Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 2, Article 44, 2005
A SAWYER DUALITY PRINCIPLE FOR RADIALLY MONOTONE FUNCTIONS
IN Rn
SORINA BARZA, MARIA JOHANSSON, AND LARS-ERIK PERSSON
D EPARTMENT OF E NGINEERING S CIENCES P HYSICS AND M ATHEMATICS
K ARLSTAD U NIVERSITY
SE - 651 88 K ARLSTAD
S WEDEN
sorina.barza@kau.se
D EPARTMENT OF M ATHEMATICS
L ULEÅ U NIVERSITY OF T ECHNOLOGY
SE - 971 87 L ULEÅ
S WEDEN
marjoh@sm.luth.se
larserik@sm.luth.se
Received 09 February, 2004; accepted 14 February, 2005
Communicated by C. Niculescu
A BSTRACT. Let f be a non-negative function on Rn , which is radially monotone (0 < f ↓ r).
For 1 < p < ∞, g ≥ 0 and v a weight function, an equivalent expression for
R
n fg
sup R R
1
f ↓r
f pv p
Rn
is proved as a generalization of the usual Sawyer duality principle. Some applications involving
boundedness of certain integral operators are also given.
Key words and phrases: Duality theorems, radially monotone functions, weighted inequalities.
2000 Mathematics Subject Classification. Primary 26D15, 47B38; Secondary 26B99, 46E30.
1. I NTRODUCTION
An explicit duality principle for positive decreasing functions of one variable was proved by
E. Sawyer in [8], and also some applications are well-known. Here we also refer to the useful
proof and ideas concerning this principle presented by V. Stepanov in [9]. See also [6] and the
proof and comments given there. Moreover, it is natural to look for extensions to functions of
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
We thank Professor Javier Soria for some good advice connected to the second proof of Theorem 3.1.
035-05
2
S ORINA BARZA , M ARIA J OHANSSON ,
AND
L ARS -E RIK P ERSSON
several variables. Such generalizations were recently obtained in [1], [2] and [3]. To be able to
describe some of these generalizations we require some notations: We write
Rn = {(x1 , x2 , . . . , xn ) : i = 1, 2, . . . , n}
and R1 = R. If f : Rn → R is decreasing (increasing) separately in each variable we write
0 ≤ f ↓ (0 ≤ f ↑). A set D ⊂ Rn is said to be decreasing if its characteristic function χD
is decreasing, and clearly if 0 ≤ h ↓ and t > 0, then the set Dh,t = {x ∈ Rn : h (x) > t} is
decreasing. For 0 < q < p < ∞, 1r = 1q − p1 , it was shown in [3] that

!− pr  p1
! rq
1q
1q
R
R
Z
Z ∞ Z
q
u
n f u
Rn
 ,
d
v
(1.1)
sup RR
u
p1 ≈ R
p1 + sup 
p
0≤f ↓
0≤h↓
D
0
D
h,t
h,t
f v
v
Rn
Rn
where u and v are weights, i.e. positive and locally integrable functions on Rn . For the case
0 < p ≤ q < ∞ c.f. [2].
If q = 1 < p < ∞ and u = g ≥ 0, then (1.1) with n = 1 is a variant of the duality theorem
given in [8] (c.f. also [9]).
An explicit form of (1.1) for n ∈ Z+ was given in [1] for g = u, but only when v is of product
type, that is weights of the form v (x) = v (x1 , x2 , . . . , xn ) = v1 (x1 ) v2 (x2 ) · · · vn (xn ), vi ≥ 0,
i = 1, 2, . . . , n.
We say that f : Rn → R is a radially decreasing (increasing) function if f (x) = f (y) when
|x| = |y| and f (x) ≥ f (y) (f (x) ≤ f (y)) when |x| < |y| and we write f ↓ r and f ↑ r ,
respectively (see also [7] for further explanations and applications of these notions).
In this paper we prove a duality formula of the type (1.1) for radially decreasing functions
and with general weights (see Theorem 2.1). We also state the corresponding result for radially increasing functions (see Theorem 3.1). In particular, these results imply that we can
describe mapping properties of operators defined on the cone of such monotone functions between weighted Lebesgue spaces. Moreover, we point out that these results can also be used to
describe mapping properties between some corresponding general weighted multidimensional
Lebesgue spaces (see Theorem 3.5 and c.f. also [9] for the case n = 1). We illustrate this useful
technique for the identity operator (see Corollary 4.3) and for the Hardy integral operator over
the n-dimensional sphere (see Corollary 4.8).
The paper is organized as follows: In Section 2 we present some known and easily derived
results required in the proofs of our statements. The announced duality theorems are stated and
proved in Section 3. Finally, in Section 4 the afore mentioned applications are given and also
some further results and remarks.
Notations and conventions: Throughout this paper all functions are assumed to be measurable. Constants, denoted a, b, c, are always positive and may be different at different places.
p
(p0 = ∞ if p = 1), v (x) and u (x) are weights
Moreover n ∈ Z+ , 1 ≤ p < ∞, p0 = p−1
(positive and measurable functions on Rn ) and
Lpv = Lpv (Rn )
(
=
f : Rn → R, measurable s.t.
Z
|f (x)|p v (x) dx
p1
)
<∞ .
Rn
Inequalities such as (2.1) are interpreted to mean that if the right hand side is finite, then so
is the left hand side and the inequality holds. The symbol ≈ (c.f. (1.1)) means that the quotient
of the right and left hand sides is bounded from above and below by positive constants, while
expressions such as 0 · ∞ = ∞ · 0 are taken as zero. Other notations will be introduced when
required.
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
3
2. P RELIMINARY R ESULTS
Consider the Hardy integral operator H of the form
Z
f (y)dy,
(Hf ) (x) =
B(x)
where B (x) is the ball in Rn centered at the origin and with radius |x|. We need the following
well-known n-dimensional form of Hardy’s inequality (see [5]):
Theorem 2.1. Let W and U be weights on Rn and 1 < p ≤ q < ∞.
(i) The inequality
Z
Z
(2.1)
W (x)
Rn
1q
q
f (y)dy
Z
≤c
dx
U (x)f (x)dx
Rn
B(x)
holds for f ≥ 0 if and only if
Z
1q Z
a := sup
W (x)dx
α>0
p1
p
|x|≥α
U
1−p0
10
p
<∞.
(x)dx
|x|≤α
Moreover, if c is the smallest constant for which (2.1) holds, then
0
1
1
a ≤ c ≤ ap p0 p q .
(ii) The inequality
Z
Z
(2.2)
W (x)
Rn
1q
q
f (y)dy
Z
≤c
dx
Rn \B(x)
holds if and only if
Z
b := sup
α>0
p1
p
U (x)f (x)dx
Rn
1q Z
W (x)dx
|x|≤α
U
1−p0
10
p
(x)dx
<∞.
|x|≥α
Moreover, if c is the smallest constant for which (2.2) holds, then
0
1
1
b ≤ c ≤ bp p0 p q .
In particular we need the following special case of Theorem 2.1 (ii):
R
Lemma 2.2. Let v be a weight function, V (x) = B(x) v(y)dy and 1 < p < ∞. Then for f ≥ 0
Z
p
Z
Z
(2.3)
v(x)
f (y)dy dx ≤ p
f p (x)V p (x) v 1−p (x)dx
Rn
Rn \B(x)
Rn
is satisfied.
Proof. Apply Theorem 2.1 (ii) with W (x) = v(x), U (x) = v 1−p (x)
q = p. Denote
Z
(2.4)
vn (s) =
sn−1 v(sσ)dσ,
R
v(y)dy
B(x)
p
and
Σn−1
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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4
S ORINA BARZA , M ARIA J OHANSSON ,
L ARS -E RIK P ERSSON
AND
where Σn−1 as usual denotes the unit sphere in Rn , s ∈ R. We note that
p1
Z
b = sup
v(x)dx
α>0
v(y)dy
|x|≤α
|x|≥α

Z
1
p
α>0
|x|≥α
Z
1
p
∞
Z
n−1
Z
∞
Z
α>0
α
Z
1
p
∞
1
(p0 − 1) p0
1
(p0 − 1)
1
p0
v (tσ) dσ dt
! 10
p
vn (s) ds
vn (t) dt
t
−p0 +1 !
Z
d
dt
α
1
p
0
= sup V (α)
α>0
t
−p0
= sup V (α)
α>0
n−1
! 10
Σn−1
t
Z
p
v (tδ) dtdδ
−p0 Z
0
∞
! 10
−p0
t
0
t
Z
α
1
p
= sup
v (x) dx
vn (s) ds
α>0
α>0
p
vn (s) ds
Σn−1
= sup V (α)
≤ sup
Z
t
α
1
p
 10
0
= sup V (α)
α>0
v(x)dx
!−p0
vn (s) ds
= sup V (α) 
p
B(x)
|x|
Z
! 10
−p0
Z
Z
vn (s) ds
0
Z
1
p
V (α)
α
! 10
p
1
dt
1 − p0
−p00+1
p
vn (t) dt
0
1
V p (α) V
1
−1
p0
1
(α) =
1
< ∞.
(p0 − 1) p0
Therefore, by Theorem 2.1 (ii), (2.3) holds with a constant c ≤
1
1
1
(p0 −1) p0
1
(p0 ) p0 p p = p and the
proof is complete.
3. T HE D UALITY P RINCIPLE FOR RADIALLY MONOTONE FUNCTIONS
R
In
the
sequel
we
sometimes
delete
the
integration
variable
and
write
e.g.
f g instead of
Rn
R
RRn f (x) g (x) dx when it cannot be misinterpreted. Moreover, as usual, kgk1 = kgkL1 =
|g (x)| dx. Our main result in this section reads:
Rn
Theorem 3.1. Suppose that v is a weight on Rn and 1 < p < ∞. If f is a positive radially
decreasing function on Rn and g a positive measurable function on Rn , then
R
n fg
(3.1)
C (g) := sup R R
1 ≈ I1 + I2 ,
pv p
f ↓r
f
Rn
where
−1
I1 = kvk1p kgk1
and
Z
I2 =
p0
−p0
G (t) V (t)
R
R
with V (t) = B(t) v (x) dx and G (t) = B(t) g (x) dx.
10
p
v (t) dt
Rn
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
5
Remark 3.2. Theorem 2.1 for the case n = 1 is simply Theorem 1 in [8]. However, our proof
below is based on the technique in [9] and our investigation in Section 3.
Proof. If f ≡ c > 0, then
R
fg
kgk1
1 =
1 = I1 .
p
p
f =c R
p
kvk
f v
1
Rn
+
R
Moreover, we use the test function f (x) = |t|>|x| h(t)dt, where h (t) ≥ 0 (note that f is
radially decreasing), Lemma 2.2 and the usual duality in Lpv -spaces to find that
R
n f (x) g (x) dx
C (g) = sup R R
1
0≤f ↓
f p (x) v (x) dx p
Rn
R R
h
(t)
dt
g (x) dx
Rn
|x|<|t|
≥ sup R
p
p1
h≥0 R
h
(t)
dt
v
(x)
dx
Rn
|x|<|t|
R
R
h
(t)
g (x) dxdt
n
R
|x|<|t|
= sup p
p1
R
h≥0 R
h (t) dt v (x) dx
Rn
|x|<|t|
R
R
h |x|<|t| g
1
Rn
≥ sup R
1
p h≥0
p V p v 1−p p
h
n
RR hV G
v
1
n
= sup R R v V 1
p h≥0
hV p
v p
C (g) ≥ sup Rn
=
1
p
Rn
+
v
Z p0 ! p10
G
v
= I2 .
V
Rn
To prove the upper bound of C (g) we use the monotonicity of f . Since f is radially decreasing
we have
Z
Z
1
gf =
gf V
V
n
Rn
Z
ZR∞ Z
1
=
f (x) g (x)
v(t)dt dx
V (x)
0
Σn−1
B(x)
Z
Z
1
=
v (t)
f (x) g (x)
dx dt
V (x)
Rn
|x|>|t|
Z
Z
1
(3.2)
≤
v (t) f (t)
g (x)
dx dt.
V (x)
Rn
|x|>|t|
To estimate the inner integral we define gn (s) and vn (s) analogously to (2.4), note that V (x) =
V (|x|) and find that
Z
Z ∞
Z
1
1
n−1
g (x)
dx =
s
g (sσ)
dsdσ
V (x)
V (sσ)
|x|>|t|
|t|
Σn−1
Z ∞ Z
1
n−1
=
s g (sσ) dσ
ds
V (s)
|t|
Σn−1
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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6
S ORINA BARZA , M ARIA J OHANSSON ,
∞
Z
=
gn (s)
|t|
AND
L ARS -E RIK P ERSSON
1
ds
V (s)
∞
Z s
1
=
gn (z) dz
V (s) 0
|t|
Z
Z s
Z ∞
1
n−1
+
s v (sσ) dσ
gn (z) dz ds
2
Σn−1
0
|t| V (s)
Z s
Z ∞
Z ∞
1
1
≤
gn (z) dz +
vn (s)
gn (z) dz ds.
2
V (∞) 0
|t| V (s)
0
|
{z
} |
{z
}
K1
K2
Hence the inner integral can be estimated by K1 + K2 and by substituting this into (3.2) and
applying Hölder’s and Minkowski’s inequalities we get
Z
Z
fg ≤
f v (K1 + K2 )
Rn
Rn
Z
p1 Z
p
≤
10
p
p0
f v
(K1 + K2 ) v
Rn
Rn
Z
f pv
≤
p1
Z
p0
Z
p
K1 v
Rn
Rn
10
10 !
p
p0
+
Rn
K2 v
.
Moreover,
Z
10
Rn
K1 v
=
Rn
=
! 10
p0
Z ∞
p
1
gn (z) dz
v (x) dx
V (∞) 0
Z ∞ Z
Z
10
p
n−1
s g (sσ) dσds
v (x) dx
Z
p
p0
1
V (∞)
0
−1+ p10
= kvk1
Rn
Σn−1
− p1
kgk1 = kvk1 kgk1 = I1 ,
and, according to Lemma 2.2,
Z
Rn
p0
K2 v
10
Z
Z
p
∞
=
Rn
|t|
Z
Z
∞
1
vn (s)
2
V (s)
Z
=
Rn
|t|
Σn−1
Z
Z
=
Rn
≤p
0
Z
Rn \B(t)
−p0
v (x)
V 2 (x)
Z
! 10
p0
p
gn (z) dz ds
v (t) dt
0
sn−1 v (sσ)
V 2 (s)
p0 ! p10
z n−1 g (zδ) dzdδ dσds
v
Z s Z
0
Σn−1
! 10
p0
Z
V (t)
Rn
s
Z
g (y) dy
p
v (t) dt
B(x)
10
p
= p0 I2 .
g v (t) dt
B(|x|)
The upper bound of C (g) follows by combining the last estimates and the proof is complete.
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
7
Remark 3.3. According to our proof we see that the duality constant C (g) in (3.1) can in fact
be estimated in the following more precise way:
max (I1 , I2 ) ≤ C (g) ≤ I1 + p0 I2 .
In particular we have the following useful information:
Corollary 3.4. Let the assumptions in Theorem 3.1 be satisfied and
R
n fg
0
(3.3)
I2 ≤ sup R R
p1 ≤ p I2 .
f ↓r
f pv
Rn
R
Rn
v = ∞. Then
The proof above is self-contained and does not depend directly on the one-dimensional result
(only on our investigations in Section 2 and similar arguments as V.D. Stepanov used when he
proved the case n = 1). Here we give another shorter proof where we directly use the (Sawyer)
one-dimensional result.
Proof. Make the following changes of variables
t = sσ and x = yτ,
(3.4)
P
where s, y ∈ (0, ∞) and σ, τ ∈ n−1 . By using the fact that f (sσ) = f (s) since f is radial,
we get:
R∞R
R
P
f (sσ) g (sσ) sn−1 dσds
f
g
0
n−1
Rn
p1 = R R
p1
R
p
∞
f
v
p (sσ) v (sσ) sn−1 dσds
P
n
f
R
0
n−1
R∞
e (s) ds
f (s) G
= 0
p1 ,
R∞
p
e
f (s) V (s) ds
0
and hence, by using the (Sawyer) one-dimensional result we find that
R
Z ∞
− p1 Z ∞
fg
Rn
e (s) ds
(3.5) R
Ve (s) ds
G
p1 ≈
p
0
0
f v
Rn
 p10

p0
Z ∞ RsG
e (y) dy
0


+
R
p0 Ve (s) ds := I.
s
0
Ve (y) dy
0
Moreover,
Z
∞
!− p1
Z
v (sσ) sn−1 dσ ds
I=
P
0
0
n−1

Z

+
0
∞
R R
s
0
R R
s
0
Z
(3.6)
=
Rn
∞
Z
− p1
v (t) dt
g (yτ ) y
P
n−1
dτ dy
!
Z
g (sσ) sn−1 dσds
P
n−1
 p10
p0
Z
n−1
p0
v (yτ ) y n−1 dτ dy

Z
Z

g (t) dt + 
P

v (sσ) sn−1 dσds
n−1
P
n−1
Rn
The proof follows by combining (3.5) and (3.6).
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
Rn
R
p0
B(t)
g (x) dx
B(t)
v (x) dx
R
 p10

p0 v (t) dt .
http://jipam.vu.edu.au/
8
S ORINA BARZA , M ARIA J OHANSSON ,
AND
L ARS -E RIK P ERSSON
For completeness and later use in our applications we also state the corresponding result for
radially increasing functions in Rn :
Theorem 3.5. Suppose that v is a weight on Rn and 1 < p < ∞. If f is a positive radially
increasing function on Rn and g a positive measurable function on Rn , then
R
n fg
D (g) := sup R R
1 ≈ I1 + I3 ,
pv p
f ↑r
f
Rn
where
−1
I1 = kvk1p kgk1 ,
and
Z
p0
I3 =
−p0
10
p
v (t) dt
,
G1 (t) V1 (t)
R
R
with G1 (t) = Rn \B(t) g (x) dx and V1 (t) = Rn \B(t) v (x) dx.
Rn
Proof. We now use Theorem 2.1 (i) (instead of (ii) as in the proof of Lemma 2.2) and obtain as
in the proof of (2.3):
Z
p
Z
Z
v(x)
f (y)dy dx ≤ p
f p (x)V1p (x) v 1−p (x)dx.
Rn
Rn
B(x)
By using this estimate the proof follows similarly as the proof of Theorem 3.1 so we leave out
the details.
Remark 3.6. In fact, similar to Remark 3.3 and Corollary 3.4, we find that
max (I1 , I3 ) ≤ D (g) ≤ I1 + p0 I3
R
and if, in addition to the assumptions in Theorem 3.5, Rn v = ∞, then
R
n fg
0
I3 ≤ sup R R
p1 ≤ p I3 .
p
f ↑r
f v
Rn
4. F URTHER R ESULTS AND A PPLICATIONS
Let T be an integral operator defined on the cone of functions f : Rn → R, which are radially
decreasing (0 < f ↓ r) and let T ∗ be the adjoint operator. Then our results imply the following
useful duality result:
R
Theorem 4.1. Let 1 < p, q < ∞, u, v be weights on Rn with Rn v (x) dx = ∞. Then the
inequality
Z
1q
Z
p1
q
p
(4.1)
(T f (x)) u (x) dx
≤c
f (x)v (x) dx
Rn
Rn
holds for all f ↓ r if and only if
Z
Z
∗
T g (y) dy
(4.2)
Rn
! 10
p0
p
V
−p0
(x) v (x) dx
B(x)
Z
≤c
q0
g (x)u
1−q 0
10
q
(x) dx
Rn
holds for every positive measurable function g.
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
9
Proof. Assume first that (4.1) holds for all 0 < f ↓ r. Then, by using Corollary 3.4, duality and
Hölder’s inequality, we find that
p0
Z Z
0
∗
T g (y) dy
V −p (x) v (x) dx
Rn
B(x)
R
f (x) T ∗ g (x) dx
Rn
≤ sup R
1
p (x) v (x) dx p
f ↓r
f
n
RR
n T f (x) g (x) dx
= sup R R
1
p (x) v (x) dx p
f ↓r
f
n
R
10
q
1q R
R
−q 0
q
q0
(T f (x)) u (x) dx
g (x) u q (x) dx
Rn
Rn
≤ sup
1
R
p (x) v (x) dx p
f ↓r
f
n
R
Z
10
q
q0
1−q 0
=c
g (x) u
(x) dx
.
Rn
On the contrary assume that (4.2) holds for all g ≥ 0. Then, by using Corollary 3.4 again, we
have
! 10
Z
10
p0
Z Z
p
p
0
0
q
1−q
0
0
∗
−p0
pc
(g (x)) (u (x))
dx
≥p
T g (t) dt
V
(x) v (x) dx
Rn
Rn
B(x)
R
f (x) T ∗ g (x) dx
T f (x) g (x) dx
Rn
≥ R
1
p1 = R
f p (x) v (x) dx p
f p (x) v (x) dx
Rn
Rn
R
Rn
for each fixed 0 < f ↓ r. Therefore we have
Z
Z
1
0
(4.3)
h (x) T f (x) u q (x) dx ≤ p c
Rn
where
p
p1
f (x) v (x) dx
,
Rn
1
g (x) u− q (x)
h (x) = .
q0 q10
R 1
g (x) u− q (x) dx
Rn
Since khkLq0 = 1 we obtain (4.1) by taking the supremum in (4.3) and usual duality in Lp spaces.
Remark 4.2. By modifying the proof above we see that a similar duality result alsoR holds
for positive radially increasing functions. In fact, in this case we just need to replace B(x) by
R
R
and V (x) by V1 (x) = Rn \B(x) v (x) dx in (4.2).
Rn \B(x)
For example when T is the identity operator we obtain the following:
n
Corollary 4.3.
R Let 1 < p ≤ q < ∞ and suppose that u, v are weights on R with
and V (x) = B(x) v (y) dy.
a) The following conditions are equivalent:
i) The inequality
Z
1q
Z
p1
q
p
(4.4)
f u
≤c
f v
Rn
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
R
Rn
v=∞
Rn
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10
S ORINA BARZA , M ARIA J OHANSSON , AND L ARS -E RIK P ERSSON
(4.5)
is satisfied for all 0 ≤ f ↓ r.
ii) The inequality
! 10
Z
p0
Z Z
p
0
−p
≤c
g (y) dy
V
(x) v (x) dx
Rn
q0
g (x)u
1−q 0
10
q
(x) dx
Rn
B(x)
holds for all g ≥ 0.
iii)
sup
Z
Rn \B(x)
v (t) dt, then for 0 ≤ f ↑ r (4.4) is equivalent to
! 10
0
Z
p
Z
g (y) dy
Rn
B(α)
B(α)
R
Rn \B(x)
10
p
0
V1−p
q0
≤c
(x) v (x) dx
g (x)u
1−q 0
q
(x) dx
Rn
which in turn is equivalent to
Z
− p1 Z
sup
v(x)dx
α>0
<∞.
u(x)dx
v(x)dx
α>0
b) If V1 (x) =
1q
− p1 Z
Z
Rn \B(α)
1q
u(x)dx
< ∞.
Rn \B(α)
Proof. a) The equivalence of i) and ii) is just a special case of Theorem 4.1. Moreover, the fact
that ii) and iii) are equivalent follows from Theorem 2.1 (i) with f replaced by g, q by p0 , p by q 0 ,
0
0
W by vV −p and U by u1−q . In fact, then (4.5) is equivalent to (note that (1 − q) (1 − q 0 ) = 1)
R
− p1
1q
Z
10 Z
Z
1q
v
p
|x|<α
−p0
u
= sup
u
< ∞.
sup
vV
1
α>0
α>0 (p0 − 1) p0
|x|<α
|x|<α
|x|>α
The proof of b) follows similarly by just using Remark 4.2 and Theorem 2.1 (ii).
Remark 4.4. The equivalence of i) and iii) can also be proved using the technique from [2].
For the case q < p cf. also [3].
The next result concerns the multidimensional Hardy operator, defined on the cone of radially
decreasing functions in Rn .
n
Proposition
4.5. Let 1 <
R
R p ≤ q < ∞ and suppose that u and v are weights on nR with
v = ∞ and V (x) = B(x) v (x) dx. If 0 ≤ f is a radially decreasing function in R , then
Rn
Z Z
q
1q
Z
p1
p
(4.6)
f (y) dy u (x) dx
≤c
f (x)v (x) dx
Rn
Rn
B(x)
is satisfied if and only if the following conditions hold:
1q
Z
− p1 Z
q
u(x) |B (x)| dx
<∞
(4.7)
sup
v(x)dx
α>0
B(α)
B(α)
and
Z
(4.8)
|B (x)| V
sup
α>0
p0
B(α)
−p0
10 Z
1q
p
(x) v (x) dx
u(x)dx
< ∞.
Rn \B(α)
Here, as usual, |B (x)| denotes the Lebesgue measure of the ball with center at 0 and with
radius |x|.
Remark 4.6. For the case n = 1 this result is due to V. Stepanov (see [9], Theorem 2).
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
11
Proposition 4.5 can be proved by using the method of reduction to the one-dimensional case
but here we present an independent proof:
R
R
Proof. Since T f (x) = B(x) f (t) dt its conjugate T ∗ is defined by T ∗ g (x) = Rn \B(x) g (t) dt.
Assume first that (4.7) and (4.8) hold. We note that, according to Theorem 4.1, (4.6) for 0 <
f ↓ r is equivalent to (4.2) for arbitrary g ≥ 0. Moreover, to be able to characterize weights for
which (4.2) is satisfied, we first compute:
Z
Z
Z
∗
T g=
g (y) dy dz
B(x)
B(x)
|y|>|z|
Z ∞ Z
Z
n−1
=
t g (tδ) dδdt dz
B(x)
|z|
Σn−1
Z ∞
Z
=
gn (t) dt dz
|z|
B(x)
|x|
Z
Z
=
s
0
n−1
gn (t) dt dσds
s
!
Z
Σn−1
Z
|x|
|x|
sn−1
= |Σn−1 |
∞
Z
|x|
t
= |Σn−1 |
s
0
|x|
Z
ds gn (t) dt + |Σn−1 |
t
dσ
Σn−1
|x|
s
n−1
0
Z
+
Z
ds
!Z
n−1
Z
t
∞
Z
|x|
Σn−1
∞
ds
gn (t) dt
|x|
g (tδ) dδ dt
Σn−1
tn−1 g (tδ) dδdt
dσds
0
|x|
Z
0
Z
=
Z
n−1
0
Z
0
gn (t) dtds
|x|
0
s
Z
!
∞
Z
gn (t) dt ds +
0
Z
|x|
Z
Σn−1
Z
Z
=
B(x)
|
|B (y)| g (y) dy + |B (x)|
{z
} |
Rn \|B(x)|
{z
I1
I2
g (y) dy
}
= I1 (x) + I2 (x) .
This means that (4.6) holds if and only if
10
Z
Z
p
p0
−p0
≤c
(4.9)
(I1 (x) + I2 (x)) V
(x) v (x) dx
q0
g (x) U
1−q 0
10
q
(x) dx
.
Rn
Rn
Moreover, by Theorem 2.1 (i) with q replaced by p0 and p replaced by q 0 , we have
Z
10
p
p0
−p0
(4.10)
(I1 (x)) V
(x) v (x) dx
Rn
|B (y)| g (y) dy
=
Rn
p
V
−p0
(x) v (x) dx
B(x)
Z
≤c
! 10
p0
Z
Z
q0
(|B (x)| g (x))
Rn
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
u (x)1−q
q0
|B (x)|
! 1q
0
dx
Z
=c
q0
1−q 0
g (x) u (x)
10
q
dx
,
Rn
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12
S ORINA BARZA , M ARIA J OHANSSON , AND L ARS -E RIK P ERSSON
which holds because, according to (4.7),
sup
α>0
−p0
Z
Z
v (x)
v (y) dy
Rn \B(α)
! 10 Z
p
dx
·
B(x)
1−q 0
u (y)
|B (y)|q
B(α)
R
≤ sup
v (y) dy
B(α)
 1q
dy 
0
− p1
Z
1q
q
u (y) |B (y)| dy
1
(p0 − 1) p0
α>0
!1−q
< ∞.
B(α)
Similarly, according to Theorem 2.1 (ii),
Z
10
p
p0
−p0
(4.11)
(I2 (x)) V
(x) v (x) dx
Rn
Z
|B (x)|
=
Rn \B(α)
Z
≤c
! 10
−p0
Z
g (y) dy
p
V
−p0
(x) v (x) dx
Rn \B(x)
q0
g (x) u
1−q 0
10
q
(x) dx
Rn
because
Z
p0
|B (x)| V
sup
α>0
−p0
1q
10 Z
p
(x) v (x) dx
u (x) dx
<∞
Rn \B(α)
B(α)
which holds by (4.8). Thus by using (4.9), Minkowski’s inequality and (4.10) – (4.11) we see
that (4.6) holds.
Now assume that (4.6) holds, i.e., that (4.9) holds. Then, in particular,
10
Z
10
Z
p
q
0
0
0
p0
(4.12)
(I1 (x)) V −p (x) v (x) dx
≤c
g q (x) u1−q (x) dx
Rn
Rn
and by using Theorem 2.1 (i) and arguing as above, we find that (4.7) holds. Moreover, (4.12)
holds also with I1 replaced by I2 so that, by using Theorem 2.1 (ii) and again arguing as in the
sufficiency part, we see that (4.8) holds. The proof is complete.
Remark 4.7. For the case when the weights are also radially decreasing or increasing some of
our results can be written in a more suitable form. Here we only state the following consequence
of Proposition 4.5:
Corollary 4.8. Let 1 < p ≤ q < ∞ and let f (x) be positive and radially decreasing in Rn .
Then the Hardy inequality
Z q
1q
Z
p1
Z
1
b
a
p
f (y) dy |B (x)| dx
≤c
f (x) |B (x)| dx
|B (x)| B(x)
Rn
Rn
holds if and only if −1 < a < p − 1, −1 < b < q − 1 and
a+1
b+1
=
.
(4.13)
p
q
Proof. Apply Proposition 4.5 with v (x) = |B (x)|a and u (x) = |B (x)|b . We note that some
straightforward calculations give
Z
− p1 Z
1q
−(a+1)n
(b+1)n
u (x) dx
≈α p + q
(4.14)
v (x) dx
B(α)
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
B(α)
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S AWYER D UALITY P RINCIPLE FOR R ADIALLY M ONOTONE F UNCTIONS
whenever a > −1, b > −1 and
10 Z
Z
p
p0
−p0
(x) v (x) dx
|B (x)| V
(4.15)
13
1q
u(x)dx
Rn \B(α)
B(α)
≈α
an−anp0 +n
+ bn−nq+n
q
p0
= αn(−
a+1
+ b+1
p
q
)
whenever a < p − 1 and b < q − 1.
Moreover, according to the estimates (4.14) and (4.15) the condition (4.13) (and only this)
gives a finite supremum. The proof is complete.
Remark 4.9. It is easy to see that Theorem 2.1 also holds if Rn is replaced by Rn+ or even some
more general cone in Rn . Therefore, by modifying our proofs, we see that all our results in this
chapter indeed hold also when Rn is replaced by Rn+ or even general cones in Rn as defined in
[4].
R EFERENCES
[1] S. BARZA, H. HEINIG AND L-E. PERSSON, Duality theorems over the cone of monotone functions and sequences in higher dimensions, J. Inequal. Appl., 7(1) (2002), 79–108.
[2] S. BARZA, L-E. PERSSON AND J. SORIA, Sharp weighted multidimensional integral inequalities
for monotone functions, Math. Nachr., 210 (2000), 43–58.
[3] S. BARZA, L-E. PERSSON AND V.D. STEPANOV, On weighted multidimensional embeddings for
monotone functions, Math. Scand., 88(2) (2001), 303–319.
[4] A. ČIŽMEŠIJA, L-E. PERSSON AND A. WEDESTIG, Weighted integral inequalities for Hardy and
geometric mean operators with kernels over cones in Rn , to appear in Italian J. Pure & Appl. Math.
[5] P. DRÁBEK, H.P. HEINIG AND A. KUFNER, Higher dimensional Hardy inequality, General Inequalities, 7 (Oberwolfach, 1995), 3–16, Internat. Ser. Numer. Math., 123, Birkhäuser, Basel, 1997.
[6] A. KUFNER AND L-E. PERSSON, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., Singapore/ New Jersey/ London/ Hong Kong, 2003.
[7] E.H. LIEB AND M. LOSS, Analysis, Graduate Studies in Mathematics, 14, AMS. (1997).
[8] E. SAWYER, Boundedness of classical operators on classical Lorentz spaces, Studia Math., 96(2)
(1990), 145–158.
[9] V.D. STEPANOV, Integral operators on the cone of monotone functions, J. London Math. Soc., 48(3)
(1993), 465–487.
J. Inequal. Pure and Appl. Math., 6(2) Art. 44, 2005
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