Publ. Mat. 61 (2017), 3–50
DOI: 10.5565/PUBLMAT 61117 01
BILINEAR WEIGHTED HARDY INEQUALITY
FOR NONINCREASING FUNCTIONS
Martin Křepela
Abstract: We characterize the validity of the bilinear Hardy inequality for nonincreasing functions
kf ∗∗ g ∗∗ kLq (w) ≤ Ckf kΛp1 (v1 ) kgkΛp2 (v2 ) ,
in terms of the weights v1 , v2 , w, covering the complete range of exponents p1 , p2 , q ∈
(0, ∞].
The problem is solved by reducing it into the iterated Hardy-type inequalities
1
∞ x
β
β
Z∞
1
Z Z
α
γ
∗∗
α


ψ(x) dx
,
(g (t)) ϕ(t) dt
≤C
(g ∗ (x))γ ω(x) dx
0
0
0
1
β
β
Z∞
1
Z∞Z∞
α
γ
∗∗
α


(g (t)) ϕ(t) dt
ψ(x) dx
≤C
(g ∗ (x))γ ω(x) dx
.

0
x
0
Validity of these inequalities is characterized here for 0 < α ≤ β < ∞ and 0 < γ < ∞.
2010 Mathematics Subject Classification: 26D10, 47G10.
Key words: Hardy operators, bilinear operators, weights, inequalities for monotone
functions.
1. Introduction
Consider the bilinear Hardy operator
1
H2 (f, g)(t) := 2
t
Zt
Zt
g(s) ds,
f (s) ds
0
0
defined for all nonnegative measurable functions f , g on (0, ∞). In this
article, we will find necessary and sufficient conditions for the boundedness
1
2
H2 : Lpdec
(v1 ) × Lpdec
(v2 ) → Lq (w)
with p1 , p2 , q ∈ (0, ∞]. In other words, the goal is to provide equivalent
estimates of the constant
(1)
C(1) = sup
f,g∈M
in terms of p1 , p2 , q, v1 , v2 , w.
kf ∗∗ g ∗∗ kLq (w)
kf kΛp1 (v1 ) kgkΛp2 (v2 )
4
M. Křepela
Let us at first summarize the used notation and symbols. Let (R, µ)
be an arbitrary totally σ-finite measure space. Then M denotes the
cone of all extended real-valued µ-measurable functions on R. Next,
M+ denotes the cone of all extended nonnegative Lebesgue-measurable
functions on (0, ∞).
p
. If p = 1, then p0 := ∞.
If p ∈ (0, 1) ∪ (1, ∞], then p0 := p−1
0
Notice that for p ∈ (0, 1) the number p is negative. Furthermore, the
conventions “ 00 = 0.∞ := 0” and “ a0 := ∞” for a ∈ (0, ∞] are used
throughout the text.
A weight is any nonnegative measurable function v on (0, ∞) such that
for all t ∈ (0, ∞) it holds 0 < V (t) < ∞, where V is defined by V (t) :=
Rt
v. If the weight is denoted by another letter, the corresponding capital
0
letter plays an analogous role.
We say that a function
R ε u ∈ M+ is integrable near the origin if there
exists ε > 0 such that 0 u < ∞. Notice that weights are integrable near
the origin by definition.
The symbol A . B means that A ≤ CB, where C is an absolute
constant independent of relevant quantities in A, B. In fact, throughout
this article such C depends only on the exponents (p, q, α, β, etc.), thus
it does not even depend on the weights. If both A . B and B . A, we
write A ' B.
By A(...) we denote the characteristic condition which appears on
the line denoted by the number in the brackets. Certain significant
optimal constants C(...) are denoted in a similar way. These symbols have
a unique meaning throughout the whole paper. Symbols B0 , B1 , etc. are
used in the proofs as an auxiliary notation for various quantities, and
their meaning may differ between the theorems. However, within the
proof of a single theorem or lemma, each symbol Bi is uniquely defined.
The text deals with various function spaces. The weighted Lebesgue
space Lp (v) consists of all extended real-valued Lebesgue-measurable
functions h on (0, ∞) such that khkLp (v) < ∞. The functional k · kLp (v)
is defined by
Z∞
khkLp (v) :=
|h(x)|p v(x) dx
1
p
,
p ∈ (0, ∞),
0
|hkL∞ (v) := ess sup |h(x)|v(x),
p = ∞.
x>0
The symbol Lpdec (v) stands for the set of all nonnegative and nonincreasing functions from Lp (v).
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
5
If f ∈ M , then f ∗ denotes its nonincreasing rearrangement and f ∗∗
the Hardy–Littlewood maximal function of f , i.e.
f
∗∗
1
(t) :=
t
Zt
f ∗ (s) ds,
t > 0.
0
For details see [3]. For the definitions of rearrangement-invariant (abbreviated r.i.) spaces and r.i. (quasi-)norms see [3, 7, 18]. If X and Y
are r.i. spaces (or just r.i. lattices), we say that X is embedded into Y
and write X ,→ Y if there exists C ∈ (0, ∞) such that for all f ∈ X it
holds
kf kY ≤ Ckf kX .
The least possible constant C in this inequality is called the optimal
constant of the embedding X ,→ Y and is equal to the norm of the
identity operator between X and Y , denoted kIdkX→Y .
Let v be a weight and p ∈ (0, ∞]. The weighted Lorentz spaces Λp (v)
and Γp (v) consist of all functions f ∈ M for which kf kΛp (v) < ∞ and
kf kΓp (v) < ∞, respectively. Here it is
kf kΛp (v) := kf ∗ kLp (v) and kf kΓp (v) := kf ∗∗ kLp (v) .
For more information about the Lorentz Λ and Γ spaces see e.g. [7] and
the references therein.
Let ϕ, ψ be weights. For g ∈ M define
kgkJ α,β (ϕ,ψ)
∞ x
1
β
β
Z Z
α
∗∗
α


:=
(g (t)) ϕ(t) dt
ψ(x) dx
,
0
kgkJ α,∞ (ϕ,ψ)
α, β ∈ (0, ∞),
0
Zx
1
α
:= ess sup
(g ∗∗ (t))α ϕ(t) dt
ψ(x),
α ∈ (0, ∞),
x>0
0
kgkK α,β (ϕ,ψ)
1
∞ ∞
β
β
Z Z
α
:= 
(g ∗∗ (t))α ϕ(t) dt
ψ(x) dx ,
0
α, β ∈ (0, ∞),
x
Z∞
1
α
kgkK α,∞ (ϕ,ψ) := ess sup
(g ∗∗ (t))α ϕ(t) dt
ψ(x),
α ∈ (0, ∞).
x>0
x
Then, as usual, it is J α,β (ϕ, ψ) := {f ∈ M ; kf kJ α,β (ϕ,ψ) < ∞} and
K (ϕ, ψ) := {f ∈ M ; kf kK α,β (ϕ,ψ) < ∞}. The “K-spaces” were
defined in [18], where they appeared as optimal spaces in certain Youngtype convolution inequalities. Besides that, in [16] it was shown that the
associate space to the generalized Γ space is also a “K-space”.
α,β
6
M. Křepela
Now, let us briefly present some background to the problems we are
about to investigate. The aforementioned operator H2 is a bilinear version of the classical Hardy operator H1 , which is defined by
1
H1 f (t) :=
t
Zt
f (s) ds
0
for all f ∈ M+ . Boundedness of H1 between weighted Lebesgue spaces
is equivalent to the validity of the weighted Hardy inequality
(2)
1
∞
q
q
Z∞
1
Z Zx
p
1

f (s) ds w(x) dx ≤ C
f p (x)v(x) dx
x
0
0
0
for all f ∈ M+ , with C being a constant independent of f . The
weights v, w for which this inequality is valid, have been characterized by
Muckenhoupt [23], Bradley [5], and Maz’ja [22]. The weighted Hardy
inequality has a broad variety of applications and represents now a basic tool in many parts of mathematical analysis, namely in the study of
weighted function inequalities. For the results, history, and applications
of this problem, see [21, 25, 20].
In the last decades, much attention has been drawn by the so-called
restricted inequalities. By this term it is meant that an inequality is not
supposed to be satisfied by the whole set of nonnegative functions, but
rather only by a certain, restricted, subset. In this way, one may ask
under which conditions the inequality (2) is satisfied for all nonincreasing
f ∈ M+ . This is equivalent to the validity of
 q1
Z∞
1
q
Z∞ Zt
p
1
∗


f (s) ds w(t) dt
≤C
(f ∗ (t))p v(t) dt ,
t

(3)
0
0
0
for all f ∈ M , with an independent C. Moreover, this corresponds to
the boundedness H1 : Lpdec (v) → Lq (w), or, in yet different words, the
existence of the embedding of the Lorentz spaces Λp (v) ,→ Γq (w).
The first results on the case Λp (v) ,→ Γp (v), 1 < p < ∞ were obtained
by Boyd [4] and in an explicit form by Ariño and Muckenhoupt [2]. The
problem with v 6= w and p 6= q, 1 < p, q < ∞ was first successfully
solved by Sawyer [26]. Many articles on this topic followed, providing
the results for a wider range of parameters, see [30, 8, 9, 28, 10, 7, 6].
In [7] the results available in 2000 were surveyed.
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
7
The restricted operator inequalities may often be handled by the socalled “reduction theorems”. These, in general, reduce a restricted inequality into certain nonrestricted inequalities. For example, the restriction to nonincreasing or quasiconcave functions may be handled in this
way, see e.g. [27, 15, 17, 12].
Let us however turn the focus to the bilinear variants of the Hardytype inequalities. Recently, Aguilar, Ortega, and Ramı́rez [1] found
necessary and sufficient conditions for the boundedness H2 : Lp1 (v1 ) ×
Lp2 (v2 ) → Lq (w),
e where w(t)
e := t2q w(t). In other words, they characterized the validity of the weighted bilinear Hardy inequality
∞ t
 q1
Z∞
1 Z∞
1
q
Z Z
Zt
p1
p2
(4) 
f p1 v1
g p2 v2
f (s) ds g(s) ds w(t) dt ≤ C
0
0
0
0
0
for all f, g ∈ M+ . The covered range of exponents in there was 1 <
p, q < ∞. For some related results see also the references in [1].
The paper [1] motivated the work presented here. Indeed, here we
consider a restricted version of (4) which may be called the bilinear
Hardy inequality for nonincreasing functions and written in the form
∞ t
q1
1 Z∞
Z∞
1
q
Z Z
Zt
p1
p2
w(t)
∗ p1
∗ p2
∗
∗

 ≤C
v
dt
(f
)
v
.
(g
)
f (s) ds g (s) ds
1
2
2q
t
0
0
0
0
0
Notice that C(1) is the least constant C for which the above inequality
holds for all f, g ∈ M .
The proofs in [1] are based on the standard technique of discretization.
Here, however, we choose a different approach. The idea is as follows. In
the first step, let g in (1) be fixed. Treating C(1) as the optimal constant
in the embedding Λp1 (v1 ) ,→ Γq ((g ∗∗ )q w), one gets
C(1) = sup
g∈M
kIdkΛp1 (v1 )→Γq ((g∗∗ )q w)
.
kgkΛp2 (v2 )
The two-side estimate of kIdkΛp1 (v1 )→Γq ((g∗∗ )q w) is known for all p1 , q ∈
(0, ∞] and it is equivalent to kgkX , a certain rearrangement-invariant
(quasi-)norm of g. Hence, in the next step, if we can find the optimal
constant kIdkΛp2 (v2 )→X , the whole problem is solved.
It will be shown that k·kX can be expressed as a sum of (quasi-)norms
in the r.i. spaces J α,β (ϕ, ψ) and K α,β (ϕ, ψ) (see Section 2 for the definitions). In Section 3 we find characterizations of the embeddings
Λγ (ω) ,→ J α,β (ϕ, ψ) and Λγ (ω) ,→ K α,β (ϕ, ψ) for 0 < α ≤ β < ∞
8
M. Křepela
and 0 < γ < ∞. In other words, we characterize the weights and exponents such that the inequalities
1
∞ x
β
β
Z∞
1
Z Z
α
γ
∗∗
α
∗
γ


(g (t)) ϕ(t) dt
(g (x)) ω(x) dx
ψ(x) dx
≤C
,
0
0
0
∞ ∞
1
β
β
Z∞
1
Z Z
α
γ
∗∗
α
∗
γ


(g (t)) ϕ(t) dt
ψ(x) dx
≤C
(g (x)) ω(x) dx
0
x
0
hold for all functions g ∈ M . These results will be then used to find
the desired estimates of the optimal constant C(1) in the bilinear Hardy
inequality (this is the matter of Section 4). However, the description of
the relation of the K-spaces to the other types of r.i. spaces, as well as
the above weighted inequalities, are of independent interest.
2. Auxiliary results
Here we present various, usually known propositions which will be useful further on. First we may recall the following simple but useful principle. Let a, b ∈ [−∞, ∞] and let f , g be nonnegative continuous functions on (a, b), f nondecreasing, and g nonincreasing. Then the derivatives f 0 (x), g 0 (x) exist at a.e. x ∈ (a, b). Denote f (a+) := limx→a+ f (x),
f (b−) := limx→b− f (x), similarly for g. Integration by parts then gives
Zb
f 0 (x)g(x) dx + f (a+)g(a+) = f (b−)g(b−) −
a
Zb
f (x)g 0 (x) dx,
a
with the convention “0.∞ := 0” taking effect if needed. Thus, if we,
for instance, consider a := 0, b := ∞, f := W α , g := V −β , and α, β ∈
(0, ∞), we get
Z∞
(5)
W α−1 (x)w(x)V −β (x) dx ' W α (∞)V −β (∞)
0
Z∞
+
W α (x)V −β−1 (x)v(x) dx.
0
R ∞ α
Analogous situations arise if we take f (x) := x w , etc. However, if
α < 1, there might appear a certain problem related to the integrability
of the involved functions (cf. [28, p. 93]). Observe that if we take α ∈
(0, 1) in (5) and a function w ∈ M+ which is not integrable near the
origin, then the equivalence in (5) fails, as the left-hand side is equal to
zero while the right-hand side is infinite. Since we originally assumed
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
9
that w was a weight, which is by definition integrable near the origin,
this problem, in fact, could not arise in (5). It may nevertheless do so in
other situations when the involved function is not a weight in this sense
and which thus require slightly more attention. We return to this issue
in Proposition 2.3 below.
Anyway, combining or splitting weighted conditions using integration
by parts in the described way is a common trick (see e.g. [30, Lemma,
p. 176]). If there is no potential danger as described above (e.g. if the
relevant exponents are greater than 1), we will use the technique throughout the text without detailed comments, and we will refer to it simply
as to integration by parts.
Another well-known principle, to which we refer as to the Lp -duality,
is expressed as follows. If f ∈ M+ , p ∈ (1, ∞) and v is a weight, then
Z∞
1
p
= sup
f p (x)v(x) dx
R∞
0
g∈M+
0
R∞
0
f (x)g(x) dx
1
.
g p0 (x)v 1−p0 (x) dx p0
We continue with other preliminary results.
Proposition 2.1. Let f, g ∈ M+ and 0 < λ < ∞. Then the identity
 x x

λ
λ−1
Z Z
Zx Zx
d 
f (t) dt g(s) ds = λf (x)
f (t) dt
g(s) ds
dx
0
s
0
s
holds for a.e. x > 0 for which the integral on the left-hand side is finite.
Analogously, the identity
∞ s

λ
λ−1
Z Z
Z∞Zs
d 
f (t) dt g(s) ds = −λf (x)
f (t) dt
g(s) ds
dx
x
x
x
x
holds for a.e. x > 0 for which the integral on the left-hand side is finite.
Proof: Let us prove the first statement, the second one is analogous. Let


λ
Zx Zx


x0 := sup x ∈ [0, ∞];
f (t) dt g(s) ds < ∞ .


0
s
Then, for any x ∈ [0, x0 ), Fubini theorem yields
Zx Zx
λ
f (t) dt
0
s


λ−1
Zx Zx Zy
g(s) ds =  λ
f (t) dt
f (y) dy  g(s) ds
0
s
=λ
s
Zy Zy
Zx
f (y)
0
λ−1
f (t) dt
0
s
g(s) ds dy.
10
M. Křepela
The expression on the second line is nondecreasing and continuous
in x, therefore its derivative with respect to x exists and is equal to
λ−1
Rx Rx
g(s) ds at a.e. point x ∈ (0, x0 ).
λf (x) 0 s f (t) dt
Proposition 2.2. Let 0 < p ≤ q < ∞ and let v, w be weights. Then it
holds
R∞
0
R∞
0
sup
ϕ∈M+
ϕ is nondecreasing
1
Z∞ 1 Z∞ − 1
q
p
ϕq (x)w(x) dx q
.
w
v
1 ' sup
x>0
p
p
ϕ (x)v(x) dx
x
x
Proof: This statement is analogous to a similar statement for nonincreasing functions (see [7, Theorem 3.1]). From there it can be also obtained
directly by the change of variables x 7→ x1 in the integrals.
Proposition 2.3. Let 1 < p < ∞ and 0 < q < p < ∞. Let v, w be
weights. Then
1
(f ∗∗ (t))q w(t) dt q
1 ' A(7) + A(8) ,
R∞
(f ∗ (t))p v(t) dt p
0
R∞
(6)
C(6) := sup
f ∈M
0
where

(7)
A(7)
Z∞
:= 
W (t)
V (t)
 p−q
q
p−q
pq
w(t) dt
0

Z∞
'
W (t)
V (t)
p
p−q
 p−q
pq
v(t) dt
1
+ W q (∞)V
1
−p
(∞)
0
and

(8)
Z∞Z∞
A(8) := 
0
 p−q
pq
q Zt
(p−1)q
0
p−q
p−q w(t)
w(s)
v(s)sp

ds
ds
dt
.
sq
tq
V p0 (s)
t
0
0
0
In particular, if C(6) < ∞, then the function s 7→ v(s)sp V −p (s) is
integrable near the origin.
0
0
Furthermore, if q > 1, or if q < 1 and the function s 7→ v(s)sp V −p (s)
is integrable near the origin, then A(8) ' A(9) , where

(9)
Z∞Z∞
A(9) := 
0
t
 p−q
pq
p Zt
(q−1)p
0
0
p−q
p−q v(t)tp
w(s)
v(s)sp

ds
ds
dt
.
sq
V p0 (s)
V p0 (t)
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
11
Proof: This assertion is stated in [7, Theorem 4.1(iii)] under the additional condition that q 6= 1. However, it is true even for q = 1, which
may be checked using [11, Theorem 3.1(iv)] and [14, Theorem 3.1].
Let us say more on the equivalence A(8) ' A(9) . If q > 1 and the
0
0
function u, defined by u(s) := v(s)sp V −p (s) for s > 0, is not integrable
near the origin (a simple example of such function u was given in [28,
p. 93]), then both A(8) and A(9) are infinite. However, if q < 1 and u is
not integrable near the origin, then A(8) = ∞ but A(9) = 0, since the
exponent (q−1)p
p−q is negative.
Proposition 2.3 will be later used e.g. in the proofs of Lemmas 3.2
and 3.3 and Theorem 4.3. In the calculations within the proofs, we will
need to use conditions in the form of A(9) . The reason is that the function
involving w appears only once in there and the resulting expression may
be understood as the (quasi-)norm in a certain space. Nevertheless, for
the final conditions which we state in the lemmas or theorems, we prefer
the “safe” form in the style of A(8) , i.e. avoiding the potentially negative
exponents. In this way, the finiteness of the condition automatically
implies the integrability of the “problematic” function near the origin.
The proposition below is a modification of [29, Proposition 2.7].
Proposition 2.4. Let k · kX be a functional acting on M+ such that for
all λ > 0 and all g, h ∈ M+ such that g ≤ h a.e. it holds kgkX ≤ khkX
and kλgkX ≤ λkgkX . Let v be a weight. Then
−1 kf ∗ kX
sup
= ess sup v(y)
.
y∈(0,•)
f ∈M kf kΛ∞ (v)
(10)
X
Proof: Let f ∈ M . Then, by the properties of k · kX , one has
∗
−1 kf ∗ kX ≤ ess sup f ∗ (x) ess sup v(y) ess sup v(y)
y∈(0,•)
x>0
y∈(0,x)
X
−1 ∗
= ess sup v(y) ess sup f (x) ess sup v(y)
y∈(0,•)
y>0
x∈(y,∞)
X
−1 ∞
= kf kΛ (v) ess sup v(y)
.
y∈(0,•)
X
Taking the supremum over f ∈ M , we get the inequality “≤” in (10).
−1
Next, there exists g ∈ M such that g ∗ = ess supy∈(0,•) v(y)
a.e. It
12
M. Křepela
is easy to observe that
!−1
kgkΛ∞ (v) = ess sup v(x) ess sup v(y)
x>0
Hence, it holds
kg ∗ kX
kgkΛ∞ (v)
= 1.
y∈(0,x)
−1 = kg ∗ kX = ess supy∈(0,•) v(y)
and thus
X
the “≥” inequality in (10) is satisfied.
3. Embeddings
In this section we characterize certain embeddings Λ ,→ J and Λ ,→ K.
These results will later form a crucial step in the proof of the bilinear
Hardy inequality.
At first, observe that the embedding Λγ (ω) → K α,∞ (ϕ, ψ) is characterized easily by rephrasing the problem as an embedding Λ ,→ Γ.
Proposition 3.1. Let ϕ, ψ, ω be weights and 0 < α, β, γ ≤ ∞. Then
kIdkΛγ (ω)→K α,∞ (ϕ,ψ) = ess sup ψ(x)kIdkΛγ (ω)→Γα (ϕχ[x,∞) ) .
x>0
Proof: We have
1
R ∞ ∗∗ α 1
(g ∗∗ )α ϕ α ψ(x)
(g ) ϕ α
x
=
ess
sup
ψ(x)
sup
1
1
R∞
R∞
x>0
g∈M
(g ∗ )γ ω γ
(g ∗ )γ ω γ
0
0
R∞
sup ess sup
g∈M
x>0
x
= ess sup ψ(x)kIdkΛγ (ω)→Γα (ϕχ[x,∞) ) .
x>0
The embeddings Λ ,→ Γ have been fully characterized (see [7, 6]).
Similarly it can be dealt with the embedding Λγ (ω) → J α,∞ (ϕ, ψ), where
the problem reduces to a characterization the boundedness of the dual
Hardy operator on the cone of nonincreasing functions. Results regarding the latter problem are also at our disposal, se e.g. [17].
Rt
Rt
Recall that if ϕ, ψ, ω are weights, then Φ(t) := 0 ϕ, Ψ(t) := 0 ψ,
Rt
Ω(t) := 0 ω for t > 0. In the couple of lemmas below there will appear
a function σ, defined by
(11)
σ(x) := sup
γα
γ−α
−1
tΩ γ (t)
,
x > 0,
t∈(0,x)
where ω is a weight and α, γ ∈ (0, ∞) are exponents specified later.
The function σ is continuous and nondecreasing on (0, ∞), hence its
derivative σ 0 exists at almost every
R x point x > 0 and, furthermore,
for all x > 0 it holds σ(x) = 0 σ 0 (t) dt + σ(0+), where σ(0+) :=
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
13
γα
γ−α
1
lim supt→0+ tΩ− γ (t)
. This notation and properties of σ are used
in the lemmas without further comment.
The lemma below brings a characterization of the embedding Λγ (ω) ,→
α,β
J (ϕ, ψ) for 0 < α ≤ β < ∞ and α < γ < ∞.
Lemma 3.2. Let ϕ, ψ, ω be weights. Denote
R∞ Rx
0
(12)
0
C(12) := sup
(g ∗∗ )α ϕ
R∞
g∈M
0
β
α
(g ∗ )γ ω
1
β
ψ(x) dx
.
1
γ
(i) Let 0 < α < γ ≤ β < ∞ and 1 < γ. Then C(12) ' A(13) + A(14) ,
where
(13) A(13) := sup Ω
1
−γ
Zx
Zx
1
β
β
(x)
+sup
Φα ψ
x>0
γ
γ
Φ γ−α Ω α−γ ω
γ−α Z∞ 1
γα
β
ψ
x>0
0
0
x
and
(14)

Zx Zx
A(14) := sup 
x>0
0
ϕ(t)
dt
tα
α
γ−α
ϕ(s)
sα
Zs
s
γ0
y ω(y)
dy
Ωγ 0 (y)
γ−α
α(γ−1)
γα
γ−α
ds
Z∞ 1
β
ψ
x
0
∞ s
1
β Zx
β
γ−1
Z Z
0
α
γ
ϕ(t)
sγ ω(s)


+sup
dt
ψ(s) ds
ds
.
tα
Ωγ 0 (s)
x>0
x
x
0
(ii) Let 0 < α < β < γ < ∞ and 1 < γ. Then C(12) ' A(15) + A(16) ,
where
(15)

Z∞
A(15) := 
Ω
β
β−γ
0
Zx
(x)
β
α
Φ ψ
β
γ−β
 γ−β
γβ
β
α
Φ (x)ψ(x) dx
0
γ−β
∞ x
γβ
γ(β−α)
Z∞ γ
Z Z
γ−β
α(γ−β)
γ
γ
γ
γ
+
Φ γ−α Ω α−γ ω
Φ γ−α (x)Ω α−γ (x)ω(x)
ψ
dx
0
0
x
14
M. Křepela
and

Z∞Zx Zx
(16) A(16) := 
0
0
ϕ(t)
dt
tα
α
γ−α
ϕ(s)
sα
s
Zs
0
y γ ω(y)
dy
Ωγ 0 (y)
×
α(γ−1)
γ−α
γ−β
γβ
β(γ−α) Z∞ β
γ−β
α(γ−β)

ds
ψ
ψ(x) dx
x
0
∞ ∞ s
β
β
Z Z Z
α
γ−β ϕ(x)
ϕ(t)
+
ψ(s)
ds
dt
α
t
xα
0
x
x
Z∞Zy
×
x
ϕ(t)
dt
tα
Zx
β−α
α
ψ(y) dy
γ−β
γβ
β(γ−1)
0
γ−β
sγ ω(s)

dx
.
ds
Ωγ 0 (s)
0
x
(iii) Let 0 < α < γ ≤ β < ∞ and γ ≤ 1. Let σ be given by (11). Then
C(12) ' A(13) + A(17) , where
(17)
A(17)
 x x
 γ−α ∞
γα Z
α
1
Z Z
γ−α ϕ(s)
β
ϕ(t)

:= sup 
dt
σ(s)
ds
ψ
α
α
t
s
x>0
0
s
x
1
∞ s
β
β
Z Z
α
γ−α
ϕ(t)
γα


.
+ sup σ
dt
ψ(s) ds
(x)
tα
x>0
x
x
(iv) Let 0 < α < β < γ ≤ 1. Let σ be given by (11). Then C(12) '
A(15) + A(18) + A(19) , where

(18)
Z∞Zx Zx
A(18) := 
0
0
ϕ(t)
dt
tα
α
γ−α
β(γ−α)
α(γ−β)
ϕ(s)
σ(s)
ds
α
s
s
 γ−β
γβ
Z∞ β
γ−β
×
ψ
ψ(x) dx
x
and

(19)
A(19)
Z∞Z∞Zs
:= 
0
x
x
Z∞Zs
×
x
β
β
α
γ−β
ϕ(t)
dt
ψ(s) ds
tα
x
ϕ(t)
dt
tα
β−α
α
 γ−β
γβ
ϕ(x) β(γ−α)
ψ(s) ds α σ α(γ−β) (x) dx
.
x
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
15
Proof: We have
(20)
C(12) = sup sup R
g∈M h∈M
+
R∞
0
×
h(x)
= sup h∈M+ R ∞
(21)
0
R∞
0
× sup
g∈M
1
∞
0
h
β
β−α
α
ψ α−β
β−α
βα
1
(g ∗∗ (t))α ϕ(t) dt dx α
1
R∞
(g ∗ )γ ω γ
0
Rx
0
1
h
β
β−α
α
ψ α−β
β−α
βα
1
R∞
(g ∗∗ (t))α ϕ(t) t h(x) dx dt α
1
R∞
(g ∗ )γ ω γ
0
=: B0 .
In step (20) we used duality of Lp -spaces and (21) follows by Fubini
theorem and changing the order of the suprema.
To make the notation shorter, define the function u by
0
(22)
u(s) :=
sγ ω(s)
,
Ωγ 0 (s)
s > 0.
Now suppose that γ > 1. Assume that u is integrable near the origin.
Then by Proposition 2.3 it holds
R
B0 ' sup
∞
0
Rs
0
ϕ(t)
γ−α
γ
γ
γα
h(x) dx dt γ−α Ω α−γ (s)ω(s) ds
R
β−α
β
α
βα
∞ β−α
α−β
h
ψ
0
R∞
t
h∈M+
1
h(x) dx dt α
+ sup β−α
β
α
1
h∈M+ R ∞
βα
β−α ψ α−β
h
Ω γ (∞)
0
R∞
0
ϕ(t)
R ∞ R ∞
0
s
R∞
t
ϕ(t)
tα
R∞
t
h(x) dx dt
γ
γ−α
Rs
0
+ sup
h∈M+
=: B1 + B2 + B3 .
R
∞
0
β
α
h β−α ψ α−β
u(y) dy
β−α
βα
γ(α−1)
γ−α
γ−α
γα
u(s) ds
16
M. Křepela
Consider now the case (i). It holds
R
(23)
∞
0
B1 ' sup
γ−α
γ
γ
γα
γ−α
α−γ (s)ω(s) ds
h(x)
dx
Ω
0
β−α
R
β
β
α
βα
∞ β−α
h
Φ α−β ψ α−β
0
Rs
h∈M+
R
∞ R∞
0
+ sup
s
h∈M+
Z∞
' sup
(24)
Ω
γ
α−γ
γ−α
γ
γ
γ
γα
h(x) dx γ−α Φγ−α (s)Ω α−γ (s)ω(s) ds
R
β−α
β
α
βα
∞ β−α
α−β
h
ψ
0
γ−α Zx
γα
ω
β
α
1
β
Φ ψ
x>0
x
0
Zx
+ sup
γ
γ
Φ γ−α Ω α−γ ω
γ−α Z∞ 1
γα
β
ψ
,
x>0
0
x
where (23) follows by Fubini theorem and (24) by Hardy inequality (see
[21, p. 3–4]). Next, Fubini theorem and Lp -duality yield
Z∞
(25)
B2 '
1
β
β
1
−γ
Ω
Φα ψ
Zx
1 γ−α
β
β
α
γα
Ω α−γ (∞)
.
(∞) = sup
Φα ψ
x>0
0
0
Therefore, we have
γ−α Zx
1
Z∞
γα
β
γ
β
α
B2 + B1 ' sup Ω α−γ (∞) + Ω α−γ ω
Φα ψ
x>0
x
Zx
+ sup
γ
0
γ
Φ γ−α Ω α−γ ω
γ−α Z∞ 1
γα
β
ψ
' A(13) .
x>0
x
0
Notice that this equivalence in fact does not involve the function u at
all, hence it holds for any u ∈ M+ . The assumption on u will be used
only in the next part. By Fubini theorem, B3 is equal to
R ∞ R ∞
0
s
h(x)
Rx
ϕ(t)
s tα
dt dx
γ
γ−α
Rs
0
sup
h∈M+
R
∞
0
β
α
h β−α ψ α−β
u(y) dy
β−α
βα
γ(α−1)
γ−α
γ−α
γα
u(s) ds
.
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
17
This expression is, by the dual version of [24, Theorem 1.1], equivalent
to
 x x
 γ−α ∞
γα Z
γ Zs
γ(α−1)
1
Z Z
γ−α
γ−α
β
ϕ(t)

sup 
u(s)
ds
dt
u(y)
dy
ψ
α
t
x>0
0
s
0

x
Z∞Zs
+ sup 
x>0
x
ϕ(t)
dt
tα
1
β
β
α
Zx
ψ(s) ds
x
γ−1
γ
u(s) ds
,
0
which is, in turn, equivalent to A(14) by Proposition 2.3, since u is integrable at the origin. Finally, observe that if u is not integrable at the
origin, then necessarily both B0 = ∞ (see the proof sketch of Proposition 2.3) and A(14) = ∞. On the other hand, if A(14) < ∞, then u is
integrable at the origin. Hence, C(12) = B0 < ∞ holds if and only if
A(13) + A(14) < ∞. Moreover, C(12) ' A(13) + A(14) , all without any
additional assumptions on the weight u.
In case (ii), using an appropriate version of Hardy inequality and
Lp -duality (cf. the analogous situation in (23), (24), and (25)), we prove
that B1 + B2 ' A(15) . To estimate B3 , we use [24, Theorem 1.2]. Then
we get
∞ x x
γ Zs γ 0
γ(α−1) γ 0
β(γ−α)
Z Z Z
γ−α
γ−α s
α(γ−β)
ϕ(t)
ω(s)
y ω(y)

B3 '
dt
dy
ds
tα
Ωγ 0 (y)
Ωγ 0 (s)
0
0
s
0
 γ−β
γβ
Z∞ β
γ−β
×
ψ
ψ(x) dx
x

Z∞Z∞Zs
+
0
x
ϕ(t)
dt
tα
β
α
γ
γ−β
ψ(s) ds
x
 γ−β
γβ
Zx γ 0
γ(β−1) γ 0
γ−β
s ω(s)
x ω(x) 
×
ds
dx
.
Ωγ 0 (s)
Ωγ 0 (x)
0
Using the assumption of integrability at the origin of u, one may show
then by integration by parts that the above expression is equivalent
to A(16) . While handling the second term in the sum, one also needs to
use Proposition 2.1. Finally, the additional assumption on u is removed
in the same way as in case (i).
18
M. Křepela
Now we assume 0 < γ ≤ 1. From [6, Theorem 3.1] it follows that
B0 = B1 + B2 + B4 , where
"
R∞
0 sup0<t≤s
#γ−α
γα γα
α
γ−α R
γ−α ϕ(s)R ∞
∞ ϕ(t)R ∞
h(x)
dx
dt
h(x)
dx
ds
α
α
1 (t)
s
t
s
t
s
t
V γ
B4 := sup
h∈M+
R∞
0
h
β
β−α
ψ
α
α−β
.
β−α
βα
Furthermore,
R∞
0
(26)
0
σ (s)
R
R∞
t
h(x) dx dt
B4 ' sup
h∈M+
R
∞
0
σ
γ−α
γα
+ sup
h∈M+
(0+)
R
R
∞
0
R∞
0
(27)
∞ ϕ(t)
tα
s
0
σ (s)
R
β
α
h β−α ψ α−β
∞ ϕ(s)
sα
0
R∞
β
α
s
h β−α ψ α−β
∞
s
h(x)
Rx
ϕ(t)
s tα
β
R
∞
0
σ
γ−α
γα
+ sup
h∈M+
(0+)
R
R
∞
0
h(x)
α
β
Rx
β−α
βα
dt dx
h β−α ψ α−β
∞
0
α
γ
γ−α
γ−α
γα
ds
β−α
βα
ϕ(s)
0 sα
h β−α ψ α−β
γ−α
γα
ds
γ
γ−α
1
α
h(x) dx ds
β−α
βα
= sup
h∈M+
1
α
ds dx
β−α
βα
=: B5 + B6 .
For (26) one uses integration by parts and (27) follows by Fubini theorem.
Next, by Lp -duality, we get

(28)
B6 = σ
γ−α
γα
Z∞Zx
(0+) 
0
0
1
β
β
α
ϕ(s)
 .
ds
ψ(x)
dx
sα
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
19
Consider now the case (iii). From the dual version of [24, Theorem 1.1]
it follows

Zx Zx
ϕ(t)
dt
tα
B5 ' sup 
x>0
0
γ
γ−α
 γ−α
γα
σ 0 (s) ds
s
Zx
+ sup
σ
Z∞ 1
β
ψ
x
0
γ−α

γα
Z∞Zs

x>0
0
x
ϕ(t)
dt
tα
β
1
β
α
ψ(s) ds .
x
Using this characterization, the expression of B6 from (28) and integrating by parts, one obtains B5 + B6 ' A(17) . Earlier (when considering
β ≥ γ > 1) we proved that B1 + B2 ' A(13) . The same is true here, as
the argument is correct even for β ≥ γ with 0 < γ ≤ 1. Hence, it follows
that C(12) ' B1 + B2 + B5 + B6 ' A(13) + A(17) and the proof of this
part is complete.
We proceed with (iv). Estimating B1 and B2 is done in the same way
as in (ii). It remains to show that B5 + B6 ' A(18) + A(19) . By the dual
version of [24, Theorem 1.2], one has

Z∞Zx Zx
(29) B5 ' 
0
0
ϕ(t)
dt
tα
γ
γ−α
γ−β
γβ
β(γ−α) Z∞ β
γ−β
α(γ−β)
σ (s) ds
ψ
ψ(x) dx
0
x
s
∞ ∞ s
γ−β
γβ
β
γ Zx γ(β−α)
Z Z Z
α
γ−β
α(γ−β)
ϕ(t)
0
0

+
dt
ψ(s)
ds
σ
σ
(x)
dx
tα
0
x
x
0
=: B7 + B8 .
Now, integration by parts provides

(30) A(18) ' B7 +σ
γ−α
γα
Z∞Zx
(0+)
0
ϕ(t)
dt
tα
γβ
α(γ−β)
0
γ−β
γβ
Z∞ β
γ−β
ψ
ψ(x) dx
.
x
Next, it holds
∞ ∞
 α(γ−β) ∞
β
γβ
Z − α
Z Z γ−β
β
sup 
ψ
ψ(t) dt
ψ
' 1,
x>0
x
t
x
20
M. Křepela
thus, by Proposition 2.2, we get
∞ x
 α(γ−β)
γβ
γβ Z∞ β
Z Z
γ−β
α(γ−β)
ϕ(t)


dt
ψ(x)
dx
ψ
tα
0
0
x

Z∞Zx
.
0
α
β
β
α
ϕ(s)

ψ(x) dx
.
ds
sα
0
Applying this in (30) (and considering (28)) we obtain
(31)
B7 . A(18) . B7 + B6 .
Furthermore, from Proposition 2.1 and integration by parts it follows
that B6 + B8 ' A(19) . Combining this estimate with (31) and (29), we
finally get B5 + B6 ' B6 + B7 + B8 ' A(18) + A(19) , which we needed to
prove.
The next lemma characterizes the embedding Λγ (ω) ,→ K α,β (ϕ, ψ)
for 0 < α ≤ β < ∞ and α < γ < ∞.
Lemma 3.3. Let ϕ, ψ, ω be weights. Denote
R∞ R∞
0
(32)
x
C(32) := sup
1
β
β
(g ∗∗ )α ϕ α ψ(x) dx
R∞
g∈M
0
(g ∗ )γ ω
1
.
γ
(i) Let 0 < α < γ ≤ β < ∞ and 1 < γ. Then C(32) ' A(33) + A(34) +
A(35) , where
(33)
A(33)
 γ−α
∞ s
γ
γα
Z Z γ−α
γ
1
α−γ


:= sup
Ω
(s)ω(s) ds
Ψ β (x)
ϕ
x>0
x
x
1
β
Zx Zx αβ


(x)
ϕ
ψ(s) ds
,

+ sup Ω
1
−γ
x>0
0

(34)
A(34)
Z∞Z∞
:= sup 
x>0
x
ϕ(t)
dt
tα
s
α
γ−α
ϕ(s)
sα
s
 γ−α
γα
Zs γ 0
α(γ−1)
γ−α
1
y ω(y)

×
dy
ds
Ψ β (x),
0
Ωγ (y)
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
21
and

(35)
Z∞Z∞
A(35) := sup 
x>0
x
1
β Zx
β
10
0
α
γ
ϕ(t)
sγ ω(s)

dt
.
ψ(s)
ds
ds
0
α
γ
t
Ω (s)
s
0
(ii) Let 0 < α < β < γ < ∞ and 1 < γ. Then C(32) ' A(36) + A(37) +
A(38) , where
(36)
A(36)
∞ ∞ s
γ
β(γ−α)
Z Z Z γ−α
α(γ−β)
γ
α−γ

:=
(s)ω(s) ds
ϕ
Ω
0
x
x
 γ−β
γβ
×Ψ

+
β
γ−β
(x)ψ(x) dx
β Zx Zx β−α
Z∞Zx Zx αβ
γ−β
α
ϕ
ψ(s) ds
ϕ
0
0
s
0
s
 γ−β
γβ
× ψ(s) ds ϕ(x) Ω

(37) A(37)
Z∞Z∞Z∞
:= 
0
x
ϕ(t)
dt
tα
α
γ−α
β
β−γ
(x) dx
,
ϕ(s)
sα
s
 γ−β
γβ
Zs γ 0
α(γ−1) β(γ−α)
γ−α
α(γ−β)
β
y ω(y)
γ−β (x)ψ(x) dx
×
dy
Ψ
ds
,
0
Ωγ (y)
0
and

(38)
Z∞Z∞Z∞
A(38) := 
0
x
Z∞
×
ϕ(t)
dt
tα
β
α
β
γ−β
ψ(s) ds
s
ϕ(t)
dt
tα
Zx
β
α
ψ(x)
x
 γ−β
γβ
β(γ−1)
0
γ−β
sγ ω(s)

ds
dx
.
Ωγ 0 (s)
0
(iii) Let 0 < α < γ ≤ β < ∞ and γ ≤ 1. Let σ be given by (11). Then
C(32) ' A(33) + A(39) + A(40) , where

(39)
Z∞Z∞
A(39) := sup 
x>0
x
s
ϕ(t)
dt
tα
α
γ−α
 γ−α
γα
1
ϕ(s)

σ(s)
ds
Ψ β (x)
sα
22
M. Křepela
and
(40)
A(40) := sup σ
γ−α
γα
x>0
∞ ∞
1
β
β
Z Z
α
ϕ(t)


(x)
dt
ψ(s) ds
.
tα
x
s
(iv) Let 0 < α < β < γ ≤ 1. Let σ be given by (11). Then C(32) '
A(36) + A(41) + A(42) , where

Z∞Z∞Z∞
(41) A(41) := 
0
x
ϕ(t)
dt
tα
α
γ−α
γ−β
γβ
β(γ−α)
α(γ−β)
β
ϕ(s)
γ−β (x)ψ(x) dx
σ(s)
ds
Ψ
sα
s
and

(42)
A(42)
Z∞Z∞Z∞
:= 
0
x
ϕ(t)
dt
tα
β
α
β
γ−β
ψ(s) ds
s
 γ−β
γβ
β
Z∞
β(γ−α)
α
ϕ(y)
α(γ−β) (x) dx
.
dy
ψ(x)
σ
×
yα
x
Proof: The proof is to a great extent analogous to that of Lemma 3.2
but there are some additional steps which we show below.
Let u be defined by (22). If 1 > γ, Lp -duality and Proposition 2.3
gives
C(12) = sup h∈M+ R
∞
0
R
1
h
β
β−α
R ∞ R s
0
0
ψ
sup
β−α g∈M
α
α−β
ϕ(t)
Rt
0
βα
γ−α
γ
γ
γα
γ−α
h(x) dx dt
Ω α−γ (s)ω(s) ds
' sup
h∈M+
1
α
h(x) dx dt
R∞
1
∗ γ
γ
0 (g ) ω
Rt
∞ ∗∗
α
0 (g (t)) ϕ(t) 0
R∞
0
β
α
β−α
βα
h β−α ψ α−β
1
α
h(x) dx dt
+ sup β−α
h∈M+ R
β
α
1
βα
∞ β−α α−β
h
ψ
Ω γ (∞)
0
R
∞
0
ϕ(t)
Rt
γ−α
α
γα
Rs
α(γ−1)
γ−α ϕ(s)R s
γ−α ds
h(x)
dx
dt
h(t)
dt
u(y)
dy
α
0
0
0
s
.
β−α
β
α
βα
R∞
β−α
α−β
ψ
0 h
R ∞ R ∞ ϕ(t)R t
0
(43)
+ sup
h∈M+
s
tα
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
23
If u is integrable near the origin, then the term (43) is equivalent to
R ∞ R ∞
0
ϕ(t)
tα
s
Rt
0
h(x) dx dt
γ
γ−α
Rs
0
u(y) dy
sup
h∈M+
R
∞
0
h
β
β−α
ψ
α
α−β
γ(α−1)
γ−α
γ−α
γα
u(s) ds
.
β−α
βα
(i) Suppose that u is integrable near the origin. As in Lemma 3.2(i),
using Hardy inequality, [24, Theorem 1.1] and the dual version of it one
shows that C(32) ' A(33) + B1 + A(35) , where
∞ ∞
γ−α
γα
γ Zs γ 0
γ(α−1) γ 0
Z Z
γ−α
γ−α s
1
ω(y)
ω(s)
y
ϕ(t)

B1 := sup 
Ψ β (x).
dt
dy
ds
tα
Ωγ 0 (y)
Ωγ 0 (s)
x>0
x
0
s
Integration by parts gives B1 + B2 ' A(34) with
Z∞
B2 := sup
x>0
10
1 Zx γ 0
α
γ
1
ϕ(t)
y ω(y)
dy
dt
Ψ β (x).
0
tα
Ωγ (y)
x
0
Using the proof idea of [13, Lemma 2.2] (a similar problem was also
treated in [19, Proposition 3.2]), one checks that B2 . B1 + A(35) . This
implies that B1 + A(35) ' A(34) + A(35) , hence C(32) ' A(33) + A(34) +
A(35) . Finally, we make the following observation, same as in Lemma 3.2.
If u is not integrable near the origin, then C(32) = ∞ (see (43)) and
A(35) = ∞. Hence, the equivalence C(32) ' A(33) + B1 + A(35) holds
even without additional assumptions on u.
(ii) Analogously to (i) we assume that u is integrable near the origin
and get C(32) ' A(36) + B3 + A(38) , where

Z∞Z∞Z∞
B3 := 
0
x
s
ϕ(t)
dt
tα
γ
γ−α
Zs
0
y γ ω(y)
dy
Ωγ 0 (y)
γ(α−1)
γ−α
β(γ−α)
0
α(γ−β)
sγ ω(s)
ds
0
γ
Ω (s)
0
 γ−β
γβ
×Ψ
β
γ−β
(x)ψ(x) dx
.
24
M. Křepela
By integration by parts it follows that B3 + B4 ' A(37) , where

Z∞Z∞
B4 := 
0
ϕ(t)
dt
tα
βγ
α(γ−β)
Zx
x
0
y γ ω(y)
dy
Ωγ 0 (y)
 γ−β
β(γ−1)
γβ
γ−β
Ψ
β
γ−β
(x)ψ(x) dx
.
0
Following the idea of [14, Theorem 3.1] (cf. [19, Proposition 3.3]) one
shows that B4 . B3 + A(38) . Then B3 + A(38) ' A(37) + A(38) and thus
C(32) ' A(36) + A(37) + A(38) . The final dropping of the integrability
assumption on u is performed in the same way as in (i).
In the remaining part of the proof we will assume that γ ∈ (0, 1],
which is the case in (iii) and (iv).
(iii) Using the same ideas as in Lemma 3.2(iii), one shows that C(32) '
A(33) + B5 + A(40) , where
∞ ∞
 γ−α
γα
γ
Z Z
γ−α
1
ϕ(t)
0

B5 := sup 
σ
(s)
ds
dt
Ψ β (x).
tα
x>0
x
s
Integration by parts yields
Z∞
B5 + sup
x>0
ϕ(t)
dt
tα
1
α
σ
γ−α
γα
1
(x)Ψ β (x) ' A(39) ,
x
hence B5 . A(39) . Moreover, it also holds
Z∞
1
α
γ−α
1
ϕ(t)
sup
dt
σ γα (x)Ψ β (x) . B5 + A(40) ,
α
t
x>0
x
which is proved by using the same argument from [13] as in (i). Combining the obtained relations, we conclude that C(32) ' A(33) +A(39) +A(40) .
(iv) In an analogy to Lemma 3.2(iv) it is proved that C(32) ' A(36) +
B6 + A(42) , where

Z∞Z∞Z∞
B6 := 
0
x
s
ϕ(t)
dt
tα
γ
γ−α
 γ−β
γβ
β(γ−α)
α(γ−β)
β
σ 0 (s) ds
Ψ γ−β (x)ψ(x) dx
.
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
25
For any x > 0, integration by parts gives
Z∞Z∞
x
ϕ(t)
dt
tα
α
γ−α
ϕ(s)
σ(s) ds '
sα
s
Z∞Z∞
x
ϕ(t)
dt
tα
γ
γ−α
σ 0 (s) ds
s
Z∞
+
γ
γ−α
ϕ(s)
σ(x).
ds
α
s
x
Hence, one gets
A(41)
∞ ∞
 γ−β
γβ
γβ
Z Z
β(γ−α)
α(γ−β)
β
ϕ(s)
α(γ−β) (x)Ψ γ−β (x)ψ(x) dx
' B6 + 
ds
σ
sα
0
x
∞ ∞
γ−β
γβ
γβ Zx β(γ−α)
Z Z
α(γ−β)
α(γ−β)
β
ϕ(s)
0
γ−β (x)ψ(x) dx
σ
Ψ
' B6 + 
ds
sα
0
0
x

+σ
γ−α
γα
Z∞Z∞
(0+) 
0
 γ−β
γβ
γβ
α(γ−β)
β
ϕ(s)
γ−β (x)ψ(x) dx
ds
Ψ
sα
x
=: B6 + B7 + B8 .
Using the same argument as in (ii) (based on [14]), we can show that
B7 . B6 + A(42) . Next, since the function s 7→ ϕ(s)
sα is nonincreasing, we
obtain

Z∞Z∞

0
α
α(γ−β)  ∞ ∞
γβ
β
γβ
β
Z Z
α
α(γ−β)
β
ϕ(s)
ϕ(s)
γ−β (x)ψ(x) dx


ds
ds
ψ(x)
dx
Ψ
.
sα
sα
x
0
x
by using the characterization of the embedding Λ ,→ Λ [7, Theorem 3.1].
Thus, since

σ
γ−α
γα
Z∞Z∞
(0+) 
0
1
β
β
α
ϕ(s)
 . A(42) ,
ds
ψ(x)
dx
sα
x
we get the inequality B8 . A(42) . Summarizing, we obtained A(41) +
A(42) ' B6 + A(42) , hence C(32) ' A(36) + A(41) + A(42) and the proof is
completed.
26
M. Křepela
Although α < γ was assumed in the above statements, the proof
method is not limited to this case. In fact, only the assumption α ≤ β is
crucial for the duality approach. We may hence consider the case 0 < γ ≤
α ≤ β < ∞ and characterize the embedding Λγ (ω) ,→ J α,β (ϕ, ψ) using
the same technique as before. The proof becomes actually considerably
simpler in this case.
Proposition 3.4. Let ϕ, ψ, ω be weights.
(i) Let 1 < γ ≤ α ≤ β < ∞. Then C(12) ' A(44) + A(45) , where
Zx
(44)
1
β
β
Ω
Φα ψ
A(44) := sup
1
−γ
Z∞ 1
β
1
−1
(x) + sup
ψ
Φ α (x)Ω γ (x)
x>0
x>0
x
0
and

(45)
Z∞Zt
A(45) := sup 
x>0
x
 β1
β
Zx γ 0
10
α
γ
ϕ(s)
t ω(t)

ds
ψ(t)
dt
dt
.
0
α
γ
s
Ω (t)
x
0
(ii) Let 0 < γ ≤ 1 and γ ≤ α ≤ β < ∞. Then C(12) ' A(44) + A(46) ,
where

(46)
Z∞Zt
A(46) := sup 
x>0
x
 β1
β
α
ϕ(s)
−1
ds
ψ(t) dt xΩ γ (x).
sα
x
Proof: Just as in (20) and (21), one has
C(12) = sup h∈M+ R ∞
0
R∞
× sup
g∈M
0
1
h
β
β−α
α
ψ α−β
β−α
βα
1
R∞
(g ∗∗ (t))α ϕ(t) t h(x) dx dt α
=: B.
1
R∞
(g ∗ )γ ω γ
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
27
Consider the case (i). Then
Rx
(47) B ' sup sup
0
h∈M+ x>0
1 −1
R∞
ϕ(t) t h(s) ds dt α Ω γ (x)
β−α
R
β
α
βα
∞ β−α
h
ψ α−β
0
R
+ sup sup
∞ ϕ(t)
tα
x
R∞
t
h∈M+ x>0
10
1 R
0
α
x γ0
γ
h(s) ds dt
t ω(t)Ω−γ (t) dt
0
R
β−α
β
α
βα
∞ β−α
h
ψ α−β
0
1 −1
hΦ α Ω γ (x)
β−α
β
α
βα
h β−α ψ α−β
Rx
(48)
' sup sup 0
x>0 h∈M+ R ∞
0
+ sup sup
x>0 h∈M+
R∞ 1 1
−1
h α Φ α (x)Ω γ (x)
x
R
β−α
β
α
βα
∞ β−α
h
ψ α−β
0
R
+ sup sup
x>0 h∈M+
(49)
∞
x
h(s)
Rs
x
10
1 R
0
α
x γ0
γ
t ω(t)Ω−γ (t) dt
dt ds
0
R
β−α
β
α
βα
∞ β−α
α−β
h
ψ
0
ϕ(t)
tα
= A(44) + A(45) .
Step (47) follows by [7, Theorem 4.1(i)], step (48) by Fubini theorem
and changing the order of the suprema, and (49) is due to Lp -duality.
Case (ii) is proved analogously, using [7, Theorem 4.1(ii)] to estimate B.
Proving an analogous proposition concerning the embedding Λγ (ω) ,→
K (ϕ, ψ), 0 < γ ≤ α ≤ β < ∞, is left to an interested reader.
α,β
4. Bilinear Hardy inequality
At this point we have all the preliminary results needed to characterize the validity of the Hardy-type inequality (4) or, in other words, to
provide equivalent estimates on C(1) . The form of the results depends
on the values of the exponents p1 , p2 , and q and their mutual relation.
28
M. Křepela
In fact, in this three-parameter setting, 23 different cases are possible
and need separate treatment. For a better orientation, we present all
the possible settings in the table below with references to the theorem in
which each particular case is presented. Note that in some cases the roles
of p1 and p2 may be switched in the corresponding theorem, compared
with the entry in the table.
Configuration of the exponents
0 < p 1 , p2 ≤ 1
0 < p1 ≤ 1 < p2
0 < p 1 , p2 ≤ q
Theorem
q<∞
q=∞
q=∞
4.4(i)
q<∞
4.1(ii)
p2 < ∞
4.4(ii)
p2 = ∞
4.4(iii)
q<∞
1 < p 1 , p2
p1 , p 2 < ∞
q=∞
0 < p1 ≤ 1
p1 < p 2 = ∞
4.4(v)
p1 = p2 = ∞
4.4(vi)
p2 ≤ 1
4.2(iii)
4.2(ii)
1 < p1
p2 = ∞
4.5(ii)
p2 < ∞
4.1(iii)
p2 = ∞
4.5(i)
1/q ≥ 1/p1 + 1/p2
4.3(v)
1/q > 1/p1 + 1/p2
0 < p2 ≤ 1 < p1
p1 < ∞
0 < q < p 1 , p2
p1 , p 2 < ∞
4.3(vi)
1/q ≥ 1/p1 +1/p2
4.3(iii)
1/q > 1/p1 +1/p2
4.3(iv)
p1 = ∞
1 < p 1 , p2
4.1(i)
4.4(iv)
1 < p2 < ∞
0 < p1 ≤ q < p2
0 < p 1 , p2 ≤ 1
4.2(i)
4.5(iv)
1/q ≥ 1/p1 +1/p2
4.3(i)
1/q > 1/p1 +1/p2
4.3(ii)
p1 < p 2 = ∞
4.5(iii)
p1 = p 2 = ∞
4.5(v)
Let us now present and prove the results. We start with the configurations in which only the “classical” spaces appear, i.e. those where all
the exponents are finite. First such case is 1 < p1 ≤ q < ∞.
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
29
Theorem 4.1. Let v1 , v2 , w be weights.
1,2
(i) Let 1 < p1 , p2 ≤ q. Then C(1) ' A(50) + A(51)
+ A2,1
(51) + A(52) , where
− p1
1
(50)
A(50) := sup W q (t)V1
t>0
(51)
Ai,j
(51) :=
Zx
sup
0<t<x<∞
t
1
− p1
2
(t)V2
(t),
1
Zt p0
10
q
− p1
p
w(s)
s j vj (s)
j
i
ds
V
(x)
ds
,
0
i
q
p
s
j
Vj (s)
0
and
Z∞
A(52) := sup
t>0
t
1 Z t p0
10 Zt p0
10
q
p1
p2
w(s)
s 2 v2 (s)
s 1 v1 (s)
ds
.
ds
ds
p01
p02
s2q
V1 (s)
V2 (s)
0
0
2,1
(ii) Let 0 < p2 ≤ 1 < p1 ≤ q. Then C(1) ' A(50) + A1,2
(52) + A(51) + A(53) ,
where
(52)
Zx
Ai,j
(52) :=
sup
0<t<x<∞
1
− p1
q
− p1
w(s)
ds
Vi i (x)tVj j (t)
q
s
t
and
Z∞
(53)
A(53) := sup
t>0
t
1 Z t p0
10
q
− p1
p1
w(s)
s 1 v1 (s)
2
ds
ds
tV
(t).
0
2
p1
s2q
V
(s)
1
0
p2 q
p2 −q .
(iii) Let 1 < p1 ≤ q < p2 < ∞. Define r2 :=
A(54) + A(55) + A(56) , where
(54)
− p1
1
A(54) := sup V1
Zx
(x)
W
x>0
r2
p2
r
− p2
(t)w(t)V2
2
(t) dt
Then C(1) '
1
r2
,
0
− p1
(55) A(55) := sup V1
x>0
1

Zx Zx
(x)
0
t
r1
r2
Zt
r20
2
0
p2 w(t)
p2
w(s)
v2 (s)sp2
 ,
ds
ds
dt
0
p
sq
tq
V2 2 (s)
0
30
M. Křepela
and

r1
10 Z∞Zt
r2
2
r
p2 w(t)
− p2
p1
w(s)
2


ds
ds
V
(t)
dt
2
p0
sq
tq
V1 1 (s)
x
x
Zx
0
v1 (s)sp1
(56) A(56) := sup
x>0
0
Zx
+sup

r2
10 Z∞Z∞
p2 w(t)
p1
w(s)

ds
ds
2q
p01
s
t2q
V1 (s)
x
t
0
v1 (s)sp1
x>0
0
 r1
Z t
r20
2
0
p2
v2 (s)sp2
 .
×
ds
dt
0
p
V2 2 (s)
0
Proof: Since 1 < p1 ≤ q < ∞, by [7, Theorem 4.1(i)], we get
C(1) '
Zx
1
q
− p1
sup
(g ∗∗ )q w V1 1 (x)kgk−1
Λp2 (v2 )
sup
g∈Λp2 (v2 ) x>0
0
10
Z∞ ∗∗
1 Zx p0
q
p1
(g (s))q w(s)
s 1 v1 (s)
ds
+ sup sup
ds
kgk−1
0
Λp2 (v2 )
q
p1
p
s
g∈Λ 2 (v2 ) x>0
V
(s)
1
x
0
− p1
1
= sup V1
Zx
(x)
sup
g∈Λp2 (v2 )
x>0
Zx
+ sup
0
sp1 v1 (s)
p0
V1 1 (s)
x>0
0
− p1
1
= sup V1
x>0
Zx
+ sup
0
1
q
kgk−1
Λp2 (v2 )
0
10
p1
ds
Z∞
sup
g∈Λp2 (v2 )
1
q
(g ∗∗ (s))q w(s)
ds
kgk−1
Λp2 (v2 )
q
s
x
(x)kIdkΛp2 (v2 )→Γq (wχ[0,x] )
0
sp1 v1 (s)
p0
x>0
(g ∗∗ )q w
V1 1 (s)
10
p1
ds
kIdkΛp2 (v2 )→Γq (s7→w(s)s−q χ
)
[x,∞) (s)
=: B1 + B2 .
Now we separate the different cases. In (i), [7, Theorem 4.1(i)] yields
2,1
B1 + B2 ' A(50) + A1,2
(51) + A(51) + A(52) . In (ii), [7, Theorem 4.1(ii)]
2,1
gives that B1 ' A(50) + A1,2
(52) and B2 ' A(51) + A(53) . Finally, in (iii),
Proposition 2.3 yields B1 + B2 ' A(54) + A(55) + A(56) .
Now we consider the case 0 < p1 ≤ 1, p1 ≤ q.
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
31
Theorem 4.2. Let v1 , v2 , w be weights.
(i) Let 0 < p1 , p2 ≤ 1 and 0 < p1 , p2 ≤ q. Then C(1) ' A(50) + A1,2
(52) +
1,2
2,1
A2,1
(52) + A(57) + A(57) , where
(57)
Ai,j
(57) :=
Z∞
sup
0<x<t<∞
1
− p1
q
− p1
w(s)
ds
tVi i (t)xVj j (x).
2q
s
t
(ii) Let 0 < p1 ≤ 1 < p2 < ∞ and p1 ≤ q < p2 . Then C(1) ' A(54) +
A(55) + A(58) + A(59) , where
(58)
∞ t
 r1
r2
Z Z
2
r
p2 w(t)
− p2
w(s)
1
2

(x) 
(t)
dt
ds
V
2
sq
tq
− p1
A(58) := sup xV1
x>0
x
x
and
(59)
∞ ∞
r2
Z Z
p2 w(t)
w(s)
1
ds
(x) 
2q
s
t2q
− p1
A(59) := sup xV1
x>0
x
t
 r1
Z t p0
r20
2
p2
s 2 v2 (s)

×
dt
.
ds
p02
V
(s)
2
0
(iii) Let 0 < p1 ≤ q < p2 ≤ 1. Then C(1) ' A(54) + A(58) + A(60) , where
 x x
r2
Z Z
p2 w(t)
w(s)
1

(x)
ds
sup
sq
tq s∈(0,t)
sr2
− p1
(60) A(60) := sup V1
x>0
0
t
+sup xV1
x>0
x
t
1
r2
dt
V2 (s)
∞ ∞
r2
Z Z
p2 w(t)
w(s)
1

(x)
ds
sup
s2q
t2q s∈(0,t)
− p1
r2
p2

sr2
r2
p2
V2 (s)

dt
1
r2
.
32
M. Křepela
Proof: Similarly as in Theorem 4.1, by [7, Theorem 4.1(ii)] (since 0 <
p1 ≤ 1, p1 ≤ q < ∞) we obtain
C(1) '
Zx
1
q
− p1
sup
(g ∗∗ )q w V1 1 (x)kgk−1
Λp2 (v2 )
sup
g∈Λp2 (v2 ) x>0
0
Z∞
+
sup
sup
g∈Λp2 (v2 ) x>0
− p1
1
= sup V1
x>0
1
q
− p1
(g ∗∗ (s))q w(s)
1
ds
xV
(x)kgk−1
1
Λp2 (v2 )
sq
x
(x)kIdkΛp2 (v2 )→Γq (wχ[0,x] )
− p1
1
+ sup xV1
(x)kIdkΛp2 (v2 )→Γq (s7→w(s)s−q χ
)
[x,∞) (s)
x>0
=: B1 + B2 .
2,1
In (i), by [7, Theorem 4.1(ii)], we have B1 + B2 ' A(50) + A1,2
(52) + A(52) +
2,1
A1,2
(57) + A(57) . In (ii) it is B1 + B2 ' A(54) + A(55) + A(58) + A(59) by
Proposition 2.3 and finally in (iii) one gets B1 +B2 ' A(54) +A(58) +A(60)
by [6, Theorem 3.1].
We continue with the case 0 < q < p1 , p2 < ∞. This case is usually
the most complicated one, especially if p1 , p2 ≤ 1. Recall that if q ∈
q
(0, 1) ∪ (1, ∞), then q 0 := q−1
, while if q = 1, then q 0 := ∞.
Theorem 4.3. Let v1 , v2 , w be weights. Let 0 < q < p1 , p2 < ∞. Define
q
p1 p2 q
ri := ppi i−q
, i ∈ {1, 2}, and R := p1 p2 −p
.
1 q−p2 q
1
q
(i) Let 1 < p1 , p2 and
1,2
2,1
A2,1
(62) + A(63) + A(63) +
(61)
Ai,j
(61)
1
p1
A1,2
(64)
≤
+
+
1
p2 . Then C(1)
A2,1
(64) , where
Zx
:= sup
W
ri
pi
x>0
r
− pi
wVi
i
1
ri
2,1
1,2
' A1,2
(61) + A(61) + A(62) +
− p1
j
Vj
(x),
0
∞ s
Z Z
(62)

Ai,j
(62) := sup
x>0
x
x
w(t)
dt
tq
rj
q
r
− qj
Vj
1
rj
Zx
(s)vj (s) ds
0
tpi vi (t)
p0
0
Vi i (t)
dt
1
p0
i
,
Bilinear Weighted Hardy Inequality for Nonincreasing Functions

(63)

Ai,j
(63) := sup
x>0
Zx Zx
0
w(t)
dt
tq
ri
pi
w(s)
sq
Zs
s
p0
Vi i (t)
0
1
ri0
0
tpi vi (t)
p
dt
i
ri
ds
− p1
Vj
j
33
(x),
and

(64)
Ai,j
(64)
Z∞Z∞
:= sup 
x>0
x
s
1
rj
rj0
rj
Zs p0
pj w(s)
p
w(t)
t j vj (t)
j

dt
ds
dt
p0
t2q
s2q
Vj j (t)
0
Zx
×
p0i
1
q
1
p1
>
10
p
i
dt
.
Vi (t)
0
(ii) Let 1 < p1 , p2 and
0
tpi vi (t)
2,1
1,2
+ p12 . Then C(1) ' A1,2
(65) + A(65) + A(66) +
1,2
2,1
A2,1
(66) + A(67) + A(67) , where

(65)
Ai,j
(65)
Z∞Zx
:= 
W
0
rj
pj
r
− pj
wVj
j
rj
pi −rj
0
×W
rj
pj
r
− pj
(x)w(x)Vj
j
(x)Vi
rj
rj −pi
1
R
(x) dx
,
(66)
∞ ∞ s
rj
rj
Z Z Z
r
q
pi −rj w(x)
− qj
w(t)
i,j

dt
A(66) :=
Vj
(s)vj (s) ds
tq
xq
0
x
x
R1
rj
Zx p0
pi (rj −1)
Z∞Zs
r
pj
pi −rj
− qj
w(t)
s i vi (s)
×
dt
Vj
(s)vj (s) ds
ds
dx
0
tq
V pi (s)
x

x
0
Z∞Zx Zx
+
0
0
s
w(t)
dt
tq
ri
pi
w(s)
sq
Zs
p0
0
ri0
0
y pi vi (y)
Vi i (y)
dy
p
i
ri
pj −ri w(x)
ds
xq
1
R
ri −pi
Zs p0
ri0
Zx Zx
ri
pi
p
w(t)
w(s)
y i vi (y)
r
−p
i
i
j

×
dt
dy
ds Vj
(x) dx ,
p0
tq
sq
Vi i (y)
0
s
0
34
M. Křepela
and

(67)
Ai,j
(67)
Z∞Z∞Z∞
:= 
0
x
w(t)
dt
t2q
ri
pi
w(s)
s2q
Zs
p0
s
0
Z∞
×
x
×
p
i
ri
pj −ri
ds
Vi i (y)
≤
1
p1
+
0
spj vj (s)
p0
0
1
q
dy
Zx p0
ri0
ri
pi w(x)
p
w(t)
y i vi (y)
i
dy
dt
0
2q
2q
p
t
x
i
Vi (y)
0
Zx
(iii) Let p2 ≤ 1 < p1 and
ri0
0
y pi vi (y)
1
p2 .
Vj j (s)
 R1
ri (pj −1)
pj −ri
ds
dx .
1,2
Then C(1) ' A(61)
+ A2,1
(61) +
1,2
A1,2
(62) + A(63) + A(68) , where

Zx Zx
(68) A(68) := sup 
x>0
0
1
r2
r2
r2
p2 w(s)
− p1
w(t)
r 2 − p2

dt
sup
y
V
(y)
V1 1 (x)
2
q
q
t
s y∈(0,s)
s
1
∞ s
r1
r1
Z Z
r
q
− p1
− q1
w(t)

(s)v
(s)
ds
dt
V
xV2 2 (x)
+ sup 
1
1
q
t
x>0
x
x
∞ ∞
1
r2
r2
Z Z
r
p2 w(s)
− p2
w(t)
r
2
2

dt
sup
y
V
(y)
ds
+ sup 
2
t2q
s2q y∈(0,s)
x>0
x
s
Zx
×
0
sp1 v1 (s)
p0
0
10
p1
ds
V1 1 (s)
∞ ∞
r1
Z Z
p1 w(s)
w(t)

+ sup
dt
2q
t
s2q
x>0
x
s
1
r1
Zs p0
r10
− 1
p1
y 1 v1 (y)
 xV2 p2 (x).
×
dy
ds
0
p1
V1 (y)
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
(iv) Let p2 ≤ 1 < p1 and
1
q
>
1
p1
+
1
p2 .
35
2,1
Then C(1) ' A1,2
(65) + A(65) +
A1,2
(66) + A(69) , where
(69)
∞ x x
r2
r2
Z Z Z
r2
p2 w(s)
p1 −r2
w(t)
r2 − p2
A(69) := 
dt
sup
y
V
(y)
ds
2
q
q
t
s y∈(0,s)
0
0
s
1
R
r2 −p2
ZxZx
r2
r2
p2
−
w(x)
w(s)
w(t)
p2
r2 −p1
r2

× q
dt
sup
y
V
(y)
ds
V
(x)
dx
2
1
x
tq
sq y∈(0,s)
0

s
Z∞Z∞Zs
r1
r1
r
q
p2 −r1
− q1
w(t)
(s)v1 (s) ds
dt
V1
tq
+
0
x
w(x)
× q
x
x
Z∞Zs
x

Z∞Z∞Z∞
x
Z∞
×
w(t)
dt
t2q
x
Z∞
x
R
r1
r1 −p2
(s)v1 (s) ds sup y V2
1
R
(y)
y∈(0,x)
r2
p2
1
R
Zx p0
r2 (p1 −1)
r2
p1 −r2
1 v (s)
−
w(x)
s
1
p2
r2

sup y V2 (y)
ds
dx
p0
x2q y∈(0,x)
V 1 (s)
0
Z∞Z∞Z∞
×
r
− 1
V1 q
s
+
0
p1
r2
p1
r2
p2 w(s)
p1 −r2
w(t)
r2 − p2
dt
sup
y
V
(y)
ds
2
2q
2q
t
s
y∈(0,s)
x

r1
x
+
0
w(t)
dt
tq
s
1
r1
r10
r1
Zs p0
p1 w(s)
p2 −r1
p1
w(t)
y 1 v1 (y)
dt
dy
ds
p01
t2q
s2q
V1 (y)
0
1
R
r1
Zx p0
r10
r1
p1 w(x)
p1
s 1 v1 (s)
w(s)
r1 −p2
R

ds
ds
sup
y
V
(y)
dx
.
2
p0
s2q
x2q
y∈(0,x)
V1 1 (s)
0
(v) Let p1 , p2 ≤ 1 and
1
q
1,2
2,1
A2,1
(70) + A(71) + A(71) +
1
1
p1 + p2 . Then C(1)
2,1
A1,2
(72) + A(72) , where
≤
2,1
1,2
' A1,2
(61) + A(61) + A(70) +
 x x
1
ri
ri
Z Z
ri
− 1
pi w(s)
−
w(t)
pi
ri

 Vj pj (x),
(y)
ds
(70) Ai,j
:=
sup
dt
sup
y
V
i
(70)
tq
sq y∈(0,s)
x>0
0
s
36
(71)
M. Křepela
Ai,j
(71)
− p1

i
:= sup xVi
x>0
Z∞Zs
(x) 
x
w(t)
dt
tq
rj
pj
1
rj
rj
w(s) − pj

V
,
(s)
ds
sq j
x
and
∞ ∞
1
rj
r
rj
Z Z
− pj
pj w(s)
w(t)
rj
j
i


(x)
(y) ds
.
dt
sup y Vj
t2q
s2q y∈(0,s)
− p1
(72) Ai,j
(72) := sup xVi
x>0
x
(vi) Let p1 , p2 ≤ 1 and
1,2
2,1
A2,1
(73) + A(74) + A(74) +
1
q
s
1
1
p1 + p2 . Then C(1)
2,1
A1,2
(75) + A(75) , where
>
2,1
1,2
' A1,2
(65) + A(65) + A(73) +
(73)


Ai,j
(73) :=
Z∞Zx Zx
0
0
w(x)
× q
x
w(t)
dt
tq
rj
pj
rj
−p
w(s)
sup y rj Vj j (y)
q
s y∈(0,s)
rj
pi −rj
s
ZxZx
0
w(t)
dt
tq
rj −pj
pj
1
R
r
ri
− pj
−
w(s)
pi
rj
j

(y) dsVi
sup y Vj
(x) dx ,
sq y∈(0,s)
s
(74)
∞ ∞ s
r
rj
rj
Z Z Z
− pj
pj w(s)
pi −rj
w(t)
i,j
j

A(74) :=
dt
V
(s) ds
tq
sq j
0
x
w(x)
× q
x
x
Z∞Zs
x
w(t)
dt
tq
rj −pj
pj
1
R
rj
rj
w(s) − pj
rj −pi
R

V
,
(s)
ds
sup
y
V
(y)
dx
i
sq j
y∈(0,x)
x
and
(75)
∞ ∞ ∞
Z Z Z

Ai,j
(75) :=
0
x
Z∞
×
x
w(t)
dt
t2q
rj
pj
r
rj
− pj
pi −rj
w(s)
rj
j
sup
y
V
(y)
ds
j
2q
s
y∈(0,s)
s
w(t)
dt
t2q
rj
pj
1
R
r
rj
− pj
w(x)
r
−p
rj
R
j
j
i
 .
sup
y
V
(y)
sup
t
V
(t)
dx
j
i
x2q y∈(0,x)
t∈(0,x)
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
37
Proof: Consider first the case 1 < p1 . Assume that the function u1
defined by
Zx
u1 (x) :=
0
sp1 v1 (s)
p0
ds,
x > 0,
V1 1 (s)
0
is integrable near the origin. Then, applying Proposition 2.3, we obtain
(76)
R∞ Rx
0
0
(g ∗∗ )q w
C(1) ' sup
r1
R∞
g∈M
0
x
1
r1
(x)v1 (x) dx
r
− q1
V1
(g ∗ )p2 v2
((g ∗∗ (s))q w(s)
sq
R ∞ R ∞
0
q
1
p2
r1 R
q
x
ds
0
+ sup
R∞
g∈M
0
R∞
0
+ sup
g∈M
0
sp1 v1 (s)
p0
V1 1 (s)
1
p2
2
(g ∗ )p2 v
r10
q
ds
!r1
1
0
xp1 v1 (x)
p0
V1 1 (x)
dx
1 − p1
(g ∗∗ )q w q V1 1 (∞)
1
R∞
(g ∗ )p2 v2 p2
0
=: B1 + B2 + B3 .
(i) We use Lemma 3.2(i) with the setting α := q, β := r1 , γ := p2 ,
−
r1
ϕ := w, ψ(t) := V1 q (t)v1 (t), ω := v2 , we obtain the characterization
of B1 , and Proposition 2.3 to get the characterization of B3 . We obtain
the equivalence
2,1
B1 + B3 ' B4 + A2,1
(62) + A(63) ,
where
Zx
B4 := sup
W
x>0
r1
q
r
− 1
V1 q
v1
1
r1
− p1
V2
2
Zx
1
r
r2
r2
− p1
− q2
q
(x) + sup
W V2
v2
V1 1 (x).
x>0
0
0
Integration by parts yields
1
− p1
2,1
q
A1,2
(61) + A(61) ' B4 + sup W (x)V1
x>0
1
− p1
(x)V2
2
(x).
38
M. Křepela
Moreover, the following series of inequalities holds true.
− p1
1
1
sup W q (x)V1
x>0
− p1
2
(x)V2
R∞
0
∗
r1
(g (t)) W
' sup
R∞
g∈M
0
R∞
0
(x)
∗∗
R∞
g∈M
0
(g ∗∗ )q w
0
≤ sup
0
1
(t) dt
1
p2
r1
r
− p1
j
(t)w(t)V1
r1
R∞
g∈M
(g ∗ (t))p2 v2 (t) dt
R ∞ R t
0
j
(t)w(t)V1
r1
p1
r1
r1
r
− p1
(g ∗ (t))p2 v2 (t) dt
(g (t)) W
. sup
r1
p1
p1
1
(t) dt
1
p2
r1
r
− p1
(g ∗∗ (t))q w(t)V1
(g ∗ (t))p2 v2 (t) dt
j
1
(t) dt
1
p2
' B1 + B3 .
The first step is due to the characterization of Λ ,→ Λ [7, Theorem 3.1(ii)]
and the last equivalence follows by integration by parts. Notice that the
resulting relation
(77)
1
− p1
sup W q (x)V1
x>0
1
− p1
(x)V2
2
(x) . B1 + B3
is established also if we consider the settings of cases (iii) and (v), i.e. if
p1 ≤ 1 or p2 ≤ 1 and the other relations between the parameters remain
unchanged. To continue, combining the obtained estimates we get
(78)
2,1
2,1
2,1
B1 + B3 ' A1,2
(61) + A(61) + A(62) + A(63) .
To deal with B2 , we use Lemma 3.3(i), setting α := q, β := r1 , γ := p2 ,
R
r10 0
q
t p0
−p01
−p0
1 v (s)V
,
ψ(t)
:=
s
(s)
ds
tp1 v1 (t)V1 1 (t), ω :=
ϕ(t) := w(t)
q
1
1
t
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
39
v2 . We obtain
1,2
B2 ' A1,2
(62) + A(64)
 x x
1
r1
r10 p0
r1 Zs p0
Z Z
q
1 v (t)
q s 1 v (s)
− p1
w(t)
t
1
1
2


+ sup
dt
ds
(x)
dt
V
0
0
2
p1
p1
tq
x>0
(t)
(s)
V
V
1
1
0
s
0
∞ ∞
1
r1
r10 p0
r1 Zs p0
Z Z
q
1 v (t)
q s 1 v (s)
w(t)
t
1
1

+ sup 
dt
ds
dt
p0
p0
t2q
x>0
V1 1 (t)
V1 1 (s)
x
s
0
Zx
×
0
tp2 v2 (t)
p02
0
dt
1
p02
.
V2 (t)
We now handle the third term in the sum by integration by parts and
the fourth one in the same way as an analogous term in the proof of
1,2
1,2
2,1
Lemma 3.3(i), concluding that B2 ' A1,2
(62) + A(63) + A(64) + A(64) . Together we get
2,1
1,2
2,1
1,2
2,1
1,2
2,1
C(1) ' A1,2
(61) + A(61) + A(62) + A(62) + A(63) + A(63) + A(64) + A(64) ,
(79)
still assuming the integrability of u1 near the origin. Now we perform
the usual final argument to drop the assumption on u1 . If u1 is not
integrable near the origin, then both A1,2
(62) = ∞ and B2 = ∞, the latter
by Proposition 2.3. Since B2 = ∞, it also holds C(1) = ∞. Then the
both sides of (79) are infinite, hence the equivalence holds trivially. The
same argument may be repeated in cases (ii)–(iv), only replacing A1,2
(62)
with another appropriate condition, when needed.
(ii) Here we use Lemmas 3.2(ii) and 3.3(ii) again, with the same respective settings of parameters as in the case (i), to estimate B1 and B2 .
Besides that, we also make use of Proposition 2.3 to estimate B3 . For
B1 and B3 we so obtain
2,1
2,1
B1 + B3 ' A1,2
(65) + A(65) + A(66) .
In order to get this equivalence, we in fact also need to prove the inequality

Z∞Zx
W

0
r1
p1
r
− p1
wV1
1
r1
p2 −r1
W
r1
p1
r
− p1
(x)w(x)V1
1
r1
r1 −p2
(x)V2
1
R
(x) dx
. B1 +B3 .
0
It is done by reusing the argument used to establish (77) (notice the
supremal condition from (77) being replaced by an integral condition
40
M. Křepela
this time, this is due to the different setting of parameters). The above
inequality is also true in case (iv). Now we continue with B2 . We get
∞ ∞ s
r2
r2
Z Z Z
r
q
p1 −r2 w(x)
− q2
w(t)
B2 ' 
dt
(s)v
(s)
ds
V
2
2
q
t
xq
0
x
x
Z∞Zy
×
x

w(t)
dt
tq
r2
p2
r
− 2
V2 q
Zx
(y)v2 (y) dy
0
x
Z∞Zx Zx
+
0
0
1
R
r2 (p1 −1)
0
p1 −r2
sp1 v1 (s)

ds
dx
0
V p1 (s)
s
r10 p0
r1
r1 Zs p0
q
p2 −r1
q s 1 v (s)
w(t)
y 1 v1 (y)
1
dy
ds
dt
p01
p01
tq
V1 (y)
V1 (s)
0
1
R
r10 p0
r1 Zs p0
ZxZx
r1
p1
q s 1 v (s)
w(t)
y 1 v1 (y)
w(x)
1
r1 −p2

dy
ds
V
(x)
dx
× q
dt
2
p0
p0
x
tq
V1 1 (y)
V1 1 (s)
0
s
0
+ B5 + B6 ,
where

Z∞Z∞Z∞
B5 := 
0
x
w(t)
dt
t2q
r2
p2
w(s)
s2q
Zs
p0
Zx
×
dy
p2
p1
p1 −r2
ds
V2 2 (y)
0
s
r20
0
y p2 v2 (y)
1
R
p1 (r2 −1) p0
p1 −r2
x 1 v1 (x) 
ds
dx
p0
p0
V1 1 (s)
V1 1 (x)
0
sp1 v1 (s)
0
and
∞ ∞ ∞
r1 Zs p0
r10 p0
r1
Z Z Z
q
p2 −r1
1 v (y)
q s 1 v (s)
y
w(t)
1
1
dt
dy
ds
B6 := 
p01
p01
t2q
V1 (y)
V1 (s)
0
0
x
s
Z∞
×
x
w(t)
dt
t2q
r1 Zx
q
p01
0
r10
0
y p1 v1 (y)
q
dy
V1 (y)
Zx
×
0
sp2 v2 (s)
p0
0
V2 2 (s)
0
xp1 v1 (x)
p0
V1 1 (x)
1
R
r1 (p2 −1)
p2 −r1
ds
dx .
Using integration by parts together with Proposition 2.1, one shows that
1,2
the first two terms in B2 are equivalent to A(66)
, hence B2 ' A1,2
(66) +
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
41
B5 + B6 . Similarly we prove that B5 ' A2,1
(67) . Next, again by integration
by parts we get
A1,2
(67)
∞ ∞
0 r1 p2
r1 p2 Zx p0
Z Z
q(p2 −r1 )
p1 (p2 −r1 )
w(t)
y 1 v1 (y)
dy
' B6 + 
dt
p01
t2q
V1 (y)
0
x
0
Zx
×
0
sp2 v2 (s)
p0
0
V2 2 (s)
1
R
p2 (r1 −1)
p2 −r1

ds
. B6 + B5 ,
dx
2,1
1,2
1,2
hence B5 + B6 ' A1,2
(67) + A(67) and therefore also B2 ' A(66) + A(67) +
A2,1
(67) . Altogether, it holds
2,1
1,2
2,1
1,2
2,1
C(1) ' B1 + B2 + B3 ' A1,2
(65) + A(65) + A(66) + A(66) + A(67) + A(67) .
Finally, the assumption of integrability of u1 is removed in a similar way
as in (i).
(iii) Using Lemmas 3.2(iii) and 3.3(iii) with the same setting as in (i)
and then repeating the argument from (i) to show (78), we get
2,1
1,2
1,2
C(1) ' B1 + B2 + B3 ' A1,2
(61) + A(61) + A(62) + A(63) + A(68) .
Then we prove that this statement holds also if u1 is not integrable near
the origin, by imitating the argument from (i).
(iv) Here we use Lemmas 3.2(iv) and 3.3(iv) to get the estimate of
B1 + B2 + B3 . Further adjustments of the conditions are made using the
corresponding arguments from (ii). We omit the details.
Now suppose that p1 ≤ 1, which is the case in (v) and (vi). For
i ∈ {1, 2} denote
r
− pi
σi (x) := sup y ri Vi
0<y≤x
i
(y),
x > 0.
42
M. Křepela
Using [6, Theorem 3.1] and integration by parts, we obtain
(80)
C(1) ' B1 + B3
R ∞ R ∞
0
(g ∗∗ (t))q w(t)
tq
x
+ sup
dt
R∞
g∈M
0
R ∞ R ∞
0
x
' B1 + B3 + sup
r1
p1
r1
(g ∗∗ (x))q w(x)
σ1 (x) dx
xq
(g ∗ )p2 v2
(g ∗∗ (t))q w(t)
tq
1
p2
dt
R∞
r1
q
(g ∗ )p2 v2
1
R
1
∗∗
q
q
∞
r
w(x)
dx
σ1 1 (0+) 0 (g (x))
xq
+ sup
1
R∞
g∈M
(g ∗ )p2 v2 p2
0
=: B1 + B3 + B7 + B8 .
g∈M
1
0
σ10 (x) dx
r1
1
1
p2
(v) We use Lemma 3.2(iii), setting α := q, β := r1 , γ := p2 , ϕ :=
−
r1
w, ψ := V1 q v1 , ω := v2 , to obtain estimates of B1 ; Lemma 3.3(iii),
0
setting α := q, β := r1 , γ := p2 , ϕ(t) := w(t)
tq , ψ := σ1 , ω := v2 , to
estimate B7 ; and [6, Theorem 3.1] to estimate B3 and B8 . Using the
obtained expressions in (80) and applying also the argument used in (i)
to show (77), we get
(81)
2,1
2,1
1,2
C(1) ' A1,2
(61) + A(61) + B9 + B10 + B11 + B12 + B13 + A(70) + A(72) ,
where
1
r2
B9 := sup σ2
x>0
∞ s
1
r1
r1
Z Z
r
q
− q1
w(t)


dt
,
(x)
V1
(s)v1 (s) ds
tq
x

1
r1
x
1
r2
r2
r
p2 w(x)
− p2
w(t)
2

dt
V
(x)
dx
,
2
tq
xq
Z∞Zx
B10 := σ1 (0+) 
0
Zx
B11 := sup
σ10
0
1
r1
0
x
Zx Zx
B12 := sup 
x>0
0
1
r2
B13 := sup σ2
x>0
Z∞Zs

x>0


w(t)
dt
tq
r2
q
r
− 2
V2 q

(s)v2 (s) ds
x
1
r1
r1
q
− 1
w(t)
0
 V2 p2 (x),
dt
σ
(s)
ds
1
q
t
s
∞ ∞
1
r1
r1
Z Z
q
w(t)
0
 .
(x) 
dt
σ
(s)
ds
1
t2q
x
s
1
r2
,
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
43
By integration by parts one verifies the following inequalities: B9 .
2,1
1,2
1,2
A2,1
(71) , B10 + B11 . A(71) , B12 . A(70) , and B13 . A(72) . From these
estimates and (81) it follows
2,1
1,2
2,1
1,2
2,1
1,2
2,1
C(1) . A1,2
(61) + A(61) + A(70) + A(70) + A(71) + A(71) + A(72) + A(72) .
Next, integration by parts yields the following: A1,2
(70) . B10 + B12 ,
2,1
2,1
2,1
1,2
A1,2
(71) . B10 + B11 + B12 , A(71) . B9 + A(71) , and A(72) . B13 + A(72) .
Using all these inequalities in (81), we get
2,1
1,2
2,1
1,2
2,1
1,2
2,1
A1,2
(61) + A(61) + A(70) + A(70) + A(71) + A(71) + A(72) + A(72) . C(1) .
The proof of this part is then completed.
(vi) Analogously to the case (v) we use Lemma 3.2(iv) to estimate B1 ,
Lemma 3.3(iv) to estimate B7 , and [6, Theorem 3.1] to get an estimate
of B3 and B8 . Inserting these expressions into (80) and merging some
of them by integration by parts (similarly to the case (ii)), we obtain
2,1
1,2
1,2
1,2
(82) C(1) ' A1,2
(65) + A(65) + A(73) + A(75) + A(75) + B10 + B14 + B15 + B16 ,
where

Z∞Z∞Zs
B14 := 
0
x
w(x)
×
xq
w(t)
dt
tq
r1
q
r
− q1
V1
r1
p2 −r1
(s)v1 (s) ds
x
Z∞Zs
x
1
R
r1
p1
r
p1
− q1
w(t)
p1 −r2

(x)
dx
V
(s)v
(s)
ds
σ
,
dt
1
2
1
tq
x
∞ ∞ s
1
R
r2
p1 Zx r1
Z Z Z
r
q
p1 −r2
p2 −r1
− q2
w(t)
0
0

B15 := 
dt
V
(s)v
(s)
ds
σ
σ
(x)
dx
,
2
1
1
2
tq
0
x
x
0
∞ x x
r1
r1
Z Z Z
q
p2 −r1
w(t)
0

B16 :=
dt
σ1 (s) ds
tq
0
0
w(x)
×
xq
s
Zx Zx
0
1
R
r1
r1
p1
w(t)
r1 −p2
0

dt
σ1 (s) dsV2
(x) dx
.
tq
s
2,1
Performing integration by parts, one gets B10 . A2,1
(74) , B14 . A(74) ,
1,2
B15 . A1,2
(74) , and B16 . A(73) . We apply these inequalities to replace
the “B-parts” in (82), and so we obtain
2,1
1,2
2,1
1,2
2,1
1,2
2,1
C(1) . A1,2
(65) + A(65) + A(73) + A(73) + A(74) + A(74) + A(75) + A(75) .
44
M. Křepela
Now observe that
(83)
A2,1
(73) ' B16
1
Z∞
q
− p1
w(t)
2
dt
(0+)
V
(∞)
2
tq
1
r1
+ σ1
0

1
r1
Z∞Zx
+σ1 (0+) 
0
1
r1
. B16 +B10 +σ1
1
R
R
p2
q
w(t)
r1 −p2

dt
V2
(x)v2 (x) dx
tq
0

Z∞Zx
(0+)
0
(84)
R
q
p2
r1 −p2
V2
1
R
(x)v2 (x) dx
0
. B16 + B10
1
r1

Z∞Zx
+σ1 (0+)
0
(85)
w(t)
dt
tq
0
1
R
r2Z∞
− r2
p2
p2
q
p1
w(t)
r1 −p2
r1 −p2

dt
V2
V2
v2
(x)v2 (x) dx
tq
x
. B16 + B10 .
Indeed, the estimates (83) and (85) follow by integration by parts, while
(84) is granted by Proposition 2.2. We proved that A2,1
(73) . B16 + B10 .
2,1
By similar means it is shown that A1,2
(74) . B10 + B15 + B16 and A(74) .
B14 + A1,2
(73) . Using these three estimates together with (82), we get
2,1
1,2
2,1
1,2
2,1
1,2
2,1
A1,2
(65) + A(65) + A(73) + A(73) + A(74) + A(74) + A(75) + A(75) . C(1) .
This completes case (vi) and thus the whole proof.
The next part deals with the “weak cases”, i.e. such configurations of
p1 , p2 , q that at least one of these exponents is infinite. The following
theorem covers the case q = ∞.
Theorem 4.4. Let v1 , v2 , w be weights. Let q = ∞.
(i) Let 0 < p1 , p2 ≤ 1. Then C(1) ' A(86) , where
(86)
A(86) := ess sup
x>0
− p1
− p1
w(x)
sup sV1 1 (s) sup tV2 2 (t).
2
x s∈(0,x)
t∈(0,x)
(ii) Let 0 < p1 ≤ 1 < p2 < ∞. Then C(1) ' A(87) , where
(87)
A(87)
Zx
10
− p1
p2
0
w(x)
1−p0
1
:= ess sup 2 sup sV1
(s)
tp2 −1 V2 2 (t) dt
.
x
x>0
s∈(0,x)
0
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
45
(iii) Let 0 < p1 ≤ 1 < p2 = ∞. Then C(1) ' A(88) , where
(88)
A(88) := ess sup
x>0
− p1
w(x)
sup sV1 1 (s)
2
x s∈(0,x)
Zx
0
dt
.
ess supy∈(0,t) v2 (y)
(iv) Let 1 < p1 , p2 < ∞. Then C(1) ' A(89) , where
w(x)
(89) A(89) := ess sup 2
x
x>0
Zx
p01 −1
s
10 Zx
1−p0
V1 1 (s) ds
p1
0
1−p02
0
tp2 −1 V2
10
(t) dt
p2
.
0
(v) Let 1 < p1 < p2 = ∞. Then C(1) ' A(90) , where
(90) A(90) := ess sup
x>0
w(x)
x2
Zx
0
1−p01
sp1 −1 V1
10 Zx
p1
(s) ds
0
0
dt
.
ess supy∈(0,t) v2 (y)
(vi) Let p1 = p2 = ∞. Then C(1) = A(91) , where
(91)
A(91)
w(x)
:= ess sup 2
x
x>0
Zx
0
Zx
ds
ess supy∈(0,s) v1 (y)
0
dt
.
ess supy∈(0,t) v2 (y)
Proof: We have
ess supx>0 f ∗∗ (x)g ∗∗ (x)w(x)
kf kΛp1 (v1 ) kgkΛp2 (v2 )
f ∈M g∈M
Rx ∗
Rx ∗
f (t) dt
g (t) dt
w(x)
0
sup 0
= ess sup 2 sup
p
x
kf
k
kgk
x>0
f ∈M
Λ 1 (v1 ) g∈M
Λp2 (v2 )
C(1) = sup sup
= ess sup
x>0
w(x)
kIdkΛp1 (v1 )→Λ1 (χ
kIdkΛp2 (v2 )→Λ1 (χ
.
(0,x) )
(0,x) )
x2
Now, in all the cases we simply
use the characterizations of the embed
ding Λp (v) ,→ Λ1 χ(0,x) provided by [7, Theorem 3.1] and Proposition 2.4.
Finally, we complete the list with the last remaining case in which
0 < q < ∞ and 0 < p2 ≤ p1 = ∞.
Theorem 4.5. Let v1 , v2 , w be weights. Let p1 = ∞ and 0 < q < ∞.
(i) Let 1 < p2 ≤ q. Then C(1) ' A(92) + A(93) , where

(92)
Zx
A(92) := sup 
x>0
0
w(s)
sq
Zs
0
dt
ess supy∈(0,t) v1 (y)
q
1
q
− p1
dt V2
2
(x)
46
M. Křepela
and
∞
1 x
q Z
10
Zs
q
Z
0
p2
w(s)
sp2 v2 (s)
dt

(93) A(93) := sup 
ds
.
dt
p02
s2q
ess supy∈(0,t) v1 (y)
x>0
V
(s)
2
x
0
0
(ii) Let 0 < p2 ≤ 1 and p2 ≤ q. Then C(1) ' A(92) + A(94) , where
(94)
A(94)
∞
1
q
Zs
q
Z
− p1
w(s)
dt
2


:= sup
(x).
dt
xV
2
s2q
ess supy∈(0,t) v1 (y)
x>0
x
0
(iii) Let 1 < p2 < ∞ and 0 < q < p2 . Then C(1) ' A(95) + A(96) , where

(95)
Z∞Zx
w(s)
sq
A(95) := 
0
Zs
0
0
w(x)
×
xq
0
dt
ess supy∈(0,t) v1 (y)
Zx
r2
q
dt
q
dt

r
− p2
V2
ess supy∈(0,t) v1 (y)
p2
2
1
r2
(x) dx
and
∞ ∞
Zs
q r2
Z Z
p2
w(s)
dt
dt
(96) A(96) := 
s2q
ess supy∈(0,t) v1 (y)
0
×
x
0
w(x)
x2q
Zx
0
dt
ess supy∈(0,t) v1 (y)
q Zx
0
1
r2
r20
p
s v2 (s)
2

ds
dx
.
p0
V2 2 (s)
p02
(iv) Let 0 < q < p2 ≤ 1. Then C(1) ' A(95) + A(97) , where

(97) A(97)
Z∞Z∞
:= 
0
w(s)
s2q
Zs
0
x
w(x)
× 2q
x
Zx
0
q
dt
ess supy∈(0,t) v1 (y)
dt
q
ess supy∈(0,t) v1 (y)
r2
p2
dt

r
− p2
sup y r2 V2
2
(x) dx
1
r2
.
y∈(0,x)
(v) Let 0 < q < p2 = ∞. Then C(1) ' A(98) , where

Z∞
(98) A(98) := 
0
w(x)
x2q
Zx
0
dt
ess supy∈(0,t) v1 (y)
Zx
0
ds
ess supy∈(0,s) v2 (y)
q
1
q
dx .
Bilinear Weighted Hardy Inequality for Nonincreasing Functions
47
Proof: From Proposition 2.4 it follows
R∞
0
C(1) = sup sup
g∈M f ∈M
1
(f ∗∗ (x))q (g ∗∗ (x))q w(x) dx q
kf kΛ∞ (v1 ) kgkΛp2 (v2 )
1
∞
q
Zx
q
Z
ds

 (g ∗∗ (x))q w(x)
dx
xq
ess supy∈(0,s) v1 (y)
' sup
0
0
kgkΛp2 (v2 )
g∈M
' kIdkΛp2 (v
q
2 )→Γ
w(x)
x7→ xq
(ess supy∈(0,s) v1 (y))−1
h
iq .
The rest is done by application of the characterization of the involved
embedding Γ ,→ Λ, which can be found in [7, Theorem 4.1] (cases (i)
and (ii)), Proposition 2.3 (for case (iii)), [6, Theorem 3.1] (case (iv)),
and finally Proposition 2.4 for case (v).
Acknowledgement
The author would like to thank the referees whose remarks helped to
correct and improve the final version of the paper.
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DOI: 10.2307/2154450.
50
M. Křepela
Karlstad University
Faculty of Health, Science and Technology
Department of Mathematics and Computer Science
651 88 Karlstad
Sweden
Charles University
Faculty of Mathematics and Physics
Department of Mathematical Analysis
Sokolovská 83
186 75 Praha 8
Czech Republic
E-mail address: [email protected]
Primera versió rebuda el 14 de gener de 2015,
darrera versió rebuda el 9 de setembre de 2015.