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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

4

and 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 onRnwith 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 subclassMconsists of those elementsgofM+which are decreasing.

The Riesz potential operatorRs

,  <s<n,n≥ is defined formally by

Rsf(x) =

f(y)|xy|sndy, fM+; ||= . (.)

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) =xRn:f(x)>λn,

| · |ndenoting the Lebesguen-measure. Note thatf∗(t) = , ift> .

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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≤gimpliesρE(g)≤ρE(g).

We use the notationsaa oraa for non-negative functions or functionals to mean that the quotienta/a is bounded; also,a≈ameans thataa andaa. We say thatais equivalent toaifa≈a.

We say that the normρEsatisfies the Minkovski inequality if for the equivalent quasi-normρp,

ρpp gj

ρpp(gj), gjM+. (.)

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∗):gM +

, 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 ≤αEβE. 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

:EGin 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:EGis investigated. LetNdstand for all domain quasi-normsρE, which are monotone, rearrangement invariant, satisfying Minkowski’s inequality,ρE(χ(,)) <∞and

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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-normρENd, the optimal target quasi-norm, denoted byρG(E), is the strongest target quasi-norm,i.e.,

ρG

gρG(E)

g∗, gM+, (.)

for any target quasi-normρGNtsuch that the coupleρE,ρGis admissible.

Definition .(Optimal domain quasi-norm) Given the target quasi-normρGNt, the optimal domain quasi-norm, denoted byρE(G), is the weakest domain quasi-norm,i.e.,

ρE(G)

gρE

g∗, gM+, (.)

for any domain quasi-normρENdsuch 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

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Theorem .(General caseβE≤) The couple(ρE,ρG)∈N is admissible if and only if ρG(χ(,)Sg)ρE(g), gM+or gM, (.)

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

, fA+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∩AAA+A,

where byXY 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()→LRn

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< ,

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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),gM+, so thatf(t) =g(t) for some positive constantc(cf.[]). Then

Rsf(x) =

|y|<|x|g

c|y|n|xy|sndy+

|y|>|x|g

c|y|n|xy|sndy,

whence

Rsf(x)|x|sn c|x|n

g(u)du+

||=

c|x|n u

s/n–g(u)duχ (,)(Sg)

c|x|n.

Note thatχ(,)Sgχ(,)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), gM, (.)

where we recall that

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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),gM. 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), gM:=

gM+,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), gM. As

ρG

χ(,)Tg

ρG(χ(,)Sg),

we have

ρG

χ(,)Tg

ρE(g).

For the reverse, it is enough to check that (.) implies (.) forgM, or ρG(χ(,)Tg)ρE(g), gM.

As

χ(,)T(,)T

ts/(,)Tg

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so

ρG(χ(,)Tg)ρE

ts/(,)Tg

ρE

ts/nQ

ts/ngρE(g).

Here we use

ρE

Q

ts/ngρ E

ts/ng, gM

,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-normρENdwe set

ρGE(χ(,)g) :=inf

ρE(h) :χ(,)gχ(,)Sh,hM+

, gM+. (.)

Then

ρG(E)(g) :=ρGE(χ(,)g) +sup

t>

t–s/ng.

Theorem . LetρENdbe 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) :=fM: 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/ngρ GE

g∗, gM+. (.)

TakegM+and consider an arbitraryhM+such that, fort< ,gS

h. By the Hardy inequalitygS(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

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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), gM

+.

Therefore ifχ(,)g∗≤χ(,)Sh,hM+, 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 . IfρENdbe a given domain quasi-norm.Then for gM+, ρGE

χ(,)g

ρχ(,)g

,

ρ(χ(,)g) :=inf

ρE(h) :χ(,)gχ(,)Th,hM

,

(.)

i.e.,

ρG(E)(g)≈ρ(χ(,)g) +sup t>

t–s/ng.

Proof Ifχ(,)g∗≤χ(,)Th,hM, 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,hM+. Then

χ(,)gχ(,)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

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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 . LetρENdbe a given domain quasi-norm.Then for gM+, ρGE

χ(,)g

ρ

χ(,)g

,

ρ(χ(,)g) :=inf

ρE(h) :χ(,)gTh,hM

,

(.)

i.e.,

ρG(E)(g)≈ρ(χ(,)g) +sup t>

t–s/ng.

Proof Ifχ(,)g∗≤χ(,)Th,hM, 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(,)gχ(,)T(h∗) +χ(,)T(h∗). As χ(,)T(,)T

ts/(,)Tg

,

we get

χ(,)gχ(,)T

h∗+ts/(,)T

h∗, whence

ρ

χ(,)g

ρE

ts/(,)T

h∗+ρE(h)

ρE

ts/nQ

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where we use

ρE

Q

ts/ngρE

ts/ng, gM,αE>s/n,t< . Therefore, taking the infimum we arrive at

ρ

gρGE

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-normρGNt, we construct an optimal domain quasi-normρE(G)by

ρE(G)(g) :=ρG

χ(,)Sg

, gM+. (.)

Theorem . IfρGNtis 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)dtfL().

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),hM+. 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) =ts/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

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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< , sincehQh, we have h∗∗=PhQPh, therefore Th∗∗TQ(Ph)

T(Ph). AlsoT(Ph)≈Th+ts/nPhandPhh∗∗. Therefore, ρGE(G)

χ(,)g

ρG(χ(,)Th) +ρG

χ(,)ts/nh∗∗

+g∗∗() ρG

χ(,)g∗∗

+ρG

χ(,)ts/nh∗∗

+g∗∗().

Fort< , sinceh(t)≤ts/ng∗∗(t) we haveh(t)ts/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=Cconsists 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-normρENdwith ρE(χ(,)(t)ts/n) <∞,we have

ρGE

χ(,)g

ρE

ts/ng∗≈ρE

ts/ng∗∗, (.)

i.e., ρG(E)

g∗≈ρGE

χ(,)g

+sup

t>

t–s/ng.

(12)

Proof Ifχ(,)g∗≤χ(,)Th,hM, then fort< ,ts/ng∗≤h∗∗, whence ρE

ts/ngρE

h∗∗≈ρE

h∗≈ρE(h). Taking the infimum, we get

ρE

ts/ngρGE

χ(,)g

.

For the reverse, we notice that χ(,)T(ts/ng∗) χ(,)g∗ =g∗, hence ρGE(χ(,)g∗)

ρE(ts/ng∗).

It remains to prove that the domain quasi-normρEis also optimal. LetρE,ρG(E)be an

admissible couple inN. Then

ρE

gρG(E)

χ(,)Sg

=ρGE

χ(,)Sg

+sup

t>

t–s/(,)Sg

ρE

ts/(,)Sg

+ 

ρE

ts/(,)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≤ ∞,

wherewandware slowly varying. Then we haveαE=α,βE=β. Now by applying the previous theorem, we get

G(E) = qs/nw

r

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:=fM:tf∗∗(t)→as t→andρE

f∗<∞,

G:=

fM: sup

<u<t

u–s/nf∗∗(u)→as t→andρG

f∗<∞

(13)

Define

ρG(g) :=ρG(χ(,)g) +sup

t>

t–s/ng.

Then the coupleρE,ρGis optimal.

Proof Note that

EL().

Indeed,ρE(f∗) =ρH(χ(,)g∗∗)f∗∗() =

f∗(u)du. Hence the above embedding follows. Consequently,ρENd. 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 thatρGNt. 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)∗∗=  iffE.

To prove thatRs:EGwe need to check that the coupleρE,ρ

Gis admissible. We write fort< ,

Tg(t) =Tg∗(t) =ts/ng∗∗(t), gM. Then

ρG

χ(,)Tg

=ρG

χ(,)Tg

+sup

t>

t–s/(,)Tg =ρH

χ(,)t–sup <u<t

u–s/nTg(u)

+sup

t>

t–s/(,)Tg =ρH

χ(,)g∗∗

=ρE(g).

To prove that the target space is optimal, notice first that

sup

<u<t

(14)

IffG, then by []

sup

<u<t

u–s/nf∗∗(u)≈

t–s/n

h(u)du (whereh, is decreasing)

t

h

v–s/nvs/ndv (by a change of variables).

Ifh(v) =h(v–s/n)vs/nthen obviouslyhMand

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 givesfts/nh∗∗, which impliesχ

(,)fχ(,)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/(,)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,

(15)

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

1. Köthe, G: Topologisch Lineare Räume. Springer, Berlin (1966)

2. Bergh, J, Löfström, J: Interpolation Spaces, an Introduction. Springer, Berlin (1976) 3. Bennett, C, Sharpley, R: Interpolation of Operators. Academic Press, Boston (1988) 4. Muckenhoupt, B: Hardy’s inequality with weights. Stud. Math.44, 31-38 (1972)

5. Brudnyˇı, YA, Krugliak, NY: Interpolation Spaces and Interpolation Functors. North-Holland, Amsterdam (1991) 6. Cianchi, A: Symmetrization and second-order Sobolev inequalities. Ann. Mat. Pura Appl.183, 45-77 (2004).

doi:10.1007/s10231-003-0080-6

10.1186/1029-242X-2014-60

References

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