Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 4, Issue 4, Article 68, 2003
WEIGHTED GEOMETRIC MEAN INEQUALITIES OVER CONES IN RN
BABITA GUPTA, PANKAJ JAIN, LARS-ERIK PERSSON, AND ANNA WEDESTIG
D EPARTMENT OF M ATHEMATICS ,
S HIVAJI C OLLEGE (U NIVERSITY OF D ELHI ),
R AJA G ARDEN ,
D ELHI -110 027 I NDIA .
babita74@hotmail.com
D EPARTMENT OF M ATHEMATICS ,
D ESHBANDHU C OLLEGE (U NIVERSITY OF D ELHI ),
N EW D ELHI -110019, I NDIA .
pankajkrjain@hotmail.com
D EPARTMENT OF M ATHEMATICS ,
L ULEÅ U NIVERSITY OF T ECHNOLOGY,
SE-971 87 L ULEÅ , S WEDEN .
larserik@sm.luth.se
annaw@sm.luth.se
Received 7 November, 2002; accepted 20 March, 2003
Communicated by B. Opić
A BSTRACT
. Let 0 < p ≤ q < ∞. Let A bea measurable subset of the unit sphere in RN , let
E = x ∈ RN : x = sσ, 0 ≤ s < ∞, σ ∈ A be a cone in RN and let Sx be the part of E with
’radius’ ≤ |x| . A characterization of the weights u and v on E is given such that the inequality
Z q
q1
Z
p1
Z
1
p
exp
ln f (y)dy
v(x)dx
≤C
f (x)u(x)dx
|Sx | Sx
E
E
holds for all f ≥ 0 and some positive and finite constant C. The inequality is obtained as a
limiting case of a corresponding new Hardy type inequality. Also the corresponding companion
inequalities are proved and the sharpness of the constant C is discussed.
Key words and phrases: Inequalities, Multidimensional inequalities, Geometric mean inequalities, Hardy type inequalities,
Cones in RN , Sharp constant.
2000 Mathematics Subject Classification. 26D15, 26D07.
ISSN (electronic): 1443-5756
c 2003 Victoria University. All rights reserved.
We thank Professor Alexandra Čižmešija for some valuable advice and the referee for pointing out an inaccuracy in our original manuscript
(see Remark 4.6) and for several suggestions which have improved the final version of this paper.
118-02
2
BABITA G UPTA , PANKAJ JAIN , L ARS -E RIK P ERSSON , AND A NNA W EDESTIG
1. I NTRODUCTION
In their paper [2] J.A. Cochran and C.S. Lee proved the inequality
Z ∞
Z x
Z ∞
a+1
−ε
ε−1
a
xa f (x)dx,
(1.1)
exp εx
y ln f (y)dy x dx ≤ e ε
0
0
0
where a, ε are real numbers with ε > 0, f is a positive function defined on (0, ∞) and the
a+1
constant e ε is the best possible. This inequality, in fact, is a generalization of what sometimes
is referred to as Knopp’s inequality1 , which is obtained by taking ε = 1 and a = 0 in (1.1).
Inequalities of the type (1.1) and its analogues have further been investigated and generalized
by many authors e.g. see [1], [5] – [11], [14] and [16] – [21].
In particular, very recently A. Čižmešija, J. Pečarić and I. Perić [1, Th. 9, formula (23)]
proved an N − dimensional analogue of (1.1) by replacing the interval (0, ∞) by RN and the
means are considered over the balls in RN centered at the origin. Their inequality reads:
Z Z
Z
a+1
−ε
ε−1
a
(1.2)
exp ε |Bx |
|By | ln f (y)dy |Bx | dx ≤ e ε
f (x) |Bx |a dx,
RN
RN
Bx
where a ∈ R, ε > 0, f is a positive function on RN , Bx is a ball in RN with radius |x| , x ∈ RN ,
centered at the origin and |Bx | is its volume.
In this paper we prove a more general result, namely we characterize the weights u and v on
N
R such that for 0 < p ≤ q < ∞
Z q
1q
Z
p1
Z
1
exp
ln f (y)dy
v(x)dx
≤C
f p (x)u(x)dx
|B
|
N
N
x
R
Bx
R
holds for some finite positive constant C (See Corollary 3.3). In the case when v(x) = |Sx |a
and u(x) = |Sx |b we obtain a genuine generalization of (1.2) (see Proposition 3.6 and Remark
3.7).
In this paper we also generalize the results in another direction, namely when the geometric
averages over spheres in RN are replaced by such averages over spherical cones in RN (see
notation below). This means in particular that our inequalities above and later on also hold
N
when RN is replaced by RN
+ or even more general cones in R .
The paper is organized in the following way. In Section 2 we collect some preliminaries
and prove a new Hardy inequality that averages functions over the cones in RN (see Theorem
2.1). In Section 3 we present and prove our main results concerning (the limiting) geometric
mean operators (see Theorem 3.1 and Proposition 3.6). Finally, in Section 4 we present the
corresponding companion inequalities (see Theorem 4.1, Corollary 4.2 and Proposition 4.4).
2. P RELIMINARIES
Let ΣN −1 be the unit sphere in RN , that is, ΣN −1 = {x ∈ RN : |x| = 1}, where |x| denotes
the Euclidean norm of the vector x ∈ RN . Let A be a measurable subset of ΣN −1 , and let
E ⊆ RN be a spherical cone, i.e.,
E = x ∈ RN : x =sσ, 0 ≤ s < ∞, σ ∈ A .
Let Sx , x ∈ RN denote the part of E with ‘radius’ ≤ |x| , i.e.,
Sx = y ∈ RN : y =sσ, 0 ≤ s ≤ |x| , σ ∈ A .
1
See e.g. [15, p. 143–144] and [12]. Note however that according to G.H. Hardy [ 4, p 156] this inequality was
pointed out to him already in 1925 by G. Polya.
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
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W EIGHTED G EOMETRIC M EAN I NEQUALITIES
3
For 0 < p < ∞ and a non-negative measurable function w on E, by Lpw := Lpw (E) we denote
the weighted Lebesgue space with the weight function w, consisting of all measurable functions
f on E such that
p1
Z
p
kf kLpw =
|f (x)| w(x)dx
<∞,
E
p
and make use of the abbreviations L and kf kLp when w(x) ≡ 1.
Let S = Sx , |x| = 1. The family of regions we shall average over is the collection of dilations
of S. For x ∈ E \ {0} denote by |Sx | the Lebesgue measure of Sx . Using polar coordinates we
obtain (dσ denotes the usual surface measure on ΣN −1 )
Z |x| Z
|x|N
N −1
|Sx | =
s
dσds =
|A|.
N
0
A
Moreover, we say that u is a weight function if it is a positive and measurable function on S.
p
.
Throughout the paper, for any p > 1 we denote p0 = p−1
For later purposes but also of independent interest we now state and prove our announced
Hardy inequality.
Theorem 2.1. Let E be a cone in RN and Sx , A be defined as above. Suppose that 1 < p ≤
q < ∞ and that u, v are weight functions on E. Then, the inequality
Z Z
q
1q
Z
p1
p
(2.1)
f (y)dy v(x)dx
≤C
f (x)u(x)dx
E
Sx
E
holds for all f ≥ 0 if and only if
Z
− p1 Z
Z
1−p0
(2.2)
D := sup
u
(x)dx
v(x)
t>0
tS
tS
u
1−p0
q
(y)dy
1q
dx
< ∞.
Sx
Moreover, the best constant C in (2.1) can be estimated as follows:
D ≤ C ≤ p0 D.
Remark 2.2. Another weight characterization of (2.1) over balls in RN was proved by P.
Drábek, H.P. Heinig and A. Kufner [3] . This result may be regarded as a generalization of
the usual (Muckenhaupt type) characterization in 1-dimension (see e.g. [13]) while our result
may be seen as a higher dimensional version of another characterization by V.D. Stepanov and
L.E. Persson (see [19] , [20]).
Proof. By the duality principle (see e.g. [13]), it can be shown that the inequality (2.1) is
equivalent to that the inequality
! 10
p0
Z
10
Z Z
p
q
0
1−p
q0
1−q 0
≤C
g (x)v
(x)dx
(2.3)
g(y)dy
u
(x)dx
E
E
E\Sx
holds for all g ≥ 0 and with the same best constant C. First assume that (2.2) holds. Using polar
coordinates and putting
Z
(2.4)
ge(t) =
g(tσ)tN −1 dσ,
t ∈ (0, ∞)
A
and
Z
(2.5)
u
e(t) =
u
1−p0
N −1
(tτ )t
1−p
dτ
,
t ∈ (0, ∞)
A
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
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4
BABITA G UPTA , PANKAJ JAIN , L ARS -E RIK P ERSSON , AND A NNA W EDESTIG
we have
p0
Z Z
0
u1−p (x)dx
g(y)dy
E
E\Sx
∞
Z
∞
Z Z
Z
=
g(sσ)s
0
A
∞
Z
t
∞
Z
=
0
t
N −1
p0
dσds
0
u1−p (tτ )tN −1 dτ dt
A
p0
0
u
e1−p (t)dt.
ge(s)ds
Thus, using this, changing the order of integration and finally using Hölder’s inequality, we get
p0
Z Z
(2.6)
I :=
0
u1−p (x)dx
g(y)dy
E
E\Sx
∞
Z
=
0
Z ∞
p0 !
d
0
−
ge(s)ds
dt u
e1−p (z)dz
dt
z
t
!
p0 −1
Z ∞ Z ∞
0
ge(s)ds
ge(t)dt u
e1−p (z)dz
0
=p
Z
∞
0
=p
0
Z
z
∞ Z
0
=p
0
Z
≤p
p0 −1
Z t
1−p0
ge(s)ds
ge(t)
u
e
(z)dz dt
∞
0
A
∞
ge(s)ds
t
Z
p0 −1 Z
∞
Z Z
0
0
t
t
∞
Z
0
u
e1−p (t)dt
∞
Z
=
0
ge(s)ds
t
∞
Z
p0
∞
Z
q0
g (tτ )v
0
1−q 0
0
1−p0
u
e
(s)ds g(tτ )tN −1 dτ dt
(tτ )t
10
q
dτ dt
∞
(p0 −1)q Z
N −1
A
∞
Z
Z Z
×
0
= p0
t
A
t
ge(s)ds
t
0
1−p0
u
e
q
(s)ds
! 1q
v(tτ )tN −1 dτ dt
10
q
1
q0
1−q 0
Jq,
g (x)v
(x)dx
Z
E
where
Z
∞
Z
J=
0
(p0 −1)q Z
∞
t
ge(s)ds
0
t
1−p0
u
e
q
(s)ds
ve(t)dt
with
Z
(2.7)
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
ve(t) =
v(tτ )tN −1 dτ.
A
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W EIGHTED G EOMETRIC M EAN I NEQUALITIES
5
Using Fubini’s theorem, (2.2), (2.5) and (2.7), we get
Z
∞
∞
Z
J=
0
Z
t
"
∞
=
0
Z
"
∞
=
0
−
z
−
−
Z Z t Z
×
u
0
Z
"
∞
=
0
A
−
0
= Dq
∞
"
0
u
zS
1−p0
A
N −1
(sσ)s
!#
1−p0
q
(y)dy
Sx
q
dσds
N −1
v(tτ )t
−
ge(s)ds
z
dτ dt dz
1−p0
pq
(x)dx dz
1−p0
pq
(t)dt dz.
u
zS
(p0 −1)q !# Z
∞
Z
v(x)dx dz
!# Z
(p0 −1)q
∞
Z
d
dz
u
e
0
q
(s)ds ve(t)dtdz
(p0 −1)q
Z
d
dz
1−p0
ge(s)ds
z
×
Z ∞"
Z
0
∞
Z
d
dz
Z
≤ Dq
0
t
ge(s)ds
z
z
z Z
(p0 −1)q !#
∞
Z
Z
0
(p0 −1)q !# Z
ge(s)ds
z
d
dz
q
(p0 −1)q ! Z t
1−p0
dz
u
e
(s)ds ve(t)dt
ge(s)ds
∞
Z
d
dz
∞
Z
d
dz
−
ge(s)ds
z
z
u
e
0
Thus, using Minkowski’s integral inequality, (2.4) and (2.5) we have

Z
q
J ≤D
∞
Z
0
Z
= Dq
∞
∞
"
t
Z
0
Z
d
dz
−
z
ge(s)ds
= Dq
g(y)dy
E
1−p0
u
e
p0
Z Z
dz
0
u
e1−p (t)dt
! pq
p0
∞
t
ge(s)ds
 pq
! pq
(p0 −1)q !#
∞
(t)dt
! pq
0
u1−p (x)dx
.
E\Sx
Assume first that in (2.6) I < ∞. Then
g(y)dy
E
! 10
p0
Z Z
E\Sx
p
u
1−p0
(x)dx
0
Z
≤pD
q0
g (x)v
1−q 0
10
q
(x)dx
E
i.e., (2.3) holds for all g ≥ 0 and also the constant C in (2.3) satisfies C ≤ p0 D. For the case
I = ∞ replace g(y) by an approximating sequence gn (y) ≤ g(y) (such that the corresponding
In < ∞) and use the Monotone Convergence Theorem to obtain the result.
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
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6
BABITA G UPTA , PANKAJ JAIN , L ARS -E RIK P ERSSON , AND A NNA W EDESTIG
Conversely, suppose that (2.1) holds for all f ≥ 0. In this inequality, taking for any fixed
0
t > 0 the function ft = χtS u1−p , we find that
q
1q Z
Z Z
− p1
p
ft (y)dy v(x)dx
C≥
ft (x)u(x)dx
Sx
E
E
Z Z
≥
u
tS
1−p0
1q Z
q
(y)dy
v(x)dx
u
Sx
1−p0
− p1
(x)dx
.
tS
By taking the supremum we find that (2.2) holds and, moreover, D ≤ C. The proof is complete.
3. G EOMETRIC M EAN I NEQUALITIES
Here we prove our main geometric mean inequality by making a limit procedure in Theorem
2.1.
Theorem 3.1. Let 0 < p ≤ q < ∞ and suppose that all other assumptions of Theorem 2.1 are
satisfied. Then the inequality
q
1q
Z
p1
Z Z
1
p
ln f (y)dy
v(x)dx
(3.1)
exp
≤C
f (x)u(x)dx
|Sx | Sx
E
E
holds for all f > 0 if and only if
− p1
1q
w(x)dx
< ∞,
Z
D1 := sup |tS|
t>0
tS
where
(3.2)
w(t) := v(x) exp
1
|Sx |
Z
1
ln
dy
u(y)
Sx
pq
< ∞.
1
Moreover, the best constant C satisfies D1 ≤ C ≤ e p D1 .
Proof. It is easy to see that (3.1) is equivalent to
Z q
1q
Z
p1
Z
1
p
exp
ln f (y)dy
w(x)dx
≤C
f (x)dx
|Sx | Sx
E
E
with w(x) defined by (3.2). Let v(x) = w(x) |Sx |−q and u(x) =1 in Theorem 2.1 and choose
an α such that 0 < α < p ≤ q < ∞. Then 1 < αp ≤ αq < ∞. Now, replacing f, p, q and v(x)
by f α , αp , αq in Theorem 2.1, we find that the inequality
! 1q
Z
p1
αq
Z Z
1
α
p
w(x)dx
≤ Cα
f (x)dx
(3.3)
f (y)dy
|Sx | Sx
E
E
holds for all functions f > 0 if and only if D1 holds. Moreover, it is easy to see that (c.f. [20])
α1
p
(3.4)
D1 ≤ Cα ≤
D1 .
p−α
α1
1
p
+
By letting α → 0 in (3.3) and (3.4) we find that p−α
→ e p and
1
|Sx |
Z
α
f (y)dy
Sx
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
α1
→ exp
1
|Sx |
Z
ln f (y)dy ,
Sx
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W EIGHTED G EOMETRIC M EAN I NEQUALITIES
7
i.e. the scale of power means converge to the geometric mean, and the proof follows.
Remark 3.2. Our proof above shows that (3.1) in Theorem 3.1 may be regarded as a natural
limiting case of Hardy’s inequality (2.1) as it is in the classical one-dimensional situation. This
fact indicates that our formulation of Hardy’s inequality in Theorem 2.1 is very natural from
this point of view.
As a special case, if we take E = RN and Sx = Bx the ball centered at the origin and with
radius |x| , and |Bx | its volume, then we immediately obtain the following corollary to Theorem
3.1 that averages functions over balls in RN :
Corollary 3.3. Let 0 < p ≤ q < ∞ and u, v be weight functions in RN . Then the inequality
Z Z
p1
q
1q
Z
1
p
exp
ln f (y)dy
v(x)dx
≤C
f (x)u(x)dx
|Bx | Bx
RN
RN
holds for all f > 0 if and only if
D2 :=
sup
− p1
Z
|Bz |
v(x) exp
z∈RN \{0}
Bz
1
|Bx |
Z
1
ln
dy
u(y)
Bx
! 1q
pq
dx
< ∞.
1
Moreover, the best constant C satisfies D2 ≤ C ≤ e p D2 .
Remark 3.4. Corollary 3.3 extends a result of P. Drábek, H.P. Heinig and A. Kufner [3, Theorem 4.1], who obtained it for the case p = q = 1 and with a completely different proof.
N
Remark 3.5. Setting E = RN
+ = (x1 , . . . , xN ) ∈ R , x1 ≥ 0, . . . , xN ≥ 0 in Theorem 3.1
N
N
we obtain that Corollary 3.3 holds also for RN
+ instead of R and Bx ∩ R+ instead of Bx .
We shall now consider the special weights discussed in our introduction and in [1].
Proposition 3.6. Let 0 < p ≤ q < ∞, a, b ∈ R, ε ∈ R+ , and E, Sx be defined as in Theorem
2.1. Then
Z q
1q
Z
p1
Z
(3.5)
exp ε |Sx |−ε
|Sy |ε−1 ln f (y)dy
|Sx |a dx
≤C
f p (x) |Sx |b dx
E
Sx
E
holds for all positive functions f for some finite constant C if and only if
a+1
b+1
=
q
p
(3.6)
and the least constant C in (3.5) satisfies
1q
1q
1
1
1 b+1
p
p
− 1q b+1
− p1
p
εp
ε
e
≤C≤
ε p − q e εp .
q
q
Proof. By writing (3.5) in polar coordinates we find that
Z
0
∞
Z "
A
εN ε
exp N ε ε
t |A|
Z tZ 0
A
|A|
N
ε−1
#q
sN ε−1 ln f (sσ)dσds
Z
∞
Z
≤
0
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
A
f p (tτ )
tN a+N −1
|A|
N
b
|A|
N
! 1q
a
dτ dt
! p1
tN b+N −1 dτ dt
.
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BABITA G UPTA , PANKAJ JAIN , L ARS -E RIK P ERSSON , AND A NNA W EDESTIG
1
1
Exchanging variables, s = r ε and t = z ε we find that this inequality can be rewritten as
Z
∞
Z exp
0
A
N
|A| z N
z
Z
Z
1
ε
ln f r σ r
0
×
N −1
q
dσdr
A
|A|
N
a
z
N ( a+1
−1) N −1 1
ε
z
Z
∞
Z
dτ dz
1
ε
fp z τ
≤C
0
ε
1q
|A| b
N
A
zN (
b+1
−1
ε
) z N −1 1 dτ dz
ε
! p1
,
that is,
Z exp
(3.7)
E
1
|Sx |
q
Z
|Sx |
ln f1 (y)dy
a+1
−1
ε
1q
dx
Sx
≤C
Z
b+1 − a+1
p1
1− 1ε )
q )(
b+1
1
1
|A| ( p
−1
p
−
εq p
f1 (x) |Sx | ε
dx ,
N
E
1
where f1 (rσ) = f (r ε σ). This means that (3.5) is equivalent to (3.7) i.e., (3.1) holds with the
a+1
b+1
weights v(x) = |Sx | ε −1 and u(x) = |Sx | ε −1 . We note that for these weights we find after a
direct calculation that the constant D1 from Theorem 3.1 is
|tS|
D1 = sup t>0
a+1
− b+1
εq
εp
a+1
ε
−
q
p
1
ep(
b+1
ε
b+1
−1
ε
)
1q
−1
so we conclude that (3.6) must hold and then
b+1
D1 = e ( ε −1)
1
p
1q
p
.
q
Thus, the proof follows from Theorem 3.1.
Remark 3.7. Setting p = q = 1, a = b, we have that (3.5) implies the estimate (1.2).
Remark 3.8 (Sharp Constant). In the above proposition, if we take p = q, then a = b. In this
situation (3.5) holds with the constant C = e(b+1)/p . Indeed, this constant is sharp. In order to
show this for δ > 0, we consider the function

N
b+1
−(b+1)

|x|− p (b+1−εδ) , x ∈ S,
 e− εp |S|
fδ (x) =

N
−(b+1)
 e− b+1
εp |S|
|x|− p (b+1+εδ) , x ∈ E\S.
By using this function in (3.5), we find that
1≤
δ
RHS
≤ ep → 1
LHS
as
δ→0
and consequently the constant is sharp. Note that the sharpness of the constant for p = q, in
Proposition 3.6 has been proved in the more general setting than that in [1].
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
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W EIGHTED G EOMETRIC M EAN I NEQUALITIES
9
4. T HE C OMPANION I NEQUALITIES
We present the following result which is a companion of Theorem 3.1:
Theorem 4.1. Let 0 < p ≤ q < ∞, ε > 0, and suppose that all other hypotheses of Theorem
3.1 are satisfied. Then the inequality
Z ε
Z
−ε−1
exp ε |Sx |
(4.1)
|Sy |
E
q
ln f (y)dy
1q
v(x)dx
E\Sx
Z
≤C
p1
f (x)u(x)dx
p
E
holds for all f > 0 if and only if
− p1
Z
D3 := sup |tS|
v∗ (x) exp
t>0
tS
1
|Sx |
Z
1
ln
dy
u∗ (y)
Sx
! 1q
pq
< ∞,
dx
where
1
1
1
v∗ (y) := v(s− ε σ) s−N (1+ ε ) .
ε
1
1
1
u∗ (y) := u(s− ε σ) s−N (1+ ε ) ,
ε
1
Moreover, the constant C satisfies D3 ≤ C ≤ e p D3 .
Proof. Note that for x ∈ RN
Z
|x|
Z
tN −1 dτ dt =
|Sx | =
0
A
|x|N
|A|.
N
Now, using polar coordinates, (4.1) can be written as
Z
∞
0
Z
A
ε |A|ε tN ε
exp
N
Z
∞
Z t
A
|A|
N
−ε−1
! 1q
!q
s−N ε−1 ln f (sσ)dσds
Z
∞
Z
≤C
p
v(tτ )tN −1 dτ dt
N −1
f (tτ )u(tτ )t
0
p1
dτ dt .
A
Using the exchange of variables s = r−1/ε and t = z −1/ε we obtain
Z
0
∞
q
1q
Z Z Z z
1
1 1
1
N
exp
ln f (r− ε σ)rN −1 dσdr
v(z − ε τ )z −N (1+ ε ) z N −1 dτ dz
|A| z N A 0
ε
A
Z ∞ Z
p1
− 1ε
−N (1+ 1ε ) 1 N −1
p − 1ε
≤C
f (z τ )u(z τ )z
z
dτ dz
ε
0
A
1
and put f∗ (tτ ) = f (t− ε τ ). (4.1) can be equivalently rewritten as
Z q
1q
Z
p1
Z
1
p
≤C
f∗ (x)u∗ (x)dx .
exp
ln f∗ (y)dy
v∗ (x)dx
|Sx | Sx
E
E
Now, the result is obtained by using Theorem 3.1.
Analogously to Corollary 3.3, we can immediately obtain a special case of Theorem 4.1 that
averages functions over balls in RN centered at origin.
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
http://jipam.vu.edu.au/
10
BABITA G UPTA , PANKAJ JAIN , L ARS -E RIK P ERSSON ,
AND
A NNA W EDESTIG
Corollary 4.2. Let 0 < p ≤ q < ∞, ε > 0, and u, v be weight functions in RN . Then the
inequality
Z
ε
Z
−ε−1
exp ε |Bx |
(4.2)
RN
|By |
q
ln f (y)dy
1q
v(x)dx
RN \Bx
p1
f (x)u(x)dx
Z
p
≤C
RN
holds for all f > 0 if and only if
− p1
Z
e := sup |Bz |
B
z∈RN
v0 (x) exp
Bz
1
|Bx |
Z
1
ln (y)dy
u0
Bx
! 1q
pq
dx
< ∞,
where
1
1
1
1
1
1
u0 (x) := u(t− ε τ ) t−N (1+ ε ) ,
v0 (x) := v(t− ε τ ) t−N (1+ ε ) .
ε
ε
1
e ≤ C ≤ e p B.
e
Moreover, the best constant C satisfies B
Remark 4.3. Note that by choosing E as in Remark 3.5 we see that Corollary 4.2 in fact holds
N
also when RN is replaced by RN
+ or more general cones in R .
The corresponding result to Proposition 3.6 reads as follows and the proof is analogous.
Proposition 4.4. Let 0 < p ≤ q < ∞, ε > 0, and a, b ∈ R, and E, Sx be defined as in Theorem
2.1. Then the inequality
Z q
1q
Z
p1
Z
ε
−ε−1
a
b
p
(4.3)
exp ε |Sx |
|Sy |
ln f (y)dy |Sx | dx
≤C
f (x) |Sx | dx
E
E\Sx
E
holds for all f > 0 and some finite positive constant C if and only if
a+1
b+1
=
q
p
and the least constant C in (4.3) satisfies
− 1q
− 1q
b+1
b+1
1
1
1
1
1
p
p
−
−
+
ε p q e ( εp p ) ≤ C ≤
ε p − q e− εp .
q
q
Remark 4.5 (Sharp Constant). Analogously to Proposition 3.6, in the above proposition we
also find that if we take p = q, then a = b. In this situation (4.3) holds with the constant
C = e−(b+1)/εp and the constant is sharp. This can be shown by considering, for δ > 0, the
function
 b+1
N
−(b+1)

|x|− p (b+1−εδ) , x ∈ S
 e εp |S|
fδ (x) =

N
−(b+1)
 e b+1
p |S|
|x|− p (b+1+εδ) , x ∈ E\S.
Remark 4.6. It is tempting to think that the results in this paper hold also in general star-shaped
regions in RN (c.f. [22]) but this is not true in general as was pointed out to us by the referee.
See also [22] and note that the results there also hold at least for cones in RN .
J. Inequal. Pure and Appl. Math., 4(4) Art. 68, 2003
http://jipam.vu.edu.au/
W EIGHTED G EOMETRIC M EAN I NEQUALITIES
11
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AND
A NNA W EDESTIG
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