On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces
UDC 517.5 In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and on generalization of the BMO-norms.
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| Дата: | 2020 |
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2020
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512222667603968 |
|---|---|
| author | Qu, M. Wang, L. Qu, M. Wang, L. |
| author_facet | Qu, M. Wang, L. Qu, M. Wang, L. |
| author_sort | Qu, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2022-03-26T11:01:57Z |
| description | UDC 517.5
In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and on generalization of the BMO-norms. |
| doi_str_mv | 10.37863/umzh.v72i7.6023 |
| first_indexed | 2026-03-24T03:25:22Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v72i7.6023
UDC 517.5
M. Qu (School Math. and Statistics, Anhui Normal Univ., Wuhu, China),
L. Wang (School Math. and Phys., Anhui Polytech. Univ., Wuhu, China)
ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS
WITH ROUGH KERNELS IN VARIABLE MORREY TYPE SPACES*
ПРО КОМУТАТОР IНТЕГРАЛIВ МАРЦИНКЕВИЧА З ГРУБИМИ ЯДРАМИ
У ЗМIННИХ ПРОСТОРАХ ТИПУ МОРРЕЯ
In the framework of variable exponent Morrey and Morrey – Herz spaces, we prove some boundedness results for the
commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and
on generalization of the BMO-norms.
У рамках змiнних експонент просторiв Моррея та Моррея – Герца доведено деякi результати щодо обмеженостi
комутатора iнтегралiв Марцинкевича з грубими ядрами. Цей пiдхiд базується на теорiї змiнних експонент та уза-
гальненнi BMO-норм.
1. Introduction. Let \BbbR n be the n-dimensional Euclidean space of points x = (x1, . . . , xn) with
norm | x| =
\Bigl( \sum n
i=1
x2i
\Bigr) 1/2
. Suppose that \BbbS n - 1 is the unit sphere in \BbbR n, n \geq 2, equipped with
the normalized Lebesgue measure d\sigma (x\prime ). Let \Omega \in L1(\BbbS n - 1) be homogeneous of degree zero and
satisfy \int
\BbbS n - 1
\Omega (x\prime )d\sigma (x\prime ) = 0, (1)
where x\prime = x/| x| for any x \not = 0. Then the Marcinkiewicz integral operator \mu \Omega of higher dimension
is defined by
\mu \Omega (f)(x) =
\left( \infty \int
0
| F\Omega ,t(f)(x)| 2
dt
t3
\right) 1
2
,
where
F\Omega ,t(f)(x) =
\int
| x - y| \leq t
\Omega (x - y)
| x - y| n - 1
f(y)dy.
A locally integrable function b is said to be a \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n) function, if it satisfies
\| b\| \ast := \mathrm{s}\mathrm{u}\mathrm{p}
x\in \BbbR n,r>0
1
| B|
\int
B
| b(y) - bB| dy < \infty ,
where B is ball centered at x and radius of r, bB =
1
| B|
\int
B
b(t)dt and \| b\| \ast is the norm in
\mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n). For b \in \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n), the commutator of the Marcinkiewicz integral operator \mu \Omega ,b, is then
* This research was supported by NNSF of China (Grant No. 11871096), Anhui Provincial Natural Science Foundation
(Grant No. 1908085MA19) and Pre-research Project of the NNSF of China (Grant No. 2019yyzr14).
c\bigcirc M. QU, L. WANG, 2020
928 ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 929
defined by
\mu \Omega ,b(f) =
\left( \infty \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
| x - y| \leq t
\Omega (x - y)
| x - y| n - 1
(b(x) - b(y))f(y)dy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
dt
t3
\right)
1
2
.
It is well-known that Stein [23] first proved that if \Omega \in \mathrm{L}\mathrm{i}\mathrm{p}\gamma (\BbbS n - 1), 0 < \gamma \leq 1, then \mu \Omega is of
type (p, p) for 1 < p \leq 2 and of weak type (1, 1). Afterwards, Ding, Fan and Pan [7] removed
the smoothness assumed on \Omega and showed that \mu \Omega is bounded on Lp(\BbbR n) for 1 < p < \infty if
\Omega \in H1(\BbbS n - 1). Here H1(\BbbS n - 1) denotes the classical Hardy space on \BbbS n - 1. On the other hand,
using a good-\lambda inequality, Torchinsky and Wang [25] established the weighted Lp-boundedness
of \mu \Omega and \mu \Omega ,b when \Omega \in \mathrm{L}\mathrm{i}\mathrm{p}\gamma (\BbbS n - 1), 0 < \gamma \leq 1. For some recent development, we refer to
[2, 8, 14 – 18] and their references.
In recent years, following the fundamental work of Kováčik and Rákosnı́k [13], function spaces
with variable exponent, such as the vaiable exponent Lebesgue, Herz and Morrey spaces etc., have
attracted a great attention due mainly to their useful applications in fluid dynamics, image restoration
and differential equations with p(x)-growth (see [1, 3, 11, 30 – 32] and the references therein). In
many applications, a crucial step has been to prove that the classical operators are bounded in variable
exponent function spaces. Ho in [9, 10] has given some sufficient conditions for the boundedness
of fractional operators and singular integral operators in variable exponent Morrey spaces \scrM p(\cdot ),u,
where u is a Morrey weight function for Lp(\cdot )(\BbbR n) (see Definition 3.1). In 2016, Tao and Li [26]
showed that if \Omega \in \mathrm{L}\mathrm{i}\mathrm{p}\gamma (\BbbS n - 1), 0 < \gamma \leq 1, then the commutator \mu \Omega ,b is bounded on \scrM p(\cdot ),u. On
the other hand, based on the extrapolation theory and some pointwise estimates, operators with rough
kernels have recently been discussed in [5, 22, 27]. These results inspire us to consider the question:
whether the variable exponent Morrey spaces estimates for \mu \Omega ,b are still true if \Omega \in Ls(\BbbS n - 1),
s > 1? The first aim of this paper is to give an affirmative answer to this question.
Morrey – Herz spaces M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n) with variable exponents p and \alpha were recently studied by
Lu and Zhu [20]. Under natural regularity assumptions on the exponent \alpha and p, either at the origin
or at infinity, they established the boundedness of a wide class of sublinear operators (including ma-
ximal, potential and Calderón – Zygmund operators) and their commutators on such spaces. In [29],
we made a further step and generalized the main theorems in [20] to the case of rough kernels. Mo-
tivated by the work of [20] and [29], the second aim of this paper is to prove that \mu \Omega ,b is bounded on
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n) provided that \lambda \geq 0 and \alpha , p are variable exponents. This result improves the corres-
ponding main theorem in [28], where the authors considered only the case \lambda = 0 and \alpha is a constant.
We usually denote cubes in \BbbR n by Q, | Q| is the Lebesgue measure of Q. \chi E is a characteristic
function of a measurable set E \subset \BbbR n. Let Bl = \{ x \in \BbbR n : | x| \leq 2l\} , l \in \BbbZ , and B := B(x, r) =
= \{ y \in \BbbR n : | x - y| < r\} . fB means the integral average of f on B, namely, fB =
1
| B|
\int
B
f(x)dx.
p\prime (\cdot ) denotes the conjugate exponent defined by 1/p(\cdot ) + 1/p\prime (\cdot ) = 1. The letter C stands for a
positive constant, which may vary from line to line. The expression f \lesssim g means that f \leq Cg, and
f \thickapprox g means f \lesssim g \lesssim f.
2. Preliminaries and lemmas. We begin with a brief and necessarily incomplete review of the
variable exponent Lebesgue spaces Lp(\cdot )(\BbbR n) (see [4, 6] for more information).
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
930 M. QU, L. WANG
Given a measurable function p, we assume that
1 < p - \leq p(x) \leq p+ < \infty , (2)
where p - := ess infx\in \BbbR np(x) and p+ := ess supx\in \BbbR np(x).
By Lp(\cdot )(\BbbR n) we denote the set of all measurable functions f on \BbbR n such that
Ip(\cdot )(f) :=
\int
\BbbR n
| f(x)| p(x)dx < \infty .
This is a Banach space with the norm (the Luxemburg – Nakano norm)
\| f\| Lp(\cdot )(\BbbR n) = inf\{ \mu > 0 : Ip(\cdot )(f/\mu ) \leq 1\} .
It is easy to see that this norm has the following property:
\| | f | \sigma \| Lp(\cdot )(\BbbR n) = \| f\| \sigma
L\sigma p(\cdot )(\BbbR n)
, \sigma \geq 1/p - . (3)
By \scrP (\BbbR n) we denote the set of variable exponents p(\cdot ) satisfying (2). When p(\cdot ) \in \scrP (\BbbR n), the
generalized H\"\mathrm{o}lder inequality holds in the form\int
\BbbR n
| f(x)g(x)| dx \leq rp\| f\| Lp(\cdot )(\BbbR n)\| g\| Lp\prime (\cdot )(\BbbR n) (4)
with rp = 1 + 1/p - - 1/p+ (see [13], Theorem 2.1).
The set \scrB (\BbbR n) consists of p(\cdot ) \in \scrP (\BbbR n) satisfying the condition that M is bounded on
Lp(\cdot )(\BbbR n), where M denotes the Hardy – Littlewood maximal operator defined by
Mf(x) = \mathrm{s}\mathrm{u}\mathrm{p}
r>0
1
| B(x, r)|
\int
B(x,r)
| f(y)| dy.
It is well-known that if p(\cdot ) \in \scrB (\BbbR n), then p\prime (\cdot ) \in \scrB (\BbbR n) (see [12], Proposition 2).
A function \phi (\cdot ) : \BbbR n \rightarrow \BbbR is called log-H\"older continuous at the origin (or has a log decay at
the origin), if there exists a constant Clog > 0 such that
| \phi (x) - \phi (0)| \leq
Clog
log(e+ 1/| x| )
, x \in \BbbR n.
If, for some \phi \infty \in \BbbR and Clog > 0, there holds
| \phi (x) - \phi \infty | \leq
Clog
log(e+ | x| )
, x \in \BbbR n,
then \phi (\cdot ) is called log-H\"older continuous at infinity (or has a log decay at the infinity).
Lemma 2.1. If p(\cdot ) \in \scrB (\BbbR n), then we have
\| \chi B\| Lp(\cdot )(\BbbR n)\| \chi B\| Lp\prime (\cdot )(\BbbR n) \lesssim | B| .
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 931
Lemma 2.2. If p(\cdot ) \in \scrB (\BbbR n), then we have, for all measurable subsets E \subset B,
\| \chi E\| Lp(\cdot )(\BbbR n)
\| \chi B\| Lp(\cdot )(\BbbR n)
\lesssim
\biggl(
| E|
| B|
\biggr) \delta 1
,
\| \chi E\| Lp\prime (\cdot )(\BbbR n)
\| \chi B\| Lp\prime (\cdot )(\BbbR n)
\lesssim
\biggl(
| E|
| B|
\biggr) \delta 2
,
where \delta 1, \delta 2 are constants with 0 < \delta 1, \delta 2 < 1.
Lemma 2.3. If p(\cdot ) \in \scrB (\BbbR n), b \in \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n), k > j, k, j \in \BbbN , then we have
\mathrm{s}\mathrm{u}\mathrm{p}
B\subset \BbbR n
1
\| \chi B\| Lp(\cdot )(\BbbR n)
\| (b - bB)\chi B\| Lp(\cdot )(\BbbR n) \thickapprox \| b\| \ast ,
\| (b - bBj )\chi Bk
\| Lp(\cdot )(\BbbR n) \lesssim (k - j)\| b\| \ast \| \chi Bk
\| Lp(\cdot )(\BbbR n).
Lemmas 2.1 – 2.3 are due to Izuki [12].
Lemma 2.4. Let p(\cdot ) \in \scrP (\BbbR n). If q > p+ and
1
p(x)
=
1\widetilde q(x) + 1
q
, then we have
\| fg\| Lp(\cdot )(\BbbR n) \lesssim \| f\| L\widetilde q(\cdot )(\BbbR n)\| g\| Lq(\BbbR n)
for all measurable functions f and g.
Lemma 2.5. Let r1 > 0. Suppose \alpha (\cdot ) \in L\infty (\BbbR n) is log-H\"older continuous both at origin and
at infinity, then we have
r
\alpha (x)
1 \lesssim r
\alpha (y)
2 \times
\left\{
\biggl(
r1
r2
\biggr) \alpha +
, 0 < r2 \leq r1/2,
1, r1/2 < r2 \leq 2r1,\biggl(
r1
r2
\biggr) \alpha -
, r2 > 2r1,
for any x \in B(0, r1)\setminus B(0, r1/2) and y \in B(0, r2)\setminus B(0, r2/2).
The proof of Lemmas 2.4 and 2.5 can be found in [21] and [1], respectively.
3. Boundedness on variable exponent Morrey spaces. We first recall the following definitions
given by Ho in [9].
Definition 3.1. Let p(\cdot ) \in L\infty (\BbbR n). A Lebesgue measurable function u(z, r) : \BbbR n\times (0,+\infty ) \rightarrow
\rightarrow (0,+\infty ) is said to be a Morrey weight function for Lp(\cdot )(\BbbR n) if u satisfies
\infty \sum
j=0
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r) \lesssim u(z, r) (5)
for any z \in \BbbR n and r > 0.
By \BbbW p(\cdot ), we denote the class of Morrey weight functions. We note that condition (5) is also
used to study the Fefferman – Stein vector-valued inequalities in weighted Morrey spaces (see [10]).
For any p(\cdot ) \in \scrB (\BbbR n), let \scrK p(\cdot ) denote the supremum of those q > 1 such that p(\cdot )/q \in \scrB (\BbbR n)
and \scrE p(\cdot ) be the conjugate of \scrK p\prime (\cdot ). The following result can be seen as a special case of the general
result in [10] for Banach function spaces.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
932 M. QU, L. WANG
Proposition 3.1. Let p(\cdot ) \in \scrB (\BbbR n). For any 1 < q < \scrK p(\cdot ) and 1 < \tau < \scrK p\prime (\cdot ), we have, for
any z \in \BbbR n and r > 0,
2jn(1 -
1
\tau
) \lesssim
\| \chi B(z,2jr)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\lesssim 2
jn
q \forall j \in \BbbN . (6)
Remark 3.1. It is easy to check that condition (5) together with (6) yields u(z, 2r) \lesssim u(z, r)
for any z \in \BbbR n and r > 0.
Definition 3.2. Let p(\cdot ) \in \scrB (\BbbR n) and u \in \BbbW p(\cdot ). The variable Morrey space \scrM p(\cdot ),u is the
collection of all Lebesgue measurable functions f satisfying
\| f\| \scrM p(\cdot ),u = \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbR n,R>0
1
u(z,R)
\| f\chi B(z,R)\| Lp(\cdot )(\BbbR n) < \infty .
Now, let us state the main result in this section.
Theorem 3.1. Suppose p(\cdot ) \in \scrB (\BbbR n) and \Omega \in Ls(\BbbS n - 1) with s > (p\prime )+ is homogeneous of
degree zero and satisfies (1). If b \in \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n) and
\infty \sum
j=0
(j + 1)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r) \lesssim u(z, r), (7)
for any z \in \BbbR n and r > 0, then we have
\| \mu \Omega ,b(f)\| \scrM p(\cdot ),u \lesssim \| b\| \ast \| f\| \scrM p(\cdot ),u .
Remark 3.2. Clearly, in comparison with the corresponding result by Tao and Li in [26, p. 56],
the smoothness condition on \Omega has been removed. More precisely, our result is an improvement of
Theorem 1.4 in [26].
Remark 3.3. There do exist some functions satisfying condition (7). For instance, if, for any
0 \leq \gamma < 1/\scrE p(\cdot ), a weight function u satisfies u(z, 2r) \leq 2n\gamma u(z, r) for any z \in \BbbR n and r > 0, then
(7) holds. In fact, for any \gamma < 1/\scrE p(\cdot ), there always exists a \tau < 1/\scrK p\prime (\cdot ) such that \gamma < 1 - 1/\tau <
< 1 - 1/\scrK p\prime (\cdot ) = 1/\scrE p(\cdot ). An application of Proposition 3.1 gives
\infty \sum
j=0
(j + 1)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r)
u(z, r)
\lesssim
\infty \sum
j=0
(j + 1)2jn(
1
\tau
+\gamma - 1) \lesssim 1.
Proof of Theorem 3.1. Let f \in \scrM p(\cdot ),u. For any z \in \BbbR n and r > 0, we decompose f = g + h,
where g = f\chi B(z,2r) and h =
\sum \infty
j=1 f\chi B(z,2j+1r)\setminus B(z,2jr). Noting that \mu \Omega ,b is a nonlinear operator,
then we have
1
u(z, r)
\| \chi B(z,r)\mu \Omega ,b(f)\| Lp(\cdot )(\BbbR n) \leq
1
u(z, r)
\| \chi B(z,r)\mu \Omega ,b(g)\| Lp(\cdot )(\BbbR n)+
+
1
u(z, r)
\| \chi B(z,r)\mu \Omega ,b(h)\| Lp(\cdot )(\BbbR n) := I + II.
For I, using u(z, 2r) \lesssim u(z, r) and the Lp(\cdot )(\BbbR n)-boundedness of \mu \Omega ,b (see [28, p. 262]), we
obtain
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 933
I \lesssim \| b\| \ast
1
u(z, 2r)
\| f\chi B(z,2r)\| Lp(\cdot )(\BbbR n) \lesssim
\lesssim \| b\| \ast \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbR n,R>0
1
u(z,R)
\| f\chi B(z,R)\| Lp(\cdot )(\BbbR n) \lesssim \| b\| \ast \| f\| \scrM p(\cdot ),u .
For II, we note that if x \in B(z, r) and y \in \widetilde Rj := B(z, 2j+1r) \setminus B(z, 2jr), then | x - y| \approx
\approx | y - z| \approx 2jr. The Minkowski inequality yields
| \mu \Omega ,b(h)(x)| \lesssim
\infty \sum
j=1
\left\{
\int
\widetilde Rj
| \Omega (x - y)|
| x - y| n - 1
| b(x) - b(y)| | f(y)|
\left( \int
| x - y| <t
dt
t3
\right)
1/2
dy
\right\} \lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n
\int
\widetilde Rj
| b(x) - b(y)| | \Omega (x - y)| | f(y)| dy \lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n
| b(x) - bB(z,r)|
\int
\widetilde Rj
| \Omega (x - y)| | f(y)| dy+
+
\infty \sum
j=1
1
(2jr)n
| bB(z,2j+1r) - bB(z,r)|
\int
\widetilde Rj
| \Omega (x - y)| | f(y)| dy+
+
\infty \sum
j=1
1
(2jr)n
\int
\widetilde Rj
| b(y) - bB(z,2j+1r)| | \Omega (x - y)| | f(y)| dy :=
:= U1 + U2 + U3.
For U1, since s > (p\prime )+, then we can choose p\ast (\cdot ) > 0 such that
1
s\prime
=
1
p(x)
+
1
p\ast (x)
, by the
property (3) and the generalized H\"\mathrm{o}lder inequality (4), we get
\| f\chi B(z,2j+1r)\| Ls\prime (\BbbR n) \lesssim \| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\ast (\cdot )(\BbbR n). (8)
In view of
1
p\ast (\cdot )
=
1
s\prime
- 1
p(\cdot )
=
1
p\prime (\cdot )
- 1
s
, Lemma 2.4 in [27, p. 178] yields
\| \chi B(z,2j+1r)\| Lp\ast (\cdot )(\BbbR n) \approx (2jr) -
n
s \| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n). (9)
Thus, from (8), (9) and the H\"\mathrm{o}lder inequality, it follows that
U1 \lesssim
\infty \sum
j=1
1
(2jr)n
| b(x) - bB(z,r)| \| f\chi B(z,2j+1r)\| Ls\prime (\BbbR n)
\left( \int \widetilde Rj
| \Omega (x - y)| sdy
\right)
1
s
\lesssim
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
934 M. QU, L. WANG
\lesssim
\infty \sum
j=1
1
(2jr)n
| b(x) - bB(z,r)| \| f\chi B(z,2j+1r)\| Ls\prime (\BbbR n)
\left( \int
| x - y| \lesssim 2jr
| \Omega (x - y)| sdy
\right)
1
s
\lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n(1 -
1
s
)
| b(x) - bB(z,r)| \| f\chi B(z,2j+1r)\| Ls\prime (\BbbR n) \lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n
| b(x) - bB(z,r)| \| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n).
For U2, noting that | bB(z,2j+1r) - bB(z,r)| \lesssim (j + 1)\| b\| \ast (see [24, p. 206]), we have
U2 \lesssim
\infty \sum
j=1
1
(2jr)n
| bB(z,2j+1r) - bB(z,r)| \| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n) \lesssim
\lesssim
\infty \sum
j=1
(j + 1)
1
(2jr)n
\| b\| \ast \| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n).
For U3, applying Lemma 2.3 with B = B(z, 2j+1r), (8) and (9), we obtain
U3 \lesssim
\infty \sum
j=1
1
(2jr)n(1 -
1
s
)
\| (b - bB(z,2j+1r))f\chi B(z,2j+1r)\| Ls\prime (\BbbR n) \lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n
\| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| (b - bB(z,2j+1r))\chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n) \lesssim
\lesssim
\infty \sum
j=1
1
(2jr)n
\| b\| \ast \| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n).
Combining the estimate of U1, U2 and U3, by Lemmas 2.3 and 2.1, we get
\| \chi B(z,r)\mu \Omega ,b(h)\| Lp(\cdot )(\BbbR n) \lesssim
\lesssim
\infty \sum
j=1
(j + 1)
1
(2jr)n
\| b\| \ast \| \chi B(z,r)\| Lp(\cdot )(\BbbR n)\| f\chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)\| \chi B(z,2j+1r)\| Lp\prime (\cdot )(\BbbR n) \lesssim
\lesssim \| b\| \ast
\infty \sum
j=1
(j + 1)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r)\times
\times \mathrm{s}\mathrm{u}\mathrm{p}
z\in \BbbR n,R>0
1
u(z,R)
\| f\chi B(z,R)\| Lp(\cdot )(\BbbR n) \lesssim
\lesssim \| b\| \ast
\infty \sum
j=1
(j + 1)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r)\| f\| \scrM p(\cdot ),u .
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ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 935
Thus, in view of the condition (7), we arrive at the desired inequality
II =
1
u(z, r)
\| \chi B(z,r)\mu \Omega ,b(h)\| Lp(\cdot )(\BbbR n) \lesssim
\lesssim \| b\| \ast \| f\| \scrM p(\cdot ),u
\infty \sum
j=1
(j + 1)
\| \chi B(z,r)\| Lp(\cdot )(\BbbR n)
\| \chi B(z,2j+1r)\| Lp(\cdot )(\BbbR n)
u(z, 2j+1r)
u(z, r)
\lesssim \| b\| \ast \| f\| \scrM p(\cdot ),u .
Theorem 3.1 is proved.
4. Boundedness on variable exponent Morrey – Herz spaces. Let Bk = \{ x \in \BbbR n : | x| \leq 2k\} ,
Rk = Bk\setminus Bk - 1 and \chi k = \chi Rk
be the characteristic function of the set Rk for k \in \BbbZ .
Definition 4.1. Let 0 < q \leq \infty , p(\cdot ) \in \scrP (\BbbR n) and \alpha (\cdot ) : \BbbR n \rightarrow \BbbR with \alpha (\cdot ) \in L\infty (\BbbR n). The
homogeneous Herz space \.K
\alpha (\cdot )
p(\cdot ),q(\BbbR
n) is defined as the class of all f \in L
p(\cdot )
loc (\BbbR
n\setminus \{ 0\} ) such that
\| f\| \.K
\alpha (\cdot )
p(\cdot ),q(\BbbR
n)
:=
\Biggl( \sum
k\in \BbbZ
\| 2k\alpha (\cdot )f\chi k\| qLp(\cdot )(\BbbR n)
\Biggr) 1/q
< \infty
with the usual modification when q = \infty .
Definition 4.2. Let 0 \leq \lambda < \infty , 0 < q \leq \infty , p(\cdot ) \in \scrP (\BbbR n) and \alpha (\cdot ) : \BbbR n \rightarrow \BbbR with
\alpha (\cdot ) \in L\infty (\BbbR n). The homogeneous Morrey – Herz space M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n) is defined as the class of all
f \in L
p(\cdot )
loc (\BbbR
n\setminus \{ 0\} ) such that
\| f\|
M \.K
\alpha (\cdot ),\lambda
q,p(\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\| qLp(\cdot )(\BbbR n)
\Biggr) 1/q
< \infty
with the usual modification when q = \infty .
Remark 4.1. It obviously follows that M \.K
\alpha (\cdot ),0
q,p(\cdot ) (\BbbR
n) = \.K
\alpha (\cdot )
p(\cdot ),q(\BbbR
n). If both \alpha (\cdot ) and p(\cdot )
are constants, then M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n) coincides with the classical Morrey – Herz spaces M \.K\alpha ,\lambda
p,q (\BbbR n)
defined in [19].
Lu and Zhu [20] obtained the following result.
Proposition 4.1. Let 0 \leq \lambda < \infty , 0 < q \leq \infty , p(\cdot ) \in \scrP (\BbbR n) and \alpha (\cdot ) \in L\infty (\BbbR n). If \alpha (\cdot ) is
log-H\"older continuous both at origin and at infinity, then
\| f\|
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
\approx \mathrm{m}\mathrm{a}\mathrm{x}
\left\{ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda
\Biggl(
k0\sum
k= - \infty
2k\alpha (0)q\| f\chi k\| qLp(\cdot )(\BbbR n)
\Biggr) 1/q
,
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0,k0\in \BbbZ
\left[ 2 - k0\lambda
\Biggl( - 1\sum
k= - \infty
2k\alpha (0)q\| f\chi k\| qLp(\cdot )(\BbbR n)
\Biggr) 1/q
+ 2 - k0\lambda
\Biggl(
k0\sum
k=0
2k\alpha \infty q\| f\chi k\| qLp(\cdot )(\BbbR n)
\Biggr) 1/q
\right] \right\} .
The results obtained in this section can be summarized as follows.
Theorem 4.1. Suppose b \in \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n), p(\cdot ) \in \scrB (\BbbR n) and \Omega \in Ls(\BbbS n - 1) with s > (p\prime )+ is
homogeneous of degree zero and satisfies (1). Let \lambda > 0, 0 < q \leq \infty and \alpha (\cdot ) \in L\infty (\BbbR n) be
log-H\"older continuous both at the origin and at infinity such that
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936 M. QU, L. WANG
\lambda - n\delta 1 < \alpha - \leq \alpha + < n\delta 2 - (n - 1)/s,
where 0 < \delta 1, \delta 2 < 1 are the constants appearing in Lemma 2.2. Then the commutator \mu \Omega ,b is
bounded on M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n).
We would like to point out that Theorem 4.1 is still true in the particular case \lambda = 0, namely,
in the framework of Herz spaces with variable exponents. By using the same method of proving
Theorem 4.1, we get the following corollary.
Corollary 4.1. Suppose b \in \mathrm{B}\mathrm{M}\mathrm{O}(\BbbR n), p(\cdot ) \in \scrB (\BbbR n) and \Omega \in Ls(\BbbS n - 1) with s > (p\prime )+ is
homogeneous of degree zero and satisfies (1). Let 0 < q \leq \infty and \alpha (\cdot ) \in L\infty (\BbbR n) be log-H\"older
continuous both at the origin and at infinity such that
- n\delta 1 < \alpha - \leq \alpha + < n\delta 2 - (n - 1)/s,
where 0 < \delta 1, \delta 2 < 1 are the constants appearing in Lemma 2.2. Then the commutator \mu \Omega ,b is
bounded on \.K
\alpha (\cdot )
p(\cdot ),q(\BbbR
n).
Proof of Theorem 4.1. Let f \in M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n). We decompose
f(x) =
\infty \sum
j= - \infty
f(x)\chi j(x) =
\infty \sum
j= - \infty
fj(x).
The Minkowski inequality implies that
\| \mu \Omega ,b(f)\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
= \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )| \mu \Omega ,b(f)| \chi k
\bigm\| \bigm\| \bigm\| q
Lp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )
\left( k - 2\sum
j= - \infty
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )
\left( k+1\sum
j=k - 1
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )
\left( \infty \sum
j=k+2
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
:=
:= V1 + V2 + V3.
For V1, noticing that | x - y| \approx | x| \approx 2k for x \in Rk, y \in Rj and j \leq k - 2, then we have
| \mu \Omega ,b(fj)(x)| \lesssim
\left( | x| \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
| x - y| \leq t
(b(x) - b(y))
\Omega (x - y)
| x - y| n - 1
fj(y)dy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
dt
t3
\right)
1
2
+
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ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 937
+
\left( \infty \int
| x|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
| x - y| \leq t
(b(x) - b(y))
\Omega (x - y)
| x - y| n - 1
fj(y)dy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
dt
t3
\right)
1
2
\lesssim
\lesssim
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)|
| x - y| n - 1
| y|
1
2
| x - y|
3
2
dy+
+
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)|
| x - y| n - 1
1
| x|
dy \lesssim
\lesssim 2 - kn
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)| dy.
This together with Lemma 2.5 yields
2k\alpha (x)
k - 2\sum
j= - \infty
| \mu \Omega ,b(fj)(x)| \chi k(x) \lesssim
\lesssim
k - 2\sum
j= - \infty
2 - kn
\int
Rj
2k\alpha (x)| \Omega (x - y)| | b(x) - b(y)| | fj(y)| dy \cdot \chi k(x) \lesssim
\lesssim
k - 2\sum
j= - \infty
2 - kn2(k - j)\alpha +
\int
Rj
2j\alpha (y)| \Omega (x - y)| | b(x) - b(y)| | fj(y)| dy \cdot \chi k(x) \lesssim
\lesssim
k - 2\sum
j= - \infty
2 - kn2(k - j)\alpha +
\left( | b(x) - bBj |
\int
Rj
2j\alpha (y)| \Omega (x - y)| | fj(y)| dy+
+
\int
Rj
2j\alpha (y)| bBj - b(y)| | \Omega (x - y)| | fj(y)| dy
\right) \cdot \chi k(x). (10)
Define a variable exponent p\ast (\cdot ) by
1
p\prime (x)
=
1
p\ast (x)
+
1
s
, since s > (p\prime )+, using the fact that
\| \chi Bj\| Lp\ast (\cdot )(\BbbR n) \approx \| \chi Bj\| Lp\prime (\cdot )(\BbbR n)| Bj | - 1/s (see [28, p. 258]) and Lemma 2.4, we obtain
\| \Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n) \lesssim \| \Omega (x - \cdot )\chi j\| Ls(\BbbR n)\| \chi j\| Lp\ast (\cdot )(\BbbR n) \lesssim
\lesssim
\left( | x| +2j\int
| x| - 2j
\int
\BbbS n - 1
| \Omega (y\prime )| sd\sigma (y\prime )\varrho n - 1d\varrho
\right)
1
s
\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)| Bj | - 1/s \lesssim
\lesssim 2(k - j)(n - 1)/s\| \chi Bj\| Lp\prime (\cdot )(\BbbR n). (11)
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938 M. QU, L. WANG
An application of Lemmas 2.3, 2.4 and (11) gives
\| (b - bBj )\Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n) \lesssim \| (b - bBj )\chi j\| Lp\ast (\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Ls(\BbbR n) \lesssim
\lesssim \| b\| \ast \| \chi Bj\| Lp\ast (\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Ls(\BbbR n) \lesssim 2(k - j)(n - 1)/s\| \chi Bj\| Lp\prime (\cdot )(\BbbR n). (12)
Then, from (10) – (12), (4) and Lemmas 2.1 – 2.3, we deduce\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )
k - 2\sum
j= - \infty
| \mu \Omega ,b(fj)| \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
Lp(\cdot )(\BbbR n)
\lesssim
\lesssim
k - 2\sum
j= - \infty
2 - kn2(k - j)\alpha +\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n)
\Bigl(
\| (b - bBj )\chi k\| Lp(\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n) +
+ \| (b - bBj )\Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n)\| \chi k\| Lp(\cdot )(\BbbR n)
\Bigr)
\lesssim
\lesssim
k - 2\sum
j= - \infty
2 - kn2(k - j)\alpha +\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n)
\Bigl(
(k - j)2(k - j)(n - 1)/s\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)\| \chi Bk
\| Lp(\cdot )(\BbbR n) +
+ 2(k - j)(n - 1)/s\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)\| \chi Bk
\| Lp(\cdot )(\BbbR n)
\Bigr)
\lesssim
\lesssim
k - 2\sum
j= - \infty
(k - j)2 - kn2(k - j)(\alpha +(n - 1)/s)\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n)\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)\| \chi Bk
\| Lp(\cdot )(\BbbR n) \lesssim
\lesssim
k - 2\sum
j= - \infty
(k - j)2(k - j)(\alpha ++(n - 1)/s)\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n)
\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)
\| \chi Bk
\| Lp\prime (\cdot )(\BbbR n)
\lesssim
\lesssim
k - 2\sum
j= - \infty
(k - j)2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n).
Therefore, we arrive at the estimate
V1 \thickapprox \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\| 2k\alpha (\cdot )
\left( k - 2\sum
j= - \infty
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( k - 2\sum
j= - \infty
(k - j)2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)\| 2j\alpha (\cdot )fj\| Lp(\cdot )(\BbbR n)
\right) q
.
Now we can distinguish two cases as follows:
Case 1\circ . If 0 < q \leq 1, using the well-known inequality\left( \infty \sum
j=1
aj
\right) q
\leq
\infty \sum
j=1
aqj , aj > 0, j = 1, 2, . . . , (13)
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ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 939
we obtain
V1 \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
k - 2\sum
j= - \infty
(k - j)q2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)q\| 2j\alpha (\cdot )fj\| qLp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0 - 2\sum
j= - \infty
\| 2j\alpha (\cdot )fj\| qLp(\cdot )(\BbbR n)
k0\sum
k=j+2
(k - j)q2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)q \lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0 - 2\sum
j= - \infty
\| 2j\alpha (\cdot )fj\| qLp(\cdot )(\BbbR n)
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
Case 2\circ . If 1 < q < \infty , the H\"\mathrm{o}lder inequality implies that
V1 \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( k - 2\sum
j= - \infty
2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)q/2\| 2j\alpha (\cdot )fj\| qLp(\cdot )(\BbbR n)
\right) \times
\times
\left( k - 2\sum
j= - \infty
(k - j)q
\prime
2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)q\prime /2
\right) q/q\prime
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0\in \BbbZ
2 - k0\lambda q
k0 - 2\sum
j= - \infty
\| 2j\alpha (\cdot )fj\| qLp(\cdot )(\BbbR n)
k0\sum
k=j+2
2(j - k)(n\delta 2 - \alpha + - (n - 1)/s)q/2 \lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
We proceed now to estimate V2. By Proposition 4.1 and the Lp(\cdot )(\BbbR n)-boundedness of the com-
mutator \mu \Omega ,b, we get
V2 \approx max
\left\{ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( k+1\sum
j=k - 1
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
,
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0,k0\in \BbbZ
\left[ 2 - k0\lambda q
- 1\sum
k= - \infty
2k\alpha (0)q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( k+1\sum
j=k - 1
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
+
+2 - k0\lambda q
k0\sum
k=0
2k\alpha \infty q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( k+1\sum
j=k - 1
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
\right]
\right\} \lesssim
\lesssim max
\Biggl\{
\mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\bigm\| \bigm\| \bigm\| 2k\alpha \infty | f\chi k|
\bigm\| \bigm\| \bigm\| q
Lp(\cdot )(\BbbR n)
,
\mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0,k0\in \BbbZ
\Biggl[
2 - k0\lambda q
- 1\sum
k= - \infty
\bigm\| \bigm\| \bigm\| 2k\alpha \infty | f\chi k|
\bigm\| \bigm\| \bigm\| q
Lp(\cdot )(\BbbR n)
+ 2 - k0\lambda q
k0\sum
k=0
\bigm\| \bigm\| \bigm\| 2k\alpha \infty | f\chi k|
\bigm\| \bigm\| \bigm\| q
Lp(\cdot )(\BbbR n)
\Biggr] \Biggr\}
\lesssim
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940 M. QU, L. WANG
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
For V3, once again by Proposition 4.1, we have
V3 \approx max\{ E,F\} ,
where
E = \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( \infty \sum
j=k+2
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
,
F = \mathrm{s}\mathrm{u}\mathrm{p}
k0\geq 0,k0\in \BbbZ
\left\{ 2 - k0\lambda q
- 1\sum
k= - \infty
2k\alpha (0)q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( \infty \sum
j=k+2
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
+
+2 - k0\lambda q
k0\sum
k=0
2k\alpha \infty q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( \infty \sum
j=k+2
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
\right\} .
For E, noticing that | x - y| \approx | y| \approx 2j for x \in Rk, y \in Rj and j \geq k + 2, then we obtain
| \mu \Omega ,b(fj)(x)| \lesssim
\left( | y| \int
0
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
| x - y| \leq t
(b(x) - b(y))
\Omega (x - y)
| x - y| n - 1
fj(y)dy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
dt
t3
\right)
1
2
+
+
\left( \infty \int
| y|
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\int
| x - y| \leq t
(b(x) - b(y))
\Omega (x - y)
| x - y| n - 1
fj(y)dy
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
2
dt
t3
\right)
1
2
\lesssim
\lesssim
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)|
| x - y| n - 1
| x|
1
2
| x - y|
3
2
dy+
+
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)|
| x - y| n - 1
1
| y|
dy \lesssim
\lesssim 2 - jn
\int
Rj
| b(x) - b(y)| | \Omega (x - y)| | fj(y)| dy \lesssim
\lesssim 2 - jn\| fj\| Lp(\cdot )(\BbbR n)
\Bigl(
| b(x) - bBk
| \| \Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n)+
+ \| \Omega (x - \cdot )(b - bBk
)\chi j\| Lp\prime (\cdot )(\BbbR n)
\Bigr)
. (14)
Similarly to (11), we get
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ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 941
\| \Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n) \lesssim \| \Omega (x - \cdot )\chi j\| Ls(\BbbR n)\| \chi j\| Lp\ast (\cdot )(\BbbR n) \lesssim
\lesssim
\left( 2j+1\int
0
\int
\BbbS n - 1
| \Omega (y\prime )| sd\sigma (y\prime )\varrho n - 1d\varrho
\right)
1
s
\| \chi Bj\| Lp\prime (\cdot )(\BbbR n)| Bj | - 1/s \lesssim
\lesssim \| \Omega \| Ls(\BbbS n - 1)\| \chi Bj\| Lp\prime (\cdot )(\BbbR n), (15)
which in conjunction with Lemma 2.4 implies
\| (b - bBk
)\Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n) \lesssim \| (b - bBk
)\chi j\| Lp\ast (\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Ls(\BbbR n) \lesssim
\lesssim (j - k)\| b\| \ast \| \chi Bj\| Lp\ast (\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Ls(\BbbR n) \lesssim
\lesssim (j - k)\| \chi Bj\| Lp\prime (\cdot )(\BbbR n). (16)
Now from (14) – (16) and Lemmas 2.1 – 2.3, we obtain
\| \mu \Omega ,b(fj)\chi k\| Lp(\cdot )(\BbbR n) \lesssim 2 - jn\| fj\| Lp(\cdot )(\BbbR n)
\Bigl(
\| (b - bBk
)\chi k\| Lp(\cdot )(\BbbR n)\| \Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n)+
+ \| (b - bBk
)\Omega (x - \cdot )\chi j\| Lp\prime (\cdot )(\BbbR n)\| \chi k\| Lp(\cdot )(\BbbR n)
\Bigr)
\lesssim
\lesssim (j - k)2 - jn\| fj\| Lp(\cdot )(\BbbR n)\| \chi Bk
\| Lp(\cdot )(\BbbR n)\| \chi Bj\| Lp\prime (\cdot )(\BbbR n) \lesssim
\lesssim (j - k)\| fj\| Lp(\cdot )(\BbbR n)
\| \chi Bk
\| Lp(\cdot )(\BbbR n)
\| \chi Bj\| Lp(\cdot )(\BbbR n)
\lesssim
\lesssim (j - k)2(k - j)n\delta 1\| fj\| Lp(\cdot )(\BbbR n).
Consequently, we have
E = \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
\left( \infty \sum
j=k+2
| \mu \Omega ,b(fj)|
\right) \chi k
\bigm\| \bigm\| \bigm\| \bigm\| \bigm\| \bigm\|
q
Lp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\left( \infty \sum
j=k+2
(j - k)2(k - j)n\delta 1\| fj\| Lp(\cdot )(\BbbR n)
\right) q
.
If 0 < q \leq 1, by (13), we get
E \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
k0 - 1\sum
j=k+2
(j - k)q2(k - j)n\delta 1q\| fj\| qLp(\cdot )(\BbbR n)
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\infty \sum
j=k0
(j - k)q2(k - j)n\delta 1q\| fj\| qLp(\cdot )(\BbbR n)
:=
:= E1 + E2.
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
942 M. QU, L. WANG
For E1, in view of n\delta 1 + \alpha (0) > n\delta 1 + \alpha - > 0, we obtain
E1 \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0 - 1\sum
j= - \infty
2j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
j - 2\sum
k= - \infty
(j - k)q2(k - j)(n\delta 1+\alpha (0))q \lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0 - 1\sum
j= - \infty
2j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
For E2, noting that \alpha (0) + n\delta 1 - \lambda > \alpha - + n\delta 1 - \lambda > 0, we get
E2 \approx \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\infty \sum
j=k0
(j - k)q2(k - j)(n\delta 1+\alpha (0))q2j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\infty \sum
j=k0
(j - k)q2(k - j)(n\delta 1+\alpha (0))q2j\lambda q\times
\times 2 - j\lambda q
j\sum
l= - \infty
2l\alpha (0)q\| fl\| qLp(\cdot )(\BbbR n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\lambda q
\infty \sum
j=k0
(j - k)q2(k - j)(n\delta 1+\alpha (0) - \lambda )q\| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
\Biggl(
k0\sum
k= - \infty
2k\lambda q
\Biggr)
\| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
If 1 < q < \infty , we have
E \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\left( k0 - 1\sum
j=k+2
(j - k)2(k - j)n\delta 1\| fj\| Lp(\cdot )(\BbbR n)
\right) q
+
+ \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\alpha (0)q
\left( \infty \sum
j=k0
(j - k)2(k - j)n\delta 1\| fj\| Lp(\cdot )(\BbbR n)
\right) q
:=
:= E3 + E4.
For E3, the H\"\mathrm{o}lder inequality yields
E3 \approx \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( k0 - 1\sum
j=k+2
(j - k)2(k - j)(n\delta 1+\alpha (0))2j\alpha (0)\| fj\| Lp(\cdot )(\BbbR n)
\right) q
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( k0 - 1\sum
j=k+2
2(k - j)(n\delta 1+\alpha (0))q/22j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
\right) \times
ISSN 1027-3190. Укр. мат. журн., 2020, т. 72, № 7
ON THE COMMUTATOR OF MARCINKIEWICZ INTEGRALS WITH ROUGH KERNELS . . . 943
\times
\left( k0 - 1\sum
j=k+2
(j - k)q
\prime
2(k - j)(n\delta 1+\alpha (0))q\prime /2
\right) q/q\prime
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0 - 1\sum
j= - \infty
2j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
j - 2\sum
k= - \infty
2(k - j)(n\delta 1+\alpha (0))q/2 \lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0 - 1\sum
j= - \infty
2j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
For E4, as argued for E2, we obtain
E4 \lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( \infty \sum
j=k0
2(k - j)(n\delta 1+\alpha (0)+\lambda )q/22j\alpha (0)q\| fj\| qLp(\cdot )(\BbbR n)
\right) \times
\times
\left( \infty \sum
j=k0
(j - k)q
\prime
2(k - j)(n\delta 1+\alpha (0) - \lambda )q\prime /2
\right) q/q\prime
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
\left( \infty \sum
j=k0
2(k - j)(n\delta 1+\alpha (0)+\lambda )q/22j\lambda q2 - j\lambda q
j\sum
l= - \infty
2l\alpha (0)q\| fl\| qLp(\cdot )(\BbbR n)
\right) \lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
k0\sum
k= - \infty
2k\lambda q
\left( \infty \sum
j=k0
2(k - j)(n\delta 1+\alpha (0) - \lambda )q/2
\right) \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
\lesssim
\lesssim \mathrm{s}\mathrm{u}\mathrm{p}
k0<0,k0\in \BbbZ
2 - k0\lambda q
\Biggl(
k0\sum
k= - \infty
2k\lambda q
\Biggr)
\| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
\lesssim \| f\| q
M \.K
\alpha (\cdot ),\lambda
q,p(\cdot ) (\BbbR
n)
.
We omit the estimation of F since it is essentially similar to that of E.
Theorem 4.1 is proved.
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| id | umjimathkievua-article-6023 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:25:22Z |
| publishDate | 2020 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/fe/c544712fe23e05d0c639be7f4e0875fe.pdf |
| spelling | umjimathkievua-article-60232022-03-26T11:01:57Z On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces Qu, M. Wang, L. Qu, M. Wang, L. UDC 517.5 In the framework of variable exponent Morrey and Morrey–Herz spaces, we prove some boundedness results for the commutator of Marcinkiewicz integrals with rough kernels. The approach is based on the theory of variable exponent and on generalization of the BMO-norms. УДК 517.5 Про комутатор iнтегралiв Марцинкевича з грубими ядрами у змiнних просторах типу Моррея У рамках змiнних експонент просторiв Морi та Морi–Герца доведено деякi результати стосовно обмеженостi комутатора iнтегралiв Марцинкевича з грубими ядрами. Цей пiдхiд базується на теорiї змiнних експонент та узагальненнi норм обмежених усереднених осциляцiй. Institute of Mathematics, NAS of Ukraine 2020-07-15 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6023 10.37863/umzh.v72i7.6023 Ukrains’kyi Matematychnyi Zhurnal; Vol. 72 No. 7 (2020); 928-944 Український математичний журнал; Том 72 № 7 (2020); 928-944 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6023/8729 |
| spellingShingle | Qu, M. Wang, L. Qu, M. Wang, L. On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_alt | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_full | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_fullStr | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_full_unstemmed | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_short | On the commutator of Marcinkiewicz integrals with rough kernels in variable Morrey type spaces |
| title_sort | on the commutator of marcinkiewicz integrals with rough kernels in variable morrey type spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6023 |
| work_keys_str_mv | AT qum onthecommutatorofmarcinkiewiczintegralswithroughkernelsinvariablemorreytypespaces AT wangl onthecommutatorofmarcinkiewiczintegralswithroughkernelsinvariablemorreytypespaces AT qum onthecommutatorofmarcinkiewiczintegralswithroughkernelsinvariablemorreytypespaces AT wangl onthecommutatorofmarcinkiewiczintegralswithroughkernelsinvariablemorreytypespaces |