Journal of Inequalities in Pure and
Applied Mathematics
ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN
OPERATOR GENERATED BY THE HARDY-TYPE
INTEGRAL TRANSFORM
volume 3, issue 4, article 48,
2002.
MARIA NASSYROVA, LARS-ERIK PERSSON AND
VLADIMIR STEPANOV
Received 27 November, 2001;
accepted 29 April, 2002.
Communicated by: B. Opić
Department of Mathematics
Luleå University of Technology
S-971 87 Luleå, SWEDEN
EMail: nassm@sm.luth.se
Abstract
EMail: larserik@sm.luth.se
URL: http://www.sm.luth.se/∼ larserik/
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Computing Centre
Russian Academy of Sciences
Far-Eastern Branch
Khabarovsk 680042, RUSSIA.
EMail: stepanov@as.khb.ru
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
084-01
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Abstract
The generalized geometric mean operator
Zx
1
GK f(x) = exp
k(x, y) log f(y)dy,
K(x) 0
Rx
with K(x) := 0 k(x, y)dy is considered. A characterization of the weights u(x)
and v(x) so that the inequality
Z
0
∞
1/q
Z
(GK f(x))q u (x) dx
≤C
∞
1/p
f(x)p v(x)dx
, f ≥ 0,
0
holds is given for all 0 < p, q < ∞ both for all GK where k(x, y) satisfies the
Oinarov condition and for Riemann-Liouville operators. The corresponding stable bounds of C = kGK kLpv →Lqu are pointed out.
2000 Mathematics Subject Classification: 26D15, 26D10
Key words: Integral inequalities, Weights, Geometric mean operator, Kernels,
Riemann-Liouville operators.
This research was supported by a grant from the Royal Swedish Academy of Sciences. The research by the first named author was also partially supported by the
RFBR grants 00-01-00239 and 01-01-06038.
The research by the third named author was partially supported also by the RFBR
grant 00-01-00239 and the grant E00-1.0-215 of the Ministry of Education of Russia.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2
General Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Hardy-type Operators. Case 0 < p ≤ q < ∞ . . . . . . . . . . . . . 10
4
Hardy-type Operators. Case 0 < q < p < ∞ . . . . . . . . . . . . . 17
5
Riemann-Liouville Operators . . . . . . . . . . . . . . . . . . . . . . . . . . 26
References
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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1.
Introduction
Let R+ := [0, ∞) and let k(x, y) ≥ 0 be a locally integrable kernel defined on
R+ × R+ and such that
Z
k(x, y)dy = 1
R+
for almost all x ∈ R+ .
Denote
Z
Kf (x) :=
∞
k(x, y)f (y)dy,
f (y) ≥ 0.
0
If Kf (x) < ∞, then there exists a limit
(1.1)
GK f (x) := lim [Kf α (x)]1/α
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
α↓0
Title Page
and
Z
(1.2)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
GK f (x) = exp
Contents
∞
k(x, y) log f (y)dy.
0
For 0 < p < ∞ and a weight function v(x) ≥ 0 we put
Z
∞
p
1/p
|f (x)| v(x)dx
kf kLpv :=
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and make use of the abbreviation kf kLp when v(x) ≡ 1.
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Suppose u(x) ≥ 0 and v(x) ≥ 0 are weight functions and 0 < p, q < ∞.
This paper deals with Lpv − Lqu inequalities of the form
Z
∞
q
(GK f ) u
(1.3)
1/q
Z
≤C
∞
p
1/p
f v
0
,
f ≥ 0,
0
where a constant C is independent on f and we always assume that C is the
least possible, that is C = kGK kLpv →Lqu , where
(1.4)
kGK kLpv →Lqu := sup
f ≥0
kGK f kLqu
.
kf kLpv
For the classical case k(x, y) = x1 χ[0,x] (y) with the Hardy averaging operator
(1.5)
1
Hf (x) :=
x
Z
x
f (y)dy
0
the inequality (1.3) was characterized in [6] (see also [7]).
In Section 2 of this paper we present a general scheme on how inequalities of the type (1.3) can be characterized via the limiting procedure in similar
characterization (in suitable forms) of some corresponding Hardy-type inequalities. In Section 3 we characterize the weights u(x) and v(x) so that (1.3) with
0 < p ≤ q < ∞ holds when
Z x
1
(1.6)
GK f (x) = exp
k(x, y) log f (y)dy
K(x) 0
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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with
Z
x
k(x, y)dy < ∞,
K(x) :=
x > 0,
0
and K(0) = 0, K(∞) = ∞, and when k (x, y) satisfies the Oinarov condition
(see Theorem 3.1). The corresponding characterization for the case 0 < q <
p < ∞ can be found in Section 4 (see Theorem 4.1). A fairly precise result in
the case k (x, y) = (x − y)γ , γ > 0, i.e. when GK is generated by the RiemannLiouville convolutional operator, can be found in Section 5 (see Theorem 5.1).
In particular, this investigation shows that inequalities of the type (1.3) can be
proved also when the Oinarov condition is violated (because for 0 < γ < 1
this kernel does not satisfy this condition). In order to prove these results we
need new characterizations of weighted inequalities with the Riemann-Liouville
operator which are of independent interest (see Theorem 5.2 and 5.3).
Finally note that the operator (1.6) was studied in a similar connection, by
E.R. Love [3], where a sufficient condition was proved for the inequality (1.3)
to be valid in the case p = q = 1 and special weights.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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2.
General Schemes
We begin with some general remarks. By using the elementary property
(2.1)
GK (f s ) = (GK f )s ,
−∞ < s < ∞,
we see that
(2.2)
kGK kLpv →Lqu = kGK kLp →Lqw = kGK k
s/p
sq/p
Ls →Lw
,
where
(2.3)
q/p
1
w := GK
u.
v
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Moreover, from (1.1) and (2.2)
Title Page
(2.4)
kGK kLpv →Lqu = lim kKk
α↓0
1/α
q/α
Lp/α →Lw
.
Contents
The last formula generates the “precise” scheme for characterization of (1.3)
provided the norm of the associated integral operator K has very accurate twosided estimates of the norm
(2.5)
c1 (p, q)F (w, p, q) ≤ kKkLp →Lqw ≤ c2 (p, q)F (w, p, q)
in the sense that there exist the limits
h p q p q i1/α
(2.6)
ci (p, q, F ) = lim ci
,
F w, ,
,
α↓0
α α
α α
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i = 1, 2,
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and
ci (p, q, F ) ∈ (0, ∞) ,
i = 1, 2.
A natural consequence of (2.2) and (2.4) is then the two-sided estimate
(2.7)
c1 (p, q, F ) ≤ kGK kLp →Lqw ≤ c2 (p, q, F ),
which characterizes (1.3) with the least possible constants provided the estimates in (2.5) are also the best.
This scheme was realized in [6] where K is the Hardy averaging operator
(1.5) and 0 < p ≤ q < ∞. For this purpose a new non-Muckenhoupt form of
(2.5) was used. We generalize this result here for the Riemann-Liouville kernel
in Section 4.
Unfortunately, the above scheme does not work for a more general operator
or even for the Hardy operator, if q < p, because the estimate (2.5) happens
to be vulnerable, when the parameters p and q tend to ∞, so that the limits
ci (p, q, F ) become either 0 or ∞. However, if the functional F is homogeneous
in the sense that
h p q i1/α
= F (w, p, q), α > 0,
(2.8)
F w, ,
α α
then an alternative “stable” scheme for (2.7) works, provided the lower bound
(2.9)
c3 (p, q)F (w, p, q) ≤ kGK kLp →Lqw
can be established. The right hand side of (2.7) follows from the upper bound
from (2.5), (2.2) and Jensen’s inequality
(2.10)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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GK f (x) ≤ Kf (x),
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
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because (2.8) implies
(2.11)
s/p
sq
F w, s,
= F (w, p, q),
p
s > 1.
To realize the “stable” scheme for the Hardy averaging operator, when 0 < q <
p < ∞ and p > 1 an alternative (non-Mazya-Rozin) functional F was found in
[6] in the form
Z
(2.12)
F (w, p, q) =
0
∞
!1/q−1/p
Z x q/(p−q)
1
w
w(x)dx
x 0
which obviously satisfies (2.11).
The idea to use (2.2) and (2.10) for the upper bound estimate of the norm
kGK kLpv →Lqu originated from the paper by Pick and Opic [8], where the authors
obtained two-sided estimates of kGH kLpv →Lqu , 0 < q < p < ∞. However, they
realized (2.5) in Muckenhoupt or Mazya-Rozin form with unstable factors and,
therefore, the estimate (2.7) was rather uncertain.
Throughout the paper, expressions of the form 0 · ∞, ∞/∞, 0/0 are taken
to be equal to zero; and the inequality A . B means A ≤ cB with a constant
c > 0 independent of weight functions. Moreover, the relationship A ≈ B is
interpreted as A . B . A or A = cB. Everywhere the constant C in explored
inequalities are considered to be the least possible one. In the case q < p the
auxiliary parameter r is defined by 1/r = 1/q − 1/p.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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3.
Hardy-type Operators. Case 0 < p ≤ q < ∞
In the sequel we let k(x, y) ≥ 0 be locally integrable in R+ × R+ and satisfies
the following (Oinarov) condition: there exists a constant d ≥ 1 independent on
x, y, z such that
1
(k(x, z)+k(z, y)) ≤ k(x, y) ≤ d(k(x, z)+k(z, y)),
d
We assume that
Z x
(3.2)
K(x) :=
k(x, y)dy < ∞, x > 0,
(3.1)
x ≥ z ≥ y ≥ 0.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
0
and K(0) = 0, K(∞) = ∞.
Without loss of generality we may and shall also assume that k(x, y) is nondecreasing in x and nonincreasing in y.
We consider the following Hardy-type operator
Z x
1
(3.3)
Kf (x) :=
k(x, y)f (y)dy, f (y) ≥ 0, x ≥ 0,
K(x) 0
and the corresponding geometric mean operator GK :
Z x
1
GK f (x) = exp
k(x, y) log f (y)dy.
K(x) 0
Our main goal in this section is to find necessary and sufficient conditions
on the weights u(x) and v(x) so that the inequality
Z ∞
1/q
Z ∞
1/p
q
p
(3.4)
(GK f (x)) u (x) dx
≤C
f (x) v(x)dx
, f ≥ 0,
0
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Lars-Erik Persson
Vladimir Stepanov
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holds for 0 < p ≤ q < ∞, and to point out the corresponding stable estimates of
C = kGK kLpv →Lqu . In view of our discussion in the previous section we first need
to have the two-sided estimates of kKkLp →Lqw in suitable form. First we observe
that it follows from ([1, Theorem 2.1]) by changing variables x → x1 , y → y1 and
by using the duality principle (see [11, Section 2.3]) that for 1 < p ≤ q < ∞
we have
c4 (d)A ≤ kKkLp →Lqw ≤ c5 (p, q, d)A,
(3.5)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
where d is defined by (3.1),
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
A = max (A0, A1 )
(3.6)
with
−1/p
Z
A0 := sup t
(3.7)
1/q
t
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w
t>0
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and
Z
p0
−1/p
k(t, y) dy
(3.8) A1 := sup
t>0
t
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0
Z t Z
×
0
0
x
0
k(x, y)p dy
q
w(x)
dx
K(x)q
1/q
.
It is easy to see that A0 satisfies (2.8) or (2.11), but not A1 . It is also known, that
neither A0 < ∞ nor A1 < ∞ alone is sufficient for kKkLp →Lqw < ∞ in general
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(see counterexamples in [4], [10] and [5]). For this reason we need to require
some additional conditions of a kernel k(x, y) for the property
A0 ≈ kKkLp →Lqw ,
(3.9)
and we will soon discuss this question in detail (see Proposition 3.2). Now we
are ready to state our main theorem of this section:
Theorem 3.1. Let 0 < p ≤ q < ∞ and the kernel k(x, y) ≥ 0 be such that
(3.9) holds. Then (3.4) holds if and only if A0 < ∞. Moreover,
A0 ≤ kGK kLpv →Lqu . A0 .
(3.10)
Proof. The sufficiency including the upper estimate in (3.10) follows from (2.2),
(2.10) and (3.9). Moreover, we note that, by (2.2),
Z
1/q
∞
q
≤ kGK kLpv →Lqu
(GK f ) w
(3.11)
Z
0
∞
f
p
.
t−1/p
w
0
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1/q
t
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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1/p
By applying this estimate with ft (x) = t−1/p χ[0,t] (x) for fixed t > 0 (obviously,
kft kp = 1) we see that
Z
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
≤ kGK kLpv →Lqu .
By taking the supremum over t we have also proved the necessity including the
lower estimate in (3.10) and the proof is complete.
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Remark 3.1. For the case when k (x, y) ≡ 1 we have that GK coincide with
the usual geometric mean operator
Z
1 x
Gf (x) = exp
log f (y)dy
x 0
and thus we see that Theorem 3.1 may be regarded as a genuine generalization
of Theorem 2 in [6] (see also[7]).
As mentioned before we shall now discuss the question of for which kernels
k (x, y) does the condition (3.9) hold. We need the following notation:
Z x
0
Kp (x) :=
k(x, y)p dy,
0
(3.12)
1/p0
α0 := sup Kp (t)
sup x
t>0
(3.13)
1/p0
1/q
k(t, x)
,
α1 := sup t1/p
0
∞
t>0
k(x, t)q K(x)−q d xq/p
1/q
Contents
,
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t
and
(3.14)
1/p0
Z
α2 := sup t
t>0
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Title Page
0<x<t
Z
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
∞
x
q/p
q
−q
k(x, t) d −K(x)
1/q
.
t
Proposition 3.2. Let 1 < p ≤ q < ∞. If a kernel k(x, y) ≥ 0 is such that for
all weights w (3.9) is true, then α0 + α1 < ∞. Conversely, if α0 + α2 < ∞,
then (3.9) holds. In particular, if α2 . α1 , then (3.9) is equivalent with the
conditions α0 < ∞ and α1 < ∞.
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Proof. It is known ([11], (see also [2, Theorem 2.13])) that
kKkLp →Lqw = max (A0 , A1 ) ,
(3.15)
where
1/q
w(x)
0
t1/p ,
A0 := sup
k(x, t)
dx
q
K(x)
t>0
t
Z ∞
1/q
w(x)
0
Kp (t)1/p .
A1 := sup
dx
q
K(x)
t>0
t
Z
∞
q
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
It follows from a more general result ([5, Theorem 4]) that
A0 ≈ kKkLp →Lqw
(3.16)
for all weights w, if and only if α0 < ∞. Moreover, if (3.9) is true for all
weights w, then (3.15) implies
A0 . A 0
(3.17)
and for w(x) = xq/p−1 it brings α1 < ∞. We observe that the inequality
1/q
t
w
0
Z t Z
=
q
x
k(x, y)dy
0
0
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is always true. Indeed, by applying Minkowski’s integral inequality we find that
Z
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A0 . pA0
(3.18)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
w(x)
dx
K(x)q
1/q
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∞
Z t Z
≤
0
y
Z
≤ A0
w(x)
k(x, y)
dx
K(x)q
q
t
1/q
dy
0
y −1/p dy
0
1/p
= A0 pt
and (3.18) follows. Thus, (3.9) implies (3.16). Consequently, α0 < ∞ and,
thus, α0 + α1 < ∞.
Now, suppose that α0 + α2 < ∞. Then (3.16) holds and it is sufficient to
prove (3.17). To this end we note that
Z ∞
w(x)
k(x, t)q
dx
K(x)q
t
Z x
Z ∞
Z ∞
w(x)
w(x)
q
q
≈ k(t, t)
dx +
dk(s, t) dx
K(x)q
K(x)q
t
t
t
Z ∞ Z ∞
−1
q
dx
= k(t, t)
w(x)
d
K(y)q
t
x
Z ∞
Z ∞
w(x)dx
q
+
dk(s, t)
K(x)q
t
s
Z y
Z ∞ −1
q
= k(t, t)
d
w(x)dx
K(y)q
t
t
Z ∞
Z ∞
Z ∞ −1
q
+
dk(s, t)
w(x)dx
d
K(y)q
t
s
x
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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∞
−1
≤
k(t, t)q
y d
q
K(y)
Z y
Z t∞
Z ∞ −1
q
+
dk(s, t)
d
w(x)dx
K(y)q
t
s
s
Z ∞
−1
q
q/p
q
≤ A0
y k(y, t) d
K(y)q
t
Aq0
Z
q/p
0
≤ Aq0 α2q t−q/p .
and (3.17) follows. Now (3.5), (3.6), (3.16) and (3.17) imply (3.9). The proof
is complete.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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4.
Hardy-type Operators. Case 0 < q < p < ∞
Put
Z
(4.1)
B0 :=
0
∞
Z t r/q !1/r
1
w
dt
.
t 0
A crucial condition for this case corresponding to the condition (3.9) for the
case 0 < p ≤ q < ∞ is the following:
(4.2)
B0 ≈ kKkLp →Lqw .
We now state our main result in this Section.
Theorem 4.1. Let 0 < q < p < ∞ and the kernel k(x, y) ≥ 0 be such that
(4.2) holds. Also we assume that
Z 1
x
1
(4.3)
k(x, xt) log dt ≤ α3 < ∞, x > 0.
K(x) 0
t
Then (3.4) holds if and only if B0 < ∞. Moreover,
(4.4)
kGK kLpv →Lqu ≈ B0 .
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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Proof. The sufficiency including the upper estimate in (4.4) follows by using
(2.2), (2.10) and (4.2). Next we show that
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(4.5)
D . kGK kLp →Lqw ,
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where
Z
∞
Z
∞
D :=
0
x
w(y)
dy
yq
!1/r
r/q
x
r/q 0
dx
.
For this purpose we consider
f0 (x) = x
r/(pq 0 )
Z
x
∞
w(y)
dy
yq
r/(pq)
.
Then
Dr/p = kf0 kp .
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
We have
kGK kLp →Lqw kf0 kp
= kGK kLp →Lqw Dr/p
Z ∞
1/q
q
≥
(GK f0 ) w
0
Z ∞ Z x
1
=
exp
k(x, y)
K(x) 0
0
#1/q
(
r/(pq) ) !q
Z ∞
w(z)
0
× log y r/(pq )
dz
dy w(x)dx
zq
y
"Z rq/(pq0 )
Z x
∞
1
≥
exp
k(x, y) log ydy
K(x) 0
0
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Z
×
x
∞
w(z)
dz
zq
#1/q
r/p
w(x)dx
=: J.
(4.6)
Moreover,
1
K(x)
Z
0
x
Z x
1
y
k(x, y) log ydy = log x +
k(x, y) log dy
K(x) 0
x
Z 1
x
1
= log x −
k(x, xt) log dt
K(x) 0
t
≥ log x − α3 ,
where α3 is a constant from (4.3). Then
Z ∞ Z ∞
r/p
−α3 rq
w(z)
0
q
w(x)xr/(pq ) dx ≈ Dr ,
(4.7) J ≥ exp
dz
0
q
(pq )
z
0
x
the last estimate is obtained by partial integration. Now (4.5) follows by just
combining (4.6) and (4.7).
It remains to show that D ≥ B0 . We have
Z t
Z t Z s
w(s)
q−1
w(x)dx = q
z dz
ds
sq
0
0
0
Z t Z t
w(s)
=q
ds z q−1 dz
q
s
0
z
Z t Z t
w(s)
q−1+q/(2p)
=q
ds z
z −q/(2p) dz
q
s
0
z
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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(by Hölder’s inequality with conjugate exponents
Z t Z
t
≤q
0
z
r/q
w(s)
0
ds
z r/q +r/(2p) dz
q
s
r
p
and )
q
q
!q/r Z
0
t
dz
√
z
q/p
.
This implies
!
r/q
w(s)
0
ds
z r/q +r/(2p) dz tr/(2p)−r/q dt
Br0 .
q
s
0
0
z
r/q
Z ∞
Z ∞ Z ∞
w(s)
r/q 0 +r/(2p)
r/(2p)−r/q
=
ds
z
t
dt dz
sq
z
0
z
r/q
Z ∞ Z ∞
w(s)
0
≈
ds
z r/q dz = Dr .
q
s
0
z
Z
∞
Z t Z
∞
Thus the lower bound in (4.4) and also the necessity is proved; so the proof is
complete.
Next we shall analyze and discuss for which kernels k (x, y) the crucial conditions (4.2) holds. First we note that it follows for the case 1 < q < p < ∞
from a well-known result ([11], see also [2, Theorem 2.19]) that
kKkLp →Lqw ≈ B0 + B1 ,
(4.8)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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where
Z
∞
Z
B0 :=
0
t
∞
k(x, t)q
w(y)
dy
K(x)q
r/q
!1/r
0
tr/q dt
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and
∞
Z
Z
∞
B1 :=
0
t
w(x)
dx
K(x)q
r/p
!1/r
0
Kp (t)r/p w(t)
dt
.
K(t)q
Moreover, it is established in ([5, Theorem 13]) that
B1 ≤ β0 B0
for all weights w for which B0 < ∞, if and only if,
Z t
−1/r
r
r/p0
1/p0
(4.9)
β0 := sup Kp (t)
k (t, x) d x
< ∞.
t>0
On Weighted Inequalities with
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Generated by the Hardy-type
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Therefore
Maria Nassyrova,
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Vladimir Stepanov
kKkLp →Lqw ≈ B0
(4.10)
under the condition (4.9).
Partly guided by our investigation in the previous section we shall now continue by comparing the constant B0 and B0 .
Proposition 4.2. Let 1 < q < p < ∞. Then
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(a) B0 . B0 ,
(b) B0 ≤ β1 B0 , if
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Z
∞
Z
(4.11) β1 :=
0
x
p/r
r r/q 0
k(x, t) t dt
xp/q
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× K(x)−p(1+1/q) d (K(x))
1/p
< ∞.
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Proof. (a) We have
Z
r
B0 :=
∞
Z t Z
q
x
k(x, y)dy
0
0
0
w(x)
dx
K(x)q
r/q
t−r/q dt
(applying Minkowski’s integral inequality)
1/q !r
Z ∞ Z t Z t
w(x)
≤
k(x, y)q
dx
dy t−r/q dt
q
K(x)
0
0
y
!r
1/q
Z ∞ Z t Z ∞
α −α
q w(x)
y y dy t−r/q dt
dx
≤
k(x, y)
q
K(x)
0
0
y
(applying Hölder’s inequality, α ∈ q10 , r10 )
!Z
r/q
r−1
t
w(x)
αr
−αr 0
q
dx
y dy
y
dy
t−r/q dt
≤
k(x, y)
q
K(x)
0
0
0
y
r/qZ ∞
Z ∞Z ∞
0 1−r
q w(x)
r−1−αr−r/q
dx
t
dt y αr dy
= (1 − αr )
k(x, y)
q
K(x)
0
y
y
Z
∞
Z t Z
∞
(1 − αr0 )1−r r
=
B .
r (α − 1/q 0 ) 0
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Vladimir Stepanov
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(b) Indeed, following the proof of Proposition 3.2, we find
Z x Z ∞ Z ∞
r/q r/q0
r
q
−q
B0 ≤
k(x, t)
w d −K(x)
t dt
0
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
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t
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(by Minkowski’s integral inequality)

Z
∞
x
Z
k(x, t)r
≤
0
=
r/q
x
w
0
!q/r
0
tr/q dt
r/q
d −K(x)−q 
t
!r/q
q/r
Z x Z x
1
0
xd −K(x)−q
w
k(x, t)r tr/q dt
x
0
0
0
!r/q
q/r
Z ∞ Z x Z x
1
0
q
w
k(x, t)r tr/q dt
xK(x)−(q+1) dK(x)
x
0
0
0
Z
≤
Z
∞
(by Hölder’s inequality applied with conjugate exponents
≤q
r/q
β1r
Z
0
∞
r
q
and pq )
Z x r/q
1
w
dx = q r/q β1r Br0
x 0
and the proof follows.
Remark 4.1. It is easy to see that the conditions (4.3), (4.9) and (4.11) are
satisfied with k(x, y) ≡ 1, i.e. when GK = G (the standard geometric mean
operator) and we conclude that Theorem 4.1 may be seen as a generalization of
Theorem 4 in [6] (see also [7]).
We finish this Section by showing that the kernel, investigated in [4], satisfies
the condition (4.3).
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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Example 4.1. Let the kernel k(x, y) be given by
y
(4.12)
k(x, y) = ϕ
,
x
where ϕ(t) ≥ 0 is decreasing function on (0, 1) satisfying
(4.13)
ϕ(ts) ≤ d (ϕ(t) + ϕ(s)) ,
0 < t, s < 1.
Then R(3.1) is obviously valid.
1
If 0 ϕ(t)dt < ∞, then the kernel k(x, y) of the form (4.12) satisfies (4.3).
Indeed,
Z 1
Z 1
1
1
k(x, xt) log dt =
ϕ(t) log dt
t
t
0
0
Z
−k
∞
2
X
1
=
ϕ(t) log dt
t
−k−1
k=0 2
Z
−k
∞
2
X
1
≤
ϕ(2−k−1 )
log dt
t
2−k−1
k=0
.
=
∞
X
k=0
∞
X
ϕ(2−k−1 ) (k + 1) 2−k−1
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ϕ(2−k )k2−k
∞
X
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k=1
≤ 2d
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Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
ϕ 2−k/2 k2−k
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≤ 2cd
∞
X
ϕ 2−k/2 2−k/2
k=1
3/2
=2
cd
∞
X
−k/2
ϕ 2
≤2
cd
∞ Z
X
dx
2−k/2
ϕ (x) dx
2(−k−1)/2
k=1
1
≤ 23/2 cd
2−k/2
2(−k−1)/2
k=1
3/2
Z
Z
ϕ(t)dt
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
0
3/2
=2
1
Z
cd
k(x, xt)dt
0
1
= 23/2 cd
x
where c = supk≥0 k2−k/2 .
Z
Maria Nassyrova,
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Vladimir Stepanov
x
k(x, z)dz,
0
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5.
Riemann-Liouville Operators
Let γ > 0 and consider the following Riemann-Liouville operators:
Z x
γ
(5.1)
Rγ f (x) := γ
(x − y)γ−1 f (y)dy,
x 0
and corresponding geometric mean operators
Z x
γ
γ−1
Gγ f (x) = exp γ
(x − y)
log f (y)dy .
x 0
In this section we shall study the question of characterization of the weights
u (x) and v (x) so that the inequality
Z
Maria Nassyrova,
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Vladimir Stepanov
1/q
∞
q
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(Gγ f (x)) u (x) dx
(5.2)
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
0
Z
≤C
∞
p
f (x) v(x)dx
Contents
1/p
,
0 < p, q < ∞
0
holds and also to point out the corresponding stable estimates of C = kGγ kLpv →Lqu .
We note that in the case 0 < γ < 1 the kernel k(x, y) = (x − y)γ−1 in (5.1)
does not satisfy the Oinarov condition (3.1) so the results in Theorems 3.1 and
4.1 cannot be applied. However, this kernel has this property for the case γ ≥ 1
so the question above can be solved by simply applying Theorems 3.1 and 4.1.
Here we unify both cases in the next theorem and give a separate proof to this
operator which gives a better estimate of the upper bound.
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Theorem 5.1. (a) Let 0 < p ≤ q < ∞. Then (5.2) holds if and only if
A0 < ∞. Moreover,
A0 ≤ kGγ kLpv →Lqu ≤ γe1/p A0,
(5.3)
1 ≤ γ < ∞.
and
A0 ≤ kGγ kLpv →Lqu . A0,
(5.4)
0 < γ < 1.
(b) Let 0 < q < p < ∞. Then (5.2) holds if and only if B0 < ∞. Moreover,
kGγ kLpv →Lqu ≈ B0 .
(5.5)
The factors of equivalence in (5.4) and (5.5) depend on p, q and γ only.
Remark 5.1. By applying Theorem 5.1 with γ = 1 we obtain Theorems 2 and
4 in [6] (see also [7]) even with the same constants in the upper estimate in
(5.3).
Partly guided by the technique used in [6], we postpone the proof of Theorem 5.1 and first prove two auxiliary results of independent interest, namely a
characterization of the inequality
Z ∞
1/q
Z ∞ 1/p
q
(5.6)
(Rγ f ) w
≤C
fp
, f ≥ 0,
0
0
in a form suitable for our purpose.
The following two theorems may be seen as a unification and generalization
of the results from ([6, Theorems 1 and 3]) and ([9, Theorems 1 and 2]).
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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Theorem 5.2. (a) Let γ ≥ 1 and 1 < p ≤ q ≤ ∞. Then
(5.7)
A0 ≤ kRγ kLp →Lqw ≤ γp0 A0 .
(b) Let 0 < γ < 1 and 1/γ < p ≤ q ≤ ∞. Then
(5.8)
A0 ≤ kRγ kLp →Lqw
#
"
1/p0
p−1
≤γ
21/p + 22/p + 21−γ p0 A0 .
pγ − 1
Remark 5.2. The lower bound A0 ≤ kRγ kLp →Lqw holds for all γ > 0 and
0 < p, q ≤ ∞.
Theorem 5.3. (a) Let γ ≥ 1, 0 < q < p < ∞ and p > 1. Then
γ
0
1/q 0
(5.9)
(p0 ) p−1/r r−1/r qB0 ≤ kRγ kLp →Lqw ≤ γq 1/q p0 B0 .
γ+1/r+1/q
2
(b) Let 0 < γ < 1, 0 < q < p < ∞ and pγ > 1. Then
1/r
1/p qr/p2
γ 2q/r − 1
2
−1
B0
23+1/r
≤ kRγ kLp →Lqw
"
1/p0
p−1
1+1/q
1/p0
≤ γB0
2
1+2
pγ − 1
!1/q 
q/r
r
q
.
(5.10)
4q + q (p0 )
+
p
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
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Vladimir Stepanov
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Proof of Theorem 5.2. For the lower bounds on (5.7) and (5.8) we replace f in
(5.6) by the test function ft (x) = χ[0,t] (x), t > 0. Then for p, q < ∞
1/p
t
Z
∞
q
C≥
1/q
(Rγ ft ) w
Z
≥
0
1/q
t
w
.
0
Hence,
kRγ kLp →Lqw ≥ A0
for all γ > 0, 0 < p, q < ∞. For p ≤ q = ∞, p = q = ∞ or q < p = ∞ the
arguments are the same.
Clearly,
Rγ f (x) ≤ γHf (x),
(5.11)
γ≥1
and the upper bound in (5.7) follows from Theorem 1 in [6].
For the upper bound in (5.8) we follow the scheme from the proof of Theorem 1 in [9]. Put
Z
J :=
0
∞
1
xγ
Z
x
(x − y)γ−1 f (y)dy
1/q
q
w(x)dx
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
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and note that, by Minkowski’s inequality,
(5.12)
J ≤ J1 + J2 + J3 ,
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where
J1q
XZ
:=
2k
k∈Z
XZ
J2q :=
and
:=
2k+1
2k
k∈Z
J3q
2k+1
XZ
2k+1
k∈Z
Z
w(x)dx
xγq
Z
γ−1
(x − y)
,
f (y)dy
2k
!q
2k
(x − y)γ−1 f (y)dy
,
2k−1
!q
2k−1
Z
w(x)dx
xγq
2k
q
x
w(x)dx
xγq
(x − y)γ−1 f (y)dy
.
0
Maria Nassyrova,
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Vladimir Stepanov
Applying Hölder’s inequality we find
J1q
≤
X Z
2k+1
f
p
!q/p Z
2k
k∈Z
2k+1
w(x)
dx
xγq
2k
Z
x
×
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q/p0
(γ−1)p0
(x − y)
dy
2k
≤
p−1
pγ − 1
q/p
≤2
q/p0 X Z
p−1
pγ − 1
k∈Z
!q/p
2k+1
fp
2k
k∈Z
Z
2k+1
2k
q/p0 X Z
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
!q/p
2k+1
f
p
!
w(x)dx 2−kq/p
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2−(k+1)q/p
×
!
2k+1
Z
w(x)dx
0
(5.13)
q/p
≤2
p−1
pγ − 1
q/p0
Aq0
∞
Z
f
p
q/p
,
0
where the last step follows by the elementary inequality
1 and the definition of A0 .
Similarly, we obtain
(5.14)
J2q
q/p
≤4
p−1
pγ − 1
q/p0
Z
Aq0
∞
f
p
P
P
aβi ≤ ( ai )β , β ≥
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
q/p
.
0
For the upper bound of J3 we note that
Z ∞
q
(1−γ)q
J3 ≤ 2
(Hf )q w
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J
so that, by Theorem 1 in [6] ,
(5.15)
J3q
(1−γ)q
≤2
Aq0
0 q
Z
(p )
∞
f
p
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q/p
.
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By combining (5.13), (5.14) and (5.15) we obtain the upper bound of (5.8) and
the proof is complete.
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(a) For the lower bound we write
Z
C
∞
fp
1/p
Z
∞
1
xγ
≥γ
0
0
Z
!1/q
!q
x/2
(x − y)γ−1 f (y)dy
w(x)dx
0
!q
!1/q
Z
1 x/2
≥ γ−1
f w(x)dx
2
x 0
0
Z ∞ Z x q
1/q
1
γ
.
≥ γ
f w(2x)dx
2
x 0
0
γ
Z
∞
By applying Theorem 3 in [6] we find
kRγ kLp →Lqw ≥
γ
c (p, q) B0 ,
2γ
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where
Z
B0 :=
0
∞
r/q !1/r
Z x
1
w (2s) ds
dx
= 2−1/r B0
x 0
0
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and c (p, q) = 2−1/q (p0 )1/q p−1/r r−1/r q and the lower bound in (5.9) is
proved. Using (5.11) and again Theorem 3 in [6] we obtain the upper
bound of (5.9).
Close
Quit
(b) It is shown in Theorem 2 in [9] that
kRγ kLp →Lqw ≥
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γ
E,
4
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J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au
where

E := 
X
Z
0
2kr/p
2k+1
w(x)dx
xq
2k
k∈Z
!r/q 1/r
.

We show that
q/r
2
1/r
1/p qr/p2
−1
2
−1
E≥
B0 .
21+1/r
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Indeed,
Br0
=
k∈Z
≤
X
2k
Z x r/q
1
w
dx
x 0
2k−1
Z k !r/q Z
XZ
2
−(k−1)r/q
2
X
=2
dx
2k−1
XZ
2−(k−1)r/p
k∈Z
r/p
m≤k
X
−kr/p
2
k∈Z
2k
w
0
k∈Z
=
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Title Page
!r/q
2m
Contents
w
JJ
J
2m−1
X
2
−mq 2
rp
Z
2m
w 2
mq 2
rp
!r/q
Go Back
2m−1
m≤k
(by applying Hölder’s inequality with
Close
and pq )
r
q
II
I
Quit
≤ 2r/p
X
k∈Z
2−kr/p
X
m≤k
2−mq/p
Z
r/q !
2m
w
2m−1
!r/p
X
2mq/r
Page 33 of 37
m≤k
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au
=
=
≤
2r/p+q/p
X
−mq/p
r/p
X
(2qr/p2 − 1) m∈Z
2r(1+1/p)
r/p
− 1)
r/q X
2m−1
22r/p
(2q/r
w
m∈Z
(2q/r − 1)
2m
2
r/p
(2q/r − 1)
Z
(2qr/p2
− 1)
2−kqr/p
2
k≥m
−mr/p
Z
r/q
2m
2
w
2m−1
Er .
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Conversely,
r
E ≤
X
−(k−1)r/p
2
≤2
w
X
−kr/p
≤ 2r/p+r/q
r/p+r/q
=2
!r/q Z
2k
Z
2
k∈Z
Br0 .
2k
Title Page
dx
2k
0
2k+1
XZ
2k+1
w
k∈Z
(5.16)
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
2k−1
k∈Z
r/p
!r/q
2k
Z
Contents
Z x r/q
1
w
dx
x 0
JJ
J
Go Back
Using the decomposition (5.12) it is shown in Theorem 2 in [9] that
J≤
p−1
pγ − 1
1/p0
1/p0
2
II
I
1/p0
1+2
1−γ
Z
E+2
∞
q
(Hf ) w
0
Close
Quit
1/q
.
Page 34 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au
Now the upper bound in (5.10) follows from (5.16) and Theorem 3 in [6].
We are now ready to complete this section by presenting
Proof of Theorem 5.1. Both sides of (5.3) follow from Theorem 5.2 (a) and
(2.4). The lower bound in (5.4) follows by using the test functions from the
proof of Theorem 3.1 and the upper bound is a consequence of (2.2), (2.10)
and (5.8). Similarly, the proof of (b) is based upon Theorem 5.3 and we use
the same test function for the lower bound as in the proof of Theorem 4.1 with
subsequent application of the inequality B0 ≤ B0 . The proof is complete.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 35 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au
References
[1] S. BLOOM AND R. KERMAN, Weighted norm inequalities for operators
of Hardy-type, Proc. Amer. Math. Soc., 113 (1991), 135–141.
[2] A. KUFNER AND L.-E. PERSSON, Integral inequalities with weights,
book manuscript, 2000.
[3] E.R. LOVE, Inequalities related to those of Hardy and of Cochran and Lee,
Math. Proc. Camb. Phil. Soc., 99 (1986), 395–408.
[4] P.J. MARTIN-REYES AND E. SAWYER, Weighted inequalities for
Riemann-Liouville fractional integrals of order one and greater, Proc.
Amer. Math. Soc., 106 (1989), 727–733.
[5] M. NASSYROVA, On characterization by a single constant of the
weighted Lp − Lq boundedness of generalized Hardy-type integral operator, Research Report 8, Dept. of Math., Luleå University, 2001.
[6] L.-E. PERSSON AND V.D. STEPANOV, Weighted integral inequalities
with the geometric mean operator, J. of Inequal. & Appl., (to appear).
[7] L.-E. PERSSON AND V.D. STEPANOV, Weighted integral inequalities
with the geometric mean operator, Russian Akad. Sci. Dokl. Math., 63(2)
(2001), 201–202.
[8] L. PICK AND B. OPIC, On geometric mean operator, J. Math. Anal. Appl.,
183 (1994), 652–662.
[9] D.V. PROKHOROV, On boundedness and compactness of a class of integral operators, J. London Math. Soc., 61(2) (2000), 617–628.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 36 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au
[10] V.D. STEPANOV, On weighted estimates for a class of integral operators,
Siberian Math. J., 34 (1993), 755–766.
[11] V.D. STEPANOV, Weighted norm inequalities of Hardy-type for a class of
integral operators, J. London Math. Soc., 50(2) (1994), 105–120.
On Weighted Inequalities with
Geometric Mean Operator
Generated by the Hardy-type
Integral Transform
Maria Nassyrova,
Lars-Erik Persson
Vladimir Stepanov
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 37 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 48, 2002
http://jipam.vu.edu.au