Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces
We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathb...
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| author | Wu, Jianglong Ву, Янглонг |
| author_facet | Wu, Jianglong Ву, Янглонг |
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| description | We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz –
Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where
$\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ with some $\gamma (x) > 0$, and $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$
when $q_1 (x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1 (x)$ satisfies the logarithmic continuity
condition both locally and at infinity and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$. |
| first_indexed | 2026-03-24T02:12:19Z |
| format | Article |
| fulltext |
UDC 517.9
J.-L. Wu (Macau Univ. Sci. and Technology and Mudanjiang Normal Univ., China)
BOUNDEDNESS FOR RIESZ-TYPE POTENTIAL OPERATORS
ON VARIABLE EXPONENT HERZ – MORREY SPACES*
ОБМЕЖЕНIСТЬ ПОТЕНЦIАЛЬНИХ ОПЕРАТОРIВ ТИПУ РIСА
НА ПРОСТОРАХ ХЕРЦА – МОРРЕЯ ЗI ЗМIННИМ ПОКАЗНИКОМ
We show the boundedness of the Riesz-type potential operator of variable order \beta (x) from the variable exponent Herz –
Morrey spaces M \.K
\alpha (\cdot ),\lambda
p
1
,q
1
(\cdot )(\BbbR
n) into the weighted space M \.K
\alpha (\cdot ),\lambda
p
2
,q
2
(\cdot )(\BbbR
n, \omega ), where \alpha (x) \in L\infty (\BbbR n) is log-Hölder
continuous both at the origin and at infinity, \omega = (1+ | x| ) - \gamma (x) with some \gamma (x) > 0, and 1/q1(x) - 1/q2(x) = \beta (x)/n
when q1(x) is not necessarily constant at infinity. It is assumed that the exponent q1(x) satisfies the logarithmic continuity
condition both locally and at infinity and 1 < (q1)\infty \leq q1(x) \leq (q1)+ < \infty , x \in \BbbR n.
Встановлено обмеженiсть потенцiального оператора типу Рiса змiнного порядку \beta (x), що дiє з просторiв Херца –
Моррея зi змiнним показником M \.K
\alpha (\cdot ),\lambda
p
1
,q
1
(\cdot )(\BbbR
n) у зважений простiр M \.K
\alpha (\cdot ),\lambda
p
2
,q
2
(\cdot )(\BbbR
n, \omega ), де \alpha (x) \in L\infty (\BbbR n) є
log-Гельдер неперервним як на початку координат, так i на нескiнченностi, \omega = (1+ | x| ) - \gamma (x) з деяким \gamma (x) > 0 та
1/q1(x) - 1/q2(x) = \beta (x)/n, коли q1(x) не обов’язково є сталою на нескiнченностi. Вважаємо, що показник q1(x)
задовольняє умову логарифмiчної неперервностi як локально, так i на нескiнченностi та 1 < (q1)\infty \leq q1(x) \leq
\leq (q1)+ < \infty , x \in \BbbR n.
1. Introduction. Last decade, there is an evident increase of investigations related to both the theory
of the variable exponent function spaces and the operator theory in these spaces. This is caused with
keen interest not in real analysis but also in partial differential equations and in applied mathematics,
because they are applicable to the modeling for electrorheological fluids, mechanics of the continuum
medium and image restoration (see, for example, [1 – 7] and references therein) etc.
The theory of function spaces with variable exponent has rapidly made progress in the past twenty
years since some elementary properties were established by Kováčik and Rákosnı́k [8]. One of the
main problems on the theory is the boundedness of the Hardy – Littlewood maximal operator on
variable Lebesgue spaces.
In 2012, Almeida and Drihem [9] discuss the boundedness of a wide class of sublinear operators
on Herz spaces K
\alpha (\cdot ),p
q(\cdot ) (\BbbR n) and \.K
\alpha (\cdot ),p
q(\cdot ) (\BbbR n) with variable exponent \alpha (\cdot ) and q(\cdot ). Meanwhile,
they also established Hardy – Littlewood – Sobolev theorems for fractional integrals on variable Herz
spaces. In 2013, Samko [10, 11] introduced a new Herz type function spaces with variable exponent,
where all the three parameters are variable, and proved the boundedness of some sublinear operators.
And in 2015, Rafeiro and Samko [12] considered the validity of Sobolev type theorem for the
Riesz potential operator in continual variable exponents Herz spaces. In recently, Wu [13, 14] also
considers the boundedness for fractional Hardy-type operator and Riesz-type potential operator on
Herz – Morrey spaces M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) with variable exponent q(\cdot ) but fixed \alpha \in \BbbR and p \in (0,\infty ).
Motivated by the above results, and based on some facts in [9, 15], the author will investigate
mapping properties of the operator I\beta (\cdot ) within the framework of the variable exponent Herz – Morrey
spaces M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n), where the Riesz-type potential operator of variable order
* Supported partially by NNSF-China (Grant No. 11571160).
c\bigcirc J.-L. WU, 2017
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9 1187
1188 J.-L. WU
I\beta (\cdot )(f)(x) =
\int
\BbbR n
f(y)
| x - y| n - \beta (x)
dy, 0 < \beta (x) < n.
2. Preliminaries. In this section, we define some function spaces with variable exponent, and
give basic properties and useful lemmas. Throughout this paper we will use the following notation:
denote by | S| the Lebesgue measure and by \chi
S
the characteristic function for a measurable set
S \subset \BbbR n;
fS denotes the mean value of f on measurable set S, namely fS :=
1
| S|
\int
S
f(x)dx;
B(x, r) is the ball cenetered at x and of radius r; B0 = B(0, 1);
C denotes a constant that is independent of the main parameters involved but whose value may
differ from line to line;
for any exponent 1 < q(x) < \infty , we denote by q\prime (x) its conjugate exponent, namely, 1/q(x) +
+ 1/q\prime (x) = 1.
2.1. Function spaces with variable exponent. Let \Omega be a measurable set in \BbbR n with | \Omega | > 0.
We first define Lebesgue spaces with variable exponent.
Definition 2.1. Let q(\cdot ) : \Omega \rightarrow (1,\infty ) be a measurable function.
(i) The variable Lebesgue spaces Lq(\cdot )(\Omega ) is defined by
Lq(\cdot )(\Omega ) =
\bigl\{
f is measurable function: Fq(f/\eta ) < \infty for some constant \eta > 0
\bigr\}
,
where Fq(f) :=
\int
\Omega
| f(x)| q(x)dx. The Lebesgue space Lq(\cdot )(\Omega ) is a Banach space when equipped
with the norm
\| f\| Lq(\cdot )(\Omega ) = \mathrm{i}\mathrm{n}\mathrm{f}
\left\{ \eta > 0 : Fq(f/\eta ) =
\int
\Omega
\Bigl( | f(x)|
\eta
\Bigr) q(x)
dx \leq 1
\right\} .
(ii) The space L
q(\cdot )
loc (\Omega ) is defined by
L
q(\cdot )
loc (\Omega ) =
\bigl\{
f is measurable function: f \in Lq(\cdot )(\Omega 0) for all compact subsets \Omega 0 \subset \Omega \} .
(iii) The weighted Lebesgue space L
q(\cdot )
\omega (\Omega ) is defined by as the set of all measurable functions
for which
\| f\|
L
q(\cdot )
\omega (\Omega )
= \| \omega 1/q(\cdot )f\| Lq(\cdot )(\Omega ) < \infty .
Next we define some classes of variable exponent functions. Given a function f \in L1
loc(\BbbR n), the
Hardy – Littlewood maximal operator M is defined by
Mf(x) = \mathrm{s}\mathrm{u}\mathrm{p}
r>0
r - n
\int
B(x,r)
| f(y)| dy \forall x \in \BbbR n,
where and what follows B(x, r) = \{ y \in \BbbR n : | x - y| < r\} .
Definition 2.2. Given a measurable function q(\cdot ) defined on \BbbR n, we write
q - := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}
x\in \BbbR n
q(x), q+ := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
q(x).
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
BOUNDEDNESS FOR RIESZ-TYPE POTENTIAL OPERATORS ON VARIABLE EXPONENT . . . 1189
(i) q\prime - = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}
x\in \BbbR n
q\prime (x) =
q+
q+ - 1
, q\prime + = \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
q\prime (x) =
q -
q - - 1
.
(ii) Denote by P0(\BbbR n) the set of all measurable functions q(\cdot ) : \BbbR n \rightarrow (0,\infty ) such that 0 <
< q - \leq q(x) \leq q+ < \infty .
(iii) Denote by P(\BbbR n) the set of all measurable functions q(\cdot ) : \BbbR n \rightarrow (1,\infty ) such that 1 <
< q - \leq q(x) \leq q+ < \infty .
(iv) B(\Omega ) =
\bigl\{
q(\cdot ) \in P(\BbbR n) : the maximal operator M is bounded on Lq(\cdot )(\Omega )
\bigr\}
.
Definition 2.3. Let q(\cdot ) : \BbbR n \rightarrow \BbbR be a real-valued function.
(1) Denote by C log
loc (\BbbR
n) the set of all local \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous functions q(\cdot ) which satisfies
| q(x) - q(y)| \leq - C
\mathrm{l}\mathrm{n}(| x - y| )
, | x - y| \leq 1/2, x, y \in \BbbR n.
(2) Denote by C log
0 (\BbbR n) the set of all \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous functions q(\cdot ) at origin satisfies
| q(x) - q(0)| \leq C
\mathrm{l}\mathrm{n}
\biggl(
e+
1
| x|
\biggr) , x \in \BbbR n. (2.1)
(3) Denote by C log
\infty (\BbbR n) the set of all \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous functions q(\cdot ) at infinity satisfies
\bigm| \bigm| q(x) - q\infty
\bigm| \bigm| \leq C\infty
\mathrm{l}\mathrm{n}(e+ | x| )
, x \in \BbbR n, (2.2)
where q\infty = \mathrm{l}\mathrm{i}\mathrm{m}| x| \rightarrow \infty q(x).
(4) Denote by C log(\BbbR n) := C log
loc (\BbbR
n) \cap C log
\infty (\BbbR n) the set of all global \mathrm{l}\mathrm{o}\mathrm{g}-Hölder continuous
functions q(\cdot ).
Remark 2.1. The C log
\infty (\BbbR n) condition is equivalent to the uniform continuity condition
| q(x) - q(y)| \leq C
\mathrm{l}\mathrm{n}(e+ | x| )
, | y| \geq | x| , x, y \in \BbbR n.
The C log
\infty (\BbbR n) condition was originally defined in this form in [16].
Now, we define variable exponent Herz – Morrey spaces M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n). Let Bk = \{ x \in \BbbR n :
| x| \leq 2k\} , Ak = Bk \setminus Bk - 1 and \chi
k
= \chi Ak
for k \in \BbbZ .
Definition 2.4. Suppose that 0 \leq \lambda < \infty , 0 < p < \infty , q(\cdot ) \in P(\BbbR n) and \alpha (\cdot ) : \BbbR n \rightarrow \BbbR with
\alpha (\cdot ) \in L\infty (\BbbR n). The variable exponent Herz – Morrey space M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) is definded by
M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) =
\Bigl\{
f \in L
q(\cdot )
loc (\BbbR
n\setminus \{ 0\} ) : \| f\|
M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n)
< \infty
\Bigr\}
,
where
\| f\|
M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n)
= \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda
\Biggl(
k0\sum
k= - \infty
\| 2k\alpha (\cdot )f\chi
k
\| p
L
q(\cdot )
(\BbbR n)
\Biggr) 1/p
.
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1190 J.-L. WU
Compare the variable Herz – Morrey space M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) with the variable Herz space [9]
\.K
\alpha (\cdot ),p
q(\cdot ) (\BbbR n), where
\.K
\alpha (\cdot ),p
q(\cdot ) (\BbbR n) =
\Biggl\{
f \in L
q(\cdot )
loc (\BbbR
n\setminus \{ 0\} ) :
\infty \sum
k= - \infty
\| 2k\alpha (\cdot )f\chi
k
\| p
Lq(\cdot )(\BbbR n)
< \infty
\Biggr\}
.
Obviously, M \.K
\alpha (\cdot ),0
p,q(\cdot ) (\BbbR
n) = \.K
\alpha (\cdot ),p
q(\cdot ) (\BbbR n). When \alpha (\cdot ) is constant, we have M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) =
= M \.K\alpha ,\lambda
p,q(\cdot )(\BbbR
n) (see [13]). If both \alpha (\cdot ) and q(\cdot ) are constants, and \lambda = 0, then M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n) =
= \.K\alpha ,p
q (\BbbR n) are classical Herz spaces.
2.2. Recent results for Riesz-type potential \bfitI \bfitbeta (\cdot ). In this subsection we recall some recent
results for Riesz-type potential operator I\beta (\cdot ). The order \beta (x) of the potential is not assumed to be
continuous. We assume that it is a measurable function on \Omega satisfying the following assumptions:
\beta 0 := \mathrm{e}\mathrm{s}\mathrm{s} \mathrm{i}\mathrm{n}\mathrm{f}
x\in \BbbR n
\beta (x) > 0,
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
p(x)\beta (x) < n, (2.3)
\mathrm{e}\mathrm{s}\mathrm{s} \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n
p\infty \beta (x) < n.
The open problem, the boundedness of the Riesz-type potential operator I\beta (\cdot ) from the variable
exponent space Lp(\cdot )(\BbbR n) into the space Lq(\cdot )(\BbbR n) with the limiting Sobolev exponent
1
q(x)
=
=
1
p(x)
- \beta (x)
n
, was first solved in the case of bounded domains \Omega \subset \BbbR n(see [17]). After Diening
[18] proved the boundedness of the maximal operator over bounded domains, the validity of the
Sobolev theorem for bounded domains became an unconditional statement.
In 2008, in the case of bounded sets, Almeida, Hasanov and Samko [19] proved the boundedness
of the maximal operator in variable exponent Morrey spaces, and in 2009, Hästö [20] used his new
“local-to-global” approach to extend the result of [19] about the maximal operator to the whole
space \BbbR n. In 2010, in the case of bounded sets, Guliyev, Hasanov and Samko [21] considered the
boundedness of the Riesz-type potential operator I\beta (\cdot ) on the generalized variable exponent Morrey
type spaces.
For the whole space \BbbR n, under the condition that the exponent p(x) is constant outside some ball
of large radius, the Sobolev theorem was proved by Diening [22].
Another version of the Sobolev theorem for the space \BbbR n was proved in [23] for the exponents
p(x) not necessarily constant in a neigbourhood of infinity, but with some extra power weight fixed
to infinity and under the assumption that p(x) takes its minimal value at infinity.
Theorem A. Let \beta (x) meet conditions (2.3) which q1(\cdot ) instead of p(\cdot ). Suppose that q1(\cdot ) \in
\in C log(\BbbR n) \cap P(\BbbR n) and
1 < (q1)\infty \leq q1(x) \leq (q1)+ < \infty . (2.4)
Then the following weighted Sobolev-type estimate is valid for the operator I\beta (\cdot ) :\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)
\bigm\| \bigm\|
Lq2(\cdot )(\BbbR n)
\leq C\| f\| Lq1(\cdot )(\BbbR n),
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
BOUNDEDNESS FOR RIESZ-TYPE POTENTIAL OPERATORS ON VARIABLE EXPONENT . . . 1191
where q2(x) is defined by
1
q2(x)
=
1
q1(x)
- \beta (x)
n
, (2.5)
and
\gamma (x) = C\infty \beta (x)
\biggl(
1 - \beta (x)
n
\biggr)
\leq n
4
C\infty , (2.6)
C\infty being the Dini – Lipschitz constant from (2.2) which q(\cdot ) is replaced by q1(\cdot ).
In 2013, in the case of unbounded sets, Guliyev and Samko [24] considered the boundedness
of the Riesz-type potential operator I\beta (\cdot ) on the generalized variable exponent Morrey type spaces.
And recently, the author [14] obtain the similar results of Theorem A on the variable exponent
Herz – Morrey space M \.K\alpha ,\lambda
p,q(\cdot )(\BbbR
n).
Remark 2.2. The fractional maximal operator is defined as
M\beta (\cdot )(f)(x) = \mathrm{s}\mathrm{u}\mathrm{p}
r>0
1
| B(x, r)| n - \beta (x)
\int
B(x,r)
| f(y)| dy. (2.7)
The pointwise estimate for (2.7) is also valid which yields Theorem A.
2.3. Auxiliary propositions and lemmas. In this subsection we state some auxiliary propositions
and lemmas which will be needed for proving our main theorems. And we only describe partial
results we need.
Proposition 2.1. Let q(\cdot ) \in P(\BbbR n).
(i) If q(\cdot ) \in C log(\BbbR n), then we have q(\cdot ) \in B(\BbbR n).
(ii) q(\cdot ) \in B(\BbbR n) if and only if q\prime (\cdot ) \in B(\BbbR n).
The first part in Proposition 2.1 is independently due to Cruz – Uribe et al. [16] and to Nekvinda
[25] respectively. The second of Proposition 2.1 belongs to Diening [26] (see Theorem 8.1 or
Theorem 1.2 in [27]).
Remark 2.3. Since \bigm| \bigm| q\prime (x) - q\prime (y)
\bigm| \bigm| \leq | q(x) - q(y)|
(q - - 1)2
,
it follows at once that if q(\cdot ) \in C log(\BbbR n), then so does q\prime (\cdot ), i.e., if the condition hold, then
M is bounded on Lq(\cdot )(\BbbR n) and Lq\prime (\cdot )(\BbbR n). Furthermore, Diening has proved general results on
Musielak – Orlicz spaces.
The next proposition is the generalization of variable exponents Herz spaces in [9], and it was
proved in [15].
Proposition 2.2. Let q(\cdot ) \in P(\BbbR n), p \in (0,\infty ), and \lambda \in [0,\infty ). If real-valued function
\alpha (\cdot ) \in L\infty (\BbbR n) \cap C log
0 (\BbbR n) \cap C log
\infty (\BbbR n), then
\| f\|
M \.K
\alpha (\cdot ),\lambda
p,q(\cdot ) (\BbbR
n)
= \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda
\Biggl(
k0\sum
k= - \infty
\| 2k\alpha (\cdot )f\chi
k
\| p
L
q(\cdot )
(\BbbR n)
\Biggr) 1/p
\approx
\approx \mathrm{m}\mathrm{a}\mathrm{x}
\left\{ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda
\left( \~k1\sum
k= - \infty
2k\alpha (0)p\| f\chi
k
\| p
L
q(\cdot )
(\BbbR n)
\right) 1/p ,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1192 J.-L. WU
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0
k0\in \BbbZ
\left( 2 - k0\lambda
\left( \~k2\sum
k= - \infty
2k\alpha (0)p\| f\chi
k
\| p
L
q(\cdot )
(\BbbR n)
\right) 1/p +
+2 - k0\lambda
\left( \~k3\sum
k=0
2k\alpha \infty p\| f\chi
k
\| p
L
q(\cdot )
(\BbbR n)
\right) 1/p
\right)
\right\} ,
where \~k1 = k0, \~k2 = - 1, \~k3 = k0.
The next lemma is known as the generalized Hölder’s inequality on Lebesgue spaces with variable
exponent, and the proof can be found in [8].
Lemma 2.1 (generalized Hölder’s inequality). Suppose that q(\cdot ) \in P(\BbbR n), then, for any f \in
\in Lq(\cdot )(\BbbR n) and any g \in Lq\prime (\cdot )(\BbbR n), we have\int
\BbbR n
\bigm| \bigm| f(x)g(x)\bigm| \bigm| dx \leq Cq\| f\| Lq(\cdot )(\BbbR n)\| g\| Lq\prime (\cdot )(\BbbR n),
where Cq = 1 + 1/q - - 1/q+.
The following lemma can be found in [28].
Lemma 2.2. (I) Let q(\cdot ) \in B(\BbbR n). Then there exist positive constants \delta \in (0, 1) and C > 0
such that
\| \chi S\| Lq(\cdot )(\BbbR n)
\| \chi B\| Lq(\cdot )(\BbbR n)
\leq C
\biggl(
| S|
| B|
\biggr) \delta
for all balls B in \BbbR n and all measurable subsets S \subset B.
(II) Let q(\cdot ) \in B(\BbbR n). Then there exists a positive constant C > 0 such that
C - 1 \leq 1
| B|
\| \chi B\| Lq(\cdot )(\BbbR n)\| \chi B\| Lq\prime (\cdot )(\BbbR n) \leq C
for all balls B in \BbbR n.
Remark 2.4. (i) If q1(\cdot ), q2(\cdot ) \in C log(\BbbR n) \cap P(\BbbR n), then we see that q\prime
1
(\cdot ), q2(\cdot ) \in B(\BbbR n).
Hence we can take positive constants 0 < \delta 1 < 1/(q\prime
1
)+, 0 < \delta 2 < 1/(q2)+ such that
\| \chi S\| Lq\prime 1(\cdot )(\BbbR n)
\| \chi B\| Lq\prime 1(\cdot )(\BbbR n)
\leq C
\biggl(
| S|
| B|
\biggr) \delta 1
,
\| \chi S\| Lq2(\cdot )(\BbbR n)
\| \chi B\| Lq2(\cdot )(\BbbR n)
\leq C
\biggl(
| S|
| B|
\biggr) \delta 2
(2.8)
hold for all balls B in \BbbR n and all measurable subsets S \subset B (see [28, 29]).
(ii) On the other hand, Kopaliani [30] has proved the conclusion: If the exponent q(\cdot ) \in P(\BbbR n)
equals to a constant outside some large ball, then q(\cdot ) \in B(\BbbR n) if and only if q(\cdot ) satisfies the
Muckenhoupt type condition
\mathrm{s}\mathrm{u}\mathrm{p}
Q : cube
1
| Q|
\| \chi Q\| Lq(\cdot )(\BbbR n)\| \chi Q\| Lq\prime (\cdot )(\BbbR n) < \infty .
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
BOUNDEDNESS FOR RIESZ-TYPE POTENTIAL OPERATORS ON VARIABLE EXPONENT . . . 1193
3. Main result and its proof. Our main result can be stated as follows.
Theorem 3.1. Suppose that q1(\cdot ) \in C log(\BbbR n) \cap P(\BbbR n) and \beta (x) meet conditions (2.3) which
q1(\cdot ) instead of p(\cdot ). Define the variable exponent q2(\cdot ) by (2.5). Let q1(\cdot ), q\prime
2
(\cdot ) satisfies condi-
tion (2.4), and 0 < p1 \leq p2 < \infty , \lambda \geq 0, and \alpha (\cdot ) \in L\infty (\BbbR n) be log-Hölder continuous both at
the origin and at infinity, with \lambda - n\delta 2 < \alpha (0) \leq \alpha \infty < \lambda + n\delta 1, where \delta 1 \in (0, 1/(q\prime 1)+) and
\delta 2 \in (0, 1/(q2)+) are the constants appearing in (2.8). Then\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)
\bigm\| \bigm\|
M \.K
\alpha (\cdot ),\lambda
p2 ,q2 (\cdot )(\BbbR
n)
\leq C\| f\|
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
,
where \gamma (x) is defined as in (2.6), and the Dini – Lipschitz constant is \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
C\infty ,
2C\infty \bigl(
(q1) - - 1
\bigr) 2\biggr\}
when the q(\cdot ) in (2.2) is replaced by q1(\cdot ).
Remark 3.1. (i) Under the assumptions of Theorem 3.1, the similar result of Theorem 3.1 is
also valid for the fractional maximal operator M\beta (\cdot )(f) defined by (2.7) (partly detail ref. [14]).
(ii) If \alpha (\cdot ) be constant exponent, then the above result can be founded in [14].
(iii) When \lambda = 0, the above result is also valid.
Proof. For any f \in M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )
(\BbbR n), if we denote fj := f\chi j = f\chi Aj for each j \in \BbbZ , then we
can write
f(x) =
\infty \sum
j= - \infty
f(x)\chi j(x) =
\infty \sum
j= - \infty
fj(x).
Because of 0 < p1/p2 \leq 1, applying inequality\Biggl( \infty \sum
i= - \infty
| ai|
\Biggr) p1/p2
\leq
\infty \sum
i= - \infty
| ai| p1/p2 , (3.1)
and Proposition 2.2, we obtain\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)
\bigm\| \bigm\| p1
M \.K
\alpha (\cdot ),\lambda
p2 ,q2 (\cdot )(\BbbR
n)
\approx
\approx \mathrm{m}\mathrm{a}\mathrm{x}
\left\{ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\alpha (0)p2\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p2
L
q2 (\cdot )
(\BbbR n)
\Biggr) p1/p2
,
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0
k0\in \BbbZ
2 - k0\lambda p1
\left[ \Biggl( - 1\sum
k= - \infty
2k\alpha (0)p2\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p2
L
q2 (\cdot )
(\BbbR n)
\Biggr) p1/p2
+
+
\Biggl(
k0\sum
k=0
2k\alpha \infty p2\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p2
L
q2 (\cdot )
(\BbbR n)
\Biggr) p1/p2
\right] \right\} \leq
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\left\{ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\alpha (0)p1\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr)
,
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1194 J.-L. WU
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl[ \Biggl( - 1\sum
k= - \infty
2k\alpha (0)p1\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr)
+
+
\Biggl(
k0\sum
k=0
2k\alpha \infty p1\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr) \Biggr] \right\} \equiv : \mathrm{m}\mathrm{a}\mathrm{x}\{ E1, E2 + E3\} ,
where
E1 = \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\alpha (0)p1
\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\bigm\| \bigm\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr)
,
E2 = \mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl( - 1\sum
k= - \infty
2k\alpha (0)p1
\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\bigm\| \bigm\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr)
,
E3 = \mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k=0
2k\alpha \infty p1
\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)\chi k
\bigm\| \bigm\| p1
L
q2 (\cdot )
(\BbbR n)
\Biggr)
.
It is not difficult to found that the estimate of E1 is analogous to that of E2, therefore, the
estimates for E1 and E3 will be considered here.
To E1, we have
E1 \leq C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( k - 2\sum
j= - \infty
\bigm\| \bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)\chi k
\bigm\| \bigm\| \bigm\|
Lq2 (\cdot )(\BbbR n)
\right) p1
\right) +
+C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( k+1\sum
j=k - 1
\bigm\| \bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)\chi k
\bigm\| \bigm\| \bigm\|
Lq2 (\cdot )(\BbbR n)
\right) p1
\right) +
+C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( \infty \sum
j=k+2
\bigm\| \bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)\chi k
\bigm\| \bigm\| \bigm\|
Lq2 (\cdot )(\BbbR n)
\right) p1
\right) \equiv
\equiv : C(E11 + E12 + E13).
First we estimate E12. Using Theorem A and Proposition 2.2, we get
E12 \leq C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( k+1\sum
j=k - 1
\| fj\chi k
\|
Lq1 (\cdot )(\BbbR n)
\right) p1
\right) \leq
\leq C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\alpha (0)p1\| f\chi
k
\| p1
Lq1 (\cdot )(\BbbR n)
\Biggr)
\leq C\| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
For E11. Note that when x \in Ak, j \leq k - 2, and y \in Aj , then | x - y| \backsim | x| , 2| y| \leq | x| .
Therefore, using the generalized Hölder’s inequality, we obtain
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
BOUNDEDNESS FOR RIESZ-TYPE POTENTIAL OPERATORS ON VARIABLE EXPONENT . . . 1195
\bigm| \bigm| I\beta (\cdot )(fj)(x)\chi k
(x)
\bigm| \bigm| \leq \int
Aj
| f(y)|
| x - y| n - \beta (x)
dy\chi
k
(x) \leq
\leq C \cdot 2 - kn\| fj\|
L
q1 (\cdot )
(\BbbR n)
\| \chi
j
\|
L
q\prime
1
(\cdot )
(\BbbR n)
| x| \beta (x)\chi
k
(x). (3.2)
Notice that the fact
I\beta (\cdot )(\chi Bk
)(x) \geq I\beta (\cdot )(\chi Bk
)(x)\chi Bk
(x) \geq C| x| \beta (x)\chi
k
(x). (3.3)
Using Theorem A, Lemma 2.2, (2.8), (3.2) and (3.3), we have\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)\chi k
(\cdot )
\bigm\| \bigm\|
L
q2 (\cdot )
(\BbbR n)
\leq
\leq C \cdot 2 - kn\| fj\|
L
q1 (\cdot )
(\BbbR n)
\| \chi
j
\|
L
q\prime
1
(\cdot )
(\BbbR n)
\bigm\| \bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(\chi Bk
)
\bigm\| \bigm\| \bigm\|
L
q2 (\cdot )
(\BbbR n)
\leq
\leq C \cdot 2 - kn\| \chi Bk
\|
L
q1 (\cdot )
(\BbbR n)
\| fj\|
L
q1 (\cdot )
(\BbbR n)
\| \chi Bj
\|
L
q\prime
1
(\cdot )
(\BbbR n)
\leq C \cdot 2(j - k)n\delta 1\| fj\|
L
q1 (\cdot )
(\BbbR n)
. (3.4)
On the other hand, note the following fact:
Case I (\~ki < 0, i = 1, 2, 3):
\| fj\|
L
q1 (\cdot )
(\BbbR n)
\leq 2 - j\alpha (0)
\Biggl(
j\sum
i= - \infty
2i\alpha (0)p1\| fi\|
p1
L
q1 (\cdot )
(\BbbR n)
\Biggr) 1/p1
\leq
\leq 2j(\lambda - \alpha (0))
\left( 2 - j\lambda
\Biggl(
j\sum
i= - \infty
\| 2i\alpha (\cdot )fi\|
p1
L
q1 (\cdot )
(\BbbR n)
\Biggr) 1/p1
\right) \leq C \cdot 2j(\lambda - \alpha (0))\| f\|
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
. (3.5)
Case II (\~ki \geq 0, i = 1, 2, 3):
\| fj\|
L
q1 (\cdot )
(\BbbR n)
\leq 2 - j\alpha \infty
\Biggl(
j\sum
i=0
2i\alpha \infty p1\| fi\|
p1
L
q1 (\cdot )
(\BbbR n)
\Biggr) 1/p1
\leq
\leq C \cdot 2j(\lambda - \alpha \infty )\| f\|
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
. (3.6)
Definition 2.4, Proposition 2.2 and the condition of \alpha (\cdot ) are used in above facts.
Thus, combining (3.4) and (3.5), and using \alpha (0) \leq \alpha \infty < \lambda + n\delta 1, it follows that
E11 \leq C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( k - 2\sum
j= - \infty
2(j - k)n\delta 1\| fj\|
L
q1 (\cdot )
(\BbbR n)
\right) p1
\right) \leq
\leq C \| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
\mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\lambda p1
\Biggr)
\leq C \| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
Now, let us turn to estimate for E13. Note that when x \in Ak, j \geq k + 2, and y \in Aj , then
| x - y| \backsim | y| , 2| x| \leq | y| . Therefore, using the generalized Hölder’s inequality, we have
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
1196 J.-L. WU
\bigm| \bigm| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)(x)\chi k
(x)
\bigm| \bigm| \leq (1 + | x| ) - \gamma (x)
\int
Aj
| f(y)|
| x - y| n - \beta (x)
dy\chi
k
(x) \leq
\leq C
\int
Aj
| f(y)| (1 + | x| ) - \gamma (x)| y| \beta (x) - n dy\chi
k
(x) \leq
\leq C \cdot 2 - jn\| fj\|
L
q1 (\cdot )
(\BbbR n)
\bigm\| \bigm\| (1 + | x| ) - \gamma (x)| \cdot | \beta (x)\chi
j
(\cdot )
\bigm\| \bigm\|
L
q\prime
1
(\cdot )
(\BbbR n)
\chi
k
(x). (3.7)
Similar to (3.3), we get
I\beta (\cdot )(\chi Bj
)(x) \geq I\beta (\cdot )(\chi Bj
)(x)\chi Bj
(x) \geq C| x| \beta (x)\chi j (x). (3.8)
Using Theorem A, Lemma 2.2, (2.8), (3.7) and (3.8), we obtain\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(fj)\chi k
\bigm\| \bigm\|
L
q2 (\cdot )
(\BbbR n)
\leq
\leq C \cdot 2 - jn\| \chi Bj
\|
L
q\prime
2
(\cdot )
(\BbbR n)
\| fj\|
L
q1 (\cdot )
(\BbbR n)
\| \chi Bk
\|
L
q2 (\cdot )
(\BbbR n)
\leq C \cdot 2(k - j)n\delta 2\| fj\|
L
q1 (\cdot )
(\BbbR n)
. (3.9)
Therefore, combining (3.5) and (3.9), and using \lambda - n\delta 2 < \alpha (0) \leq \alpha \infty , it follows that
E13 \leq C \mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\left( k0\sum
k= - \infty
2k\alpha (0)p1
\left( \infty \sum
j=k+2
2
(k - j)n\delta 2\| fj\|
L
q1 (\cdot )
(\BbbR n)
\right) p1
\right) \leq
\leq C\| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
\mathrm{s}\mathrm{u}\mathrm{p}
k0<0
k0\in \BbbZ
2 - k0\lambda p1
\Biggl(
k0\sum
k= - \infty
2k\lambda p1
\Biggr)
\leq C\| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
Combining the estimates for E11, E12 and E13 yields
E1 \leq C\| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
For E3, similar to the estimate of E1, using Theorem A, Proposition 2.2, (2.8), (3.1) – (3.4),
(3.6) – (3.9), we have
E3 \leq C\| f\| p1
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
Joint the estimate for E1, E2 and E3 yields\bigm\| \bigm\| (1 + | x| ) - \gamma (x)I\beta (\cdot )(f)
\bigm\| \bigm\|
M \.K
\alpha (\cdot ),\lambda
p2 ,q2 (\cdot )(\BbbR
n)
\leq C\| f\|
M \.K
\alpha (\cdot ),\lambda
p1 ,q1 (\cdot )(\BbbR
n)
.
Theorem 3.1 is proved.
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Received 21.12.15,
after revision — 21.07.16
ISSN 1027-3190. Укр. мат. журн., 2017, т. 69, № 9
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| id | umjimathkievua-article-1769 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:12:19Z |
| publishDate | 2017 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/35/70e911f2ee8e9fcce52f90ab09069b35.pdf |
| spelling | umjimathkievua-article-17692019-12-05T09:26:20Z Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces Обмеженiсть потенцiальних операторiв типу Рiса на просторах Херца–Моррея зi змiнним показником Wu, Jianglong Ву, Янглонг We show the boundedness of the Riesz-type potential operator of variable order $\beta (x)$ from the variable exponent Herz – Morrey spaces $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ into the weighted space $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, where $\alpha (x) \in L^{\infty} (\mathbb{R}^n) is log-Holder continuous both at the origin and at infinity, $\omega = (1+| x| ) \gamma (x)$ with some $\gamma (x) > 0$, and $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$ when $q_1 (x)$ is not necessarily constant at infinity. It is assumed that the exponent $q_1 (x)$ satisfies the logarithmic continuity condition both locally and at infinity and $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$. Встановлено обмеженiсть потенцiального оператора типу Рiса змiнного порядку $\beta (x)$, що дiє з просторiв Херца – Моррея зi змiнним показником $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_1 ,q_1 (\cdot )}(\mathbb{R}^n)$ у зважений простiр $M \dot{K}^{\alpha (\cdot ),\lambda}_{p_2 ,q_2 (\cdot )}(\mathbb{R}^n, \omega )$, де $\alpha (x) \in L^{\infty} (\mathbb{R}^n)$ є log-Гельдер неперервним як на початку координат, так i на нескiнченностi, $\omega = (1+| x| ) \gamma (x)$ з деяким $\gamma (x) > 0$ та $1/q_1 (x) 1/q_2 (x) = \beta (x)/n$, коли $q_1 (x)$ не обов’язково є сталою на нескiнченностi. Вважаємо, що показник $q_1 (x)$ задовольняє умову логарифмiчної неперервностi як локально, так i на нескiнченностi та $1 < (q_1)_{\infty} \leq q_1(x) \leq (q_1)_+ < \infty, \;x \in \mathbb{R}$. Institute of Mathematics, NAS of Ukraine 2017-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1769 Ukrains’kyi Matematychnyi Zhurnal; Vol. 69 No. 9 (2017); 1187-1197 Український математичний журнал; Том 69 № 9 (2017); 1187-1197 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1769/751 Copyright (c) 2017 Wu Jianglong |
| spellingShingle | Wu, Jianglong Ву, Янглонг Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey spaces |
| title | Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey
spaces |
| title_alt | Обмеженiсть потенцiальних операторiв типу Рiса
на просторах Херца–Моррея зi змiнним показником |
| title_full | Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey
spaces |
| title_fullStr | Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey
spaces |
| title_full_unstemmed | Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey
spaces |
| title_short | Boundedness of Riesz-type potential operators on variable exponent Herz – Morrey
spaces |
| title_sort | boundedness of riesz-type potential operators on variable exponent herz – morrey
spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1769 |
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