Journal of Inequalities in Pure and
Applied Mathematics
WEIGHTED MODULAR INEQUALITIES FOR HARDY-TYPE
OPERATORS ON MONOTONE FUNCTIONS
volume 1, issue 1, article 10,
2000.
HANS P. HEINIG AND QINSHENG LAI
Department of Mathematics and Statistics
McMaster University
Hamilton, Ontario
L8S 4K1, CANADA
EMail: heinig@mcmail.cis.mcmaster.ca
EMail: jlai@comnetix.com
Received 3 November, 1999;
accepted 31 January, 2000.
Communicated by: B. Opic
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School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
010-99
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Abstract
Rx
If (Kf)(x) = 0 k(x, y)f(y) dy, x > 0, is a Hardy-type operator defined on
the cone of monotone functions, then weight characterizations for which the
modular inequality
Z ∞
Z ∞
Q−1
Q[θ(Kf)]w ≤ P −1
P [Cf]v
0
0
holds, are given for a large class of modular functions P, Q. Specifically, these
functions need not both be N-functions, and the class includes the case where
Q ◦ P −1 is concave. Our results generalize those in [7, 24], where the case
Q ◦ P −1 convex, with P, Q, N-function was studied. Applications involving the
Hardy averaging operator, its dual, the Hardy-Littlewood maximal function, and
the Hilbert transform are also given.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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2000 Mathematics Subject Classification: 26D15, 42B25, 26A33, 46E30
Key words: Hardy-type operators, modular inequalities, weights, N -functions, characterizations, Orlicz-Lorentz spaces.
The first author was supported in part by NSERC Grant A-4837. The second author’s
research was supported by NSERC Grant PDF 181814.
Contents
1
2
3
4
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Hardy-type operators on increasing functions . . . . . . . . . . . . . 39
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1.
Introduction
An integral operator K defined by
Z x
(Kf )(x) =
k(x, y)f (y) dy,
x > 0, f ≥ 0
0
is called a Hardy type operator, if the kernel k satisfies
(1.1)(i)
(ii)
k(x, y) > 0, x > y > 0, k is increasing in x and decreasing in y.
k(x, y) ≤ D[k(x, z) + k(z, y)], 0 < y < z < x,
for some constant D > 0.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
k(x, y) = 1; k(x, y) = φ(x − y), φ increasing, φ(a + b) ≤ D[φ(a) + φ(b)]
0 < a, b < ∞; and k(x, y) = ψ(y/x), ψ decreasing, ψ(ab) ≤ D[ψ(a) + ψ(b)]
0 < a, b < 1; are examples of kernels satisfying (1.1) and hence define Hardytype operators.
If k(x, y) has no monotonicity properties, satisfies (ii) and its reverse, then k
is said to satisfy the Oinarov condition ([22]) and we write k(x, y) ≈ k(x, z) +
k(z, y), 0 < y < z < x.
In this paper we study Hardy-type operators (and its duals) defined on the
cone of monotone functions. Specifically, weight functions θ, w, v are characterized for which the modular inequality
(1.2) Z
Z
∞
−1
Q
∞
Q[θ(x)(Kf )(x)]w(x) dx
0
≤P
−1
P [Cf (x)]v(x) dx
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0
is satisfied for a large class of modular functions P, Q, and f ≥ 0, monotone.
J. Ineq. Pure and Appl. Math. 1(1) Art. 10, 2000
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For example, if K = I, the identity operator and 0 ≤ f↓, then the weights are
characterized for which (1.2) holds with P , Q increasing and P weakly convex
(cf. Theorem 3.1). For general K, defined on 0 ≤ f↓, weight characterizations
are given for which (1.2) holds with P an N -function, P , P̃ ∈ ∆2 and Q
weakly convex (cf. Theorem 3.4). Specifically, Q ◦ P −1 may be concave. These
results together with the corresponding results where K is defined on the cone
of increasing functions are new. The case 0 < q < 1 < p for the general K,
defined on 0 ≤ f↑, was unknown until this paper.
If P (x) = xp , Q(x) = xq , 0 < p, q < ∞, θ(x) = 1, then our results reduce
to weighted Lebesgue space inequalities and in particular if k(x, y) = 1, to the
weight characterizations of Ariño-Muckenhoupt [1] (p = q > 1 w = v), Sawyer
[21] (1 < p, q < ∞) and Stepanov [23] (0 < q < p, p > 1). The general case
where P and Q are N -functions, such that P and its complementary function P̃
satisfy ∆2 with Q ◦ P −1 convex (more precisely P Q) was studied by Sun
[24] with k(x, y) the convolution kernel.
To explain the scope of our results we require some definitions and known
facts.
A non-negative function P on R+ is called an N -function if it has the form
Z x
(1.3)
P (x) =
p(t) dt,
x > 0,
0
where p is non-decreasing, right continuous on (0, ∞), p(0+) = 0, p(∞) = ∞
and p(t) > 0 if t > 0. Clearly
lim
x→0+
P (x)
x
= lim
= 0.
x→∞
x
P (x)
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Given an N -function P , then its complementary function P̃ is defined by P̃ (y) =
supx>0 {xy − P (x)} and
(1.4)
t ≤ P −1 (t)P̃ −1 (t) ≤ 2t,
p(t/2)/2 ≤ P (t)/t ≤ p(t),
t>0
holds. It is easily seen that if P is an N -function so is P̃ , and the complement
relation is symmetric.
If (X, µ) is a σ-finite measure space, then a µ-measurable function f belongs
to the Orlicz-space LP (µ) if the Luxemburg norm
Z
|f (x)|
kf kP (µ) = inf λ > 0 :
P
dµ(x) ≤ 1
λ
X
is finite. The Orlicz norm in LP (µ) is defined by
Z
Z
0
kf kP (µ) = sup f g dµ :
P̃ (g)dµ ≤ 1 .
X
X
We note that the Luxemburg and Orlicz norm are equivalent and
Z
(1.5)
kf kP (µ) ≤ 1 if and only if
P (f )dµ ≤ 1.
X
Given an N -function P , we always use the Luxemburg norm in LP (µ) and
define that associate space LP̃ (µ) of LP (µ) consists of those µ-measurable g, for
which the Orlicz norm
Z
kgkP̃ (µ) = sup f g dµ : kf kP (µ) ≤ 1
X
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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is finite.
A weight function u (u 6≡ 0, u 6≡ ∞) is a non-negative measurable and
locally integrable function on R+ , and if dµ(x) = u(x)dx, then we write
P (µ) = P (u). The standard duality principle in Orlicz spaces may be written as
R∞
g
fg
0
sup
= ,
g ≥ 0.
u P̃ (u)
0≤f kf kP (u)
For these and other facts see [13, 14, 20].
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Definition 1.1.
a) An increasing function P : R+ → R+ is said to satisfy ∆2 , (P ∈ ∆2 ), if
there is a constant C > 1, such that P (2t) ≤ CP (t), t ≥ 0.
b) A strictly increasing function Q : R+ → R+ is weakly convex, (Q ∈ ∆2 ),
if Q(0) = 0, Q(∞) = ∞ and 2Q(t) ≤ Q(M t), t ≥ 0, for some constant
M > 1.
c) ([16]) If P and Q are increasing, then we write P Q, if there is a
constant A > 0, such that
!
X
X
Q ◦ P −1 (aj ) ≤ Q ◦ P −1 A
aj
j
Hans P. Heinig and Qinsheng Lai
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is satisfied for all non-negative sequences {aj }j∈Z .
A convex function Q satisfying Q(0) = 0, Q(∞) = ∞ is weakly convex
(with M = 2). However, the weakly convex function Q(t) = tα , t ≥ 0,
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0 < α < 1, is not convex, and Q(t) = ln(1 + t), t ≥ 0 is not weakly convex.
Observe also that if Q ◦ P −1 is convex, then P Q.
The main result of this paper (Theorem 3.4) characterizes the weights θ, w, v
for which (1.2) is satisfied for decreasing f ≥ 0 with P an N -function, P ,
P̃ ∈ ∆2 and Q weakly convex. This characterization is expressed in terms of
estimates involving covering sequences.
Definition 1.2. A strictly increasing positive sequence {xj }j∈Z is called a covM
ering sequence if the sequence is of the form {xj }∞
j=−∞ or of the form {xj }j=N ,
where M and/or N is finite. In the latter case we define xN −1 = 0 and/or
xM +1 = ∞.
R xj
In some instances covering
sequences
satisfy
v = 2k , k ∈
0
R∞
R ∞Z, where v is a
N
N +1
weight function. If 2 < 0 v < 2
then in the case 2N < 0 v < 3 · 2N −1
−1
we set xN = ∞ and the covering sequence is {xj }N
j=−∞ . In the remaining
case we set xN +1 =R∞ and the covering sequence is {xj }N
j=−∞ . Under these
xj+1
k−1
k−1
conventions 2
≤ xj v ≤ 3 · 2
for 0 < xj < ∞.
The manuscript is divided into four sections. The next section contains the
weight characterization of a modular Hardy-type inequality for Young’s and
weakly convex functions by Qinsheng Lai [19]. As a consequence a corresponding result for the dual operator follows. In addition, modular Hardy and
conjugate Hardy inequalities (Lemma 2.3) are given. Section 3, the main results, contain the weighted modular inequalities for the identity operator (Theorem 3.1) and Hardy-type operator (Theorem 3.4) defined on decreasing functions. Some special cases given there are needed in Section 4 and seem to
be new even in the Lebesgue space case. In the last section results for the
Hardy operator on increasing functions are given. Moreover, the bounded-
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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ness of the Hardy-Littlewood maximal function and the Hilbert transform in
weighted Orlicz-Lorentz spaces are characterized.
The notation is standard, R+ and R denote the non-negative real and real
numbers respectively, while Z denotes the set of integers. The symbol χE
stands for the characteristic function of a set E. All functions are assumed
measurable
functions.
R and u, v, w, θ denote
R x weight
R ∞ If u is a weight function
∗
u(E) = E u(x) dx, U (x) = 0 u and U (x) = x u, (x > 0). Instead of nonincreasing, non-decreasing we shall say decreasing and increasing respectively,
otherwise we shall prefix it by “strictly”. If f ≥ 0 is increasing (decreasing) we
shall write 0 ≤ f ↑ (0 ≤ f ↓) and similarly for sequences. Expressions of the
form A ≈ B are interpreted to mean that A/B are bounded above and below
by positive constants. Constants are (with the exception of those of Definition
1.1) denoted by B and C and they may have different values at different places.
Inequalities, such as (1.2), are interpreted to mean that if the right side is finite,
so is the left side and the inequality holds.
Other notations and concepts are introduced when needed.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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2.
Preliminary Results
In order to prove weighted modular inequalities for Hardy type operators defined on the cone of monotone functions, a number of results are required. The
first result (Theorem 2.1) by Q. Lai [19] is a weight characterization of the
Hardy-type operator for which a weighted modular inequality is satisfied. This
theorem extends corresponding work of [3, 4, 18, 22, 24] to Young’s functions
P and weakly convex functions Q without the assumption that Q◦P −1 (or more
precisely P Q) is convex.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Theorem 2.1. ([19, Thm. 1]) Suppose K is a Hardy-type operator, P a Young’s
function and Q weakly convex. Let θ, w, ρ and v be weight functions, then the
modular inequality
Z ∞
Z ∞
−1
−1
Q
Q[θ(x)Kf (x)]w(x) dx ≤ P
P [Cρ(x)f (x)]v(x) dx
0
Hans P. Heinig and Qinsheng Lai
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Contents
is satisfied for all f ≥ 0, if and only if there are constants B > 0, such that,
"
#
!
!
X Z xj+1
X
k(x
θ(x) j , ·)χ(xj−1 ,xj ) −1
−1
Q
Q
w(x) dx ≤ P
1/εj
B
ε
vρ
j
x
j
P̃ (εj v)
j
j
and
−1
Q
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XZ
j
xj+1
xj
"
#
χ(xj−1 ,xj ) θ(x)k(x, xj ) Q
w(x) dx
εj vρ B
P̃ (εj v)
!
!
≤ P −1
X
hold for all positive sequences {εj }j∈Z and all covering sequences {xj }j∈Z .
j
1/εj
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A corresponding result for the conjugate Hardy-type operator
Z ∞
∗
(K h)(x) =
k(y, x)h(y) dy,
x > 0, h ≥ 0,
x
¯ (y) =
where k satisfies (1.1), also holds. In fact, writing k̃(x, y) = k( y1 , x1 ) and h̄
h(1/y)/y 2 , then a change of variables shows that
Z x
∗
¯ (y) dy ≡ (K h̄
¯ )(x)
(K h)(1/x) =
k̃(x, y)h̄
0
is a Hardy-type operator since k̃(x, y) satisfies the same conditions as k(x, y).
Writing ḡ(x) = g(1/x) and ḡ¯(x) = g(1/x)/x2 it follows that
Z ∞
Z ∞
−1
∗
−1
¯
Q
Q[θ(x)K h(x)]w(x) dx = Q
Q[θ̄(x)K h̄(x)]w̄¯ (x) dx
0
P −1
∞
¯ (x)]v̄¯(x) dx
P [Cx2 ρ̄(x)h̄
= P −1
Z
∞
P [Cρ(x)h(x)]v(x) dx .
0
0
Also
Hans P. Heinig and Qinsheng Lai
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0
and
Z
Weighted modular inequalities
for Hardy-type operators on
monotone functions
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k(·, )χ(1/xj ,1/xj−1 ) xj
=
ε
ρv
j
¯)
P̃ (εj v¯
k̃(x , ·)χ
j
(xj−1 ,xj ) εj ρ̄v̄
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P̃ (εj v)
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and
χ(xj−1 ,xj ) χ(1/xj ,1/xj−1 ) =
.
εj ρ̄v̄ ε
ρv
¯
j
P̃ (εj v¯)
P̃ (εj v)
Therefore, if 1/xj = y−k , k ∈ Z, then {yj }j∈Z is also a covering sequence,
whenever {xj }j∈Z is. Thus, the following characterization follows from Theorem 2.1.
Proposition 2.2. If K ∗ is the conjugate Hardy-type operator, P a Young’s function and Q weakly convex, then
Z ∞
Z ∞
−1
∗
−1
Q
Q[θ(x)(K h)(x)]w(x) dx ≤ P
P [Cρ(x)h(x)]v(x) dx
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
0
is satisfied for all h ≥ 0, if and only if there is a constant B > 0, such that
#
!
!
"
X Z yj
X
k(·,
y
)χ
)
θ(x)
j
(y
,y
j j+1 Q−1
Q
w(x) dx ≤ P −1
1/εj
B
ε
ρv
j
y
j−1
P̃
(ε
v)
j
j
j
and
Q−1
#
!
χ
θ(x)k(yj , x) (yj ,yj+1 ) Q
w(x) dx ≤ P −1
εj ρv B
yj−1
P̃ (εj v)
XZ
j
yj
!
"
X
1/εj
j
holds for all positive sequences {εj }j∈Z and all covering sequences {yj }j∈Z .
Note that if Q is an N -function, then Q is convex and in particular, weakly
convex. Hence Theorem 2.1 and Proposition 2.2 hold in this case.
The following result is required in the next section.
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Lemma 2.3. Suppose P and P̃ are N -functions, V (x) =
and v is a weight function.
Rx
0
v, V ∗ (x) =
R∞
x
v
(i) If V (∞) = ∞, then there exists a constant C > 0, such that
Z ∞ Z x Z ∞
1
(2.1)
P
f v v(x) dx ≤
P [Cf (x)]v(x) dx,
V (x) 0
0
0
is satisfied for all f ≥ 0 if and only if P̃ ∈ ∆2 , and
Z ∞ Z ∞
Z ∞
fv
(2.2)
v(x) dx ≤
P [Cf (x)]v(x) dx,
P
V
0
x
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
is satisfied for all f ≥ 0 if and only if P ∈ ∆2 .
(ii) If V ∗ (0) = ∞, then
Z ∞ Z ∞ Z ∞
1
(2.3)
P
f v v(x) dx ≤
P [Cf (x)]v(x) dx,
V ∗ (x) x
0
0
is satisfied for all f ≥ 0 if and only if P̃ ∈ ∆2 , and
Z ∞
Z ∞ Z x
fv
(2.4)
P
v(x) dx ≤
P [Cf (x)]v(x) dx,
∗
0
0 V
0
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is satisfied for all f ≥ 0 if and only if P ∈ ∆2 .
Page 12 of 51
∗
The conditions V (∞) = ∞ and V (0) = ∞ are only required in the necessity part of the proof.
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Proof. First observe that if f¯(x) = f (1/x), v̄¯(x) = v(1/x)/x2 then via obvious changes of variables, (2.3) reduces to (2.1) with f replaced by f¯ and v by
v̄¯. A similar
R t change of variable shows that (2.4) reduces to (2.2). Note that
∗
V (1/t) = 0 v̄¯. Therefore it suffices to prove only part (i) of Lemma 2.3.
Next we observe that (2.1) is equivalent to
Z ∞ Z ∞
Z ∞
fv
(2.5)
P̃
v(x) dx ≤
P̃ [Cf (x)]v(x) dx.
V
0
x
0
To see this, recall that by [4, Prop. 2.5] (see also [11]) that (2.1) holds if and
only if for every ε > 0,
Z x
1
kT f kP (εv) ≤ Ckf kP (εv) where T f (x) =
f v.
V (x) 0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
Title Page
But by the standard duality principle in Orlicz spaces this is equivalent to
∗ Z ∞
g
T g g(t)
∗
≤C , where T g(x) = v(x)
dt
εv εv P̃ (εv)
V (t)
x
P̃ (εv)
is the conjugate operator of T . By homogeneity of the norm and again applying
[4, Prop. 2.5] it follows that this inequality is equivalent to
Z ∞ Z ∞ 1
Cg(x)
∗
(T g)(x) v(x) dx ≤
P̃
v(x) dx,
P̃
v(x)
v(x)
0
0
which is (2.5) with g = f v. Hence we only need to show that (2.1) is satisfied,
if and only if P̃ ∈ ∆2 .
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Let P̃ ∈ ∆2 and define f + (x) = f (x) if x ≥ 0 and zero otherwise,
v(|x|)dx = dµ(x), then
Z x
Z
1
1
+
f v ≤ (Mµ f )(x) := sup
f + dµ, I ∈ R.
V (x) 0
x∈I µ(I) I
Clearly Mµ is sublinear and of type (∞, ∞), and weak type (1, 1), with respect
to dµ. Now the argument of [5, p. 149-150] shows that P̃ ∈ ∆2 is sufficient for
Z ∞
Z ∞
+
P Mµ f (x) dµ(x) ≤
P (Cf (x)) dµ(x),
0
0
from which (2.1) follows.
To prove that (2.1) implies P̃ ∈ ∆2 , it suffices (see [5, Prop. 3]) to prove
that there exists a δ > 0, such that p(δx) ≤ 1/2 p(x), where p(x) = P 0 (x) with
p(0) = 0.
By Theorem 2.1, with Q = P , k(x, y) = 1, θ = 1/V , ρ = 1/v, f replaced
by f v, xj = r > 0, xj−1 = 0 and xj+1 = ∞, (2.1) implies
Z ∞ 1 χ(0,r) v(t) dt ≤ 1/ε
P
B V (t)
ε P̃ (εv)
r
for all ε > 0 and r > 0. But by the definition of the Luxemburg norm and (1.4)
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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with t = 1/(ε V (r))
Z r χ
1
1
(0,r) = inf λ > 0 :
P̃
εv(t) dt ≤ 1
ε P̃ (εv) ε
λ
0
1
=
εP̃ −1 εV1(r)
V (r) −1
1
≥
P
.
2
εV (r)
Hence (2.1) implies
Z ∞
If x =
V (r)
2BV (t)
1
V (r)
−1
P
v(t) dt ≤ 1/ε.
P
2BV (t)
εV (r)
r
1
−1
P
, this inequality is
εV (r)
Z
P −1 (
1
)/(2B)
εV (r)
0
P (x)
2B
.
dx
≤
1
x2
−1
εV (r)P
εV (r)
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Writing
y = P −1
1
εV (r)
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one obtains
Page 15 of 51
Z
(2.6)
0
y/(2B)
P (x)
dx ≤ 2B P (y)/y,
x2
y > 0.
J. Ineq. Pure and Appl. Math. 1(1) Art. 10, 2000
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Then it follows from (1.4) that
then on integrating by parts
R y/(4B)
0
Z
p(x)
x
dx ≤ 4Bp(y). Now let 0 < η < 1,
y/(4B)
p(x)
dx
x
0
Z y/(4B)
y ≥
log
dp(t)
4Bt
0
Z ηy/(4B)
y ≥
log
dp(t)
4Bt
0
≥ log(1/η)p(ηy/(4B)).
4Bp(y) ≥
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
Choose η so that log(1/η) ≥ 8B and δ = η/(4B), then p(δy) ≤
as was noted, implies by [5, Prop. 3] that P̃ ∈ ∆2 .
1
2
p(y). This,
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3.
Main Results
Our first result concerns the identity operator defined on monotone functions.
Theorem 3.1. Suppose P and Q are increasing and P is weakly convex. Then
Z ∞
Z ∞
−1
−1
(3.1) Q
Q[θ(x)f (x)]w(x) dx ≤ P
P [Cf (x)]v(x) dx
0
0
holds for all 0 ≤ f↓, if and only if there is a constant B > 0, such that,
(3.2)
!
!
Z xj+1
i
X Z xj+1 h εj
X
−1
−1
θ(x) w(x) dx ≤ P
P (εj )
v(x) dx
Q
Q
B
xj
xj
j
j
is satisfied for all non-negative decreasing sequences {εj }j∈Z and the covering
sequence
Rx
{xj }j∈Z such that 0 j v = 2k , k ∈ Z.
Similarly, (3.1) holds for all 0 ≤ f ↑, if and only if (3.2) is satisfied for all
non-negative
sequences {εj }j∈Z and the covering sequence {xj }j∈Z
R ∞increasing
−k
satisfying xj = 2 .
Proof. We only prove the first part of the theorem since the argument for the
second part is similar.
P
Let {εj }j∈Z be any decreasing sequence, then f (x) = j εj χ(xj ,xj+1 ) (x) is
decreasing and substituting f into (3.1), (3.2) follows with B = C.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Conversely if (3.2) holds then, since
M >1
Z ∞
−1
Q
Q[θ(x)f (x)]w(x) dx ≤ Q−1
R xj
v = 2k and 2P (x) ≤ P (M x),
0
XZ
0
Q[θ(x)f (xj )]w(x) dx
xj
j
≤ P −1
!
xj+1
X
Z
P (Bf (xj ))
=P
v
xj
j
−1
X
Z
≤P
XZ
=P
Z
Weighted modular inequalities
for Hardy-type operators on
monotone functions
v
xj−1
!
xj
Hans P. Heinig and Qinsheng Lai
P (M Bf (x))v(x) dx
xj−1
j
−1
!
xj
2P (Bf (xj ))
j
−1
!
xj+1
∞
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P [M Bf (x)]v(x) dx .
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0
This proves Theorem 3.1.
If Q ◦ P −1 is convex, Theorem 3.1 has the following form:
Corollary 3.2. Suppose P and Q are increasing, P is weakly convex and P Q. Then (3.1) holds for all 0 ≤ f↓, if and only if for all ε > 0 and r > 0, there
is a constant B > 0, such that,
Z r ε
θ(x) −1
−1
Rr
P
(3.3)
Q
Q
w(x) dx ≤ P −1 (ε).
B
v
0
0
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Similarly, (3.1) is satisfied for all 0 ≤ f↑, if and only if
Z ∞ ε
θ(x) −1
−1
R∞
P
(3.4)
Q
Q
w(x) dx ≤ P −1 (ε)
B
v
r
r
is satisfied.
Proof. By Theorem 3.1 is suffices to show that (3.2) with increasing (decreasing) sequence {εj }j∈Z is equivalent to (3.3) (respectively (3.4)).
First Rfix j = k0 ∈ Z and let xj0 = r > 0. Then for fixed ε > 0 define εm =
r
−1
P (ε/ 0 v), if m < k0 and zero otherwise. Clearly {εm }m∈Z is decreasing
and by (3.2)
Z r θ(x) −1
ε
−1
R
Q
P
Q
w(x) dx
r
B
v
0
0
!
X Z xj+1 θ(x)
ε
= Q−1
Q
P −1 R r
w(x) dx
B
v
x
j
0
j<k0
!
X Z xj+1 θ(x)εj w(x) dx
= Q−1
Q
B
j<k0 xj
!
Z xj+1
X ε
v(x) dx = P −1 (ε).
≤ P −1
P P −1 R r
v
xj
0
j<k
0
To prove the converse, recall that since P is weakly convex, there is an M ≥ 1,
such that 2P (x) ≤ P (M x). Hence with y = P (M x)
(3.5)
P −1 (y) ≤ M P −1 (y/2),
y > 0.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Rx
If {xj }j∈Z is a covering sequence satisfying 0 j v = 2k and ηj > 0, to be
determined later, then by (3.5) and (3.3) with ε = ηj and r = xj+1 ,
!#
Z xj+1 "
θ(x) −1
ηj
R xj+1
P
Q
w(x) dx
BM
v
xj
xj
Z xj+1 ηj
θ(x) −1
R xj+1
≤
Q
P
w(x) dx ≤ Q ◦ P −1 (ηj ).
B
v
0
0
Since P Q, summing over k ∈ Z yields
!#
"
X Z xj+1
ηj
θ(x) −1
R xj+1
P
w(x) dx
Q
BM
v
x
j
xj
j
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
!
≤
X
j
Q ◦ P −1 (ηj ) ≤ Q ◦ P −1
X
Aηj
,
j
where A is the constant arising from condition P Q (cf. Defn.
R x 1.1c) ). Now
choose ηj so that {ηj /2k } is decreasing, hence εj = P −1 (Aηj / xjj+1 v) defines
a decreasing sequence. Therefore
!
Z xj+1 !
X Z xj+1 θ(x)
X
P
(ε
)
j
Q−1
Q
P −1
w(x) dx ≤ P −1
P (εj )
v ,
M
B
A
x
x
j
j
j
j
and applying (3.5) α-times so that 2α /A ≥ 1, then
2P (εj )
1 −1 2
1
−1
P
≥
P
P (εj ) ≥ · · · ≥ α εj
2A
M
A
M
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and the result follows.
If 0 ≤ f↑, fix k0 ∈ Z such that xj0 = r > 0 and define
Z ∞
−1
εm = P (ε/
v) if m ≥ k0 and zero otherwise.
r
Then {εm }m∈Z is increasing and the previous argument shows that
R ∞ (3.4) follows
from (3.2). Also if {xj }j∈Z is a covering sequence such that xj v = 2−k and
Rx
ηj > 0, then by (3.5) and (3.4), since 2 xjj+1 v = 2−k ,
Z
"
xj+1
Q
xj
θ(x) −1
P
MB
Z
≤
ηj
R xj+1
xj
"
∞
xj
Q
!#
v
w(x) dx
θ(x) −1
P
B
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
η
R ∞j
xj
!#
v
w(x) dx ≤ Q ◦ P −1 (ηj ).
η
j
Summing over k, and choosing ηj so that { 2−k
} is an increasing sequence, then
with
!
Aη
j
εj = P −1 R xj+1
,
k ∈ Z,
v
xj
defines an increasing sequence, where A is the constant arising from the condition P Q. The inequality (3.2) now follows as before.
Corollary 3.2 was proved by J. Q. Sun [24, Lemma 3.1] in the case when P
and Q are N -functions (hence convex). If P (x) = xp , Q(x) = xq , 0 < p ≤ q <
∞, one obtains (with θ(x) = 1) the well known weight conditions ([21, 23])
which characterize (3.1). If 0 < q < p < ∞ we have:
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Corollary 3.3. Let 0 < q < p < ∞ and 1/r = 1/q − 1/p, then the following
are equivalent:
Z
∞
q
1/q
∞
≤C
f w
(3.6)
Z
p
1/p
f v
0
0
is satisfied for all 0 ≤ f↓.
Z ∞
(3.7)
[W 1/p V −1/p ]r w ≡ B0r < ∞,
Weighted modular inequalities
for Hardy-type operators on
monotone functions
0
X
r
w(Ej )1/q (Ej )−1/p ≡ B1r < ∞,
(3.8)
Hans P. Heinig and Qinsheng Lai
j
Rx
Rx
where w(Ej ) = xjj+1 w, v(Ej ) = xjj+1 v and the covering sequence {xj }
satisfies V (xj ) = 2k .
#1/q
"
(3.9)
X
j
εqj w(Ej )
#1/p
"
≤B
X
εpj v(Ej )
j
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holds for all decreasing sequences {εj }j∈Z and covering sequences {xj } with
Rt
Rt
V (xj ) = 2k . (Recall: W (t) = 0 w, V (t) = 0 v.)
Close
If 0 ≤ f ↑ the above statement holds with W and V replaced by W ∗ and
V , respectively, the covering sequence {xj } satisfies V ∗ (xj ) = 2−k and {εj }
is taken to be increasing.
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Proof. We only prove the corollary in the case 0 ≤ f↓ since the case 0 ≤ f↑ is
proved, with obvious modifications, in the same way.
The equivalence of (3.6) and (3.9) follows at once from Theorem 3.1 with
Q(x) = xq , P (x) = xp , θ(x) = 1. Since the equivalence of (3.6) and (3.7) was
proved in [21, 23], it remains to prove (3.7)
R xj+1⇒ (3.8) ⇒ (3.9).
Since r/q = r/p + 1 and w(Ej ) = xj w, it follows that
r/q
w(Ej )
r
=
q
≤
r
q
Z
xj+1
Z
t
!
w w(t) dt
xj
Z xj+1
xj
W (t)r/p w(t) dt
xj
on integrating. Since v(Ej ) = 2k = V (xj ) it follows therefore that
Z
X
r X xj+1 −rk/p
1/q
−1/p r
[w(Ej ) v(Ej )
] ≤
2
W (t)r/p v(t) dt
q
x
j
j
j
r/p X Z xj+1
r2
≤
V (t)−r/p W (t)r/p w(t) dt
q
xj
j
r2r/p r
=
B0 .
q
Hence (3.7) ⇒ (3.8).
Since the dual of the sequence space `p/q is `r/q , it follows that
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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−q/p r/q
]
j [w(Ej )v(Ej )
P
= B1r < ∞ implies
!q/p
X
ηj w(Ej )v(Ej )−q/p ≤ B1q
X
j
p/q
ηj
j
1/q
for any positive sequence {ηj } in `p/q . Now choose {ηj } so that ηj = εj v(Ej )1/p =
εj 2k/p with {εj }j∈Z decreasing. Thus (3.8) ⇒ (3.9), which completes the proof.
α
Note that if 0 < q < p = 1, then with q = α+1
, α > 0, r = α and one shows
that
B1
≤ B0 ≤ 2(1 + α)1/α B1 .
2(1 + α)1/α
B0 =
Hans P. Heinig and Qinsheng Lai
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Here of course
Z
Weighted modular inequalities
for Hardy-type operators on
monotone functions
∞
W α V −α w
Contents
!1/α
1/α
and B1 =
0
X
w(Ej )α+1 v(Ej )−α
.
j
We now give the main result of this section.
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Theorem 3.4. Suppose P is an N -function, P and P̃ satisfy the ∆2 condition
and Q weakly convex. If K is a Hardy-type operator defined on the cone of
decreasing functions, then
Z ∞
Z ∞
−1
−1
(3.10) Q
Q[θ(x)Kf (x)]w(x) dx ≤ P
P [Cf (x)]v(x) dx
0
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is satisfied, if and only if there is a constant B > 0, such that for all positive
sequences {εj }j∈Z and all covering sequences {xj }j∈Z with i(x) = x
(3.11)
!
"
#
!
X Z xj+1
X
k(x
,
·)χ
i
1
θ(x)
j
(xj−1 ,xj )
Q−1
Q
w(x) dx ≤ P −1
B εj V
εj
j
xj
P̃ (εj v)
j
(3.12)
Q−1
XZ
j
xj+1
xj
#
!
χ
i
θ(x)k(x, xj ) (xj−1 ,xj ) w(x) dx ≤ P −1
Q
εj V B
P̃ (εj v)
"
X 1
εj
j
!
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
(3.13)
−1
Q
XZ
j
xj+1
xj
"
#
(K1)χ
θ(x) (xj−1 ,xj ) w(x) dx
Q
B εj V
P̃ (εj v)
!
≤ P −1
X 1
εj
j
!
are satisfied, and for all positive
sequences {εj }j∈Z and the covering
R xj decreasing
k
sequence {xj }j∈Z satisfying 0 v = 2
(3.14)
!
Z xj+1 !
X Z xj+1 εj θ(x)(K1)(x) X
Q−1
Q
w(x) dx ≤ P −1
P (εj )
v
B
x
x
j
j
j
j
is satisfied.
Proof. (Sufficiency.)
R ∞ The idea comes from [23]. We may assume that f has
the form f (x) = x h, h > 0, for once the result has been proved for such
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f , a limitingR argument (see e.g. [24]) gives the general case. Clearly since
x
(K1)(x) = 0 k(x, y) dy
Z x
Z ∞
(Kf )(x) =
k(x, y)
h(t) dtdy
0
y
Z t
Z x
= (K1)(x)f (x) +
h(t)
k(x, y) dy dt.
0
But since
Z x
1
1
−
=
V (y)−2 v(y) dy
V (t) V (x)
t
0
Z
x
Z
h(s)V (s) ds ≤
and
0
x
f (t)v(t) dt,
0
it follows again on interchanging the order of integration that
Z x
Z t
h(t)
k(x, y) dydt
0
0
Z xZ t
Z x
1
−2
+
V (s) v(s) ds dydt
=
k(x, y)h(t)V (t)
V (x)
t
0
0
Z x
Z x
1
=
k(x, y)
h(t)V (t) dtdy
V (x) 0
y
Z t
Z x
Z s
−2
+
V (s) v(s)
h(t)V (t)
k(x, y) dy dtds
0
0
0
Z x
Z x
1
≤
k(x, y)
f (t)v(t) dtdy
V (x) 0
0
Z s
Z s
Z x
−2
+
V (s) v(s)
k(x, y) dy
f (t)v(t) dtds
0
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
0
Hans P. Heinig and Qinsheng Lai
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Z x
1
≤ (K1)(x)
f (t)v(t) dt + I(x)
V (x) 0
(by definition of I(x)) respectively.
Now since k(x, y) ≤ D[k(x, s) + k(s, y)], y < s < x,
Z
I(x) ≤ D
0
x
−2
Z
s
k(x, s)V (s) v(s)s
f (t)v(t) dtds
0
Z s
Z x
−2
+
V (s) v(s)
f (t)v(t) dt (K1)(s) ds
0
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
and writing
F (s) = V (s)−2 v(s)s
Z
s
f v,
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one obtains
Contents
Z
x
1
f (t)v(t) dt
(Kf )(x) ≤ (K1)(x)f (x) + (K1)(x)
V (x) 0
Z x
Z x
(K1)(s)
+D
k(x, s)F (s) ds + D
F (s) ds
s
0
0
≡ (I1 + I2 + I3 + I4 )(x),
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respectively. Now
θ(x)(Kf )(x) ≤ θ(x)
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X
i=1
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Ii (x) ≤ 4θ(x) max Is (x) = 4θ(x)Is(x) (x),
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where s(x) ∈ {1, 2, 3, 4}, and since Q is increasing and satisfies 2Q(x) ≤
Q(M x), M > 1, we have
Q[θ(x)(Kf )(x)] ≤ Q[4θ(x)Is(x) (x)]
≤
≤
4
X
s=1
4
X
i=1
Q[4θ(x)Is (x)]
1
Q[4M 2 θ(x)Is (x)].
4
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Integration yields
Z
∞
Q[θ(x)(Kf )(x)]w(x) dx ≤
0
Hans P. Heinig and Qinsheng Lai
4
X
s=1
1
4
Z
∞
Q[4M 2 θ(x)Is (x)]w(x) dx
and therefore it suffices to prove that
Z
Z ∞
2
−1
(3.15)
Q[4M θ(x)Is (x)]w(x) dx ≤ Q ◦ P
0
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∞
P [Cf (x)]v(x) dx
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s = 1, 2, 3, 4 is satisfied.
Since I1 (x) = (K1)(x)f (x), then by Theorem 3.1 with θ replaced by θ(x)(K1)(x),
(3.15) holds if and only if (3.14) Ris satisfied.
Rx
x
1
1
Next, since 0 ≤ f↓ so is V (x)
f
v,
and
since
I
(x)
=
(K1)(x)
f v,
2
V (x) 0
0
Theorem 3.1 shows (with θ replaced by θ(x)(K1)(x)) that (3.14) is equivalent
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to
Z ∞
2
Q[4M θ(x)I2 (x)]w(x) dx ≤ Q ◦ P
0
−1
Z
∞
P C
1
V (x)
Z
x
f v v(x) dx
Z ∞
−1
≤Q◦P
P [Cf (x)]v(x) dx .
0
0
0
Here the last inequality follows from (2.1) of Lemma 2.3.
Next, I3 (x) = D(KF )(x), so that by Theorem 2.1 with ρ(x) = V (x)/(xv(x))
Z ∞ Z x Z ∞
1
2
−1
f v v(x) dx
Q[4M θ(x)I3 (x)]w(x) ≤ Q ◦ P
P C
V (x) 0
0
0
Z ∞
−1
≤Q◦P
P [Cf (x)]v(x) dx .
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
Title Page
Here the first inequality holds if (3.11) and (3.12) are satisfied and the second
follows from (2.1) ofR Lemma 2.3.
x
Since I4 (x) = D 0 (K1)(s) F (s)
ds we apply Theorem 2.1 with k(x, y) = 1
s
and ρ(x) = V (x)/(v(x)(K1)(x)), so that
Z ∞ Z ∞
Z x 1
2
−1
Q[4M θ(x)I4 (x)]w(x) dx ≤ Q ◦ P
P C
f v v(x) dx
V (x) 0
0
0
Z ∞
−1
≤Q◦P
P [Cf (x)]v(x) dx
0
is satisfied if and only if (3.13) holds. The last inequality follows of course
again from (2.1) of Lemma 2.3.
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(Necessity.) Since 0 ≤ f↓, (Kf )(x) ≥ (K1)(x)f (x) so that (3.10) implies
(3.1) with θ replaced by θ(x)(K1)(x). Now Theorem 3.1 applies if P is an
N -function and Q weakly convex and so (3.14) follows.
To prove that (3.10) implies (3.11) observe first that for fixed k,
ik(xj , ·)χ(xj−1 ,xj ) (3.16)
εj V
P̃ (εj v)
is bounded. If this is not the case, then there is a sequence {fn } of non-negative
functions satisfying kCfn kP (εj v) ≤ 1, with C the constant of (3.10), and a
sequence {αn } with αn → ∞, n → ∞, such that (by definition of Orlicz norm)
for each n
Z xj
xk(xj , x)fn (x)v(x)
αn < C
dx
V (x)
xj−1
Z x Z xj
k(xj , x)fn (x)v(x)
dy dx
≤C
V (x)
0
0
Z xj Z xj
k(xj , x)fn (x)v(x)
=C
dxdy
V (x)
0
y
Z xj
≤C
k(xj , y)Fn (y) dy,
0
since k(xj , ·) is decreasing. Here Fn (y) =
Z
R∞
y
fn (x)v(x)
V (x)
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∞
P [Cfn (x)]εj v(x) dx ≤ 1,
0
dx. But since
Weighted modular inequalities
for Hardy-type operators on
monotone functions
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(2.2) of Lemma 2.3 and (3.10) show that
Z ∞
1
−1
−1
P
≥P
P [Cfn (x)]v(x) dx
εj
0
Z ∞ C
−1
≥P
Fn (x) v(x) dx
P
C1
0
Z ∞ θ(x)
−1
≥Q
Q
(KFn )(x) w(x) dx
C1
0
!
Z xj+1 Z
θ(x) xj
−1
≥Q
Q
k(xj , y)Fn (y) dy w(x) dx
C1 0
xj
!
Z xj+1 θ(x)αn
−1
>Q
Q
w(x) dx ,
C C1
xj
where C1 is the constant of (2.2). But this is a contradiction since αn → ∞.
Hence (3.16) is bounded.
Now suppose (3.11) fails to be satisfied. Then for any B > 0 there exists a
covering sequence {xj }j∈Z and a positive sequence {εj }j∈Z such that
!
"
#
!
X 1
X Z xj+1
k(x
,
·)χ
i
θ(x)
j
(x
,x
)
j−1 j
P −1
< Q−1
Q
w(x) dx
ε
2BC
ε
V
j
1
j
x
j
P̃
(ε
v)
j
j
j
where C1 is taken to be the constant of (2.2). Now for k ∈ Z, choose fj ≥ 0,
such that suppfj ⊂ (xj−1 , xj ) with
Z ∞
(3.17)
P [BC1 fj (x)]εj v(x) dx ≤ 1
0
Weighted modular inequalities
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and
1
2BC1
Z xj
k(xj , ·)χ(xj−1 ,xj ) i xk(xj , x)fj (x)v(x)
≤
dx
εj V
V (x)
xj−1
P̃ (εj v)
Z xj
≤
k(xj , y)Fj (y) dy,
0
where
Z
∞
fj (x)v(x)
dx.
V (x)
y
R∞
P
Let f (x) = j fj (x) and F (x) = x f (t)v(t)
dt, then by (2.2) of Lemma
V (t)
2.3, (3.17) and our assumption
Z ∞
−1
P
P [BF (x)]v(x) dx
0
Z ∞
−1
≤P
P [BC1 f (x)]v(x) dx
0
!
X Z xj
= P −1
P [BC1 fj (x)]v(x) dx
Fj (y) =
j
≤ P −1
−1
<Q
Hans P. Heinig and Qinsheng Lai
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xj−1
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!
Close
X 1
εj
j
X Z xj+1
j
Weighted modular inequalities
for Hardy-type operators on
monotone functions
xj
Quit
"
θ(x)
Q
2BC1
#
k(xj , ·)χ(xj−1 ,xj ) i w(x) dx
εj V
P̃ (εj v)
!
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≤ Q−1
XZ
xj
j
−1
Z
xj+1
Z
Q θ(x)
xj
!
k(xj , y)Fj (y) dy w(x) dx
0
∞
≤Q
Q [θ(x)(KF )(x)] w(x) dx
Z ∞
−1
≤P
P [CF (x)]v(x) dx ,
0
0
where the last inequality is (3.10). But this is impossible for B > C and hence
(3.11) must be satisfied.
To show that (3.12) and (3.13) are satisfied one proceeds as before. First one
shows that both
χ(xj−1 ,xj ) i (K1)χ(xj−1 ,xj ) and εj V ε
V
j
P̃ (εj v)
P̃ (εj v)
Weighted modular inequalities
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monotone functions
Hans P. Heinig and Qinsheng Lai
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are bounded for fixed k. Then with f (x) and F (x) defined as above one has
Z xj
Z xj Z ∞
x fj (x)v(x)
fj (x)v(x)
dx ≤
dxdy
V (x)
V (x)
xj−1
0
y
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and for (3.13)
Z xj
Z xj Z x
(K1)(x)fj (x)v(x)
fj (x)v(x)
dx ≤
k(x, y) dy
dx
V (x)
V (x)
xj−1
0
0
Z xj
Z ∞
fj (x)v(x)
≤
k(xj , y)
dxdy.
V (x)
0
y
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Inequality (3.12) is then obtained as (3.11) was shown to hold. To prove (3.13)
assume to the contrary that (3.13) fails. Then for any B > 0, we have
Z ∞
−1
P
P [BF (x)]v(x) dx
0
Z ∞
−1
≤P
P [BC1 f (x)]v(x) dx
0
!
X 1
≤ P −1
εj
j
#
!
"
X Z xj+1
(K1)χ
θ(x)
(x
,x
)
j−1 j < Q−1
w(x) dx
Q
2BC
ε
V
1
j
x
j
P̃
(ε
v)
j
j
!
Z
Z
Z
x
x
∞
j+1
j
X
fj (t)v(t)
≤ Q−1
dtdy w(x) dx
Q θ(x)
k(x, y)
V (t)
xj
0
y
j
!
Z xj
X Z xj+1 ≤ Q−1
Q θ(x)
k(x, y)F (y) dy w(x) dx
xj
j
−1
Z
0
∞
≤Q
Q [θ(x)(KF )(x)] w(x) dx
Z ∞
−1
≤P
P [CF (x)]v(x) dx
0
0
from which the contradiction follows for B > C. This proves Theorem 3.4.
If k(x, y) = 1, θ(x) = xa , −1 ≤ a < ∞, the conditions (3.11), (3.12),
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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(3.13) coincide and since (K1)(x) = x we get the following corollary.
Corollary 3.5. Let P and Q be as in Theorem 3.4 and a ≥ −1. Then
Z ∞ Z x Z ∞
−1
a
−1
Q
Q x
f w(x) dx ≤ P
P [Cf (x)]v(x) dx
0
0
0
is satisfied for all 0 ≤ f ↓, if and only if for
decreasing sequences {εj }j∈Z
R xall
j
and the covering sequence {xj } satisfying 0 v = 2k ,
!
Z xj+1 !
i
X Z xj+1 h εj
X
(3.18) Q−1
Q
xa+1 w(x) dx ≤ P −1
P (εj )
v
B
x
x
j
j
j
j
holds, and
(3.19)
XZ
−1
Q
j
Hans P. Heinig and Qinsheng Lai
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xj+1
xj
#
!
iχ
x (xj−1 ,xj ) w(x) dx ≤ P −1
Q
B εj V P̃ (εj v)
"
a
X 1
εj
j
!
is satisfied for all positive sequences {εj } and all covering sequences {xj }.
If Q is also an N -function, then a result corresponding to Corollary 3.5 holds
also for the dual operator.
Corollary 3.6. Let P and Q be N -functions and P , P̃ ∈ ∆2 . If a ≥ −1 then
(3.20) Z
Z ∞
Z ∞
∞
−1
a
−1
Q
Q
t f (t) dt w(x) dx ≤ P
P [Cf (x)]v(x) dx
0
Weighted modular inequalities
for Hardy-type operators on
monotone functions
x
0
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is satisfied for all 0 ≤ f↓, if and only if
(3.21)
#
!
" X Z yj
k(·,
y
1 j )χ(yj ,yj+1 ) −1
Q
w(x) dx ≤ P −1
Q
B
εj V
yj−1
P̃ (εj v)
j
X 1
εj
j
!
and
(3.22)
Q−1
#
!
χ
k(yj , x) (yj ,yj+1 ) Q
w(x) dx ≤ P −1
εj V B
yj−1
P̃ (εj v)
XZ
j
"
yj
X 1
εj
j
!
holds for all positive sequences {εj }j∈Z and all covering sequences {yj }j∈Z .
Here
ln(y/x)
if a = −1,
k(y, x) =
a+1
a+1
y
−x
if a > −1.
Proof. By [7, Thm. 2.2], (3.20) is equivalent to
Z ∞ Z ∞ Z ∞
−1
a
Q
Q
t
h(s) dsdt w(x) dx
0
x
t
Z ∞ V (x)h(x)
−1
≤P
v(x) dx ,
P C
v(x)
0
h ≥ 0. However, since
Z ∞ Z
ta
x
t
∞
Z
h(s) dsdt =
Weighted modular inequalities
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monotone functions
Hans P. Heinig and Qinsheng Lai
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∞
k(y, x)h(y) dy,
Page 36 of 51
x
the result follows from Proposition 2.2 with θ(x) = 1, ρ(x) = V (x)/v(x).
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Remark 3.1.
(i) Let P (x) = xp , Q(x) = xq , 0 < q < p < ∞, p > 1 and a = −1 then
(3.18) is
!1/q
!1/p
X q
X p
εj w(Ej )
≤C
εj v(Ej )
,
j
where w(Ej ) =
equivalent to
Z
j
R xj+1
xj
w and v(Ej ) =
R xj+1
xj
v. But by Corollary 3.3 this is
∞
[W 1/p V −1/p ]r w < ∞,
0
−q/p
Also, if ηj = εj
X
Z
xj+1
ηj
xj
j
Weighted modular inequalities
for Hardy-type operators on
monotone functions
1
1 1
= − .
r
q p
Hans P. Heinig and Qinsheng Lai
then (3.19) takes the form
x−q w(x) dx
Title Page
! Z
xj
xj−1
!q/p0
0
0
tp V (t)−p v(t) dt
!q/p
≤C
X
p/q
ηj
j
But the dual space of `p/q is `r/q , where 1r = 1q − p1 and hence (3.19) is in
this case

!r/q Z
!r/p0 1/r
X Z xj+1

xj
0
0
x−q w(x) dx
tp V (t)−p v(t) dt
≤ C.


xj
xj−1
j
(Cf. [21, Thm. 2], where this was proved in case 1 < q < p < ∞ and [23]
in the remaining case.)
.
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(ii) Considering Corollary 3.6 in the case P (x) = xp , Q(x) = xq , 1 < q <
p < ∞, a = −1 we see that (3.21) takes the form
!q/p0
!q/p
Z yj ! Z yj+1
X
X p/q
0
0
ηj
w
lnp (t/yj )V (t)−p v(t) dt
≤C
ηj
,
yj−1
j
yj
j
−q/p
where again ηj = εj . But again since `r/q is the dual of `p/q it follows
that in this case (3.21) is equivalent to

!r/q Z
!r/p0 1/r
X Z yj

yj+1
0
0
w
lnp (t/yj )V (t)−p v(t) dt
≤ C,


yj−1
yj
j
where
1
r
=

X Z

1
r
− p1 . Similarly, (3.22) takes the form
!r/q
yj
lnq (yj /x)w(x) dx
yj−1
j
=
1
q
1
q
Z
yj+1
yj
!r/p0 1/r

0
V (t)−p v(t) dt
≤ C,

− p1 , for all covering sequences {yj }j∈Z .
Hence these two conditions are necessary and sufficient for the inequality
Z ∞
Z ∞
q 1/q
Z ∞
1/p
f (t)
p
dt dx
≤C
f (x) v(x) dx
w(x)
t
0
x
0
Weighted modular inequalities
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to be satisfied for all 0 ≤ f↓.
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4.
Hardy-type operators on increasing functions
In order to obtain weight characterizations for which modular inequalities for
the Hardy-type operator
Z x
(Kf )(x) =
k(x, y)f (y) dy,
0 ≤ f↑
0
are satisfied, we require also that the kernel k̄ defined by
Z x
(4.1)
k̄(x, y) =
k(x, t) dt
Weighted modular inequalities
for Hardy-type operators on
monotone functions
y
Hans P. Heinig and Qinsheng Lai
satisfies also conditions (i) and (ii) of (1.1). That is,
k̄(x, y) ≤ D[k̄(x, z) + k̄(z, y)],
(4.2)
0 < y < z < x.
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Note that if k(x, t) = (x − t)α , α ≥ 0 then k̄ satisfies (4.2). On the other
hand if k(x, t) = ln(x/t) then k̄ does not satisfy (4.2) for any D ≥ 1.
The principal result for Hardy-type operators defined on the cone of increasing functions is the following:
Theorem 4.1. Suppose K is a Hardy-type operator and k̄ defined by (4.1) satisfies (4.2). Let P be an N -function with P , P̃ ∈ ∆2 and Q weakly convex.
Then the modular inequality
(4.3) Z
Z
∞
−1
Q
∞
Q[θ(x)(Kf )(x)]w(x) dx
0
≤P
−1
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P [Cf (x)]v(x) dx
0
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is satisfied for all 0 ≤ f↑, if and only if there is a constant B > 0, such that,
(4.4)
!
#
!
"
X Z xj+1
X 1
k̄(x
θ(x) j , ·)χ(xj−1 ,xj ) −1
−1
Q
w(x) dx ≤ P
Q
∗
B
ε
V
εj
j
x
j
P̃ (εj v)
j
j
and
(4.5)
−1
Q
XZ
j
xj+1
xj
#
!
χ
k̄(x, xj )θ(x) (xj−1 ,xj ) w(x) dx ≤ P −1
Q
∗
B
εj V
P̃ (εj v)
"
X 1
εj
j
holds for all positiveRsequences {εj }j∈Z and all covering sequences {xj }j∈Z .
∞
Here again V ∗ (x) = x v with V ∗ (0) = ∞.
Proof.
Without loss of generality we may assume that
the form f (x) =
Rx
R xf has
∗
−1
∗
−2
h,
h
≥
0
(cf.
[24,
Lemma
3.2]).
Since
V
(x)
=
V
(t)
v(t) dt, chang0
0
ing the order of integration we show that
Z x
Z y
(Kf )(x) =
k(x, y)
h(s) dsdy
0
0
Z x
V ∗ (s)
=
h(s)k̄(x, s) ∗
ds
V (s)
0
Z x
Z s
∗
=
h(s)k̄(x, s)V (s)
V ∗ (t)−2 v(t) dtds
0
Z0 x
Z x
∗
−2
≤
V (t) v(t)k̄(x, t)
h(s)V ∗ (s) dsdt
0
t
!
Weighted modular inequalities
for Hardy-type operators on
monotone functions
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Z
≤
x
∗
−2
Z
V (t) v(t)k̄(x, t)
0
∞
v(y)f (y) dydt.
t
R∞
Rx
Hence if F (t) = V ∗ (t)−2 v(t) t f v, then Kf (x) ≤ 0 k̄(x, t)F (t) dt and by
Theorem 2.1 with ρ(x) = V ∗ (x)/v(x)
Z ∞
Z ∞ Z x
Q[θ(x)(Kf )(x)]w(x) dx ≤
Q θ(x)
k̄(x, t)F (t) dt w(x) dx
0
0
0
Z ∞ Z ∞ C
−1
≤Q◦P
P
vf v(x) dx
V ∗ (x) x
0
if and only if (4.4) and (4.5) are satisfied. Now (4.3) follows from (2.3) of
Lemma 2.3.
To prove necessity one proves first that for fixed k
k̄(xj , ·)χ(xj−1 ,xj ) εj V ∗
P̃ (εj v)
is bounded. But this is proved (via contradiction) in the same way as the boundedness of (3.16) in the proof of Theorem 3.4, only now k and V are replaced by
k̄ and V ∗ , respectively. To prove that (4.4) is satisfied assume to the contrary
that for every B > 0 there exist {xj } and {εj } such that
"
#
!
!
X Z xj+1
X
k̄(x
,
·)χ
θ(x)
1
j
(xj−1 ,xj )
Q−1
Q
w(x) dx > P −1
.
∗
2BC
ε
V
ε
1
j
j
x
j
P̃
(ε
v)
j
j
j
Weighted modular inequalities
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By
R ∞ duality of Orlicz spaces there exists fj ≥ 0 such that suppfj ⊂ (xj−1 , xj ),
P [BC1 fj ]εj v ≤ 1 and
0
Z xj
k̄(x
1 k̄(xj , x)fj (x)v(x)
j , ·)χ(xj−1 ,xj ) <
dx.
∗
2BC1
εj V
V ∗ (x)
xj−1
P̃ (εj v)
Rx
P
Now let f = fj and F (x) = 0 Vf v∗ , so F↑. Also
Z xj
Z xj
Z
k̄(xj , x)fj (x)v(x)
fj (x)v(x) xj
dx ≤
k(xj , s) dsdx
V ∗ (x)
V ∗ (x)
xj−1
0
x
Z xj
Z s
fj (x)v(x)
dxds
=
k(xj , s)
V ∗ (x)
0
0
Z xj
≤
k(xj , s)F (s) ds
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
0
Title Page
and therefore by (2.4) of Lemma 2.3
Z ∞
−1
P
P [BF (x)]v(x) dx
0
Z ∞
−1
≤P
P [BCf (x)]v(x) dx
0
!
X 1
−1
≤P
εj
j
#
!
"
X Z xj+1
k̄(x
,
·)χ
θ(x)
j
(x
,x
)
j−1 j w(x) dx
< Q−1
Q
∗
2BC
ε
V
1
j
xj
P̃ (εj v)
j
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XZ
≤ Q−1
Z
Z
!
xj
Q[θ(x)
xj
j
−1
xj+1
k(xj , s)F (s) ds]w(x) dx
0
∞
≤Q
Q[θ(x)(KF )(x)]w(x) dx
Z ∞
−1
≤P
P [CF (x)]v(x) dx .
0
0
Here the last inequality is (4.3). But this is a contradiction for B > C. Hence
(4.4) is satisfied.
The proof of (4.5) is similar, only now fj is chosen so that
Z xj
χ
1 fj (y)v(y)
(xj−1 ,xj ) <
dy
2C1
εj V ∗
V ∗ (y)
xj−1
P̃ (εj v)
Weighted modular inequalities
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Hans P. Heinig and Qinsheng Lai
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and
Z
k̄(x, xj )
0
xj
Z
xj
fj (y)v(y)
dy
V ∗ (y)
0
Z x
fj (y)v(y)
dy
≤
k̄(x, y)
V ∗ (y)
0
Z x
Z s
fj (y)v(y)
=
k(x, s)
dyds,
V ∗ (y)
0
0
fj (y)v(y)
dy ≤
V ∗ (y)
k̄(x, y)
x ∈ (xj , xj+1 ). We omit the details. This proves the theorem.
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Remark 4.1. (i) If V ∗ (0) < ∞, Theorem 4.1 still holds, provided that in addition to (4.4) and (4.5) the weight condition
(4.6) Z
∞
1 −1
1
1
−1
−1
Q
Q
P
θ(x)
k̄(x,
0)
w(x)
dx
≤
P
B
εV ∗ (0)
ε
0
is also satisfied for all ε > 0.
(ii) If Q is an N -function and hence convex, the result may also be proved via
the duality principle given in [7, Thm. 2.2].
A consequence of Theorem 4.1 is the following:
Corollary 4.2.
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
(i) ([10, Thm. 2.1]) If 1 < p ≤ q < ∞, then
Z ∞
1/p
Z ∞ Z x q
1/q
1
p
≤C
f v
(4.7)
f w(x) dx
x 0
0
0
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JJ
J
is satisfied for all 0 ≤ f↑, if and only if, for all t > 0
Z ∞
1/q
q −q
(x − t) x w(x) dx
V ∗ (t)−1/p
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and
Z
∞
−q
1/q Z
are bounded.
p0
∗
−p0
(t − x) V (x)
x w(x) dx
t
t
1/p0
v(x) dx
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(ii) If 0 < q < p < ∞, p > 1 then (4.7) is satisfied for all 0 ≤ f↑ if and only
if for all covering sequences {xj }
(4.8)

!r/q Z
!r/p0 1/r
xj
X Z xj+1
0
0

 ≤C
x−q w(x) dx
(xj − x)p V ∗ (x)−p v(x) dx
xj
j
and

X Z

=
!r/q
(x − xj )q x−q w(x) dx
xj
j
1
r
xj+1
xj−1
1
q
∗
Z
xj
!r/p0 1/r
0
V ∗ (x)−p v(x) dx
 ≤C
xj−1
Hans P. Heinig and Qinsheng Lai
− p1 , are satisfied.
(If V (0) < ∞ the condition (4.6) must also be taken into account.)
Proof. Let Q(x) = xq , P (x) = xp , 1 < p ≤ q < ∞, θ(x) = x1 , k(x, y) = 1
in Theorem 4.1. Since P Q we may take in Theorem 4.1 xj = t > 0,
xj−1 = 0, xj+1 = ∞ and the result (i) follows. If 0 < q < p < ∞, p > 1, then
Q is weakly convex and by Theorem 4.1, (4.7) is satisfied for all 0 ≤ f↑, if and
only if for all covering sequences {xj }

! Z
!q/p0 
Z
x
x
j+1
j
X
0
0


ηj
x−q w(x) dx
(xj − x)p V ∗ (x)−p v(x) dx
j
xj
Weighted modular inequalities
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xj−1
!q/p
≤ C1
X
j
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p/q
ηj
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and

X

ηj
j
Z
xj+1
(x − xj )q xq w(x) dx
xj
! Z
xj
!q/p0 
0
V ∗ (x)−p v(x) dx

xj−1
!q/p
≤ C2
X
p/q
ηj
j
−q/p
where we have taken ηj = εj
in (4.4) and (4.5). But since the dual of `p/q
is `r/q , 1r = 1q − p1 , the previous two estimates are equivalent to (4.8) and (4.9)
respectively.
The result of Corollary 4.2 (ii) in the case 1 < q < p < ∞ was also proved
in [10, Thm. 2.2], but the case 0 < q < 1 < p seems to be new.
In the remaining portion of this section we apply the results of the previous
section to show that the Hardy-Littlewood maximal function and the Hilbert
transform are bounded in weighted Orlicz-Lorentz spaces. This, in particular,
extends the Lorentz space results of Ariño-Muckenhoupt [1] and Sawyer [21]
to this general setting.
If P V
is an increasing function of R+ with P (0) = 0, then the Orlicz-Lorentz
spaces P R(v), with weight v consist
of all Lebesgue measurable f on Rn such
∞
that P −1 0 P (f ∗ (x))v(x) dx < ∞. Here f ∗ (t) = inf{s > 0 : |{x :
|f (x)| > s}| ≤ t} denotes the equimeasurable decreasing rearrangement of
|f |.
R
1
Recall that if (M f )(x) = supx∈Q |Q|
|f (y)| dy is the Hardy-Littlewood
Q
Rx
maximal function, then it is well known (cf. [2]) that (M f )∗ (x) ≈ x1 0 f ∗ . It
Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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follows therefore from Corollary 3.5 with a = −1 that the following proposition
holds:
Proposition 4.3.
Q weakly con
V SupposeVP is an N -function, P , P̃ ∈−1∆2R and
∞
∗
vex. Then M : P (v) → Q (w) is bounded, that is Q
Q((M
f
)
)w
≤
0
R∞
−1
∗
CP
P (Cf )v , if and if there are constants B > 0, such that
0
!
Z xj+1 !
X εj Z xj+1
X
−1
−1
(4.9)
Q
Q
w ≤P
P (εj )
v
B xj
xj
j
j
is satisfiedRfor all decreasing sequences {εj } and the covering sequence {xj }
x
satisfying 0 j v = 2k , and
(4.10)
#
!
!
"
X Z xj+1
X
i
χ
1
1
(x
,x
)
j−1 j Q−1
w(x) dx ≤ P −1
Q
εj V
xB
εj
xj
P̃ (εj v)
j
j
is satisfied for all positive sequences {εj }j∈Z and all covering sequences {xj }.
Rx
Here again V (x) = 0 v and i(x) = x.
Another illustration involves the Hilbert transform defined by the principle
value integral
Z
1 ∞ f (t)
dt.
(Hf )(x) = P.V.
π −∞ x − t
Then (see [21, (1.15)]) the rearrangement inequality
Z x
Z ∞ ∗
1
f (t)
∗
∗
(Hf ) (x) ≤ C1
f (t) dt +
dt ≤ C2 (Hf ∗ )∗ (x)
x 0
t
x
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is satisfied.
But this implies that the Hilbert transform is bounded from
V
to Q (w) if and only if
−1
Z
∞
Q
Q[T f (x)]w(x) dx
≤P
−1
0
Z
∞
V
P (v)
P [Cf (x)]v(x) dx
0
is satisfied, where
(4.11)
T f (x) = x
−1
Z
x
Z
f (t) dt +
0
x
∞
f (t)
dt,
t
0 ≤ f↓ .
However, (4.11) will be satisfied if and only if it is satisfied for the averaging
operator and its conjugate defined on decreasing functions. Hence Corollaries
3.5 and 3.6 apply with a = −1 and we have:
Proposition
V
V4.4. Suppose P and Q are N -functions and P , P̃ ∈ ∆2 . Then H :
(v) → Q (w) is bounded if and only if (4.9) (with {εj }↓, {xj } satisfying
R xPj
v = 2k ), (4.10) and (see (3.18), (3.19))
0
#
!
!
" X
X Z yj
ln(·/y
)χ
1
1
j
(yj ,yj+1 )
w(x) dx ≤ P −1
,
Q−1
Q
B
ε
V
ε
j
j
y
j−1
P̃
(ε
v)
j
j
j
#
!
!
"
Z
X yj
X 1
χ
ln(y
/x)
(y
,y
)
j
j
j+1
w(x) dx ≤ P −1
,
Q−1
Q
εj V B
εj
y
j−1
P̃ (εj v)
j
j
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are satisfied for all positive sequences {εj }j∈Z and all covering sequences {xj }.
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References
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Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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Weighted modular inequalities
for Hardy-type operators on
monotone functions
Hans P. Heinig and Qinsheng Lai
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