coupleρE,ρ
Gis admissible. We write fort< ,
Tg(t) =Tg∗(t) =ts/ng∗∗(t), g∈M. Then
ρG
χ(,)Tg
=ρG
χ(,)Tg
+sup
t>
t–s/nχ(,)Tg =ρH
χ(,)t–sup <u<t
u–s/nTg(u)
+sup
t>
t–s/nχ(,)Tg =ρH
χ(,)g∗∗
=ρE(g).
To prove that the target space is optimal, notice first that
sup
<u<t
Iff ∈G, then by []
sup
<u<t
u–s/nf∗∗(u)≈
t–s/n
h(u)du (whereh, is decreasing)
≈
t
h
v–s/nv–s/ndv (by a change of variables).
Ifh(v) =h(v–s/n)v–s/nthen obviouslyh∈Mand
sup
<u<t
u–s/nf∗∗(u)≈
t
h(v)dv=th∗∗(t),
whence
ρE(h)≈ρH
χ(,)h∗∗
≈ρH
χ(,)t–sup <u<t
u–s/nf∗∗(u)
≈ρG
χ(,)f∗
.
On the other hand,
sup
<u<t
u–s/nf∗∗(u)≈th∗∗(t)
impliest–s/nf∗(t)th∗∗(t), which givesf∗ts/nh∗∗, which impliesχ
(,)f∗χ(,)Th, and therefore
ρGE
χ(,)f∗
ρE(h)ρG
χ(,)f∗
,
proving optimality ofG. To check optimality ofE, we notice that
ρE(G)(h) =ρG
χ(,)Sh∗
ρG
χ(,)Th∗∗
≈ρH
t–sup
<u<t
u–s/nχ(,)Th∗∗(u)
ρH
χ(,)h∗∗
.
Hence
ρE(G)(h)ρE(h).
Example . LetE=∞(tw)(), consisting of allf ∈∞(tw)() such thattf∗∗(t)→ ast→,wis slowly varying. ThenβE= . IfG=∞(t–s/nv)∩∞(tw), wherev(t) =
supu>tw(u) and
∞ (v) :=f∈∞(v) : sup
<u<t
u–s/nf∗(u)→ ast→,
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Author details
1Department of Mathematics and RINS, Gyeongsang National University, Jinju, 660-701, Korea.2Department of
Mathematics, Lahore Leads University, Lahore, 54810, Pakistan.3Department of Mathematics, Government College
University, Faisalabad, Pakistan.4Centre for Advanced Studies in Pure and Applied Mathematic, Bahauddin Zakariya
University, Multan, 54000, Pakistan. 5Department of Mathematics, Dong-A University, Pusan, 614-714, Korea.
Acknowledgements
This study was supported by research funds from Dong-A University.
Received: 15 October 2013 Accepted: 20 January 2014 Published:10 Feb 2014 References
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doi:10.1007/s10231-003-0080-6
10.1186/1029-242X-2014-60