Two-weighted inequalities for Riesz potential in $p$-convex weighted modular Banach function spaces
The main goal of this paper is to prove a two-weight boundedness for Riesz potential from one weighted Banach function space to another weighted Banach function space. In particular, we obtain a two-weight boundedness for Riesz potential and find sufficient conditions on the weights for boundedness...
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| author | Bandaliyev, R. A. Guliyev, V. S. Hasanov, S. G. Бандалієв, Р. А. Гулієв, В. С. Хасанов, С. Г. |
| author_facet | Bandaliyev, R. A. Guliyev, V. S. Hasanov, S. G. Бандалієв, Р. А. Гулієв, В. С. Хасанов, С. Г. |
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| description | The main goal of this paper is to prove a two-weight boundedness for Riesz potential from one weighted Banach function
space to another weighted Banach function space. In particular, we obtain a two-weight boundedness for Riesz potential
and find sufficient conditions on the weights for boundedness of Riesz potential in weighted Musielak – Orlicz spaces. |
| first_indexed | 2026-03-24T02:12:48Z |
| format | Article |
| fulltext |
UDC 517.5
R. A. Bandaliyev (Inst. Math. and Mech. Nat. Acad. Sci. Azerbaijan, Baku),
V. S. Guliyev (Ahi Evran Univ., Kirsehir, Turkey),
S. G. Hasanov (Ganja State Univ., Ganja)
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL
IN \bfitp -CONVEX WEIGHTED MODULAR BANACH FUNCTION SPACES
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL
IN \bfitp -CONVEX WEIGHTED MODULAR BANACH FUNCTION SPACES
The main goal of this paper is to prove a two-weight boundedness for Riesz potential from one weighted Banach function
space to another weighted Banach function space. In particular, we obtain a two-weight boundedness for Riesz potential
and find sufficient conditions on the weights for boundedness of Riesz potential in weighted Musielak – Orlicz spaces.
Основна мета роботи — встановити двохвагову обмеженiсть потенцiалу Рiса з одного вагового Банахового простору
в iнший ваговий Банахiв простiр. Зокрема, встановлено двохвагову обмеженiсть потенцiалу Рiса та отримано
достатнi умови, що треба накласти на вагу з метою гарантувати обмеженiсть потенцiалу Рiса у вагових просторах
Мусiляка – Орлiча.
1. Introduction. The investigation of Riesz operator in weighted Banach function spaces (BFS)
have recent history. The goal of this investigations were closely connected with founding the criterion
on the geometry and on the weights of BFS for validity of boundedness of Riesz operator in BFS.
Characterization of the mapping properties such as boundedness and compactness was considered
in the papers [9, 10, 14, 33] and etc. More precisely, in [9, 10] were considered the boundedness
of certain integral operator in ideal Banach spaces. In [14] the boundedness of Hardy operator was
proved in Orlicz spaces. Also, in [33] the compactness and measure of noncompactness of Hardy
type operator in BFS was proved. But in this paper we used the boundedness of Hardy operator
in p-convex BFS. Note that the notion of BFS was introduced in [35]. In particular, the weighted
Lebesgue spaces, weighted Lorentz spaces, weighted variable Lebesgue spaces, variable Lebesgue
spaces with mixed norm, Musielak – Orlicz spaces, etc are BFS.
In this paper, we establish an integral-type sufficient condition on weights, which provides the
boundedness of the Riesz operator from one weighted BFS to another weighted BFS.
2. Preliminaries. Let (\Omega , \mu ) be a complete \sigma -finite measure space. By L0 = L0(\Omega , \mu ) we
denote the collection of all real-valued \mu -measurable functions on \Omega .
Definition 2.1 [20]. Let L be a real vector space. A function \rho : L \mapsto \rightarrow [0,\infty ] is called a
semimodular on L if the following properties hold:
(a) \rho (0) = 0.
(b) \rho (\lambda x) = \rho (x) for all x \in L and \lambda \in \BbbR with | \lambda | = 1.
(c) \rho is convex.
(d) \rho is left-continuous.
(e) \rho (\lambda x) = 0 for all \lambda > 0 implies x = 0.
A semimodular \rho is called modular if
(f) \rho (x) = 0 implies x = 0.
A semimodular \rho is called continuous if
(g) the mapping \lambda \mapsto \rightarrow \rho (\lambda x) is continuous on [0,\infty ) for every x \in L.
c\bigcirc R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11 1443
1444 R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV
If \rho is semimodular or modular on L, then
L\rho :=
\biggl\{
x \in L : \mathrm{l}\mathrm{i}\mathrm{m}
\lambda \rightarrow 0
\rho (\lambda x) = 0
\biggr\}
is called a semimodular space or modular space, respectively. The limit \lambda \rightarrow 0 takes place in \BbbR .
Theorem 2.1 [20]. Let \rho be semimodular on L. Then L\rho is a normed real vector space. The
norm called the Luxemburg norm, is defined by
\| x\| \rho := \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\lambda > 0 : \rho
\biggl(
1
\lambda
x
\biggr)
\leq 1
\biggr\}
.
Definition 2.2 [8, 32, 35]. We say that real normed space X is a Banach function space (BFS)
if :
(P1) the norm \| f\| X is defined for every \mu -measurable function f, and f \in X if and only if
\| f\| X <\infty ; \| f\| X = 0 if and only if f = 0 a.e.;
(P2) \| f\| X = \| | f | \| X for all f \in X;
(P3) if 0 \leq fn \uparrow f \leq g a.e., then \| fn\| X \uparrow \| f\| X (Fatou property);
(P4) if E is a measurable subset of \Omega such that \mu (E) <\infty , then \| \chi E\| X <\infty , where \chi E is the
characteristic function of the set E;
(P5) for every measurable set E \subset \Omega with \mu (E) < \infty , there is a constant CE > 0 such that\int
E
f(x) dx \leq CE \| f\| X .
Recall that condition (P3) immediately yields the following property:
if 0 \leq f \leq g, then \| f\| X \leq \| g\| X .
Given a BFS X we can always consider its associate space X \prime consisting of those g \in L0 that
f \cdot g \in L1 for every f \in X with usual order and the norm \| g\| X\prime = \mathrm{s}\mathrm{u}\mathrm{p} \{ \| f \cdot g\| L1 : \| g\| X\prime \leq 1\} .
Note that X \prime is a BFS in (\Omega , \mu ) and a closed norming subspace.
Let X be a BFS and \omega be a weight, that is, positive Lebesgue measurable and a.e. finite
function on \Omega . Let X\omega = \{ f \in L0 : f\omega \in X\} . This space is a weighted BFS equipped with the
norm \| f\| X\omega = \| f \omega \| X . (For more detail and proofs of results about BFS we refer the reader to
[8, 32].)
Let us recall the notion of p-convexity and p-concavity of BFS.
Definition 2.3 [42]. Let X be a BFS. Then X is called p-convex for 1 \leq p \leq \infty if there exists
a constant M > 0 such that for all f1, . . . , fn \in X\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl(
n\sum
k=1
| fk| p
\Biggr) 1/p
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
X
\leq M
\Biggl(
n\sum
k=1
\| fk\| pX
\Biggr) 1/p
if 1 \leq p <\infty ,
or
\bigm\| \bigm\| \mathrm{s}\mathrm{u}\mathrm{p}1\leq k\leq n | fk| \bigm\| \bigm\| X \leq M \mathrm{m}\mathrm{a}\mathrm{x}1\leq k\leq n \| fk\| X if p = \infty . Similarly, X is called p-concave for 1 \leq
\leq p \leq \infty if there exists a constant M > 0 such that for all f1, . . . , fn \in X\Biggl(
n\sum
k=1
\| fk\| pX
\Biggr) 1/p
\leq M
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\Biggl(
n\sum
k=1
| fk| p
\Biggr) 1/p
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
X
if 1 \leq p <\infty ,
or \mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq n
\| fk\| X \leq M
\bigm\| \bigm\| \mathrm{s}\mathrm{u}\mathrm{p}1\leq k\leq n | fk| \bigm\| \bigm\| X if p = \infty .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL IN p-CONVEX WEIGHTED MODULAR 1445
Remark 2.1. Note that the notion of p-convexity, respectively p-concavity are closely related
to the notion of upper p-estimate (strong \ell p-composition property), respectively lower p-estimate
(strong \ell p-decomposition property) as can be found in [32].
Now we show some examples of p-convex and respectively p-concave BFS.
Let \BbbR n be n-dimensional Euclidean space of the points x =
\bigl(
x1, . . . , xn
\bigr)
and let \Omega be a Lebesgue
measurable subset in \BbbR n and | x| =
\Bigl( \sum n
i=1
x2i
\Bigr) 1/2
. The Lebesgue measure of a set \Omega will be denoted
by | \Omega | . It is well known that | B(0, 1)| = \pi n/2
\Gamma
\Bigl( n
2
+ 1
\Bigr) , where B(0, 1) = \{ x : x \in \BbbR n | x| < 1\} . As-
sume that \delta : \Omega \rightarrow [1,\infty ). Throughout this paper, assume \delta = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}x\in \Omega \delta (x) and \delta = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in \Omega \delta (x)
and p\prime =
p
p - 1
be conjugate exponent of p > 1.
Example 2.1. Let 1 \leq q \leq \infty and X = Lq. Then the space Lq is p-convex (p-concave)
modular BFS if and only if 1 \leq p \leq q \leq \infty (1 \leq q \leq p \leq \infty .)
The proof implies from Minkowski inequality in Lebesgue spaces.
Example 2.2. The following lemma shows that the variable Lebesgue space Lq(\cdot )(\Omega ) is a p-
convex modular BFS.
Lemma 2.1 [1]. Let 1 \leq p \leq q(x) \leq q <\infty for all x \in \Omega 2 \subset \BbbR m. Then the inequality\bigm\| \bigm\| \| f\| Lp(\Omega 1)
\bigm\| \bigm\|
Lq(\cdot )(\Omega 2)
\leq C2/p
p,q
\bigm\| \bigm\| \bigm\| \| f\| Lq(\cdot )(\Omega 2)
\bigm\| \bigm\| \bigm\|
Lp(\Omega 1)
is valid, where
Cp,q =
\biggl(
\| \chi \Delta 1\| \infty + \| \chi \Delta 2\| \infty + p
\biggl(
1
q
- 1
q
\biggr) \biggr) \bigl(
\| \chi \Delta 1\| \infty + \| \chi \Delta 2\| \infty
\bigr)
,
\Delta 1 = \{ (x, y) \in \Omega 1 \times \Omega 2 : q(y) = p\} , \Delta 2 = \Omega 1\times \Omega 2 \setminus \Delta 1 and f : \Omega 1\times \Omega 2 \rightarrow \BbbR is any measurable
function such that
\bigm\| \bigm\| \| f\| Lp(\Omega 1)
\bigm\| \bigm\|
Lq(\cdot )(\Omega 2)
= \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \mu > 0 :
\int
\Omega 2
\biggl( \| f(\cdot , y)\| Lp(\Omega 1)
\mu
\biggr) q(y)
dy \leq 1
\right\} <\infty
and \| f(\cdot , y)\| Lp(\Omega 1) =
\biggl( \int
\Omega 1
| f(x, y)| p dx
\biggr) 1/p
.
Analogously, if 1 \leq q(x) \leq p <\infty , then Lq(x)(\Omega ) is a p-concave BFS.
Definition 2.4 [20, 40]. Let \Omega \subset \BbbR n be a Lebesgue measurable set. A real function \varphi : \Omega \times
\times [0,\infty ) \mapsto \rightarrow [0,\infty ) is called a generalized \varphi -function if it satisfies:
(a) \varphi (x, \cdot ) is a \varphi -function for all x \in \Omega , i.e., \varphi (x, \cdot ) : [0,\infty ) \mapsto \rightarrow [0,\infty ) is convex and satisfies
\varphi (x, 0) = 0, \mathrm{l}\mathrm{i}\mathrm{m}t\rightarrow +0 \varphi (x, t) = 0;
(b) \psi : x \mapsto \rightarrow \varphi (x, t) is measurable for all t \geq 0.
If \varphi is a generalized \varphi -function on \Omega , we briefly write \varphi \in \Phi .
Definition 2.5 [20, 40]. Let \varphi \in \Phi and be \rho \varphi defined by the expression
\rho \varphi (f) :=
\int
\Omega
\varphi (x, | f(x)| ) dx for all f \in L0(\Omega ).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1446 R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV
We put L\varphi = \{ f \in L0(\Omega ) : \rho \varphi (\lambda 0f) <\infty for some \lambda 0 > 0\} and
\| f\| L\varphi = \mathrm{i}\mathrm{n}\mathrm{f}
\biggl\{
\lambda > 0 : \rho \varphi
\biggl(
f
\lambda
\biggr)
\leq 1
\biggr\}
.
The space L\varphi is called Musielak – Orlicz space.
Let \omega be a weight function on \Omega , i.e., \omega be a nonnegative, almost everywhere positive function
on \Omega . We denote
L\varphi , \omega = \{ f \in L0(\Omega ) : f\omega \in L\varphi \} .
It is obvious that the norm in this space is given by
\| f\| L\varphi , \omega = \| f\omega \| L\varphi .
Remark 2.2. Let \varphi (x, t) = tq(x) in Definition 2.4, where 1 \leq q(x) < \infty and x \in \Omega . Then we
have the definition of variable exponent weighted Lebesgue spaces Lq(x)(\Omega ). About detail information
on variable exponent Lebesgue spaces we refer to [18].
Example 2.3. The following lemma shows that the Musielak – Orlicz space L\varphi is a p-convex
modular BFS.
Lemma 2.2 [6]. Let \Omega 1 \subset \BbbR n and \Omega 2 \subset Rm. Let (x, t) \in \Omega 1 \times [0,\infty ) and \varphi
\bigl(
x, t1/p
\bigr)
\in \Phi for
some 1 \leq p <\infty . Suppose f : \Omega 1 \times \Omega 2 \mapsto \rightarrow R. Then the inequality\bigm\| \bigm\| \| f(x, \cdot )\| Lp(\Omega 2)
\bigm\| \bigm\|
L\varphi
\leq 21/p
\bigm\| \bigm\| \| f(\cdot , y)\| L\varphi \bigm\| \bigm\| Lp(\Omega 2)
is valid.
We note that the Lebesgue spaces with mixed norm, weighted Lorentz spaces, etc are p-convex
(p-concave) modular BFS. Now we reduce a more general result connected with Minkowski’s integral
inequality.
Let X and Y be BFS on (\Omega 1, \mu ) and (\Omega 2, \nu ), respectively. By X[Y ] and Y [X] we denote
the spaces with mixed norm and consisting of all functions g \in L0 (\Omega 1 \times \Omega 2, \mu \times \nu ) such that\bigm\| \bigm\| g(x, \cdot )\bigm\| \bigm\|
Y
\in X and
\bigm\| \bigm\| g(\cdot , y)\bigm\| \bigm\|
X
\in Y. The norms in this spaces define are as follows:
\| g\| X[Y ] = \| \| g(x, \cdot )\| Y \| X , \| g\| Y [X] = \| \| g(\cdot , y)\| X\| Y .
It is known that X[Y ] and Y [X] are BFS on \Omega 1 \times \Omega 2 (see [32].)
Definition 2.6 [40]. We say that modular BFS X satisfies the \Delta 2-condition if there exists K \geq 2
such that
\rho (2f) \leq K \rho (f)
for all f \in X and all t > 0. The smallest such K is called the \Delta 2-constant of \rho .
Lemma 2.3. Let X modular BFS, \gamma \geq 1 and 1 \leq q(x) \leq q <\infty . Further, let
\mathrm{m}\mathrm{i}\mathrm{n}
s>0
\{ s, s\gamma \} \rho (f) \leq \rho (sf) \leq \mathrm{m}\mathrm{a}\mathrm{x}
s>0
\bigl\{
s, sq(x)
\bigr\}
\rho (f) (2.1)
for almost all x \in \Omega and all f \in X\rho . Then \rho
\biggl(
f
\| f\| \rho
\biggr)
= 1 and
\mathrm{m}\mathrm{i}\mathrm{n}
\| f\| \rho
\bigl\{
\| f\| \rho , \| f\| \gamma \rho
\bigr\}
\leq \rho (f) \leq \mathrm{m}\mathrm{a}\mathrm{x}
\| f\| \rho
\Bigl\{
\| f\| \rho , \| f\| q(x)\rho
\Bigr\}
for any x \in \Omega .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL IN p-CONVEX WEIGHTED MODULAR 1447
Proof. Let 0 < \| f\| \rho < \infty and \rho
\biggl(
f
\| f\| \rho
\biggr)
< 1. We choose a positive number \lambda \leq \| f\| \rho
such that \rho
\biggl(
f
\lambda
\biggr)
< 1. Indeed, we put \lambda = \| f\| \rho \rho 1/q
\biggl(
f
\| f\| \rho
\biggr)
. Then \lambda < \| f\| \rho and by virtue of
condition (2.1) for s > 1 we have
\rho
\biggl(
f
\lambda
\biggr)
= \rho
\left( f
\| f\| \rho \rho 1/q
\Bigl(
f
\| f\| \rho
\Bigr)
\right) \leq \rho - q(x)/q
\biggl(
f
\| f\| \rho
\biggr)
\rho
\biggl(
f
\| f\| \rho
\biggr)
\leq
\leq \rho - 1
\biggl(
f
\| f\| \rho
\biggr)
\rho
\biggl(
f
\| f\| \rho
\biggr)
= 1.
Lemma 2.3 is proved.
We consider the multidimensional Hardy type operator and its dual operator
Hf(x) =
\int
| y| <| x|
f(y) dy and H\ast f(x) =
\int
| y| >| x|
f(y) dy,
where f \geq 0 and x \in \BbbR n.
Now we reduce a two-weight criterion for multidimensional Hardy type operator acting from
the p-concave weighted BFS to weighted Lebesgue spaces. Suppose that M > 0 the constant in
Definition 2.3.
Theorem 2.2 [7]. Let v(x) and w(x) be weights on \BbbR n. Suppose that Xw is a p-convex weighted
BFSs for 1 \leq p <\infty on \BbbR n. Then the inequality
\| Hf\| Xw \leq C\| f\| Lp, v (2.2)
holds for every f \geq 0 if and only if there is a \alpha \in (0, 1) such that
A(\alpha ) = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\left( \int
| y| <t
v(y) - p
\prime
dy
\right)
\alpha /p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \chi \{ | z| >t\} (\cdot )
\left( \int
| y| <| \cdot |
v(y) - p
\prime
dy
\right)
(1 - \alpha )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Xw
<\infty .
Moreover, if C > 0 is the best possible constant in (2.2), then
\mathrm{s}\mathrm{u}\mathrm{p}
0<\alpha <1
p\prime A(\alpha )
(1 - \alpha )
\biggl( \biggl(
p\prime
1 - \alpha
\biggr) p
+
1
\alpha (p - 1)
\biggr) 1/p
\leq C \leq M \mathrm{i}\mathrm{n}\mathrm{f}
0<\alpha <1
A(\alpha )
(1 - \alpha )1/p\prime
.
For the dual operator, the below stated theorem is proved analogously.
Theorem 2.3 [7]. Let v(x) and w(x) be weights on \BbbR n. Suppose that Xw is a p-convex weighted
BFS for 1 \leq p <\infty on \BbbR n. Then the inequality
\| H\ast f\| Xw \leq C \| f\| Lp, v (2.3)
holds for every f \geq 0 if and only if there is a \gamma \in (0, 1)
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1448 R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV
B(\gamma ) = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\left( \int
| y| >t
v(y) - p
\prime
dy
\right)
\gamma /p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \chi \{ | z| <t\} (\cdot )
\left( \int
| y| >| \cdot |
v(y) - p
\prime
dy
\right)
(1 - \gamma )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Xw
<\infty .
Moreover, if C > 0 is the best possible constant in (2.3), then
\mathrm{s}\mathrm{u}\mathrm{p}
0<\gamma <1
p\prime B(\gamma )
(1 - \gamma )
\biggl( \biggl(
p\prime
1 - \gamma
\biggr) p
+
1
\gamma (p - 1)
\biggr) 1/p
\leq C \leq M \mathrm{i}\mathrm{n}\mathrm{f}
0<\gamma <1
B(\gamma )
(1 - \gamma )1/p\prime
.
Corollary 2.1. Note that Theorems 2.2 and 2.3 in the case Xw = L\varphi ,w, \varphi
\bigl(
x, t1/p
\bigr)
\in \Phi for
some 1 \leq p < \infty , x \in \BbbR n were proved in [6]. In the case Xw = Lq, w, 1 < p \leq q < \infty , for
x \in (0,\infty ), \alpha =
s - 1
p - 1
and s \in (1, p) Theorems 2.2 and 2.3 were proved in [44]. For x \in \BbbR n in
the case Xw = Lq(x), w and 1 < p \leq q(x) \leq \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbR n q(x) < \infty Theorems 2.2 and 2.3 were
proved in [3] (see also [2]).
Remark 2.3. In the case n = 1, Xw = Lq, w, 1 < p \leq q \leq \infty , at x \in (0,\infty ), for classical
Lebesgue spaces the various variants of Theorems 2.2 and 2.3 were proved in [12, 23, 25 – 28, 30,
31, 34, 38, 39, 43] etc. In particular, in the Lebesgue spaces with variable exponent the boundedness
of Hardy type operator was proved in [15 – 17, 19, 21, 24, 29, 36, 37] etc. For Xw = Lq(x), w, 1 <
< p \leq q(x) \leq \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in [0,1] q(x) <\infty and x \in [0, 1] the two-weighted criterion for one-dimensional
Hardy operator was proved in [29]. Also, other type two-weighted criterion for multidimensional
Hardy type operator in the case Xw = Lq(x), w, 1 < p \leq q(x) \leq \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}x\in \BbbR n q(x) <\infty and x \in \BbbR n
was proved in [36] (see also [37] and [17]). In the case Lq(x), w for 0 < q \leq q < 1 the boundedness
of classical Hardy operator was proved in [5]. In the papers [11] and [41] the inequalities of modular
type for more general operators were proved. Also, in [13] the Hardy type inequalities with special
power-type weights in Orlicz spaces were proved.
3. Main result. Now we consider the Riesz potential \scrR sf(x) =
\int
\BbbR n
f(y)
| x - y| n - s
dy, where
0 < s < n.
The sufficient conditions for general weights ensuring the validity of the two-weight strong type
inequalities for the Riesz potential in BFS are given in the following theorem.
Theorem 3.1. Suppose that v(x) and w(x) be weight functions on \BbbR n. Let Yw be a modular
p-convex weighted BFS for 1 \leq p < \infty and x \in \BbbR n. Let 0 < s < n, \scrR s is bounded from X into
Y and let Lp,v(\BbbR n) \lhook \rightarrow Xv. Let there exists r(x) : 1 < p \leq r(x) \leq r < \infty such that, for all C > 0
\rho (Cf) \leq C1(r) \rho (f), where C1(r) = \mathrm{m}\mathrm{a}\mathrm{x}
\bigl\{
Cr, Cr
\bigr\}
.
Moreover, let v(x) and w(x) satisfy the following three conditions:
1) A = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| <t
v(y) - p
\prime
dy
\Biggr) \alpha /p\prime \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \chi \{ | x| >t\}
| x| n - s
\Biggl( \int
| y| <| x|
v(y) - p
\prime
dy
\Biggr) (1 - \alpha )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
<\infty ; (3.1)
2) B = \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| >t
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) \beta /p\prime
\times
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TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL IN p-CONVEX WEIGHTED MODULAR 1449
\times
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \chi \{ | x| <t\}
\Biggl( \int
| y| >| x|
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) (1 - \beta )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
<\infty , (3.2)
where 0 < \alpha , \beta < 1;
3) there exists M > 0 such that
\mathrm{s}\mathrm{u}\mathrm{p}
| x| /2<| y| \leq 4 | x|
w(y) \leq M \mathrm{i}\mathrm{n}\mathrm{f}
| x| /2<| y| \leq 4 | x|
v(y). (3.3)
Then there exists a positive constant C, independent of f, such that for all f \in Xv
\| \scrR sf\| Yw \leq C\| f\| Xv .
Proof. Let Z = \{ 0,\pm 1,\pm 2, . . .\} . For k \in Z we define Ek =
\bigl\{
x \in \BbbR n : 2k < | x| \leq 2k+1
\bigr\}
,
Ek,1 =
\bigl\{
x \in \BbbR n : | x| \leq 2k - 1
\bigr\}
, Ek,2 =
\bigl\{
x \in \BbbR n : 2k - 1 < | x| \leq 2k+2
\bigr\}
, Ek,3 =
\bigl\{
x \in \BbbR n :
| x| > 2k - 1
\bigr\}
. Then Ek,2 = Ek - 1 \cup Ek \cup Ek+1 and the multiplicity of the covering \{ Ek,2\} k\in Z is
equal to 3.
Given f \in Lp,v(\BbbR n), we write\bigm| \bigm| \scrR sf(x)
\bigm| \bigm| =\sum
k\in Z
| \scrR sf(x)| \chi Ek(x) \leq
\leq
\sum
k\in Z
| \scrR sfk,1(x)| \chi Ek(x) +
\sum
k\in Z
| \scrR sfk,2(x)| \chi Ek(x) +
\sum
k\in Z
| \scrR sfk,3(x)| \chi Ek(x) =
= \scrR s
1f(x) +\scrR s
2f(x) +\scrR s
3f(x),
where \chi Ek is the characteristic function of the set Ek, fk,i = f\chi Ek,i , i = 1, 2, 3.
First we shall estimate \| \scrR s
1f\| Yw . Note that for x \in Ek, y \in Ek,1 we have | y| < 2k - 1 \leq | x| /2.
Moreover, Ek \cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} fk,1 = \varnothing and | x - y| \geq | x| - | y| \geq | x| - | x| /2 = | x| /2. Hence we have
\bigm| \bigm| \scrR s
1f(x)
\bigm| \bigm| \leq C
\sum
k\in Z
\left( \int
\BbbR n
| fk,1(y)|
| x - y| n - s
dy
\right) \chi Ek \leq C
\int
| y| <| x| /2
| f(y)|
| x - y| n - s
dy \leq
\leq C
\int
| y| <| x|
| f(y)|
| x - y| n - s
dy \leq 2nC
1
| x| n - s
\int
| y| <| x|
| f(y)| dy
for any x \in Ek. Hence we get
\| \scrR s
1f\| Yw \leq 2nC
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
1
| x| n - s
\int
| y| <| x|
| f(y)| dy
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
=
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
| y| <| x|
| f(y)| dy
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw/| x| n - s
.
By condition (3.1) and Theorem 2.2, we obtain
\| \scrR s
1f\| Yw \leq C1\| f\| Lp,v(\BbbR n) \leq C2\| f\| Xv , (3.4)
where C1 > 0 is independent of f and x \in \BbbR n.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1450 R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV
Next we estimate \| \scrR s
3f\| Yw . It is obvious that, for x \in Ek, y \in Ek,3 we have | y| > 2 | x| and
| x - y| \geq | y| - | x| \geq | y| - | y| /2 = | y| /2. Since Ek \cap \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} fk,3 = \varnothing for x \in Ek we have
| \scrR s
3f(x)| \leq C
\int
| y| >2| x|
| f(y)|
| x - y| n - s
dy \leq 2nC
\int
| y| >2| x|
| f(y)|
| y| n - s
dy.
Hence we get
\| \scrR s
3f\| Yw \leq 2nC
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
| y| >2| x|
| f(y)|
| y| n - s
dy
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
\leq
\leq 2nC
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\int
| y| >| x|
| f(y)|
| y| n - s
dy
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
.
By condition (3.2) and Theorem 2.3, we obtain\bigm\| \bigm\| \scrR s
3f
\bigm\| \bigm\|
Yw
\leq C2 \| f\| Lp,v(\BbbR n) \leq C3\| f\| Xv , (3.5)
where C2 > 0 is independent of f and x \in \BbbR n.
Finally we estimate \| \scrR sfk,2\| Yw , where
\| \scrR sfk,2\| Yw =
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \sum
k\in Z
| \scrR sfk,2| \chi Ek
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Yw
.
By virtue of Lemma 2.3 it suffices to prove that from \| f\| Xv \leq 1 implies
\rho
\Biggl(
w
\sum
k\in Z
| \scrR sfk,2| \chi Ek
\Biggr)
\leq C,
where C > 0 is independent of k \in Z.
By the boundedness of \scrR s from X to Y and condition (3.3), we have
\rho
\Biggl(
w(y)
\sum
k\in Z
| \scrR sfk,2(y)| \chi Ek(y)
\Biggr)
=
\sum
m\in Z
\rho
\Biggl(
w(y)
\sum
k\in Z
| \scrR sfk,2(y)| \chi Ek(y)
\Biggr)
=
\sum
k\in Z
\rho (w(y) | \scrR sfk,2(y)| ) =
\sum
k\in Z
\rho
\biggl(
C w(y) \| fk,2\| X
| \scrR sfk,2|
C \| fk,2\| X
\biggr)
\leq
\leq
\sum
k\in Z
\bigl(
C w(y) \| fk,2\| X
\bigr) r(y)
\rho
\biggl(
| \scrR sfk,2|
C \| fk,2\| X
\biggr)
\leq
\leq C2
\sum
k\in Z
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Ek
\Bigl(
w(y)\| f\| X(Ek,2)
\Bigr) r(y)
\rho
\biggl(
| \scrR sfk,2|
C \| fk,2\| X
\biggr)
\leq
\leq C2
\sum
k\in Z
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Ek
(w(y)\| f\| X)r(y) = C2
\sum
k\in Z
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Ek
\Bigl(
\| f w\| X(Ek,2)
\Bigr) r(y)
\leq
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL IN p-CONVEX WEIGHTED MODULAR 1451
\leq C3
\sum
k\in Z
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Ek
\biggl(
\| f \mathrm{i}\mathrm{n}\mathrm{f}
y\in Ek,2
v(y)\| X(Ek,2)
\biggr) r(y)
\leq C3
\sum
k\in Z
\mathrm{s}\mathrm{u}\mathrm{p}
y\in Ek
\Bigl(
\| f v\| X(Ek,2)
\Bigr) r(y)
=
= C3
\sum
k\in Z
\Bigl(
\| f\| Xv(Ek,2)
\Bigr) inf
y\in Ek
r(y)
\leq C3
\sum
k\in Z
\Bigl(
\| f\| Xv(Ek,2)
\Bigr) r
\leq
\leq C3
\sum
k\in Z
\rho
\bigl(
| f(y)| v(y)\chi Ek,2
\bigr) r/\gamma
=
= C3
\sum
k\in Z
\bigl[
\rho
\bigl(
| f(y)| v(y)
\bigl(
\chi Ek - 1
+ \chi Ek + \chi Ek+1
\bigr) \bigr) \bigr] r/\gamma \leq
\leq C3 [\rho (| f(y)| v(y))]r/\gamma
\Biggl( \sum
k\in Z
\chi Ek - 1
+
\sum
k\in Z
\chi Ek +
\sum
k\in Z
\chi Ek+1
\Biggr) r/\gamma
=
= C3 (3 \rho (| f(y)| v(y)))r/\gamma \leq 3r/\gamma C3 \leq C4.
Thus
\| \scrR s
2f\| Yw \leq C5, (3.6)
where C > 0 is independent of f and x \in \BbbR n.
Combining the inequalities (3.4), (3.5) and (3.6), we obtain the proof of Theorem 3.4.
Theorem 3.2 [40]. Let \psi \in \Phi and \delta \geq 1. Then L\psi (\BbbR n) \lhook \rightarrow L\delta (\BbbR n) if and only if there exists
C > 0 and h \in L1(\BbbR n) with \| h\| L1(\BbbR n) \leq 1 such that\biggl(
t
C
\biggr) \delta
\leq \psi (x, t) + h(x) (3.7)
for almost all x \in \BbbR n and all t \geq 0.
Lemma 3.1. Let \psi \in \Phi , \gamma \geq 1 and 1 \leq q(x) \leq q <\infty . Further, let
\mathrm{m}\mathrm{i}\mathrm{n}
s>0
\{ s, s\gamma \} \psi (x, t) \leq \psi (x, st) \leq \mathrm{m}\mathrm{a}\mathrm{x}
s>0
\bigl\{
s, sq(x)
\bigr\}
\psi (x, t) (3.8)
for almost all x \in \Omega and all t \geq 0. Then \rho \psi
\biggl(
f
\| f\| L\psi
\biggr)
= 1 and
\mathrm{m}\mathrm{i}\mathrm{n}
\| f\| L\psi
\Bigl\{
\| f\| L\psi , \| f\|
\gamma
L\psi
\Bigr\}
\leq \rho \psi (f) \leq \mathrm{m}\mathrm{a}\mathrm{x}
\| f\| L\psi
\Bigl\{
\| f\| L\psi , \| f\|
q(x)
L\psi
\Bigr\}
.
From Theorem 3.1 we have the following corollary.
Corollary 3.1. Let for some 1 < p <\infty , \varphi (x, t1/p) \in \Phi and a function \psi \in \Phi satisfy conditions
(3.7) and (3.8), where x \in \BbbR n. Suppose that v(x) and w(x) be weight functions on \BbbR n. Let \scrR s is
bounded from L\psi (\BbbR n) to L\varphi (\BbbR n). Let there exists r(x) : 1 < \theta \leq r(x) \leq r < \infty such that, for all
C > 0 \varphi (x,Ct) \leq Cr(x) \varphi (x, t).
Moreover, let v(x) and w(x) satisfy the following three conditions:
1) \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| <t
v(y) - p
\prime
dy
\Biggr) \alpha /p\prime \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| w(\cdot )
| \cdot | n - s
\Biggl( \int
| y| <| \cdot |
v(y) - p
\prime
dy
\Biggr) (1 - \alpha )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L\varphi (| \cdot | >t)
<\infty ;
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
1452 R. A. BANDALIYEV, V. S. GULIYEV, S. G. HASANOV
2) \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| >t
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) \beta /p\prime \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| w(\cdot )
\Biggl( \int
| y| >| \cdot |
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) (1 - \beta )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
L\varphi (| \cdot | <t)
<\infty ,
where 0 < \alpha , \beta < 1.
3) there exists M > 0 such that
\mathrm{s}\mathrm{u}\mathrm{p}
| x| /2<| y| \leq 4 | x|
w(y) \leq M \mathrm{i}\mathrm{n}\mathrm{f}
| x| /2<| y| \leq 4 | x|
v(y).
Then there exists a positive constant C, independent of f, such that for all f \in L\psi ,v(\BbbR n)
\| \scrR sf\| L\varphi ,w(\BbbR n) \leq C\| f\| L\psi ,v(\BbbR n).
Further, we assume that the exponent p(x) satisfies the standard conditions\bigm| \bigm| p(x) - p(y)
\bigm| \bigm| \leq M1
- \mathrm{l}\mathrm{n} | x - y|
, 0 < | x - y| \leq 1
2
, x, y \in \BbbR n, (3.9)
together with the following conditions at infinity:
| p(x) - p(y)| \leq M2
\mathrm{l}\mathrm{n}(e+ | x| )
, | x| \geq | y| , x, y \in \BbbR n, (3.10)
where the positive constants M1 and M2 are independent of x and y. Note that, from condition (3.10)
implies that there is some number p\infty such that p(x) \rightarrow p\infty as | x| \rightarrow \infty , and this limit holds
uniformly in all directions. It is known that if p(x) satisfies (3.10), p\infty = p and
1
r(x)
=
1
p
- 1
p(x)
,
then
1
r(x)
satisfies (3.10), \mathrm{l}\mathrm{i}\mathrm{m}| x| \rightarrow \infty r(x) = \infty and Lp(x)(\BbbR n) \lhook \rightarrow Lp(\BbbR n). In particular, for Xv =
= Lp(x),v(\BbbR n) and Yw = Lq(x),w(\BbbR n) from Theorem 3.1 we have the following corollary.
Corollary 3.2. Let
1
p(x)
- 1
q(x)
=
s
n
, p > 1, p < n/s, q \geq p and p(x) satisfy conditions (3.9)
and (3.10) with p\infty = p. Moreover, let v(x) and w(x) be weight functions on \BbbR n and satisfy the
following three conditions:
1) \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| <t
v(y) - p
\prime
dy
\Biggr) \alpha /p\prime \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| w(\cdot )
| \cdot | n - s
\Biggl( \int
| y| <| \cdot |
v(y) - p
\prime
dy
\Biggr) (1 - \alpha )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lq(\cdot )(| \cdot | >t)
<\infty ,
2) \mathrm{s}\mathrm{u}\mathrm{p}
t>0
\Biggl( \int
| y| >t
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) \beta /p\prime \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| w(\cdot )
\Biggl( \int
| y| >| \cdot |
\bigl(
v(y)| y| n - s
\bigr) - p\prime
dy
\Biggr) (1 - \beta )/p\prime
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lq(\cdot )(| \cdot | <t)
<
<\infty , where 0 < \alpha , \beta < 1;
3) there exists a constant M > 0 such that
\mathrm{s}\mathrm{u}\mathrm{p}
| x| /4<| y| \leq 4 | x|
w(y) \leq M \mathrm{i}\mathrm{n}\mathrm{f}
| x| /4<| y| \leq 4 | x|
v(y) for a.e. x \in \BbbR n.
Then there exists a positive constant C independent of f such that for all f \in Lp(x),v(\BbbR n)\bigm\| \bigm\| \scrR sf
\bigm\| \bigm\|
Lq(\cdot ),w(\BbbR n)
\leq C\| f\| Lp(\cdot ),v(\BbbR n).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
TWO-WEIGHTED INEQUALITIES FOR RIESZ POTENTIAL IN p-CONVEX WEIGHTED MODULAR 1453
Remark 3.1. In the case Xv = Lp, v, Yw = Lq, w, 1 < p \leq q \leq \infty for classical Lebesgue spaces
various variants of Theorem 3.1 were proved in [4, 22, 45] etc.
The research of R. A. Bandaliyev and V. S. Guliyev was partially supported by the grant of
Science Development Foundation under the President of the Republic of Azerbaijan, Grant EIF-
2013-9(15)-46/10/1 and by the grant of the Presidium of Azerbaijan National Academy of Science
2015. The authors would like to express their gratitude to the referees for his (her) very valuable
comments and suggestions.
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Received 28.02.16,
after revision — 13.06.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 11
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| id | umjimathkievua-article-1794 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:48Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/20/736cbcbd9e52629a18d69d89b82f1320.pdf |
| spelling | umjimathkievua-article-17942019-12-05T09:27:02Z Two-weighted inequalities for Riesz potential in $p$-convex weighted modular Banach function spaces Two-weighted inequalities for Riesz potential in $p$-convex weighted modular Banach function spaces Bandaliyev, R. A. Guliyev, V. S. Hasanov, S. G. Бандалієв, Р. А. Гулієв, В. С. Хасанов, С. Г. The main goal of this paper is to prove a two-weight boundedness for Riesz potential from one weighted Banach function space to another weighted Banach function space. In particular, we obtain a two-weight boundedness for Riesz potential and find sufficient conditions on the weights for boundedness of Riesz potential in weighted Musielak – Orlicz spaces. Основна мета роботи — встановити двохвагову обмеженiсть потенцiалу Рiса з одного вагового Банахового простору в iнший ваговий Банахiв простiр. Зокрема, встановлено двохвагову обмеженiсть потенцiалу Рiса та отримано достатнi умови, що треба накласти на вагу з метою гарантувати обмеженiсть потенцiалу Рiса у вагових просторах Мусiляка – Орлiча. Institute of Mathematics, NAS of Ukraine 2017-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1794 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 11 (2017); 1443-1454 Український математичний журнал; Том 69 № 11 (2017); 1443-1454 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1794/776 Copyright (c) 2017 Bandaliyev R. A.; Guliyev V. S.; Hasanov S. G. |
| spellingShingle | Bandaliyev, R. A. Guliyev, V. S. Hasanov, S. G. Бандалієв, Р. А. Гулієв, В. С. Хасанов, С. Г. Two-weighted inequalities for Riesz potential in $p$-convex weighted modular Banach function spaces |
| title | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_alt | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_full | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_fullStr | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_full_unstemmed | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_short | Two-weighted inequalities for Riesz potential
in $p$-convex weighted modular Banach function spaces |
| title_sort | two-weighted inequalities for riesz potential
in $p$-convex weighted modular banach function spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1794 |
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