Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications
The main aim of the paper is to prove a two-weight criterion for the multidimensional Hardy-type operator from weighted Lebesgue spaces into p-convex weighted Banach function spaces. The problem for the dual operator is also considered. As an application, we prove a two-weight criterion of boundedne...
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Bandaliev, R.A. 2020-02-13T17:17:18Z 2020-02-13T17:17:18Z 2015 Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications / R.A. Bandaliev // Український математичний журнал. — 2015. — Т. 67, № 3. — С. 313–325. — Бібліогр.: 44 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165493 517.5 The main aim of the paper is to prove a two-weight criterion for the multidimensional Hardy-type operator from weighted Lebesgue spaces into p-convex weighted Banach function spaces. The problem for the dual operator is also considered. As an application, we prove a two-weight criterion of boundedness of the multidimensional geometric mean operator from weighted Lebesgue spaces into weighted Musielak–Orlicz spaces. Головною метою даної статті є доведення двовагового критерію для багатовимiрного оператора типу Хардi із вагових лебегових просторів в p-опуклі вагові банахові простори функцій. Також розглянуто задачу для дуального оператора. Як застосування, встановлено двоваговий критерій обмеженості багатовимірного оператора геометричного середнього з вагових лебегових просторів у вагові простори Мусєляка-Орліча. This paper was partially supported by the Science Development Foundation under the President of the Republic of Azerbaijan (Grant EIF-2014-9(15)-46/10/1) and by the grant of Presidium of Azerbaijan National Academy of Sciences 2015. en Інститут математики НАН України Український математичний журнал Статті Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications Про двоваговий критерій для багатовимірного оператора типу Харді в p-опуклих банахових просторах функцій та деякі застосування Article published earlier |
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Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications |
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Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications Bandaliev, R.A. Статті |
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Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications |
| title_full |
Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications |
| title_fullStr |
Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications |
| title_full_unstemmed |
Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications |
| title_sort |
two-weight criteria for the multidimensional hardy-type operator in p-convex banach function spaces and some applications |
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Bandaliev, R.A. |
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Bandaliev, R.A. |
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Статті |
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2015 |
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English |
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Український математичний журнал |
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Інститут математики НАН України |
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Про двоваговий критерій для багатовимірного оператора типу Харді в p-опуклих банахових просторах функцій та деякі застосування |
| description |
The main aim of the paper is to prove a two-weight criterion for the multidimensional Hardy-type operator from weighted Lebesgue spaces into p-convex weighted Banach function spaces. The problem for the dual operator is also considered. As an application, we prove a two-weight criterion of boundedness of the multidimensional geometric mean operator from weighted Lebesgue spaces into weighted Musielak–Orlicz spaces.
Головною метою даної статті є доведення двовагового критерію для багатовимiрного оператора типу Хардi із вагових лебегових просторів в p-опуклі вагові банахові простори функцій. Також розглянуто задачу для дуального оператора. Як застосування, встановлено двоваговий критерій обмеженості багатовимірного оператора геометричного середнього з вагових лебегових просторів у вагові простори Мусєляка-Орліча.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/165493 |
| citation_txt |
Two-Weight Criteria for the Multidimensional Hardy-Type Operator in p-Convex Banach Function Spaces and Some Applications / R.A. Bandaliev // Український математичний журнал. — 2015. — Т. 67, № 3. — С. 313–325. — Бібліогр.: 44 назв. — англ. |
| work_keys_str_mv |
AT bandalievra twoweightcriteriaforthemultidimensionalhardytypeoperatorinpconvexbanachfunctionspacesandsomeapplications AT bandalievra prodvovagoviikriteríidlâbagatovimírnogooperatoratipuhardívpopuklihbanahovihprostorahfunkcíitadeâkízastosuvannâ |
| first_indexed |
2025-11-26T23:51:30Z |
| last_indexed |
2025-11-26T23:51:30Z |
| _version_ |
1850783962073399296 |
| fulltext |
UDC 517.5
R. A. Bandaliyev (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku)
ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE
OPERATOR IN p-CONVEX BANACH FUNCTION SPACES
AND SOME APPLICATION*
ПРО ДВОВАГОВИЙ КРИТЕРIЙ ДЛЯ БАГАТОВИМIРНОГО ОПЕРАТОРА
ТИПУ ХАРДI В p-ОПУКЛИХ БАНАХОВИХ ПРОСТОРАХ ФУНКЦIЙ
ТА ДЕЯКI ЗАСТОСУВАННЯ
The main goal of this paper is to prove a two-weight criterion for multidimensional Hardy type operator from weighted
Lebesgue spaces into p-convex weighted Banach function spaces. The problem for the dual operator is also considered. As
an application, we prove a two-weight criterion for the boundedness of multidimensional geometric mean operator from
weighted Lebesgue spaces into weighted Musielak – Orlicz spaces.
Головною метою даної статтi є доведення двовагового критерiю для багатовимiрного оператора типу Хардi iз
вагових лебегових просторiв в p-опуклi ваговi банаховi простори функцiй. Також розглянуто задачу для дуального
оператора. Як застосування, встановлено двоваговий критерiй обмеженостi багатовимiрного оператора геометрич-
ного середнього з вагових лебегових просторiв у ваговi простори Мусєляка – Орлiча.
1. Introduction. The investigation of Hardy operator in weighted Banach function spaces (BFS) have
recently history. The goal of this investigations were closely connected with the found of criterion
on the geometry and on the weights of BFS for validity of boundedness of Hardy operator in BFS.
Characterization of the mapping properties such as boundedness and compactness were considered in
the papers [10, 11, 14, 30] and etc. More precisely, in [10] and [11] were considered the boundedness
of certain integral operator in ideal Banach spaces. In [14] was proved the boundedness of Hardy
operator in Orlicz spaces. Also, in [30] the compactness and measure of noncompactness of Hardy
type operator in BFS was proved. But in this paper we consider the boundedness of Hardy operator
in p-convex BFS and find a new type criterion on the weights for validity of Hardy inequality. Note
that the notion of BFS was introduced in [32]. In particular, the weighted Lebesgue spaces, weighted
Lorentz spaces, weighted variable Lebesgue spaces, variable Lebesgue spaces with mixed norm,
Musielak – Orlicz spaces and etc. is BFS.
In this paper, we establish an integral-type necessary and sufficient condition on weights, which
provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue
spaces into p-convex weighted BFS. We also investigate the corresponding problems for the dual
operator. It is well known that the classical two-weight inequality for geometric mean operator is
closely connected with the one-dimensional Hardy inequality (see [20]). Analogously, the Pólya –
Knopp type inequalities with multidimensional geometric mean operator are connected with the
multidimensional Hardy type operator. Therefore, in this paper, as an application of Hardy inequality
we prove the boundedness of multidimensional geometric mean operator and boundedness of certain
sublinear operator from weighted Lebesgue spaces into weighted Musielak – Orlicz spaces.
* This paper was partially supported by the Science Development Foundation under the President of the Republic of
Azerbaijan (Grant EIF-2014-9(15)-46/10/1) and by the grant of Presidium of Azerbaijan National Academy of Sciences
2015.
c© R. A. BANDALIYEV, 2015
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3 313
314 R. A. BANDALIYEV
2. Preliminaries. Let (Ω, µ) be a complete σ-finite measure space. By L0 = L0(Ω, µ) we
denote the collection of all real-valued µ-measurable functions on Ω.
Definition 1 [9, 29, 32]. We say that real normed space X is a BFS if :
(P1) the norm ‖f‖X is defined for every µ-measurable function f, and f ∈ X if and only if
‖f‖X <∞; ‖f‖X = 0 if and only if f = 0 a.e.;
(P2) ‖f‖X = ‖|f |‖X for all f ∈ X;
(P3) if 0 ≤ fn ↑ f ≤ g a.e., then ‖fn‖X ↑ ‖f‖X (Fatou property);
(P4) if E is a measurable subset of Ω such that µ(E) <∞, then ‖χE‖X <∞, where χE is the
characteristic function of the set E;
(P5) for every measurable set E ⊂ Ω with µ(E) < ∞, there is a constant CE > 0 such that∫
E
f(x) dx ≤ CE‖f‖X .
Given a BFS X we can always consider its associate space X ′ consisting of those g ∈ L0
that f · g ∈ L1 for every f ∈ X with the usual order and the norm ‖g‖X′ = sup
{
‖f · g‖L1 :
‖g‖X′ ≤ 1
}
. Note that X ′ is a BFS in (Ω, µ) and a closed norming subspaces.
Let X be a BFS and ω be a weight, that is, positive Lebesgue measurable and a.e. finite
functions on Ω. Let Xω = {f ∈ L0 : fω ∈ X} . This space is a weighted BFS equipped with the
norm ‖f‖Xω = ‖f ω‖X . (For more detail and proofs of results about BFS we refer the reader to [9]
and [29].)
Note that the notion of BFS was introduced by W. A. J. Luxemburg in [32].
Let us recall the notion of p-convexity and p-concavity of BFS’s.
Definition 2 [42]. Let X is a BFS. Then X is called p-convex for 1 ≤ p ≤ ∞ if there exists a
constant M > 0 such that for all f1, . . . , fn ∈ X∥∥∥∥∥∥∥
(
n∑
k=1
|fk|p
)1
p
∥∥∥∥∥∥∥
X
≤M
(
n∑
k=1
‖fk‖pX
)1
p
if 1 ≤ p <∞,
or
∥∥sup1≤k≤n |fk|
∥∥
X
≤M max1≤k≤n ‖fk‖X if p = ∞. Similarly X is called p-concave for 1 ≤
≤ p ≤ ∞ if there exists a constant M > 0 such that for all f1, . . . , fn ∈ X(
n∑
k=1
‖fk‖pX
)1
p
≤M
∥∥∥∥∥∥∥
(
n∑
k=1
|fk|p
)1
p
∥∥∥∥∥∥∥
X
if 1 ≤ p <∞,
or max1≤k≤n ‖fk‖X ≤M
∥∥sup1≤k≤n |fk|
∥∥
X
if p =∞.
Remark 1. Note that the notions of p-convexity, respectively p-concavity are closely related
to the notions of upper p-estimate (strong `p-composition property), respectively lower p-estimate
(strong `p-decomposition property) as can be found in [29].
Now we reduce some examples of p-convex and respectively p-concave BFS. Let Rn be the
n-dimensional Euclidean space of points x = (x1, . . . , xn) and Ω be a Lebesgue measurable subset
in Rn and |x| =
(∑n
i=1
x2
i
)1/2
. The Lebesgue measure of a set Ω will be denoted by |Ω|. It is well
known that |B(0, 1)| = πn/2
Γ
(n
2
+ 1
) , where B(0, 1) = {x : x ∈ Rn ; |x| < 1} .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE OPERATOR IN P -CONVEX . . . 315
Example 1. Let 1 ≤ q ≤ ∞ and X = Lq. Then the space Lq is p-convex (p-concave) BFS if
and only if 1 ≤ p ≤ q ≤ ∞ (1 ≤ q ≤ p ≤ ∞.)
The proof implies from usual Minkowski inequality in Lebesgue spaces.
Example 2. The following lemma shows that the variable Lebesgue spaces Lq(y)(Ω) is p-
convex BFS.
Lemma 1 [1]. Let 1 ≤ p ≤ q(x) ≤ q <∞ for all y ∈ Ω2 ⊂ Rm. Then the inequality∥∥‖f‖Lp(Ω1)
∥∥
Lq(·)(Ω2)
≤ Cp,q
∥∥∥‖f‖Lq(·)(Ω2)
∥∥∥
Lp(Ω1)
is valid, where
Cp,q =
(
‖χ∆1‖∞ + ‖χ∆2‖∞ + p
(
1
q
− 1
q
))
(‖χ∆1‖∞ + ‖χ∆2‖∞), q = ess inf
Ω2
q(x),
q = ess sup
Ω2
q(x), ∆1 = {(x, y) ∈ Ω1 × Ω2 : q(y) = p} , ∆2 = Ω1 × Ω2 \∆1
and f : Ω1 × Ω2 → R is any measurable function such that
∥∥‖f‖Lp(Ω1)
∥∥
Lq(·)(Ω2)
= inf
µ > 0 :
∫
Ω2
(‖f(·, y)‖Lp(Ω1)
µ
)q(y)
dy ≤ 1
<∞
and ‖f(·, y)‖Lp(Ω1) =
(∫
Ω1
|f(x, y)|p dx
)1/p
.
Analogously, if 1 ≤ q(x) ≤ p <∞, then Lq(x)(Ω) is p-concave BFS.
Definition 3 [18, 37]. Let Ω ⊂ Rn be a Lebesgue measurable set. A real function ϕ : Ω ×
× [0,∞) 7→ [0,∞) is called a generalized ϕ-function if it satisfies:
(a) ϕ(x, ·) is a ϕ-function for all x ∈ Ω, i.e., ϕ(x, ·) : [0,∞) 7→ [0,∞) is convex and satisfies
ϕ(x, 0) = 0, limt→+0 ϕ(x, t) = 0;
(b) ψ : x 7→ ϕ(x, t) is measurable for all t ≥ 0.
If ϕ is a generalized ϕ-function on Ω, we shortly write ϕ ∈ Φ.
Definition 4 [18, 37]. Let ϕ ∈ Φ and be ρϕ defined by the expression
ρϕ(f) :=
∫
Ω
ϕ(y, |f(y)|) dy for all f ∈ L0(Ω).
We put Lϕ =
{
f ∈ L0(Ω) : ρϕ(λ0f) <∞ for some λ0 > 0
}
and
‖f‖Lϕ = inf
{
λ > 0 : ρϕ
(
f
λ
)
≤ 1
}
.
The space Lϕ is called Musielak – Orlicz space.
Let ω be a weight function on Ω, i.e., ω is a nonnegative, almost everywhere positive function
on Ω. In this work we considered the weighted Musielak – Orlicz spaces. We denote
Lϕ, ω =
{
f ∈ L0(Ω) : fω ∈ Lϕ
}
.
It is obvious that the norm in this spaces is given by
‖f‖Lϕ, ω = ‖fω‖Lϕ .
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
316 R. A. BANDALIYEV
Remark 2. Let ϕ(x, t) = tq(x) in the Definition 4, where 1 ≤ q(x) < ∞ and x ∈ Ω. Then we
have the definition of variable exponent weighted Lebesgue spaces Lq(x)(Ω) (see [18]).
Example 3. The following lemma shows that the Musielak – Orlicz spaces Lϕ is p-convex BFS.
Lemma 2 [6]. Let Ω1 ⊂ Rn and Ω2 ⊂ Rm. Let (x, t) ∈ Ω1 × [0,∞) and ϕ
(
x, t1/p
)
∈ Φ for
some 1 ≤ p <∞. Suppose f : Ω1 × Ω2 7→ R. Then the inequality∥∥‖f(x, ·)‖Lp(Ω2)
∥∥
Lϕ
≤ 21/p
∥∥‖f(·, y)‖Lϕ
∥∥
Lp(Ω2)
is valid.
We note that the Lebesgue spaces with mixed norm, weighted Lorentz spaces and etc. is p-convex
(p-concave) BFS. Now we reduce more general result connected with Minkowski’s integral inequality.
Let X and Y be BFS’s on (Ω1, µ) and (Ω2, ν) respectively. By X[Y ] and Y [X] we denote
the spaces with mixed norm and consisting of all functions g ∈ L0 (Ω1 × Ω2, µ× ν) such that
‖g(x, ·)‖Y ∈ X and ‖g(·, y)‖X ∈ Y. The norms in this spaces is defined as
‖g‖X[Y ] = ‖‖g(x, ·)‖Y ‖X , ‖g‖Y [X] = ‖‖g(·, y)‖X‖Y .
Theorem 1 [42]. Let X and Y be BFS’s with the Fatou property. Then the generalized
Minkowski integral inequality
‖f‖X[Y ] ≤M ‖f‖Y [X]
holds for all measurable functions f(x, y) if and only if there exists 1 ≤ p ≤ ∞ such that X is
p-convex and Y is p-concave.
It is known that X[Y ] and Y [X] are BFS’s on Ω1 × Ω2 (see [29]).
3. Main results. We consider the multidimensional Hardy type operator and its dual operator
Hf(x) =
∫
|y|<|x|
f(y) dy and H∗f(x) =
∫
|y|>|x|
f(y) dy,
where f ≥ 0 and x ∈ Rn.
Now we prove a two-weight criterion for multidimensional Hardy type operator acting from the
p-concave weighted BFS to weighted Lebesgue spaces.
Theorem 2. Let v(x) and w(x) are weights on Rn. Suppose that Xw be a p-convex weighted
BFS’s for 1 ≤ p <∞ on Rn. Then the inequality
‖Hf‖Xw ≤ C ‖f‖Lp, v (1)
holds for every f ≥ 0 and for all α ∈ (0, 1) if and only if
A(α) = sup
t>0
∫
|y|<t
[v(y)]−p
′
dy
α
p′
∥∥∥∥∥∥∥∥∥χ{|z|>t}(·)
∫
|y|<|·|
[v(y)]−p
′
dy
1−α
p′
∥∥∥∥∥∥∥∥∥
Xw
<∞. (2)
Moreover, if C > 0 is the best possible constant in (1), then
sup
0<α<1
p′A(α)
(1− α)
[(
p′
1− α
)p
+
1
α (p− 1)
]1/p
≤ C ≤M inf
0<α<1
A(α)
(1− α)1/p′
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE OPERATOR IN P -CONVEX . . . 317
Proof. Sufficiency. Passing to the polar coordinates, we have
h(y) =
∫
|z|<|y|
[v(z)]−p
′
dz
α
p′
=
|y|∫
0
sn−1
∫
|ξ|=1
[v(sξ)]−p
′
dξ
ds
α
p′
,
where dξ is the surface element on the unit sphere. Obviously, h(y) = h(|y|), i.e., h(y) is a radial
function.
Applying Hölder’s inequality for Lp(Rn) spaces and after some standard transformations, we
obtain
‖Hf‖Xw =
∥∥∥∥∥∥∥w(·)
∫
|y|<|·|
f(y) dy
∥∥∥∥∥∥∥
X
=
∥∥∥∥∥∥∥w(·)
∫
|y|<|·|
[f(y)h(y)v(y)] [h(y)v(y)]−1 dy
∥∥∥∥∥∥∥
X
≤
≤
∥∥∥∥w(·) ‖f h v‖Lp(|y|<|·|)
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|·|)
∥∥∥∥
X
=
=
∥∥∥∥∥
∥∥∥∥w(·) f h v χ{|y|<|·|}(y)
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|·|)
∥∥∥∥
Lp
∥∥∥∥∥
X
=
=
∥∥∥∥w f h v χ{|·|<|x|}(·) ∥∥∥[h v]−1
∥∥∥
Lp′ (|·|<|x|)
∥∥∥∥
X[Lp]
.
Applying Theorem 1, we get∥∥∥∥w f h v χ{|·|<|x|}(·) ∥∥∥[h v]−1
∥∥∥
Lp′ (|·|<|x|)
∥∥∥∥
X[Lp]
≤
≤M
∥∥∥∥w f h v χ{|·|<|x|}(·)∥∥∥[h v]−1
∥∥∥
Lp′ (|·|<|x|)
∥∥∥∥
Lp[X]
=
= M
∥∥∥∥∥∥∥∥w(·) fh v χ{|y|<|·|}(y)
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|·|)
∥∥∥∥
X
∥∥∥∥
Lp
=
= M
∥∥∥∥fh v ∥∥∥∥w(·)χ{|y|<|·|}(y)
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|·|)
∥∥∥∥
X
∥∥∥∥
Lp
.
By switching to polar coordinates and after some calculations, we have
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|x|)
=
∫
|y|<|x|
[h(|y|) v(y)]−p
′
dy
1/p′
=
=
|x|∫
0
rn−1 [h(r)]−p
′
∫
|ξ|=1
[v(rξ)]−p
′
dξ
dr
1/p′
=
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
318 R. A. BANDALIYEV
=
|x|∫
0
r∫
0
sn−1
∫
|ξ|=1
[v(sξ)]−p
′
dξ
ds
−α ∫
|ξ|=1
[v(rξ)]−p
′
dξ
rn−1dr
1/p′
=
=
1
(1− α)1/p′
|x|∫
0
d
dr
r∫
0
sn−1
∫
|ξ|=1
[v(sξ)]−p
′
dξ
ds
1−α dr
1/p′
=
=
1
(1− α)1/p′
|x|∫
0
sn−1
∫
|ξ|=1
[v(sξ)]−p
′
dξ
ds
1−α
p′
=
=
1
(1− α)1/p′
∫
|z|<|x|
[v(z)]−p
′
dz
1−α
p′
.
Therefore from condition (2), we obtain∥∥∥∥fh v ∥∥∥∥w(·)χ{|y|<|·|}(y)
∥∥∥[h v]−1
∥∥∥
Lp′ (|y|<|·|)
∥∥∥∥
X
∥∥∥∥
Lp
=
=
1
(1− α)1/p′
∥∥∥∥∥∥∥∥∥f v
h
∥∥∥∥∥∥∥∥∥χ{|·|>|y|}
∫
|z|<|·|
[v(z)]−p
′
dz
1−α
p′
∥∥∥∥∥∥∥∥∥
Xw
∥∥∥∥∥∥∥∥∥
Lp
≤ A(α)
(1− α)1/p′
‖f v‖Lp .
Thus
‖Hf‖Xw ≤M
A(α)
(1− α)1/p′
‖f‖Lp, v for all α ∈ (0, 1).
Necessity. Let f ∈ Lp,v (Rn) , f ≥ 0, and inequality (1) is valid. We choose the test function as
f(x) =
p′
1− α
[g(t)]
− αp′−
1
p v−p
′
(x)χ{|x|<t}(x) + [g(|x|)]−
α
p′−
1
p v−p
′
(x)χ{|x|>t}(x),
where t > 0 is a fixed number and
g(t) =
∫
|y|<t
v−p
′
(y) dy =
t∫
0
sn−1
∫
|η|=1
v−p
′
(sη) dη
ds.
It is obvious that
dg
dt
= tn−1
∫
|η|=1
v−p
′
(tη) dη. Again by switching to polar coordinates, from the
right-hand side of inequality (1) we get
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ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE OPERATOR IN P -CONVEX . . . 319
‖f‖Lp,v =
∫
|x|<t
(
p′
1− α
)p
[g(t)]−α(p−1)−1 v−p
′
(x) dx+
∫
|x|>t
[g(|x|)]−α(p−1)−1 v−p
′
(x) dx
1/p
=
=
( p′
1− α
)p
[g(t)]α(1−p) +
∞∫
t
rn−1 [g(r)]−α(p−1)−1
∫
|ξ|=1
v−p
′
(rξ) dξ
dr
1/p
=
=
( p′
1− α
)p
[g(t)]α(1−p) − 1
α(p− 1)
∞∫
t
d
dr
[g(r)]−α(p−1) dr
1/p
=
=
( p′
1− α
)p
[g(t)]α(1−p) +
1
α(p− 1)
[g(t)]−α(p−1) −
∫
Rn
v−p
′
(y) dy
−α(p−1)
1/p
≤
≤
[(
p′
1− α
)p
+
1
α(p− 1)
]1/p
[g(t)]−α/p
′
=
[(
p′
1− α
)p
+
1
α(p− 1)
]1/p
[h(t)]−1.
After some calculations, from the left-hand side of inequality (1), we have
‖Hf‖Xw =
∥∥∥∥∥∥∥
∫
|y|<|·|
f(y) dy
∥∥∥∥∥∥∥
Xw
≥
∥∥∥∥∥∥∥χ{|·|>t}
∫
|y|<|·|
f(y) dy
∥∥∥∥∥∥∥
Xw
=
=
∥∥∥∥∥∥∥χ{|·|>t}
p′
1− α
∫
|y|<t
[g(t)]
− αp′−
1
p v−p
′
(y) dy +
∫
t<|y|<|·|
[g(|y|)]−
α
p′−
1
p v−p
′
(y) dy
∥∥∥∥∥∥∥
Xw
=
=
∥∥∥∥∥∥∥χ{|·|>t}
p′
1− α
[g(t)]
1−α
p′ +
|·|∫
t
rn−1 [g(r)]
− αp′−
1
p
∫
|η|=1
v−p
′
(rη) dη
dr
∥∥∥∥∥∥∥
Xw
=
=
∥∥∥∥∥∥∥χ{|·|>t}
p′
1− α
[g(t)]
1−α
p′ +
p′
1− α
|·|∫
t
d
dr
[g(r)]
1−α
p′ dr
∥∥∥∥∥∥∥
Xw
=
=
∥∥∥∥χ{|·|>t} [ p′
1− α
[g(t)]
1−α
p′ +
p′
1− α
(
[g(| · |)]
1−α
p′ − [g(t)]
1−α
p′
)]∥∥∥∥
Xw
=
=
p′
1− α
∥∥∥∥χ{|·|>t} [g(·)]
1−α
p′
∥∥∥∥
Xw
.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
320 R. A. BANDALIYEV
Hence, this implies that
p′
1− α
[(
p′
1− α
)p
+
1
α(p− 1)
]−1/p
[g(t)]α/p
′
∥∥∥∥χ{|·|>t} [g(·)]
1−α
p′
∥∥∥∥
Xw
≤ C,
i.e.,
p′A(α)
(1− α)
[(
p′
1− α
)p
+
1
α (p− 1)
]1/p
≤ C for all α ∈ (0, 1).
Theorem 2 is proved.
For the dual operator, the below stated theorem is proved analogously.
Theorem 3. Let v(x) and w(x) are weights on Rn. Suppose that Xw be a p-convex weighted
BFS’s for 1 ≤ p <∞ on Rn. Then the inequality
‖H∗f‖Xw ≤ C ‖f‖Lp, v (3)
holds for every f ≥ 0 and for all γ ∈ (0, 1) if and only if
B(γ) = sup
t>0
∫
|y|>t
[v(y)]−p
′
dy
γ
p′
∥∥∥∥∥∥∥∥∥χ{|z|<t}(·)
∫
|y|>|·|
[v(y)]−p
′
dy
1−γ
p′
∥∥∥∥∥∥∥∥∥
Xw
<∞.
Moreover, if C > 0 is the best possible constant in (3), then
sup
0<γ<1
p′B(γ)
(1− γ)
[(
p′
1− γ
)p
+
1
γ (p− 1)
]1/p
≤ C ≤M inf
0<γ<1
B(γ)
(1− γ)1/p′
.
Corollary 1. Note that Theorems 2 and 3 in the case Xw = Lϕ,w, ϕ
(
x, t1/p
)
∈ Φ for some
1 ≤ p < ∞, x ∈ Rn was proved in [6]. In the case Xw = Lq, w, 1 < p ≤ q < ∞, for
x ∈ (0,∞), α =
s− 1
p− 1
and s ∈ (1, p) Theorems 2 and 3 was proved in [44]. For x ∈ Rn in the
case Xw = Lq(x), w and 1 < p ≤ q(x) ≤ ess supx∈Rn q(x) < ∞ Theorems 2 and 3 was proved in
[3] (see also [2]).
Remark 3. In the case n = 1, Xw = Lq, w, 1 < p ≤ q ≤ ∞, at x ∈ (0,∞), for classical
Lebesgue spaces the various variants of Theorems 2 and 3 were proved in [13, 20, 27, 28, 35, 36,
43] and etc. In particular, in the Lebesgue spaces with variable exponent the boundedness of Hardy
type operator was proved in [16, 17, 19, 21, 26, 33, 34] and etc. For Xw = Lq(x), w, 1 < p ≤
≤ q(x) ≤ ess supx∈[0,1] q(x) < ∞ and x ∈ [0, 1] the two-weighted criterion for one-dimensional
Hardy operator was proved in [26]. Also, other type two-weighted criterion for multidimensional
Hardy type operator in the case Xw = Lq(x), w, 1 < p ≤ q(x) ≤ ess supx∈Rn q(x) <∞ and x ∈ Rn
was proved in [33] (see also [34]). In the papers [12] and [41] the inequalities of modular type for
more general operators was proved. Also, in [14] the Hardy type inequalities with special power-type
weights in Orlicz spaces was proved.
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ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE OPERATOR IN P -CONVEX . . . 321
4. Application. Now we consider the multidimensional geometric mean operator defined as
Gf(x) = exp
1
|B(0, |x|)|
∫
B(0, |x|)
ln f(y) dy
,
where f > 0 and |B(0, |x|)| = |B(0, 1)| |x|n. It is obvious that G (f1 · f2) (x) = Gf1(x) ·Gf2(x).
We formulate a two-weighted criterion on boundedness of multidimensional geometric mean
operator in weighted Musielak – Orlicz spaces.
Theorem 4. Let ϕ
(
x, t1/p
)
∈ Φ for some 0 < p <∞ and x ∈ Rn. Suppose that v(x) and w(x)
are weight functions on Rn. Then the inequality
‖Gf‖Lϕ,w ≤ C ‖f‖Lp, v (4)
holds for every f > 0 and for all s ∈ (1, p) if and only if
D(s) =sup
t>0
|B(0, t)|
s−1
p
∥∥∥∥∥∥∥
χ{|z|>t}(·)
|B(0, | · |)|s/p
exp
1
|B(0, | · |)|
∫
B(0,|·|)
ln
1
v(y)
dy
∥∥∥∥∥∥∥
Lϕ,w
<∞. (5)
Moreover, if C > 0 is the best possible constant in (4), then
sup
s>1
es/p(
es +
1
s− 1
)1/p
D(s) ≤ C ≤ 21/p inf
s>1
e
s−1
p D(s).
Proof. Let α =
s− 1
p− 1
, where 1 < s < p. We replace f with fβ, v with vβ, w with
wβ(x)
|B(0, |x|)|
,
0 < β < p, and p with
p
β
and ϕ(x, t) with ϕ
(
x, t1/β
)
in (1), (2), we find that for 1 < s <
p
β
∥∥∥∥ wβ
|B(0, | · |)|
H(fβ)
∥∥∥∥
L
ϕ(·,t1/β)
=
∥∥∥∥∥∥∥∥
1
|B(0, | · |)|
∫
B(0, |·|)
fβ(y) dy
1/β
∥∥∥∥∥∥∥∥
β
Lϕ,w(Rn)
≤
≤ Cβ
∫
Rn
[f(y)v(y)]p dy
β/p .
Then the inequality∥∥∥∥∥∥∥∥
1
|B(0, | · |)|
∫
B(0, |·|)
fβ(y) dy
1/β
∥∥∥∥∥∥∥∥
Lϕ,w(Rn)
≤ C1/β
β
∫
Rn
[f(y)v(y)]p dy
1/p
(6)
holds if and only if
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
322 R. A. BANDALIYEV
A
(
s− 1
p− 1
)
=
=
sup
t>0
∫
|y|<t
[v(y)]
− β p
p−β dy
s−1
p
∥∥∥∥∥∥∥∥∥
χ{|z|>t}(·)
|B(0, | · |)|
p
p−βs
∫
|y|<|·|
[v(y)]
− βp
p−β dy
p−βs
βp
∥∥∥∥∥∥∥∥∥
Lϕ,w
β
=
= Bβ(s, β) <∞
and
sup
1<s<p/β
(
p
p− sβ
)p/β
(
p
p− sβ
)p/β
+
1
s− 1
β/p
Bβ(s, β) ≤ Cβ ≤ 2β/p inf
1<s<p/β
(
p− β
p− sβ
)p−β
p
Bβ (s, β) .
(7)
By the L’Hospital rule, we get
lim
β→+0
1
|B(0, |x|)|
p
p−βs
∫
|y|<|x|
[v(y)]
− βp
p−β dy
p−βs
βp
=
= lim
β→+0
exp
p ln
1
|B(0, |x|)|
+ (p− βs) ln
( ∫
|y|<|x|
[v(y)]
− βp
p−β dy
)
p β
=
= lim
β→+0
exp
−s
p
ln
∫
|y|<|x|
[v(y)]
− βp
p−β dy
+
+
(p− βs)
(
p
p− β
)2 ∫
|y|<|x|
[v(y)]
− βp
p−β ln
1
v(y)
dy
p
∫
|y|<|x|
[v(y)]
− βp
p−β dy
=
= exp
sp ln
1
|B(0, |x|)|
+
∫
|y|<|x|
ln
1
v(y)
dy
|B(0, |x|)|
=
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
ON A TWO-WEIGHT CRITERIA FOR MULTIDIMENSIONAL HARDY TYPE OPERATOR IN P -CONVEX . . . 323
=
1
|B(0, |x|)|s/p
exp
1
|B(0, |x|)|
∫
B(0,|x|)
ln
1
v(y)
dy
.
Therefore
lim
β→+0
B (s, β) = sup
t>0
|B(0, t)|
s−1
p
∥∥∥∥∥∥∥
χ{|z|>t}(·)
|B(0, | · |)|s/p
exp
1
|B(0, | · |)|
∫
B(0,|·|)
ln
1
v(y)
dy
∥∥∥∥∥∥∥
Lϕ,w
=
= D(s) <∞
and
sup
s>1
es/p(
es +
1
s− 1
)1/p
D(s) ≤ lim
β→+0
C
1/β
β ≤ 21/p inf
s>1
e(s−1)/pD(s). (8)
Further, we have
lim
β→+0
1
|B(0, |x|)|
∫
B(0, |x|)
fβ(y) dy
1/β
= exp
1
|B(0, |x|)|
∫
B(0, |x|)
ln f(y) dy
= Gf(x).
From (7) it follows that limβ→+0Cβ = 1, and according to (5) and (8) limβ→+0C
1/β
β = C <∞.
Therefore the inequality (8) is valid. Moreover, from (6) for β → +0 we obtain that
‖Gf‖Lq(·), w(Rn) ≤ C ‖f‖Lp, v(Rn)
and by (8)
sup
s>1
es/p(
es +
1
s− 1
)1/p
D(s) ≤ C ≤ 21/p inf
s>1
e(s−1)/pD(s).
Theorem 4 is proved.
Remark 4. Let ϕ(x, t) = tq and n = 1. Note that the simplest case of the condition ϕ(y, Ct) ≤
≤ Cq(y)ϕ(y, t) with v = w = 1 and p = q = 1 was considered in [20] and in [25]. Later this
inequality was generalized in various ways by many authors in [15, 22 – 24, 31, 38 – 40, 44] and etc.
Corollary 2. Let ϕ(x, t) = tq, 0 < p ≤ q <∞ and f be a positive function on Rn. Then ∫
Rn
[Gf(x)]q |x|δ q dx
1/q
≤ C
∫
Rn
fp(x) |x|µ p dx
1/p
(9)
holds with a finite constant C if and only if
δ +
n
q
=
µ
n
+
n
p
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
324 R. A. BANDALIYEV
and the best constant C has the following condition:
q
√
p
nq
eµ/n
2 |B(0, 1)|1/q−1/p sup
s>1
es/p (s− 1)1/p−1/q
[(s− 1)es + 1]1/p
≤ C ≤ |B(0, 1)|1/q−1/p eµ/n
2+1/q
q
√
n
.
Remark 5. Let ϕ(x, t) = tq and q = p. Then inequality (9) is sharp with the constant
C =
eµ/n
2+1/p
p
√
n
.
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Received 08.12.13,
after revision — 18.11.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
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