Journal of Inequalities in Pure and
Applied Mathematics
A SAWYER DUALITY PRINCIPLE FOR RADIALLY MONOTONE
FUNCTIONS IN Rn
volume 6, issue 2, article 44,
2005.
S. BARZA, M. JOHANSSON AND L.-E. PERSSON
Department of Engineering Sciences Physics and Mathematics
Karlstad University
SE - 651 88 Karlstad
Sweden
EMail: sorina.barza@kau.se
Department of Mathematics
Luleå University of Technology
SE - 971 87 Luleå
Sweden
EMail: marjoh@sm.luth.se
EMail: larserik@sm.luth.se
Received 09 February, 2004;
accepted 14 February, 2005.
Communicated by: C. Niculescu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
035-05
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Abstract
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
p p
f↓r
Rn f v
is proved as a generalization of the usual Sawyer duality principle. Some applications involving boundedness of certain integral operators are also given.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
2000 Mathematics Subject Classification: Primary 26D15, 47B38; Secondary
26B99, 46E30.
Key words: Duality theorems, radially monotone functions, weighted inequalities.
We thank Professor Javier Soria for some good advice connected to the second
proof of Theorem 3.1.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
The Duality Principle for Radially Monotone Functions . . . . 10
4
Further Results and Applications . . . . . . . . . . . . . . . . . . . . . . . . 19
References
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1.
Introduction
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 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 < ∞,
1
= 1q − p1 , it was shown in [3] that
r
R
1q

u
p1 + sup 
0≤h↓
v
Rn
≈ RR
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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1
f qu q
Rn
sup R
1
pv p
0≤f ↓
f
n
R
R
(1.1)
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
n
Z
∞
! rq
Z
u
0
Dh,t
!− pr  p1
Z
d
v
 ,
Dh,t
where u and v are weights, i.e. positive and locally integrable functions on Rn .
For the case 0 < p ≤ q < ∞ c.f. [2].
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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 3.3 and c.f. also [9] for the
case n = 1). We illustrate this useful technique for the identity operator (see
Corollary 4.2) and for the Hardy integral operator over the n-dimensional sphere
(see Corollary 4.4).
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
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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to be measurable. Constants, denoted a, b, c, are always positive and may be
p
different at different places. Moreover n ∈ Z+ , 1 ≤ p < ∞, p0 = p−1
(p0 = ∞
if p = 1), v (x) and u (x) are weights (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.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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2.
Preliminary Results
Consider the Hardy integral operator H of the form
Z
(Hf ) (x) =
f (y)dy,
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]):
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Theorem 2.1. Let W and U be weights on Rn and 1 < p ≤ q < ∞.
(i) The inequality
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Z
W (x)
(2.1)
Rn
1q
q
Z
f (y)dy
Z
≤c
dx
p
U (x)f (x)dx
Rn
B(x)
a := sup
α>0
1q Z
W (x)dx
|x|≥α
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holds for f ≥ 0 if and only if
Z
p1
U
1−p0
10
p
(x)dx
<∞.
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Moreover, if c is the smallest constant for which (2.1) holds, then
0 p10
1
q
a ≤ c ≤ ap p .
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(ii) The inequality
Z
W (x)
(2.2)
Rn
1q
q
Z
dx
f (y)dy
Rn \B(x)
Z
p1
p
≤c
U (x)f (x)dx
Rn
holds if and only if
1
q
Z
Z
b := sup
α>0
W (x)dx
|x|≤α
U
1−p0
(x)dx
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
1
p0
<∞.
|x|≥α
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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
is satisfied.
Rn \B(x)
Rn
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Proof. Apply Theorem 2.1 (ii) with W (x) = v(x), U (x) = v 1−p (x)
and q = p. Denote
Z
(2.4)
vn (s) =
sn−1 v(sσ)dσ,
R
v(y)dy
B(x)
p
Σn−1
where Σn−1 as usual denotes the unit sphere in Rn , s ∈ R. We note that
p1
Z
v(x)dx
b = sup
α>0
v(y)dy
|x|≤α
|x|≥α

1
p
Z
α>0
|x|≥α
1
p
Z
∞
α
1
Z
tn−1
Z
∞
Z
∞
Z
α
1
p
Z
∞
= sup V (α)
α>0
v (x) dx
vn (s) ds
−p0 Z
0
Z
α
p
Contents
v (tδ) dtdδ
−p0
t
! 10
tn−1 v (tσ) dσ dt
Σn−1
p
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! 10
p
vn (s) ds
vn (t) dt
t
−p0 +1 !
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0
d
dt
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! 10
−p0
t
0
t
= sup V p (α)
α>0
Sorina Barza, Maria Johansson
and Lars-Erik Persson
p
vn (s) ds
α
1
Z
Σn−1
= sup V p (α)
α>0
 10
0
= sup V (α)
α>0
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
v(x)dx
!−p0
vn (s) ds
= sup V (α) 
p
B(x)
|x|
Z
! 10
−p0
Z
Z
Z
vn (s) ds
0
! 10
1
dt
1 − p0
p
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≤ sup
α>0
= sup
α>0
1
1
(p0 − 1) p0
1
(p0 − 1)
1
p0
1
p
Z
V (α)
α
−p00+1
p
vn (t) dt
0
1
V p (α) V
1
−1
p0
(α) =
1
1
< ∞.
(p0 − 1) p0
Therefore, by Theorem 2.1 (ii), (2.3) holds with a constant c ≤
(p0 −1)
p and the proof is complete.
1
1
1
p0
1
(p0 ) p0 p p =
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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3.
The Duality Principle for Radially Monotone
Functions
R
In the sequel
we
sometimes
delete
the
integration
variable
and
write
e.g.
fg
Rn
R
instead of Rn f (x) g (x)
dx
when
it
cannot
be
misinterpreted.
Moreover,
as
R
usual, kgk1 = kgkL1 = Rn |g (x)| dx. Our main result in this section reads:
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
I1 = kvk1 kgk1
and
I2 =
p0
−p0
G (t) V (t)
10
p
v (t) dt
Rn
with V (t) =
R
v (x) dx and G (t) =
B(t)
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Title Page
−1
p
Z
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
R
B(t)
g (x) dx.
Remark 1. 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.
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Proof. If f ≡ c > 0, then
R
C (g) ≥ sup f =c
R
fg
kgk1
1 = I1 .
p1 =
p
kvk1
f pv
Rn
+
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
p (x) v (x) dx p
0≤f ↓
f
n
R
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
R
R
Z p0 ! p10
hV G
v
1
1
G
n
= sup R R v V 1 =
v
= I2 .
p h≥0
p
V
hV p
Rn
v p
Rn
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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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
=
v(t)dt dx
f (x) g (x)
V (x)
B(x)
0
Σn−1
Z
Z
1
f (x) g (x)
=
v (t)
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
1
g (x)
dx
V (x)
|x|>|t|
Z ∞
Z
1
n−1
=
s
g (sσ)
dsdσ
V (sσ)
|t|
Σn−1
Z ∞ Z
1
n−1
=
s g (sσ) dσ
ds
V (s)
|t|
Σn−1
Z ∞
1
=
gn (s)
ds
V (s)
|t|
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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∞
Z s
1
=
gn (z) dz
V (s) 0
|t|
Z s
Z
Z ∞
1
n−1
+
s v (sσ) dσ
gn (z) dz ds
2
|t| V (s)
Σn−1
0
Z s
Z ∞
Z ∞
1
1
≤
gn (z) dz +
vn (s)
gn (z) dz ds.
2
V (∞) 0
0
|t| V (s)
|
{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 )
Z
p1 Z
p
≤
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p
(K1 + K2 ) v
Contents
Rn
Rn
f pv
≤
10
p0
f v
Z
p1
Z
Rn
0
Rn
K1p v
10
Z
p
+
Rn
0
K2p v
10 !
p
.
Moreover,
Rn
p0
K1 v
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Rn
Rn
Z
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
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10
Z
p
=
Rn
1
V (∞)
Z
gn (z) dz
0
! 10
p0
∞
p
v (x) dx
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1
=
V (∞)
∞
Z
Z
s
0
−1+ p10
n−1
10
Z
p
g (sσ) dσds
v (x) dx
Rn
Σn−1
− p1
= kvk1
kgk1 = kvk1 kgk1 = I1 ,
and, according to Lemma 2.2,
Z
10
p
p0
K2 v
Rn
Z
Z
∞
=
Rn
|t|
Z
Z
∞
1
vn (s)
2
V (s)
Z
s
=
Rn
|t|
Σn−1
Z
Z
=
Rn
≤ p0
Z
Rn
Rn \B(t)
0
V (t)−p
s
Z
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
0
n−1
v (sσ)
V (s)
2
v (x)
V 2 (x)
Z
! 10
p0
p
gn (z) dz ds
v (t) dt
p0 ! p10
z n−1 g (zδ) dzdδ dσds
v
Z s Z
0
Σn−1
! 10
p0
Z
g (y) dy
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Title Page
p
v (t) dt
B(x)
10
p
g v (t) dt
= p0 I2 .
B(|x|)
The upper bound of C (g) follows by combining the last estimates and the proof
is complete.
Remark 2. 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:
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max (I1 , I2 ) ≤ C (g) ≤ I1 + p0 I2 .
J. Ineq. Pure and Appl. Math. 6(2) Art. 44, 2005
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In particular we have the following useful information:
Corollary 3.2. Let the assumptions in Theorem 3.1 be satisfied and
Then
R
n fg
0
(3.3)
I2 ≤ sup R R
p1 ≤ p I2 .
p
f ↓r
f v
Rn
R
Rn
v = ∞.
The proof above is self-contained and does not depend directly on the onedimensional 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.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Proof. Make the following changes of variables
Title Page
t = sσ and x = yτ,
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
n−1
R
1 = R R
p1
R
pv p
∞
f
p (sσ) v (sσ) sn−1 dσds
P
f
Rn
0
n−1
R∞
e (s) ds
f (s) G
= 0
p1 ,
R∞
p
e
f (s) V (s) ds
0
(3.4)
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and hence, by using the (Sawyer) one-dimensional result we find that
R
(3.5)
fg
1 ≈
R
pv p
f
n
R
Rn
Z
− p1 Z
∞
∞
Ve (s) ds
0
e
G (s) ds
0

Z

+
∞
R
R
0
s
0
e (y) dy
G
s
0
Ve (y) dy
 p10
p0

p0 Ve (s) ds
:= I.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Moreover,
Z ∞Z
I=
v (sσ) s
0

Z

+
n−1
!− p1 Z
dσ ds
g (sσ) s
P
0
n−1
R R
s
dτ dy
0
n−1
R R
p0
s
0
n−1 dτ dy
P
v
(yτ
)
y
0
n−1
Z
− p1 Z
v (t) dt
=
g (t) dt
∞
g (yτ ) y
P
Rn
(3.6)
Rn
R
p0
B(t)
g (x) dx
B(t)
v (x) dx
R
 p10
Z
P

v (sσ) sn−1 dσds
n−1
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 p10

p0 v (t) dt .
The proof follows by combining (3.5) and (3.6).
dσds
n−1
Rn

Z

+
n−1
P
p0
n−1
Sorina Barza, Maria Johansson
and Lars-Erik Persson
!
∞Z
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For completeness and later use in our applications we also state the corresponding result for radially increasing functions in Rn :
Theorem 3.3. 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
p1 ≈ I1 + I3 ,
p
f ↑r
f v
Rn
where
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
−1
p
I1 = kvk1 kgk1 ,
and
Z
p0
−p0
G1 (t) V1 (t)
I3 =
10
p
v (t) dt
,
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Rn
with G1 (t) =
R
Rn \B(t)
g (x) dx and V1 (t) =
R
Rn \B(t)
v (x) dx.
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
B(x)
Rn
By using this estimate the proof follows similarly as the proof of Theorem 3.1
so we leave out the details.
Remark 3. In fact, similar to Remark 2 and Corollary 3.2, we find that
max (I1 , I3 ) ≤ D (g) ≤ I1 + p0 I3
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R
and if, in addition to the assumptions in Theorem 3.3, Rn v = ∞, then
R
n fg
0
I3 ≤ sup R R
p1 ≤ p I3 .
p
f ↑r
f v
Rn
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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4.
Further Results and Applications
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
q
p1
p
≤c
(T f (x)) u (x) dx
(4.1)
Z
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
f (x)v (x) dx
Rn
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
holds for all f ↓ r if and only if
Z
Z
∗
T g (y) dy
(4.2)
Rn
! 10
p0
p
V
−p0
Title Page
(x) v (x) dx
B(x)
Contents
Z
≤c
0
0
g q (x)u1−q (x) dx
10
q
Rn
holds for every positive measurable function g.
Proof. Assume first that (4.1) holds for all 0 < f ↓ r. Then, by using Corollary
3.2, duality and Hölder’s inequality, we find that
Z
Rn
Z
T ∗ g (y) dy
p0
0
V −p (x) v (x) dx
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B(x)
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∗
n f (x) T g (x) dx
≤ sup RR
1
f ↓r
f p (x) v (x) dx p
Rn
R
n T f (x) g (x) dx
= sup R R
1
f ↓r
f p (x) v (x) dx p
Rn
R
q
R
Rn
(T f (x)) u (x) dx
1q
q0
R
Rn
g (x) u
≤ sup
R
f p (x) v (x) dx
10
q
q0
1−q 0
g (x) u
(x) dx
.
f ↓r
−q 0
q
10
q
(x) dx
p1
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Rn
Z
=c
Rn
On the contrary assume that (4.2) holds for all g ≥ 0. Then, by using Corollary
3.2 again, we have
Z
10
p
0
0
q
1−q
p0 c
(g (x)) (u (x))
dx
Rn
≥ p0
Z
Rn
Z
! 10
p0
p
0
∗
−p
T g (t) dt
V
(x) v (x) dx
B(x)
R
T f (x) g (x) dx
f (x) T ∗ g (x) dx
Rn
≥ R
p1 = R
1
f p (x) v (x) dx
f p (x) v (x) dx p
Rn
Rn
R
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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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
Rn
p
f (x) v (x) dx
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,
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where
1
g (x) u− q (x)
h (x) = .
q0 q10
R − 1q
g (x) u (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. By modifying the proof above we see that a similar duality result
also holds for positive
R radially
R increasing functions. In fact,
R in this case we
just need to replace B(x) by Rn \B(x) and V (x) by V1 (x) = Rn \B(x) v (x) dx in
(4.2).
For example when T is the identity operator we obtain the following:
Corollary
4.2. Let 1 < p ≤ qR < ∞ and suppose that u, v are weights on Rn
R
with Rn v = ∞ and V (x) = B(x) v (y) dy.
a) The following conditions are equivalent:
q
f u
(4.4)
Rn
is satisfied for all 0 ≤ f ↓ r.
1q
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Title Page
Contents
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i) The inequality
Z
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Z
≤c
p
f v
Rn
p1
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ii) The inequality
g (y) dy
(4.5)
Rn
! 10
p0
Z
Z
p
V
−p0
(x) v (x) dx
B(x)
Z
q0
≤c
g (x)u
1−q 0
10
q
(x) dx
Rn
holds for all g ≥ 0.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
iii)
sup
v(x)dx
α>0
b) If V1 (x) =
Z
R
Rn \B(x)
Sorina Barza, Maria Johansson
and Lars-Erik Persson
<∞.
u(x)dx
B(α)
B(α)
v (t) dt, then for 0 ≤ f ↑ r (4.4) is equivalent to
p0
Z
g (y) dy
Rn
1q
− p1 Z
Z
Rn \B(x)
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Contents
! 10
p
0
V1−p
(x) v (x) dx
Z
q0
≤c
g (x)u
1−q 0
10
q
(x) dx
Rn
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which in turn is equivalent to
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− p1 Z
Z
sup
α>0
v(x)dx
Rn \B(α)
1q
u(x)dx
< ∞.
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Rn \B(α)
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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)
0
0
with f replaced by g, q by p0 , p by q 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
Z
1q
10 Z
Z
1q
v
p
|x|<α
0
sup
vV −p
u
= sup
u
< ∞.
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 and Theorem 2.1
(ii).
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Remark 5. The equivalence of i) and iii) can also be proved using the technique
from [2]. For the case q < p cf. also [3].
Sorina Barza, Maria Johansson
and Lars-Erik Persson
The next result concerns the multidimensional Hardy operator, defined on
the cone of radially decreasing functions in Rn .
Title Page
Proposition R4.3. Let 1 < p ≤ q < ∞ and
R suppose that u and v are weights
n
on R with Rn v = ∞ and V (x) = B(x) v (x) dx. If 0 ≤ f is a radially
decreasing function in Rn , then
Z Z
q
1q
p1
Z
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:
Z
− p1 Z
1q
q
v(x)dx
(4.7)
sup
u(x) |B (x)| dx
<∞
α>0
B(α)
B(α)
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and
Z
|B (x)| V
(4.8) sup
α>0
p0
−p0
10 Z
p
(x) v (x) dx
1q
u(x)dx
< ∞.
Rn \B(α)
B(α)
Here, as usual, |B (x)| denotes the Lebesgue measure of the ball with center
at 0 and with radius |x|.
Remark 6. For the case n = 1 this result is due to V. Stepanov (see [9], Theorem 2).
Proposition 4.3 can be proved by using the method of reduction to the onedimensional case but here we present an independent proof:
R
Proof. Since T f (x) = B(x) f (t) dt its conjugate T ∗ is defined by T ∗ g (x) =
R
g (t) dt. Assume first that (4.7) and (4.8) hold. We note that, according
Rn \B(x)
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
B(x)
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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|z|
J. Ineq. Pure and Appl. Math. 6(2) Art. 44, 2005
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|x|
Z
Z
=
s
0
n−1
Σn−1
Z
|x|
|x|
sn−1
= |Σn−1 |
∞
gn (t) dt dσds
s
!
Z
Z
|x|
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
gn (t) dt
|x|
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
tn−1 g (tδ) dδdt
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Σn−1
∞
Z
|x|
Σn−1
∞
ds
g (tδ) dδ dt
t
dσds
0
|x|
Z
0
Z
=
Z
0
Z
0
n−1
!
∞
|x|
0
t
= |Σn−1 |
Z
gn (t) dtds
s
Z
|x|
gn (t) dt ds +
0
Z
Z
Z
Σn−1
Z
=
B(x)
|
|B (y)| g (y) dy + |B (x)|
{z
} |
Rn \|B(x)|
{z
I1
I2
g (y) dy
}
Title Page
Contents
= I1 (x) + I2 (x) .
JJ
J
This means that (4.6) holds if and only if
Z
p0
(I1 (x) + I2 (x)) V
(4.9)
−p0
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10
p
(x) v (x) dx
Close
Rn
Z
≤c
II
I
q0
g (x) U
1−q 0
10
Quit
q
(x) dx
.
Rn
Moreover, by Theorem 2.1 (i) with q replaced by p0 and p replaced by q 0 , we
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have
Z
p0
−p0
(I1 (x)) V
(4.10)
10
p
(x) v (x) dx
Rn
|B (y)| g (y) dy
=
Rn
! 10
p0
Z
Z
p
V
−p0
(x) v (x) dx
B(x)
Z
q0
≤c
(|B (x)| g (x))
Rn
Z
=c
! 1q
1−q 0
u (x)
|B (x)|q
10
q
q0
1−q 0
g (x) u (x)
dx
,
0
dx
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Rn
which holds because, according to (4.7),
sup
α>0
−p0
Z
Z
v (x)
Rn \B(α)
Title Page
v (y) dy
!
1
p0
Contents
dx
JJ
J
B(x)

Z
×
1−q 0
u (y)
!1−q
0
|B (y)|q
R
− p1
Z
v (y) dy
B(α)
 1q
dy 
II
I
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B(α)
≤ sup
α>0
1
(p0 − 1) p0
B(α)
Close
q
u (y) |B (y)| dy
1q
Quit
< ∞.
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Similarly, according to Theorem 2.1 (ii),
Z
p0
(I2 (x)) V
(4.11)
−p0
10
p
(x) v (x) dx
Rn
Z
|B (x)|
=
≤c
g (y) dy
p
V
−p0
(x) v (x) dx
Rn \B(x)
Rn \B(α)
Z
! 10
−p0
Z
q0
g (x) u
1−q 0
10
q
(x) dx
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Rn
because
Z
0
α>0
0
|B (x)|p V −p (x) v (x) dx
sup
10 Z
Sorina Barza, Maria Johansson
and Lars-Erik Persson
1q
p
u (x) dx
<∞
Rn \B(α)
B(α)
Title Page
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,
Z
p0
(I1 (x)) V
(4.12)
Rn
−p0
10
Z
p
(x) v (x) dx
≤c
q0
g (x) u
1−q 0
10
q
(x) dx
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.
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Remark 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 4.3:
Corollary 4.4. Let 1 < p ≤ q < ∞ and let f (x) be positive and radially
decreasing in Rn . Then the Hardy inequality
Z
Rn
1
|B (x)|
q
Z
f (y) dy
b
1q
Z
|B (x)| dx
a
p
≤c
p1
f (x) |B (x)| dx
Rn
B(x)
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
holds if and only if −1 < a < p − 1, −1 < b < q − 1 and
a+1
b+1
=
.
p
q
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Proof. Apply Proposition 4.3 with v (x) = |B (x)|a and u (x) = |B (x)|b . We
note that some straightforward calculations give
Title Page
(4.13)
− p1 Z
Z
1q
v (x) dx
(4.14)
u (x) dx
B(α)
≈α
−(a+1)n
(b+1)n
+ q
p
B(α)
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whenever a > −1, b > −1 and
Close
Z
(4.15)
0
0
|B (x)|p V −p (x) v (x) dx
10 Z
1q
p
u(x)dx
Rn \B(α)
B(α)
≈α
an−anp0 +n
+ bn−nq+n
q
p0
+ b+1
− a+1
p
q
= αn(
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J. Ineq. Pure and Appl. Math. 6(2) Art. 44, 2005
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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 8. 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].
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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References
[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.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
Title Page
Contents
[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).
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[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.
A Sawyer Duality Principle for
Radially Monotone Functions in
Rn
Sorina Barza, Maria Johansson
and Lars-Erik Persson
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