Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 3, Issue 4, Article 48, 2002
ON WEIGHTED INEQUALITIES WITH GEOMETRIC MEAN OPERATOR
GENERATED BY THE HARDY-TYPE INTEGRAL TRANSFORM
1
MARIA NASSYROVA, 1 LARS-ERIK PERSSON, AND 2 VLADIMIR STEPANOV
1
D EPARTMENT OF M ATHEMATICS
L ULEÅ U NIVERSITY OF T ECHNOLOGY
S-971 87 L ULEÅ , SWEDEN
nassm@sm.luth.se
larserik@sm.luth.se
2
C OMPUTING C ENTRE
RUSSIAN ACADEMY OF S CIENCES
FAR -E ASTERN B RANCH
K HABAROVSK 680042, RUSSIA.
stepanov@as.khb.ru
Received 27 November, 2001; accepted 29 April, 2002
Communicated by B. Opic
A BSTRACT. The generalized geometric mean operator
Z x
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 ∞
1/q
Z ∞
1/p
q
p
(GK f (x)) u (x) dx
≤C
f (x) v(x)dx
, f ≥ 0,
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.
Key words and phrases: Integral inequalities, Weights, Geometric mean operator, Kernels, Riemann-Liouville operators.
2000 Mathematics Subject Classification. 26D15, 26D10.
ISSN (electronic): 1443-5756
c 2002 Victoria University. All rights reserved.
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.
084-01
2
M ARIA NASSYROVA , L ARS -E RIK P ERSSON ,
AND
V LADIMIR S TEPANOV
1. I NTRODUCTION
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
GK f (x) := lim [Kf α (x)]1/α
(1.1)
α↓0
and
Z
GK f (x) = exp
(1.2)
∞
k(x, y) log f (y)dy.
0
For 0 < p < ∞ and a weight function v(x) ≥ 0 we put
1/p
Z ∞
p
kf kLpv :=
|f (x)| v(x)dx
0
and make use of the abbreviation kf kLp when v(x) ≡ 1.
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
1/p
Z ∞
1/q
Z ∞
q
p
f v
, f ≥ 0,
≤C
(GK f ) u
(1.3)
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
Z
1 x
(1.5)
Hf (x) :=
f (y)dy
x 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
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
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
3
GK is generated by the Riemann-Liouville 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 5.3 and 5.5).
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.
2. G ENERAL S CHEMES
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)
s/p
kGK kLpv →Lqu = kGK kLp →Lqw = kGK k
sq/p
Ls →Lw
,
where
q/p
1
w := GK
u.
v
(2.3)
Moreover, from (1.1) and (2.2)
(2.4)
kGK kLpv →Lqu = lim kKk
α↓0
1/α
q/α
Lp/α →Lw
.
The last formula generates the “precise” scheme for characterization of (1.3) provided the norm
of the associated integral operator K has very accurate two-sided 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
and
ci (p, q, F ) ∈ (0, ∞) , i = 1, 2.
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
(2.8)
F w, ,
= F (w, p, q), α > 0,
α α
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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M ARIA NASSYROVA , L ARS -E RIK P ERSSON ,
AND
V LADIMIR S TEPANOV
then an alternative “stable” scheme for (2.7) works, provided the lower bound
c3 (p, q)F (w, p, q) ≤ kGK kLp →Lqw
(2.9)
can be established. The right hand side of (2.7) follows from the upper bound from (2.5), (2.2)
and Jensen’s inequality
GK f (x) ≤ Kf (x),
(2.10)
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
!1/q−1/p
Z ∞ Z x q/(p−q)
1
(2.12)
F (w, p, q) =
w
w(x)dx
x 0
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.
3. H ARDY- TYPE O PERATORS . C ASE 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)), x ≥ z ≥ y ≥ 0.
d
We assume that
Z x
(3.2)
K(x) :=
k(x, y)dy < ∞, x > 0,
(3.1)
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
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
5
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
1/q
Z ∞
1/p
Z ∞
q
p
(GK f (x)) u (x) dx
≤C
f (x) v(x)dx
, f ≥ 0,
(3.4)
0
0
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 twosided 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)
where d is defined by (3.1),
A = max (A0, A1 )
(3.6)
with
−1/p
w
A0 := sup t
(3.7)
1/q
t
Z
t>0
0
and
t
Z
(3.8)
k(t, y) dy
A1 := sup
t>0
−1/p Z t Z
p0
x
p0
k(x, y) dy
0
0
0
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 (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 3.3). 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
Z ∞ 1/p
q
(3.11)
(GK f ) w
≤ kGK kLpv →Lqu
fp
.
0
0
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 t 1/q
−1/p
t
w
≤ kGK kLpv →Lqu .
0
By taking the supremum over t we have also proved the necessity including the lower estimate
in (3.10) and the proof is complete.
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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M ARIA NASSYROVA , L ARS -E RIK P ERSSON ,
AND
V LADIMIR S TEPANOV
Remark 3.2. 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
sup x
α0 := sup Kp (t)
1/q
k(t, x)
,
0<x<t
t>0
(3.13)
1/p0
1/p0
∞
Z
−q
q
α1 := sup t
k(x, t) K(x) d x
t>0
q/p
1/q
,
t
and
(3.14)
1/p0
Z
∞
α2 := sup t
x
t>0
q/p
q
−q
k(x, t) d −K(x)
1/q
.
t
Proposition 3.3. 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 < ∞.
Proof. It is known ([11], (see also [2, Theorem 2.13])) that
(3.15)
kKkLp →Lqw = max (A0 , A1 ) ,
where
1/q
w(x)
0
A0 := sup
k(x, t)
dx
t1/p ,
q
K(x)
t>0
t
Z ∞
1/q
w(x)
0
A1 := sup
dx
Kp (t)1/p .
q
K(x)
t>0
t
Z
∞
q
It follows from a more general result ([5, Theorem 4]) that
(3.16)
A0 ≈ kKkLp →Lqw
for all weights w, if and only if α0 < ∞. Moreover, if (3.9) is true for all weights w, then (3.15)
implies
(3.17)
A0 . A 0
and for w(x) = xq/p−1 it brings α1 < ∞. We observe that the inequality
(3.18)
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
A0 . pA0
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
7
is always true. Indeed, by applying Minkowski’s integral inequality we find that
1/q
Z t 1/q Z t Z x
q
w(x)
dx
w
=
k(x, y)dy
K(x)q
0
0
0
1/q
Z t Z ∞
q w(x)
≤
k(x, y)
dx
dy
K(x)q
0
y
Z t
0
≤ A0
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)
dx
k(x, t)q
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 ∞
Z ∞
Z ∞
−1
w(x)dx
q
q
= k(t, t)
w(x)
d
dx +
dk(s, t)
K(y)q
K(x)q
t
x
t
s
Z y
Z ∞
Z ∞ Z ∞
Z ∞ −1
−1
q
q
= k(t, t)
d
w(x)dx +
dk(s, t)
w(x)dx
d
K(y)q
K(y)q
t
t
t
s
x
Z y
Z ∞
Z ∞
Z ∞ −1
−1
q
q
≤ Aq0
y q/p d
k(t,
t)
+
dk(s,
t)
d
w(x)dx
K(y)q
K(y)q
t
t
s
s
Z ∞
−1
q
q/p
q
≤ A0
y k(y, t) d
K(y)q
t
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.
4. H ARDY- TYPE O PERATORS . C ASE 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)
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
kGK kLpv →Lqu ≈ B0 .
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M ARIA NASSYROVA , L ARS -E RIK P ERSSON ,
AND
V LADIMIR S TEPANOV
Proof. The sufficiency including the upper estimate in (4.4) follows by using (2.2), (2.10) and
(4.2). Next we show that
D . kGK kLp →Lqw ,
(4.5)
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 .
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
Z ∞
r/(pq) ) !q
w(z)
0
× log y r/(pq )
dz
dy w(x)dx
zq
y
"Z #1/q
rq/(pq0 ) Z ∞
r/p
Z x
∞
1
w(z)
≥
exp
k(x, y) log ydy
dz
w(x)dx
K(x) 0
zq
0
x
(4.6)
=: J.
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)
q
r/(pq 0 )
(4.7)
J ≥ exp
dz
w(x)x
dx ≈ Dr ,
q
(pq 0 )
z
0
x
the last estimate is obtained by partial integration. Now (4.5) follows by just combining (4.6)
and (4.7).
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
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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)
ds z
z −q/(2p) dz
=q
q
s
0
z
r
p
(by Hölder’s inequality with conjugate exponents and )
q
q
!q/r Z
r/q
q/p
Z t Z t
t
w(s)
dz
0
r/q +r/(2p)
√
≤q
ds
z
dz
.
sq
z
0
z
0
This implies
!
r/q
w(s)
0
Br0 .
ds
z r/q +r/(2p) dz tr/(2p)−r/q dt
sq
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
0
z
z
r/q
Z ∞ Z ∞
w(s)
0
≈
ds
z r/q dz = Dr .
q
s
0
z
Thus the lower bound in (4.4) and also the necessity is proved; so the proof is complete.
Z
∞
∞
Z t Z
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)
where
Z
∞
Z
B0 :=
0
t
∞
w(y)
k(x, t)
dy
K(x)q
q
r/q
t
and
Z
∞
Z
B1 :=
0
t
∞
w(x)
dx
K(x)q
r/p
!1/r
r/q 0
r/p0
dt
!1/r
Kp (t) 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
1/p0
r/p0
(4.9)
β0 := sup Kp (t)
k (t, x) d x
< ∞.
t>0
0
Therefore
(4.10)
kKkLp →Lqw ≈ B0
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 .
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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10
M ARIA NASSYROVA , L ARS -E RIK P ERSSON , AND V LADIMIR S TEPANOV
Proposition 4.2. Let 1 < q < p < ∞. Then
(a) B0 . B0 ,
(b) B0 ≤ β1 B0 , if
!1/p
p/r
Z ∞ Z x
0
(4.11)
β1 :=
k(x, t)r tr/q dt
xp/q K(x)−p(1+1/q) d (K(x))
< ∞.
0
Proof.
0
(a) We have
Br0
Z
∞
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 ∞
w(x)
≤
k(x, y)q
dx
y α y −α dy t−r/q dt
K(x)q
0
0
y
(applying Hölder’s inequality, α ∈ q10 , r10 )
! Z
r/q
r−1
Z ∞ Z t Z ∞
t
w(x)
q
αr
−αr 0
≤
k(x, y)
dx
y dy
y
dy
t−r/q dt
q
K(x)
0
0
y
0
r/q Z ∞
Z ∞ Z ∞
0 1−r
q w(x)
r−1−αr−r/q
= (1 − αr )
k(x, y)
dx
t
dt y αr dy
q
K(x)
0
y
y
(1 − αr0 )1−r r
B .
r (α − 1/q 0 ) 0
(b) Indeed, following the proof of Proposition 3.3, we find
Z x Z ∞ Z ∞
r/q r/q0
r
q
−q
B0 ≤
k(x, t)
w d −K(x)
t dt
=
0
t
t
(by Minkowski’s integral inequality)

r/q
!q/r
Z x r/q
Z ∞ Z x
0
≤
k(x, t)r
w
tr/q dt
d −K(x)−q 
0
0
∞
Z
≤
0
Z
=
q
0
t
!r/q
q/r
Z x Z x
1
r r/q 0
−q
w
k(x, t) t dt
xd −K(x)
x 0
0
∞
!r/q
Z x Z x
q/r
1
0
w
k(x, t)r tr/q dt
xK(x)−(q+1) dK(x)
x 0
0
(by Hölder’s inequality applied with conjugate exponents qr and pq )
Z ∞ Z x r/q
1
r/q r
≤ q β1
w
dx = q r/q β1r Br0
x
0
0
and the proof follows.
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
11
Remark 4.3. 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).
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.
ThenR(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
∞
X 2
1
=
ϕ(t) log dt
t
−k−1
k=0 2
Z 2−k
∞
X
1
−k−1
≤
ϕ(2
)
log dt
t
2−k−1
k=0
.
=
∞
X
k=0
∞
X
ϕ(2−k−1 ) (k + 1) 2−k−1
ϕ(2−k )k2−k
k=1
≤ 2d
∞
X
ϕ 2−k/2 k2−k
k=1
∞
X
≤ 2cd
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
0
3/2
=2
1
Z
cd
k(x, xt)dt
0
3/2
=2
1
cd
x
Z
x
k(x, z)dz,
0
where c = supk≥0 k2−k/2 .
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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12
M ARIA NASSYROVA , L ARS -E RIK P ERSSON , AND V LADIMIR S TEPANOV
5. R IEMANN -L IOUVILLE O PERATORS
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 ∞
1/q
Z ∞
1/p
q
p
(5.2)
(Gγ f (x)) u (x) dx
≤C
f (x) v(x)dx
, 0 < p, q < ∞
0
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.
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 5.2. 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
(Rγ f ) w
≤C
fp
, f ≥ 0,
(5.6)
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]).
Theorem 5.3.
(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
"
#
1/p0
p−1
(5.8)
A0 ≤ kRγ kLp →Lqw ≤ γ
21/p + 22/p + 21−γ p0 A0 .
pγ − 1
Remark 5.4. The lower bound A0 ≤ kRγ kLp →Lqw holds for all γ > 0 and 0 < p, q ≤ ∞.
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
Theorem 5.5.
(5.9)
13
(a) Let γ ≥ 1, 0 < q < p < ∞ and p > 1. Then
γ
1/q 0
2γ+1/r+1/q
(p0 )
0
p−1/r r−1/r qB0 ≤ kRγ kLp →Lqw ≤ γq 1/q p0 B0 .
(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

2
1+2
+
(5.10)
≤ γB0
pγ − 1
!1/q 
q/r
r
q
.
4q + q (p0 )
p
Proof of Theorem 5.3. 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
1/q
∞
Z
q
C≥
≥
(Rγ ft ) w
1/q
t
Z
w
0
.
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 ∞ Z x
q
1/q
1
γ−1
(x − y)
f (y)dy w(x)dx
J :=
xγ 0
0
and note that, by Minkowski’s inequality,
J ≤ J1 + J2 + J3 ,
(5.12)
where
J1q
:=
XZ
2k
k∈Z
J2q
:=
2k+1
XZ
2k+1
2k
k∈Z
q
x
w(x)dx
xγq
Z
w(x)dx
xγq
Z
w(x)dx
xγq
Z
γ−1
(x − y)
f (y)dy
,
2k
!q
2k
γ−1
(x − y)
f (y)dy
,
2k−1
and
J3q
:=
XZ
k∈Z
2k+1
2k
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
!q
2k−1
γ−1
(x − y)
f (y)dy
.
0
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14
M ARIA NASSYROVA , L ARS -E RIK P ERSSON , AND V LADIMIR S TEPANOV
Applying Hölder’s inequality we find
Z 2k+1 !q/p Z
X
J1q ≤
fp
Z x
q/p0
w(x)dx
(γ−1)p0
(x − y)
dy
γq
x
k
k
k
2
2
2
k∈Z
!
q/p0 X Z 2k+1 !q/p Z 2k+1
p−1
fp
≤
w(x)dx 2−kq/p
pγ − 1
k
k
2
2
k∈Z
!
!
q/p
q/p0 X Z 2k+1
Z 2k+1
p
−
1
fp
2−(k+1)q/p
≤ 2q/p
w(x)dx
pγ − 1
k
2
0
k∈Z
q/p0
Z ∞ q/p
p−1
q
q/p
(5.13)
≤2
A0
fp
,
pγ − 1
0
P β
P
where the last step follows by the elementary inequality
ai ≤ ( ai )β , β ≥ 1 and the
definition of A0 .
Similarly, we obtain
q/p0
Z ∞ q/p
p−1
q
q
q/p
A0
fp
.
(5.14)
J2 ≤ 4
pγ − 1
0
2k+1
For the upper bound of J3 we note that
J3q
(1−γ)q
Z
∞
≤2
(Hf )q w
0
so that, by Theorem 1 in [6] ,
(5.15)
J3q
(1−γ)q
≤2
Aq0
0 q
Z
(p )
∞
f
p
q/p
.
0
By combining (5.13), (5.14) and (5.15) we obtain the upper bound of (5.8) and the proof is
complete.
(a) For the lower bound we write
!q
!1/q
Z ∞ 1/p
Z ∞
Z x/2
1
C
fp
≥γ
(x − y)γ−1 f (y)dy w(x)dx
γ
x
0
0
0
!q
!1/q
Z ∞
Z
γ
1 x/2
≥ γ−1
f w(x)dx
2
x 0
0
Z ∞ Z x q
1/q
γ
1
≥ γ
f w(2x)dx
.
2
x 0
0
By applying Theorem 3 in [6] we find
γ
kRγ kLp →Lqw ≥ γ c (p, q) B0 ,
2
where
r/q !1/r
Z ∞ Z x
1
B0 :=
w (2s) ds
dx
= 2−1/r B0
x
0
0
0
0
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).
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
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W EIGHTED I NEQUALITIES WITH G EOMETRIC M EAN O PERATOR
15
(b) It is shown in Theorem 2 in [9] that
γ
E,
4
kRγ kLp →Lqw ≥
where

X
0
E := 
2kr/p
2k+1
Z
w(x)dx
xq
2k
k∈Z
!r/q 1/r
.

We show that
1/r
1/p qr/p2
2q/r − 1
2
−1
E≥
B0 .
21+1/r
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
w
=
2−(k−1)r/p
X −mq2 Z
2 rp
2−kr/p
k∈Z
≤ 2r/p
2−kr/p
k∈Z
=
=
≤
X
X
(2q/r − 1)
r/p
(2q/r
r/p
− 1)
2m
w
X
(2qr/p2 − 1) m∈Z
(2qr/p2
− 1)
2mq/r
m≤k
r/q X
2m−1
2r(1+1/p)
!r/q
!r/p
w
Z
22r/p
mq 2
rp
X
2m−1
m∈Z
(2q/r − 1)
w 2
r/q !
2m
Z
2−mq/p
2−mq/p
r/p
and pq )
r
q
m≤k
2r/p+q/p
2m
2m−1
m≤k
(by applying Hölder’s inequality with
X
w
2m−1
m≤k
X
!r/q
2m
XZ
k∈Z
= 2r/p
dx
2k−1
0
k∈Z
X
2k
2−kqr/p
2
k≥m
−mr/p
Z
r/q
2m
2
w
2m−1
Er .
Conversely,
Er ≤
X
2−(k−1)r/p
≤2
X
k∈Z
J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
w
2k−1
k∈Z
r/p
!r/q
2k
Z
−kr/p
Z
2
!r/q Z
2k
2k+1
w
0
dx
2k
http://jipam.vu.edu.au/
16
M ARIA NASSYROVA , L ARS -E RIK P ERSSON , AND V LADIMIR S TEPANOV
r/p+r/q
≤2
XZ
k∈Z
(5.16)
2k+1
2k
Z x r/q
1
w
dx
x 0
= 2r/p+r/q Br0 .
Using the decomposition (5.12) it is shown in Theorem 2 in [9] that
1/p0
Z ∞
1/q
p−1
q
1/p0
1/p0
1−γ
2
1+2
E+2
.
J≤
(Hf ) w
pγ − 1
0
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 5.3 (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 5.5 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.
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J. Inequal. Pure and Appl. Math., 3(4) Art. 48, 2002
http://jipam.vu.edu.au/