R E S E A R C H
Open Access
Estimates of the modular-type operator
norm of the general geometric mean
operator
Chang-Pao Chen
1*and Jin-Wen Lan
2*Correspondence:
1Center for General Education,
Hsuan Chuang University, Hsinchu, 30092, Taiwan, Republic of China Full list of author information is available at the end of the article
Abstract
In this paper, the modular-type operator norm of the general geometric mean operator over spherical cones is investigated. We give two applications of a new limit process, introduced by the present authors, to the establishment of Pólya-Knopp-type inequalities. We not only partially generalize the sufficient parts of Persson-Stepanov’s and Wedestig’s results, but we also provide new proofs to these results.
MSC: 47A30; 26D10; 26D15
Keywords: operator norm; integral operator; Hardy-Knopp-type inequalities; Pólya-Knopp-type inequalities
1 Introduction
LetEbe a spherical cone inRn. By this, we mean thatE=
s>sAfor some Borel
measur-able subsetAof the unit spheren–. LetKDK∩Lp (v dx)→L
q
(u dx)(in brief,K∗) denote
the smallest constantCin (.):
E
◦Kf(x)qu(x)dx
/q
≤C
E
◦f(x)pv(x)dx
/p
(.)
for allf ∈DK∩Lp(v dx), wherep,q> ,u(x)≥,v(x) > ,∈CV+(I),◦f(x) =(f(x)), andKf(x) is of the form
Kf(x) :=
˜ Sx
k(x,t)f(t)dt (x∈E). (.)
HereCV+(I) denotes the set of all nonnegative convex functions defined on an open
in-tervalIinR,DKis the space of thosef such thatKf(x) is well defined for almost allx∈E, andLp(v dx) is the set of all real-valued Borel measurablef with
f,p,v:=
E
◦f(x)pv(x)dx
/p <∞.
Moreover,S˜x=
<s≤xsA,Sx=S˜x\ xA, andk(x,t)≥ is locally integrable overE×E.
We writeLp(v dx) andf
p,vinstead ofLp(v dx) andf,p,v, respectively, for the case
(s) =|s|. We also writeLp(E,v dx) forLp(v dx), whenever the integral regionEis empha-sized.
Clearly,
K∗=sup f
◦Kfq,u
◦fp,v ,
where the supremum is taken over allf ∈DK∩Lp(v dx) with◦fp,v= . This number reduces to the operator norm ofKfor the case(s) =|s|. The investigation of the value
K∗ has a long history in the literature. In [], the present authors introduced a gener-alized Muckenhoupt constantAM(p,q) and established the following Muckenhoupt-type estimate forK∗:
K∗≤
q p∗ +
q
η
/q +p
∗
η
η∗/(p∗q∗)
AM(p,q), (.)
where ≤p,q≤ ∞,η=max(p,q), and (·)∗is the conjugate exponent of (·) in the sense that /(·) + /(·)∗= . For the particular case that
(s) =|s|, k(x,t) = , (.)
there are two other types of estimates. They are
K∗≤p∗APS(p,q) (.)
and
K∗≤AW(p,q) := inf
<s<pAW(s,p,q)
p– p–s
/p∗
. (.a)
These two inequalities were proved in [] and [], Theorem . and Lemma ., for the case <p≤q<∞(see also [], Theorem .). We refer the readers to Section for details.
In this paper, we focus on the evaluation ofK∗for the following case of (.):
(s) =es, k(x,t) =g(t)/G(x), f(t)−→logf(t), wheref(t) > ,g(t) > , and
G(x) =
˜ Sx
g(t)dt (x∈E). (.)
The corresponding inequality to (.) takes the form
E
exp
G(x)
˜ Sx
g(t)logf(t)dt q
u(x)dx /q
≤C
E
f(x)pv(x)dx
/p
, (.)
In [], Theorem ., [, ], and [], Theorem ., the particular caseg(t) = of (.) was considered. They obtained the following estimates by means of the formula (GKf)(x) =
lim→+[K(f)]/(x):
K∗≤e/pD∗PS and K∗≤infs>e(s–)/pD∗OG(s), (.)
where <p≤q<∞. The definitions ofD∗PSandD∗OG(s) are given in Section .
The purpose of this paper is two-fold. We not only extend the aforementioned sufficient parts of [, , ], and [] fromu(x) > andg(t) = tou(x)≥ and
min
sup x∈E
g(x),sup x∈E
gv((xx)) <∞, (.)
but we also provide a new proof of (.) from the viewpoint of (.):
K∗≤ inf ∈F+
(Ap/,q/)/≤lim inf →+
(Ap/,q/)/
, (.)
where <p,q<∞,F+={> :∈CV+(I)}, andA
p,qare absolute constants subject to the condition
E
Kf(x)qu(x)dx /q
≤Ap,q
E
f(x)pv(x)dx /p
(f≥). (.)
It is clear that (.) is applicable to the case(s) =es. In this case,F+
={> }and the second inequality in (.) holds. We remark that it may not be an equality (cf.[]). On the other hand, we havep/→ ∞andq/→ ∞as→+. This indicates that the infimum
in (.) can be estimated by evaluating thoseAp,qwithp,qlarge enough.
The limit process (.) differs from the scheme by means of the formula (GKf)(x) =
lim→+[K(f)]/(x). It was introduced in [] to get different types of Pólya-Knopp in-equalities, including the n-dimensional extensions of the LevCochran-Lee-type in-equalities and Carleson’s result. We showed that the infimum in (.) can easily be eval-uated by applying the following choice ofAp,qfor <p,q<∞:
Ap,q≤
q p∗ +
q
η
/q +p
∗
η
η∗/(p∗q∗)
AM(p,q).
2 General forms of (1.5) and (1.5a)
Let <p≤q<∞,g(t) > ,u(x)≥, andv(x) > . Consider the inequality:
E
G(x)
˜ Sx
g(t)f(t)dt
q u(x)dx
/q
≤C
E
f(x)pv(x)dx /p
(f ≥), (.)
whereG(x) is defined by (.). This corresponds to the case(s) =|s|andk(x,t) =g(t)/G(x) of (.). Inequality (.) reduces to the form (.) for the caseg(t) = :
E
˜ Sx
f(t)dt
q
˜
u(x)dx /q
≤C
E
f(x)pv(x)dx /p
(f ≥), (.)
whereu˜(x) =u(x)/G(x)q. In [], Theorem ., [] and [], Lemma .(a), it was proved that under the conditionsu(x) > andAPS(p,q) <∞, (.) holds, in other words, (.) withu˜(x) replaced byu(x) is true forC=p∗APS(p,q), where
APS(p,q) :=sup x∈E
˜ Sx
v(t)–p∗dt –/p
˜ Sx
˜ St
v(y)–p∗dy
q u(t)dt
/q .
This result will be extended below fromg(t) = andu(x) > tog(t) > andu(x)≥. We shall see its application in the proof of Theorem ..
Theorem . Let <p≤q<∞,u(x)≥,v(x) > ,g(t) > ,and <G(x) <∞,where G(x) is defined by(.).IfA˜PS(p,q) <∞,then(.)holds for C≤p∗A˜PS(p,q),where
˜
APS(p,q) =sup x∈E
˜ Sx
g(t) v(t)
p∗ v(t)dt
–
p
˜ Sx
G(t)
˜ St
g(y) v(y)
p∗ v(y)dy
q u(t)dt
q
.
Proof The caseu(x) > follows from [], Theorem ., or [], Lemma .(a), under the following substitutions:
f(t)−→g(t)f(t), u(x)−→ u(x)
(G(x))q, v(x)−→ v(x)
(g(x))p. (.)
As foru(x)≥, letuτ(x) =u(x) +ρτ(x), where <τ < andρτ(x) > is subject to the condition
˜ Sx
G(t)
˜ St
g(y) v(y)
p∗ v(y)dy
q
ρτ(t)dt≤τ
˜ Sx
g(t) v(t)
p∗ v(t)dt
q/p
. (.)
Suchρτ(x) exists. We haveuτ(x) > onE. Moreover, the condition /q< implies that (a+b)/q≤a/q+b/qfor alla,b≥. Putting this together with (.) yields
˜ Sx
G(t)
˜ St
g(y) v(y)
p∗ v(y)dy
q uτ(t)dt
/q
≤
˜ Sx
G(t)
˜ St
g(y) v(y)
p∗ v(y)dy
q u(t)dt
q
+τq
˜ Sx
g(t) v(t)
p∗ v(t)dt
p
This leads us to
˜
APS(p,q,τ)≤ ˜APS(p,q) +τ/q<∞, (.)
where A˜PS(p,q,τ) is the number obtained fromA˜PS(p,q) by replacingu(t) byur(t). We haveuτ(x) >u(x) onE. By the result of the caseu(x) > , the following inequality holds for f ≥:
E
G(x)
˜ Sx
g(t)f(t)dt
q u(x)dx
q
≤
E
G(x)
˜ Sx
g(t)f(t)dt
q uτ(x)dx
q
≤p∗A˜PS(p,q,τ)
E
f(x)pv(x)dx
p
. (.)
It follows from (.) that lim infτ→+A˜PS(p,q,τ)≤ ˜APS(p,q). Putting this together with
(.) yields the desired inequality. The proof is complete.
Next, consider (.a). The numberAW(s,p,q) in (.a) is defined by the formula:
AW(s,p,q) =sup x∈E
˜ Sx
v(t)–p∗dt
s–
p
E\Sx
˜ St
v(y)–p∗dy
q(p–s)
p
u(t)dt
q
.
In [], Lemma .(b),AW(s,p,q) is replaced by another notationA∗W(s). Like (.), (.a) can be generalized in the following way, in whichg(t) = andu(x) > are relaxed tog(t) > andu(x)≥. We shall see its application in the proof of Theorem ..
Theorem . Let <p≤q<∞,u(x)≥,v(x) > ,g(t) > ,and <G(x) <∞,where G(x) is defined by(.).IfA˜W(s,p,q) <∞for some <s<p,then(.)holds for C≤ ˜AW(p,q), where
˜
AW(p,q) := inf
<s<pA˜W(s,p,q)
p– p–s
/p∗
(.)
and
˜
AW(s,p,q) =sup x∈E
˜ Sx
g(t) v(t)
p∗ v(t)dt
s–
p
×
E\Sx
˜ St
g(y) v(y)
p∗ v(y)dy
q(p–s)
p u(t)dt
(G(t))q
q
. (.)
Proof The caseu(x) > follows from [], Lemma .(b), under the substitutions (.). For the caseu(x)≥, we modify the proof of Theorem . in the following way. Let <s<p and <τ< . Setuτ(x,s) =u(x) +ρτ(x,s), whereρτ(x,s) > and satisfies the condition
E\Sx
˜ St
g(y) v(y)
p∗ v(y)dy
q(p–s)
p ρτ(t,s)
(G(t))qdt≤τ
p– p–s
–q p∗
˜ Sx
g(t) v(t)
p∗ v(t)dt
q(–s)
p
Suchρτ(x,s) exists. We haveuτ(x,s) > onx∈E. Moreover,
˜
AτW(s,p,q)≤ ˜AW(s,p,q) +τ/q
p– p–s
–/p∗
, (.)
where A˜τW(s,p,q) is obtained from A˜W(s,p,q) by making the change in (.): u(t)−→ uτ(t,s). Obviously,uτ(x,s) >u(x). Applying the preceding result of the caseu(x) > to uτ(x,s), we get
E
G(x)
˜ Sx
g(t)f(t)dt
q u(x)dx
/q
≤
E
G(x)
˜ Sx
g(t)f(t)dt
q
uτ(x,s)dx
/q
≤
inf <s<p
˜
AτWs,p,qp– p–s
/p∗
E
f(x)pv(x)dx /p
≤ ˜AτW(s,p,q)
p– p–s
/p∗
E
f(x)pv(x)dx /p
. (.)
Taking ‘inf<s<p’ for both sides of (.), we get
E
G(x)
˜ Sx
g(t)f(t)dt
q u(x)dx
/q
≤ ˜Aτ W(p,q)
E
f(x)pv(x)dx /p
. (.)
Here
˜
Aτ
W(p,q) =<inf s<p
˜
Aτ W(s,p,q)
p– p–s
/p∗ .
From (.), we obtainA˜τW(p,q)≤ ˜AW(p,q) +τ/q. Takingτ →+for both sides of (.),
we get the desired inequality. This completes the proof.
3 Extensions and new proofs of (1.8)
To derive the extensions of (.), we need the following lemma.
Lemma . Let <p<∞,v(x) > ,g(t) > ,and <G(x) <∞,where G(x)is defined by (.).Ifsupx∈E{g(x)/v(x)}<∞,then,for all t∈E,
lim →+
G(t)
˜ St
g(y) v(y)
p–
g(y)dy
=
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
p
. (.)
Proof Letα≥supx∈E{g(x)/v(x)}. Without loss of generality, we may assumeα> . We first consider the case thatS˜tg(y)|log(
g(y)
v(y))|dy<∞. Let
h() = G(t)
˜ St
g(y) v(y)
/(p–)
We have
˜ St
g(y) v(y)
/(p–)
g(y)dy≤α/(p–)G(t) <∞,
soh() is well defined and has a finite value. For∈[,p/) and <τ<min(p/ –,), it follows from the mean value theorem that
h(+τ) –h()
τ =
G(t)
˜ St
τ
g(y) v(y)
+τ p––τ
–
g(y) v(y)
p–
g(y)dy
= p
G(t)
˜ St
(p–)
g(y) v(y)
/(p–) g(y)
logg(y)
v(y) dy, (.)
where:=(y) lies betweenand+τ. We know that
χS˜
t(y)
(p–)
g(y) v(y)
/(p–) g(y)log
g(y) v(y) ≤
αχS˜
t(y)g(y)
(p–)
log
g(y) v(y) ∈L
(E,dy).
By (.) and the Lebesgue dominated convergence theorem,his differentiable on [,p/). In addition,
h() = lim τ→+
h(+τ) –h()
τ =
p (p–)G(t)
˜ St
g(y) v(y)
/(p–) g(y)
logg(y) v(y) dy.
Thus,
lim →+log
G(t)
˜ St
g(y) v(y)
/(p–) g(y)dy
/
= lim →+
logh() –logh()
= d
d
logh() ==
h() h() =
pG(t)
˜ St
g(y)
logg(y) v(y) dy.
We get the desired result for the caseS˜
tg(y)|log(
g(y)
v(y))|dy<∞. Next, consider the case
˜
Stg(y)|log(
g(y)
v(y))|dy=∞. This implies
∞=
g(y)log
g(y) v(y) dy+
g(y)log
g(y)
v(y) dy, (.)
where={y∈ ˜St:g(y)/v(y)≤}and={y∈ ˜St:g(y)/v(y) > }. We have
g(y)log
g(y)
v(y) dy≤(logα)G(t) <∞.
Combining this with (.), we find thatg(y)|log(gv((yy)))|dy=∞. This leads us to
˜ St
g(y)
logg(y)
v(y) dy= –
g(y)log
g(y) v(y) dy+
g(y)log
g(y)
We shall show
lim →+
G(t)
˜ St
g(y) v(y)
/(p–) g(y)dy
/ = .
If so, the desired equality follows. Let <<p/ andy∈ ˜St. By the mean value theorem, we get
g(y) v(y)
/(p–)
– = p
(p–)
g(y) v(y)
/(p–) logg(y)
v(y)
for some∈(,). This implies
G(t)
˜ St
g(y) v(y)
/(p–) g(y)dy
= +
p G(t)
˜ St
(p–)
g(y) v(y)
/(p–) g(y)
logg(y)
v(y) dy . (.)
By Fatou’s lemma, we get
lim inf →+
p G(t)
˜ St
(p–)
g(y) v(y)
/(p–) g(y)log
g(y) v(y) dy
≥
pG(t)
˜ St
lim inf →+
g(y) v(y)
/(p–) g(y)log
g(y) v(y) dy
=
pG(t)
˜ St
g(y)log
g(y)
v(y) dy=∞.
Like (.), decompose the integralS˜t(· · ·) as the sum
(· · ·) +
(· · ·). For theterm, we have
p G(t)
(p–)
g(y) v(y)
/(p–) g(y)log
g(y) v(y) dy
≤αlogα
pG(t)
g(y)dy≤αlogα p <∞,
which implies
lim →+
p G(t)
˜ St
(p–)
g(y) v(y)
/(p–) g(y)
logg(y)
v(y) dy= –∞.
From (.) and the fact thatlim→( +θ)/=eθfor anyθ∈R, we get
lim sup →+
G(t)
˜ St
g(y) v(y)
/(p–) g(y)dy
/
≤lim sup →+ ( +
θ)/=eθ
Lemma . may be false for the case that supx∈Eg(x)/v(x) =∞. A counterexample is given as follows. Consider n= , t= , g(t) = , and v(x) =∞m=e–mχ
(m–
m,
m](x) +
χR\
m≥(m–m,m](x). We have
g(y) v(y)
/(p–)
g(y)dy=
v(y)/(–p)dy≥ ∞
m=
me
m
p– =∞ ( <<p/)
and
g(y)
logg(y) v(y) dy=
log
v(y)dy= ∞
m=
m<∞.
From these, we know that (.) is false for this example.
Now, we go back to the investigation of the first part of (.). Set
˜
DPS:=sup x∈E
G(x)p
˜ Sx
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q p
u(t)dt
q
,
whereG(x) is defined by (.). The caseg(t) = ofD˜PSreduces toD∗PSmentioned in (.). We shall establish the following result, which extends the first inequality in (.) from u(x) > andg(t) = tou(x)≥ and thoseg(t) subject to the condition (.). This extension gives the Persson-Stepanov-type estimate of the modular-type operator norm of the gen-eral geometric mean operator corresponding tog(t). In particular,g(t) can be of the form g(t) =|˜St|s–. An elementary calculation of this case will lead us to the Levin-Cochran-Lee-type inequality. We leave such a calculation to the readers. Our result partially generalizes the sufficient parts of [], Theorem ., [], and [], Theorem .(a).
Theorem . Let <p≤q<∞,u(x)≥,v(x) > ,g(t) > ,and <G(x) <∞,where G(x)
is defined by(.).If(.)is true andD˜PS<∞,then(.)holds for C≤e/pD˜PS.
Proof Let(s) =es,k(x,t) =g(t)/G(x), andf(t)−→logf(t). The proof is the same as to prove thatK∗≤e/pD˜PS. We first assume thatsupx∈E{g(x)/v(x)}<∞. Consider the case thatuis bounded on˜randu(x) = onE\ ˜r, wherer≥ and˜r={x∈E: /r≤ x ≤r}. By (.)-(.) and Theorem ., we know that
K∗≤lim inf →+
(p/)∗A˜PS(p/,q/)/, (.)
provided that the term (· · ·)/in (.) is finite for all sufficiently small> . By an elemen-tary calculation, we obtainlim→+((p/)∗)/=lim→+( p
p–)/
=e/p. On the other hand,
let <<p. Thenp/> andq/> . Moreover, we have (p/)∗=p/(p–), so
g(t) v(t)
(p/)∗ v(t) =
g(t) v(t)
p/(p–) v(t) =
g(t) v(t)
It follows from the definition ofA˜PS(p/,q/) that
˜
APS(p/,q/)/=sup x∈E
˜ Sx
g(t) v(t)
/(p–) g(t)dt
–/p
×
˜ Sx
G(t)
˜ St
g(y) v(y)
/(p–) g(y)dy
q/ u(t)dt
/q
. (.)
We have assumed thatu(x) = onE\ ˜r. Moreover, fort∈ ˜Sx, we have
G(t)
˜ St
g(y) v(y)
/(p–)
g(y)dy≤
sup y∈˜Sx
g(y) v(y)
/(p–)
G(t)
˜ St
g(y)dy
=
sup y∈˜Sx
g(y) v(y)
/(p–)
.
These imply
˜
APS(p/,q/)
/
≤
˜ B/r
g(t) v(t)
/(p–) g(t)dt
–/p
×
sup y∈E
g(y) v(y)
/(p–)
˜ r
u(t)dt /q
<∞, (.)
whereB˜ρ={x∈E:x ≤ρ}. The above argument guarantees the validity of (.). Now, we try to estimate the limit infimum given in (.). It suffices to show that
lim inf →+
˜
APS(p/,q/)/≤ ˜DPS. (.)
Clearly, the term (S˜
x(· · ·))
–/pin (.) becomes bigger wheneverxwithx>ris replaced
by rx/x. Moreover, the term (S˜
x{· · · }
q/u(t)dt)/q in (.) is zero for x< /rand it
keeps the same value for the change:xwithx>r−→rx/x. Hence, the term ‘supx∈E’ in (.) can be replaced by ‘supx∈ ˜
r’. By the Heine-Borel theorem, we can choose <m<p/,
αm> , andx,xm∈ ˜r, such thatm→,αm→,xm→x, and the following inequality
holds for allm:
˜
APS(p/m,q/m)
/m
≤
˜ Sxm
g(t) v(t)
m/(p–m) g(t)dt
–/p
×
˜ Sxm
G(t)
˜ St
g(y) v(y)
m/(p–m) g(y)dy
q/m
u(t)dt /q
+αm. (.)
We have
χS˜xm(t)
g(t) v(t)
m/(p–m)
g(t)≤χB˜r(t)
sup y∈E
g(y) v(y) +
By the Lebesgue dominated convergence theorem, we infer that
lim
m→∞
˜ Sxm
g(t) v(t)
m/(p–m)
g(t)dt –/p
=
˜ Sx
lim m→∞
g(t) v(t)
m/(p–m)
g(t)dt –/p
=G(x)
–/p
. (.)
Similarly, the hypotheses onu(t) andg(t)/v(t) imply
χS˜xm(t)
G(t)
˜ St
g(y) v(y)
m/(p–m)
g(y)dy
q/m
u(t)
≤χB˜
r(t)
sup y∈E
g(y) v(y)
q/(p–m)
G(t)
˜ St
g(y)dy
q/m
u(t)
≤χB˜r(t)
sup y∈E
g(y) v(y) +
q/p
u(t)∈L(E,dt).
Applying the Lebesgue dominated convergence theorem again, it follows from Lemma . that
lim
m→∞
˜ Sxm
G(t)
˜ St
g(y) v(y)
m/(p–m) g(y)dy
q/m
u(t)dt /q
=
˜ Sx
lim m→∞
G(t)
˜ St
g(y) v(y)
m/(p–m)
g(y)dy
q/m
u(t)dt /q
=
˜ Sx
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q/p u(t)dt
/q
. (.)
Putting (.)-(.) together yields (.). This finishes the proof for thoseuandvwith the restrictions stated above. Now, we come back to the proof of the case u≥ and supx∈E{g(x)/v(x)}<∞. Letur(x) =min{u(x),r}χ˜r(x), wherer= , , . . . . By the preceding
result,
E
exp
G(x)
˜ Sx
g(t)logf(t)dt q
ur(x)dx /q
≤e/pD˜PS(r)
E
f(x)pv(x)dx /p
(f> ), (.)
where
˜
DPS(r) =sup x∈E
G(x)–
p
˜ Sx
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q p
ur(t)dt
q
.
Next, we deal with the case supx∈Eg(x) <∞. Letv(x) =v(x) + /, where= , , . . . . Thensupx∈E{g(x)/v(x)}<∞for each. By the preceding result,
E
exp
G(x)
˜ Sx
g(t)logf(t)dt q
u(x)dx /q
≤e/pD˜ PS
E
f(x)pv(x)dx /p
(f> ), (.)
where
˜
DPS=sup x∈E
(G(x))p
˜ Sx
exp
G(t)
˜ St
g(y)log
g(y) v(y) dy
q p
u(t)dt
q
.
We havev(x)≥v(x), soD˜PS≤ ˜DPS. This says that (.) can be replaced by (.):
E
exp
G(x)
˜ Sx
g(t)logf(t)dt q
u(x)dx /q
≤e/pD˜PS
E
f(x)pv(x)dx /p
(f> ). (.)
We shall claim thatv(x) in (.) can be replaced byv(x). Without loss of generality, we may assumeE(f(x))pv(x)dx<∞. Set
fr(x) =χB˜
r(x)min
f(x),r+χE\ ˜B
r(x)h(x) (r= , , . . .),
whereB˜ρis defined before andh:E→(,∞) is chosen so that
h(x)≤minf(x), and
E
h(x)pv(x)dx<∞.
Replacingf in (.) byfr, we get
E
exp
G(x)
˜ Sx
g(t)logfr(t)dt q
u(x)dx /q
≤e/pD˜PS
E
fr(x)pv(x)dx /p
. (.)
For eachr, we have
E
fr(x)
p
v(x)dx=
˜ Br
minf(x),rpv(x)dx+
E\ ˜Br
h(x)pv(x)dx
≤
E
f(x)pv(x)dx+
˜ Br
rpdx+
E
and|fr(x)|pv(x)≤(fr(x))pv
(x) for= , , . . . . Applying the Lebesgue dominated
conver-gence theorem to the right hand side of (.), we get
E
exp
G(x)
˜ Sx
g(t)logfr(t)dt q
u(x)dx /q
≤e/pD˜PS
E
fr(x)pv(x)dx /p
. (.)
By definition,fr(x)↑f(x) asr→ ∞. Applying the monotone convergence theorem to both sides of (.), the right hand side tends to
e/pD˜PS
E
f(x)pv(x)dx /p
(asr→ ∞)
and the left hand side has the limit
E
exp
G(x)rlim→∞
˜ Sx
g(t)logfr(t)dt q
u(x)dx /q
. (.)
Letx∈E. SinceS˜
xg(t)logf(t)dtis well defined, the following equality makes sense:
˜ Sx
g(t)logf(t)dt=
˜ Sx
g(t)logf(t)+dt–
˜ Sx
g(t)logf(t)–dt,
whereξ+=max(ξ, ) andξ–=min(–ξ, ). Considerr≥max(x, ). By the monotone con-vergence theorem,
˜ Sx
g(t)logfr(t)dt=
˜ Sx
g(t)logminf(t),rdt
=
˜ Sx
g(t)minlogf(t)+,logrdt–
˜ Sx
g(t)logf(t)–dt
−→
˜ Sx
g(t)logf(t)+dt–
˜ Sx
g(t)logf(t)–dt=
˜ Sx
g(t)logf(t)dt.
Inserting this limit in (.) yields the desired inequality. This finishes the proof.
Theorem . gives a new proof of [], Theorem .(a). In the following, we shall display another example to show how (.) works well for the estimate of Opic-Gurka type. Set
˜
DOG(s) :=sup x∈E
G(x)s–p
×
E\Sx
G(t) –sq
p
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q p
u(t)dt
q
,
corresponding tog(t). In particular,g(t) can be of the formg(t) =|˜St|s–, which leads us to the Levin-Cochran-Lee-type inequality. Our result partially generalizes the sufficient parts of [] and [], Theorem .(b).
Theorem . Let <p≤q<∞,u(x)≥,v(x) > ,g(t) > , and <G(x) <∞,where G(x)is defined by(.).If(.)is true andD˜OG(s) <∞for some s> ,then(.)holds for C≤infs>e(s–)/pD˜
OG(s).
Proof Let(s) =es,k(x,t) =g(t)/G(x), andf(t)−→logf(t). The proof is similar to The-orem .. We shall show thatK∗≤infs>e(s–)/pD˜OG(s). To observe the proof of
Theo-rem ., we find that it suffices to prove this inequality for the case:uis bounded on˜r, u(x) = onE\ ˜r, andsupx∈E{g(x)/v(x)}<∞, where˜ris defined in the proof of Theo-rem .. It follows from (.)-(.) and TheoTheo-rem . that
K∗≤ inf <<p
˜
AW(p/,q/)/
= inf <<p
inf <s<p/
p–
p–s
/–/p
˜
AW(s,p/,q/)
/
≤inf s>
lim inf →+
p–
p–s
/–/p
˜
AW(s,p/,q/)
/
. (.)
Fors> , we havelim→+(p– p–s)
/–/p=e(s–)/p. We shall prove
lim inf →+
˜
AW(s,p/,q/)
/
≤ ˜DOG(s).
If so, the desired inequality follows from (.). Let <<p/s. We have
˜
AW(s,p/,q/)
/ =sup
x∈E
˜ Sx
g(t) v(t)
p–
g(t)dt
s–
p
×
E\Sx
˜ St
g(y) v(y)
p–
g(y)dy
q(p–s)
p u(t)dt
(G(t))q/
/q
. (.)
The term ‘(S˜
x(· · ·)) s–
p ’ in (.) increases in x. On the other hand, the term
‘(E\Sx{· · · }q(p–s)/(p) u(t)dt
(G(t))q/)/q’ in (.) is zero for x>r and it keeps the same value
for the change:xwithx< /r−→(/r)x/x. These imply that the term ‘supx∈E’ in (.) can be replaced by ‘supx∈ ˜
r’. By the Heine-Borel theorem, we can choose <m <p/s,
αm> , andx,xm∈ ˜rsuch thatm→,αm→,xm→x, and the following inequality
holds for allm:
˜
AW(s,p/m,q/m)
/m
≤
˜ Sxm
g(t) v(t)
m p–m
g(t)dt
s–
p
×
E\Sxm
˜ St
g(y) v(y)
m p–m
g(y)dy
q(p–ms)
mp u(t)dt
(G(t))q/m
/q
For the first integral in (.), we have
lim
m→∞
˜ Sxm
g(t) v(t)
m p–m
g(t)dt
s–
p
=
˜ Sx
lim m→∞
g(t) v(t)
m p–m
g(t)dt
s–
p
=G(x)
s–
p . (.)
As for the second integral, it follows from Lemma . that
˜ St
g(y) v(y)
m p–m
g(y)dy
q(p–ms)
mp
(G(t))q/m
=
˜ St
g(y) v(y)
m p–m
g(y)dy
–qs/p
G(t)
˜ St
g(y) v(y)
m p–m
g(y)dy q/m
−→G(t) –qs
p
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q p
asm→ ∞. (.)
Moreover, formlarge enough,
χE\Sxm(t)
˜ St
g(y) v(y)
m p–m
g(y)dy
q(p–ms)
mp u(t)
(G(t))q/m
≤
sup x∈E
g(y) v(y) +
q/p
χ˜
r(t)G(t)
–qs/pu(t)∈L(E,dt).
Integrating the left hand side of (.) with respect tou(t)dtfirst and then applying the Lebesgue dominated convergence theorem, we obtain
lim
m→∞
E\Sxm
˜ St
g(y) v(y)
m p–m
g(y)dy
q(p–ms)
mp u(t)dt
(G(t))q/m
/q
=
E\Sx lim m→∞
(G(t))q/m
˜ St
g(y) v(y)
m p–m
g(y)dy
q(p–ms)
mp
u(t)dt /q
=
E\Sx (G(t))qs/p
exp
G(t)
˜ St
g(y)
logg(y) v(y) dy
q p
u(t)dt
q
. (.)
Putting (.), (.), and (.) together yields the desired inequality. This finishes the
proof.
For other estimates of Hardy-type inequalities, we may use a similar limit process to Theorems . and . to get the corresponding Pólya-Knopp inequalities.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Author details
1Center for General Education, Hsuan Chuang University, Hsinchu, 30092, Taiwan, Republic of China.2Municipal Jianguo
High School, Taipei, 10066, Taiwan, Republic of China.
Acknowledgements
The first author was supported in part by the Ministry of Science and Technology, Taipei, ROC, under Grants
Most103-2115-M-364-001 and Most104-2115-M-364-001. We express our gratitude to Professor Lars-Erik Persson and the reviewers for their valued comments in developing the final version of the article.
Received: 18 May 2015 Accepted: 12 October 2015
References
1. Chen, C-P, Lan, J-W, Luor, D-C: The best constants for multidimensional modular inequalities over spherical cones. Linear Multilinear Algebra62(5), 683-713 (2014). doi:10.1080/03081087.2013.777438
2. Persson, L-E, Stepanov, VD: Weighted integral inequalities with the geometric mean operator. J. Inequal. Appl.7(5), 727-746 (2002)
3. Wedestig, A: Weighted Inequalities of Hardy-type and their Limiting Inequalities. Dissertation, Luleå University of Technology, Luleå (2003)
4. Gupta, B, Jain, P, Persson, L-E, Wedestig, A: Weighted geometric mean inequalities over cones inRN. J. Inequal. Pure
Appl. Math.4(4), 68 (2003)
5. Opic, B, Gurka, P: Weighted inequalities for geometric means. Proc. Am. Math. Soc.120(3), 771-779 (1994) 6. Chen, C-P, Lan, J-W, Luor, D-C: Multidimensional extensions of Polya-Knopp-type inequalities over spherical cones.
