Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators
UDC 517.9 We establish the sharp boundedness of $p$-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of $p$-adic multilinear Hausdorff opera...
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| author | Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. |
| author_facet | Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. |
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| description | UDC 517.9
We establish the sharp boundedness of $p$-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of $p$-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained. |
| doi_str_mv | 10.37863/umzh.v73i7.441 |
| first_indexed | 2026-03-24T02:02:45Z |
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DOI: 10.37863/umzh.v73i7.441
UDC 517.9
N. M. Chuong (Inst. Math., Vietnam. Acad. Sci. and Technology, Hanoi, Vietnam),
D. V. Duong (School Math., Mientrung Univ. Civil Engineering, Phuyen, Vietnam),
K. H. Dung (Van Lang Univ., Ho Chi Minh City, Vietnam)
WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES
FOR P-ADIC MULTILINEAR HAUSDORFF OPERATORS
AND ITS COMMUTATORS
ЗВАЖЕНI ОЦIНКИ ДЛЯ \bfitP -АДИЧНИХ БАГАТОЛIНIЙНИХ
ГАУСДОРФОВИХ ОПЕРАТОРIВ ТА ЇХНIХ КОМУТАТОРIВ
НА ПРОСТОРАХ ЛЕБЕГА I ЦЕНТРАЛЬНИХ ПРОСТОРАХ МОРРI
We establish the sharp boundedness of p-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey
spaces associated with both power weights and Muckenhoupt weights. Moreover, the boundedness for the commutators of
p-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained.
Встановлено точну обмеженiсть p-адичних багатолiнiйних гаусдорфових операторiв на добутку просторiв Лебега
i центральних просторiв Моррi, асоцiйованих як з вагами степенiв, так i з вагами Макенхаупта. Також доведено
обмеженiсть комутаторiв p-адичних багатолiнiйних гаусдорфових операторiв на таких просторах iз символами в
центральному BMO-просторi.
1. Introduction. The p-adic analysis in the past decades has received a lot of attention due to its
important applications in mathematical physics as well as its necessity in sciences and technologies
(see, e.g., [2 – 4, 10, 20 – 22, 28 – 31] and the references therein). It is known that the theory of
functions from \BbbQ p into \BbbC play an important role in p-adic quantum mechanics, the theory of p-
adic probability in which real-valued random variables have to be considered to solve covariance
problems. In recent years, there is an increasing interest in the study of harmonic analysis and
wavelet analysis over the p-adic fields (see, e.g., [1, 4, 8, 18, 19, 22]).
It is crucial that the Hausdorff operator is one of the important operators in harmonic analysis. It
is closely related to the summability of the classical Fourier series (see, for instance, [11, 13, 15] and
the references therein). Let \Phi be a locally integrable function on \BbbR n. The matrix Hausdorff operator
H\Phi ,A associated to the kernel function \Phi is then defined by
H\Phi ,A(f)(x) =
\int
\BbbR n
\Phi (y)
| y| n
f(A(y)x)dy, x \in \BbbR n,
where A(y) is an n \times n invertible matrix for almost everywhere y in the support of \Phi . It is worth
pointing out that if the kernel function \Phi is chosen appropriately, then the Hausdorff operator reduces
to many classcial operators in analysis such as the Hardy operator, the Cesàro operator, the Riemann –
Liouville fractional integral operator and the Hardy – Littlewood average operator.
In 2010, Volosivets [32] introduced the matrix Hausdorff operator on the p-adic numbers field as
follows:
\scrH \varphi ,A(f)(x) =
\int
\BbbQ n
p
\varphi (t)f(A(t)x)dt, x \in \BbbQ n
p ,
c\bigcirc N. M. CHUONG, D. V. DUONG, K. H. DUNG, 2021
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7 979
980 N. M. CHUONG, D. V. DUONG, K. H. DUNG
where \varphi (t) is a locally integrable function on \BbbQ n
p and A(t) is an n\times n invertible matrix for almost
everywhere t in the support of \varphi . It is easy to see that if \varphi (t) = \psi (t1)\chi \BbbZ \ast
p
n(t) and A(t) = t1.In
(In is an identity matrix) for t = (t1, t2, . . . , tn), where \psi : \BbbQ p \rightarrow \BbbC is a measurable function, then
\scrH \varphi ,A reduces to the p-adic weighted Hardy – Littlewood average operator due to Rim and Lee [26].
In recent years, the theory of the Hardy operators, the Hausdorff operators over the p-adic
numbers field has been significantly developed into different contexts, and they are actually useful
for p-adic analysis (see, e.g., [5, 6, 14, 33]). It is known that the authors in [7] also introduced and
studied a general class of multilinear Hausdorff operators on the real field defined by
\scrH \Phi , \vec{}A(
\vec{}f)(x) =
\int
\BbbR n
\Phi (y)
| y| n
m\prod
i=1
fi(Ai(y)x)dy, x \in \BbbR n,
for \vec{}f = (f1, . . . , fm) and \vec{}A = (A1, . . . , Am).
Motivated by above results, in this paper we shall introduce and study a class of p-adic multilinear
(matrix) Hausdorff operators defined as follows.
Definition 1.1. Let \Phi be a measurable complex-valued function on \BbbQ n
p . The p-adic multilinear
Hausdorff operator is defined by
\scrH p
\Phi , \vec{}A
(\vec{}f)(x) =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
fi(Ai(y)x)dy, x \in \BbbQ n
p ,
where \vec{}f = (f1, . . . , fm) and f1, f2, . . . , fm are measurable complex-valued functions on \BbbQ n
p .
Note that in this paper we will confine our attention to the case, where \Phi is the nonnegative
function.
Let b be a measurable function. The operator \scrM b is defined by \scrM bf(x) = b(x)f(x) for any
measurable function f. If \scrH is a linear operator on some measurable function space, the com-
mutator of Coifman – Rochberg – Weiss type formed by \scrM b and \scrH is defined by [\scrM b,\scrH ]f(x) =
= (\scrM b\scrH - \scrH \scrM b)f(x). Similarly, the commutators of p-adic multilinear Hausdorff operator is
defined as follows.
Definition 1.2. Let \Phi , \vec{}A be as above. The Coifman – Rochberg – Weiss type commutator of p-
adic multilinear Hausdorff operator is defined by
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr)
(x) =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
(bi(x) - bi(Ai(y)x))
m\prod
i=1
fi(Ai(y)x)dy,
where x \in \BbbQ n
p ,
\vec{}b = (b1, . . . , bm) and bi are locally integrable functions on \BbbQ n
p for all i = 1, . . . ,m.
The main purpose of this paper is to study the p-adic multilinear Hausdorff operators and its
commutators on the p-adic numbers field. More precisely, we obtain the necessary and sufficient
conditions for the boundedness of \scrH p
\Phi , \vec{}A
and \scrH p
\Phi , \vec{}A,\vec{}b
on the product of Lebesgue and central Morrey
spaces with weights on p-adic field. In each case, we estimate the corresponding operator norms.
Moreover, the boundedness of \scrH p
\Phi , \vec{}A,\vec{}b
on the such spaces with symbols in central BMO space is also
established. It should be pointed out that all our results are new even in the case of p-adic linear
Hausdorff operators.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 981
This paper is organized as follows. In Section 2, we present some notations and preliminaries
about p-adic analysis as well as give some definitions of the Lebesgue and central Morrey spaces
associated with power weights and Muckenhoupt weights. Main theorems are given and proved in
Sections 3 and 4.
2. Some notations and definitions. For a prime number p, let \BbbQ p be the field of p-adic
numbers. This field is the completion of the field of rational numbers \BbbQ with respect to the non-
Archimedean p-adic norm | \cdot | p. This norm is defined as follows: if x = 0, then | 0| p = 0; if x \not = 0 is
an arbitrary rational number with the unique representation x = p\alpha
m
n
, where m, n are not divisible
by p, \alpha = \alpha (x) \in \BbbZ , then | x| p = p - \alpha . This norm satisfies the following properties: | x| p \geq 0
\forall x \in \BbbQ p and | x| p = 0 \leftrightarrow x = 0; | xy| p = | x| p| y| p \forall x, y \in \BbbQ p; and | x + y| p \leq \mathrm{m}\mathrm{a}\mathrm{x}(| x| p, | y| p)
\forall x, y \in \BbbQ p, and when | x| p \not = | y| p, we have | x+ y| p = \mathrm{m}\mathrm{a}\mathrm{x}(| x| p, | y| p).
It is also known that any non-zero p-adic number x \in \BbbQ p can be uniquely represented in the
canonical series
x = p\alpha
\bigl(
x0 + x1p+ x2p
2 + . . .
\bigr)
,
where \alpha = \alpha (x) \in \BbbZ , xk \in \{ 0, 1, . . . , p - 1\} , x0 \not = 0, k = 0, 1, . . . . This series converges
in the p-adic norm since
\bigm| \bigm| xkpk\bigm| \bigm| p \leq p - k. The space \BbbQ n
p = \BbbQ p \times . . . \times \BbbQ p consists of all points
x = (x1, . . . , xn), where xi \in \BbbQ p, i = 1, . . . , n, n \geq 1. The p-adic norm of \BbbQ n
p is defined by
| x| p = \mathrm{m}\mathrm{a}\mathrm{x}1\leq j\leq n | xj | p. Let A be an n\times n matrix with entries aij \in \BbbQ p. For x = (x1, . . . , xn) \in \BbbQ n
p ,
we denote
Ax =
\left( n\sum
j=1
a1jxj , . . . ,
n\sum
j=1
anjxj
\right) .
By Lemma 2 in paper [33], the norm of A is \| A\| p := \mathrm{m}\mathrm{a}\mathrm{x}1\leq i\leq n\mathrm{m}\mathrm{a}\mathrm{x}1\leq j\leq n | aij | p. For simplicity
of notation, we write kA = \mathrm{l}\mathrm{o}\mathrm{g}p\| A\| p. It is clear to see that kA \in \BbbZ . It is easy to show that
| Ax| p \leq \| A\| p.| x| p for any x \in \BbbQ n
p . In addition, if A is invertible, by the same arguments as the real
setting (see also Lemma 3.1 [25] for the setting of the Heisenberg group), we get
\| A\| - n
p \leq
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t} \bigl( A - 1
\bigr) \bigm| \bigm|
p
\leq
\bigm\| \bigm\| A - 1
\bigm\| \bigm\| n
p
. (2.1)
Let B\alpha (a) =
\bigl\{
x \in \BbbQ n
p : | x - a| p \leq p\alpha
\bigr\}
be a ball of radius p\alpha with center at a \in \BbbQ n
p . Similarly,
denote by S\alpha (a) =
\bigl\{
x \in \BbbQ n
p : | x - a| p = p\alpha
\bigr\}
the sphere with center at a \in \BbbQ n
p and radius p\alpha .
If B\alpha = B\alpha (0), S\alpha = S\alpha (0), then, for any x0 \in \BbbQ n
p , we have x0 + B\alpha = B\alpha (x0) and x0 +
+ S\alpha = S\alpha (x0). Since \BbbQ n
p is a locally compact commutative group under addition, it follows from
the standard theory that there exists a Haar measure dx on \BbbQ n
p , which is unique up to positive
constant multiple and is translation invariant. This measure is unique by normalizing dx such that\int
B0
dx = | B0| = 1, where | B| denotes the Haar measure of a measurable subset B of \BbbQ n
p . By
simple calculation, it is easy to obtain that | B\alpha (a)| = pn\alpha , | S\alpha (a)| = pn\alpha (1 - p - n) \simeq pn\alpha for any
a \in \BbbQ n
p . For f \in L1
loc
\bigl(
\BbbQ n
p
\bigr)
, we have\int
\BbbQ n
p
f(x)dx = \mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow +\infty
\int
B\alpha
f(x)dx = \mathrm{l}\mathrm{i}\mathrm{m}
\alpha \rightarrow +\infty
\sum
- \infty <\gamma \leq \alpha
\int
S\gamma
f(x)dx.
In particular, if f \in L1
\bigl(
\BbbQ n
p
\bigr)
, we can write
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
982 N. M. CHUONG, D. V. DUONG, K. H. DUNG
\int
\BbbQ n
p
f(x)dx =
+\infty \sum
\alpha = - \infty
\int
S\alpha
f(x)dx and
\int
\BbbQ n
p
f(tx)dx =
1
| t| np
\int
\BbbQ n
p
f(x)dx,
where t \in \BbbQ p \setminus \{ 0\} . For a more complete introduction to the p-adic analysis, we refer the readers to
[20, 31] and the references therein.
Let \omega be a weighted function, that is a nonnegative locally integrable measurable function on \BbbQ n
p .
The weighted Lebesgue space Lq
\omega
\bigl(
\BbbQ n
p
\bigr)
, 0 < q < \infty , is defined to be the space of all measurable
functions f on \BbbQ n
p such that
\| f\| Lq
\omega (\BbbQ n
p )
=
\left( \int
\BbbQ n
p
| f(x)| q\omega (x)dx
\right)
1/q
<\infty .
The space Lq
\omega ,loc
\bigl(
\BbbQ n
p
\bigr)
is defined as the set of all measurable functions f on \BbbQ n
p satisfying\int
K
| f(x)| q\omega (x)dx <\infty
for any compact subset K of \BbbQ n
p . The space Lq
\omega ,loc
\bigl(
\BbbQ n
p \setminus \{ 0\}
\bigr)
is also defined in a similar way as the
space Lq
\omega ,loc
\bigl(
\BbbQ n
p
\bigr)
. Throught the whole paper, we denote by C a positive constant that is independent
of the main parameters, but can change from line to line. We also write a \lesssim b to mean that there is
a positive constant C, independent of the main parameters, such that a \leq Cb. The symbol f \simeq g
means that f is equivalent to g (i.e., C - 1f \leq g \leq Cf ). For any real number \ell > 1, denote by \ell \prime
conjugate real number of \ell , i.e.,
1
\ell
+
1
\ell \prime
= 1. Denote \omega (B)\lambda =
\biggl( \int
B
\omega (x)dx
\biggr) \lambda
for \lambda \in \BbbR . Remark
that if \omega (x) = | x| \alpha p for \alpha > - n, then we have
\omega (B\gamma ) =
\int
B\gamma
| x| \alpha pdx =
\sum
k\leq \gamma
\int
Sk
pk\alpha dx =
\sum
k\leq \gamma
pk(\alpha +n)
\bigl(
1 - p - n
\bigr)
\simeq p\gamma (\alpha +n). (2.2)
Next, let us give the definition of weighted \lambda -central Morrey spaces on p-adic numbers field as
follows.
Definition 2.1. Let \lambda \in \BbbR and 1 < q < \infty . The weighted \lambda -central Morrey p-adic spaces
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
consists of all functions f \in Lq
\omega ,loc
\bigl(
\BbbQ n
p
\bigr)
satisfying \| f\| .
B
q,\lambda
\omega (\BbbQ n
p)
<\infty , where
\| f\| .
B
q,\lambda
\omega (\BbbQ n
p)
= \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
\left( 1
\omega (B\gamma )1+\lambda q
\int
B\gamma
| f(x)| q\omega (x)dx
\right)
1/q
.
Remark that
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
is a Banach space and reduces to \{ 0\} when \lambda < - 1
q
.
Let us recall the definition of the weighted central BMO p-adic space of John – Nirenberg type.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 983
Definition 2.2. Let 1 \leq q < \infty and \omega be a weight function. The weighted central bounded
mean oscillation space CMOq
\omega
\bigl(
\BbbQ n
p
\bigr)
is defined as the set of all functions f \in Lq
\omega ,loc
\bigl(
\BbbQ n
p
\bigr)
such that
\| f\| CMOq
\omega (\BbbQ n
p)
= \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
\left( 1
\omega (B\gamma )
\int
B\gamma
| f(x) - fB\gamma | q\omega (x)dx
\right)
1
q
<\infty ,
where
fB\gamma =
1
| B\gamma |
\int
B\gamma
f(x)dx.
The theory of A\ell weight was first introduced by Benjamin Muckenhoupt on the Euclidean spaces
in order to characterise the boundedness of Hardy – Littlewood maximal functions on the weighted
L\ell spaces (see [24]). For A\ell weights on the p-adic fields, more generally, on the local fields or
homogeneous type spaces, one can refer to [9, 16] for more details. Let us now recall the definition
of A\ell weights.
Definition 2.3. Let 1 < \ell < \infty . It is said that a nonnegative locally integrable function \omega \in
\in A\ell
\bigl(
\BbbQ n
p
\bigr)
if there exists a constant C such that, for all balls B, we have\left( 1
| B|
\int
B
\omega (x)dx
\right) \left( 1
| B|
\int
B
\omega (x) - 1/(\ell - 1)dx
\right) \ell - 1
\leq C.
It is said that a weight \omega \in A1
\bigl(
\BbbQ n
p
\bigr)
if there is a constant C such that, for all balls B, we get
1
| B|
\int
B
\omega (x)dx \leq C \mathrm{e}\mathrm{s}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{f}
x\in B
\omega (x).
We denote by A\infty
\bigl(
\BbbQ n
p
\bigr)
=
\bigcup
1\leq \ell <\infty A\ell
\bigl(
\BbbQ n
p
\bigr)
. Let us give the following standard result related
to the Muckenhoupt weights.
Proposition 2.1. (i) A\ell
\bigl(
\BbbQ n
p
\bigr)
\subsetneq Aq
\bigl(
\BbbQ n
p
\bigr)
for 1 \leq \ell < q <\infty .
(ii) If \omega \in A\ell
\bigl(
\BbbQ n
p
\bigr)
for 1 < \ell < \infty , then there is an \varepsilon > 0 such that \ell - \varepsilon > 1 and
\omega \in A\ell - \varepsilon
\bigl(
\BbbQ n
p
\bigr)
.
We note that the class A\infty
\bigl(
\BbbQ n
p
\bigr)
is closely connected with the reverse Hölder condition. More
precisely, if there exist r > 1 and a fixed constant C such that\left( 1
| B|
\int
B
\omega (x)rdx
\right) 1/r
\leq C
| B|
\int
B
\omega (x)dx
for all balls B \subset \BbbQ n
p , then we say that \omega satisfies the reverse Hölder condition of order r and write
\omega \in RHr
\bigl(
\BbbQ n
p
\bigr)
. According to Theorem 19 and Corollary 21 in [17], \omega \in A\infty
\bigl(
\BbbQ n
p
\bigr)
if and only
if there exists some r > 1 such that \omega \in RHr
\bigl(
\BbbQ n
p
\bigr)
. Moreover, if \omega \in RHr
\bigl(
\BbbQ n
p
\bigr)
, r > 1, then
\omega \in RHr+\varepsilon
\bigl(
\BbbQ n
p
\bigr)
for some \varepsilon > 0. We thus write r\omega = \mathrm{s}\mathrm{u}\mathrm{p}
\bigl\{
r > 1 : \omega \in RHr
\bigl(
\BbbQ n
p
\bigr) \bigr\}
to denote the
critical index of \omega for the reverse Hölder condition. It is worth noticing that an important example
of A\ell
\bigl(
\BbbQ n
p
\bigr)
weight is the power function | x| \alpha p . By the similar arguments as Propositions 1.4.3 and
1.4.4 in [23], we obtain the following properties of power weights.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
984 N. M. CHUONG, D. V. DUONG, K. H. DUNG
Proposition 2.2. Let x \in \BbbQ n
p . Then we have:
(i) | \cdot | \alpha p \in A1
\bigl(
\BbbQ n
p
\bigr)
if and only if - n < \alpha \leq 0;
(ii) | \cdot | \alpha p \in A\ell
\bigl(
\BbbQ n
p
\bigr)
for 1 < \ell <\infty if and only if - n < \alpha < n(\ell - 1).
Let us give the following standard characterization of A\ell weights which it is proved in the similar
way as the real setting (see [12, 27] for more details).
Proposition 2.3. Let \omega \in A\ell
\bigl(
\BbbQ n
p
\bigr)
\cap RHr
\bigl(
\BbbQ n
p
\bigr)
, \ell \geq 1 and r > 1. Then there exist constants
C1, C2 > 0 such that
C1
\biggl(
| E|
| B|
\biggr) \ell
\leq \omega (E)
\omega (B)
\leq C2
\biggl(
| E|
| B|
\biggr) (r - 1)/r
for any measurable subset E of a ball B.
Proposition 2.4. If \omega \in A\ell
\bigl(
\BbbQ n
p
\bigr)
, 1 \leq \ell < \infty , then, for any f \in L1
loc
\bigl(
\BbbQ n
p
\bigr)
and any ball
B \subset \BbbQ n
p , we have
1
| B|
\int
B
| f(x)| dx \leq C
\left( 1
\omega (B)
\int
B
| f(x)| \ell \omega (x)dx
\right) 1/\ell
.
Let us recall the definition of the Hardy – Littlewood maximal operator
\scrM f(x) = \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
1
pn\gamma
\int
B\gamma (x)
| f(y)| dy.
It is useful to remark that the Hardy – Littlewood maximal operator \scrM is bounded on L\ell
\omega
\bigl(
\BbbQ n
p
\bigr)
if
and only if \omega \in A\ell
\bigl(
\BbbQ n
p
\bigr)
for all \ell > 1. Finally, we introduce a new maximal operator which is used
in the sequel, that is,
\scrM modf(x) = \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
| x| p\leq p\gamma
1
pn\gamma
\int
B\gamma (x)
| f(y)| dy.
3. Main results about the boundness of \bfscrH \bfitp
\bfPhi , \vec{}\bfitA
. Let us now assume that q and qi \in [1,\infty ),
\alpha , \alpha i are real numbers such that \alpha i \in ( - n,\infty ) for i = 1, 2, . . . ,m and
1
q1
+
1
q2
+ . . .+
1
qm
=
1
q
,
\alpha 1
q1
+
\alpha 2
q2
+ . . .+
\alpha m
qm
=
\alpha
q
.
In this section, we will investigate the boundedness of multilinear Hausdorff operators on weighted
Lebesgue spaces and weighted central Morrey spaces associated to the case of matrices having the
important property as follows: there exists \nu \vec{}A \in \BbbN such that
\| Ai(y)\| p.
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\|
p
\leq p\nu \vec{}A for all i = 1, . . . ,m (3.1)
and for almost everywhere y \in \BbbQ n
p . Thus, by the property of invertible matrice, it is easy to show
that
\| Ai(y)\| \sigma p \lesssim
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - \sigma
p
for all \sigma \in \BbbR (3.2)
and
| Ai(y)x| \sigma p \gtrsim
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - \sigma
p
| x| \sigma p for all \sigma \in \BbbR , x \in \BbbQ n
p \setminus \{ 0\} . (3.3)
First main result of this paper is the following.
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 985
Theorem 3.1. Let \omega 1(x) = | x| \alpha 1
p , . . . , \omega m(x) = | x| \alpha m
p and \omega (x) = | x| \alpha p . Then \scrH p
\Phi , \vec{}A
is bounded
from Lq1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times Lqm
\omega m
\bigl(
\BbbQ n
p
\bigr)
to Lq
\omega
\bigl(
\BbbQ n
p
\bigr)
if and only if
\scrC 1 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i+n
qi
p
dy <\infty .
Furthermore,
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\bigm\| \bigm\| \bigm\|
L
q1
\omega 1(\BbbQ n
p)\times ...\times Lqm
\omega m(\BbbQ n
p)\rightarrow Lq
\omega (\BbbQ n
p)
\simeq \scrC 1.
Proof. Firstly, we will prove the sufficiency of the condition \scrC 1 <\infty . By applying the Minkowski
inequality and the Hölder inequality, we have\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (\BbbQ n
p)
\leq
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\| fi(Ai(y).)\| Lqi
\omega i(\BbbQ n
p)
dy.
By using the change of variables, we get
\| fi(Ai(y).)\| Lqi
\omega i(\BbbQ n
p)
\leq \mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} 1
qi
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\| fi\| Lqi
\omega i(\BbbQ n
p)
.
Thus, \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (\BbbQ n
p)
\leq
\leq
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} 1
qi
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
dy
\right) m\prod
i=1
\| fi\| Lqi
\omega i(\BbbQ n
p)
. (3.4)
Note that, by (2.1) and (3.2), we obtain
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} 1
qi
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\lesssim
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| (\alpha i+n)
qi
p
. (3.5)
This shows that \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (\BbbQ n
p)
\lesssim \scrC 1
m\prod
i=1
\| fi\| Lqi
\omega i(\BbbQ n
p)
.
Next, to prove that the condition \scrC 1 <\infty is necessary, let us now take \vec{}fr = (f1r, . . . , fmr), where
fi,r(x) =
\left\{ 0, if | x| p \leq p - \nu \vec{}A
- 1,
| x|
- n
qi
- \alpha i
qi
- p - r
p , otherwise,
for i = 1, . . . ,m and r \in \BbbZ +. By a simple calculation, we have
\| fi,r\| Lqi
\omega i(\BbbQ n
p)
=
\left( \int
\BbbQ n
p
| x| - n - \alpha i - qip
- r
p \chi Bc
- \nu \vec{}A
- 1
(x)| x| \alpha i
p dx
\right)
1
qi
=
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
986 N. M. CHUONG, D. V. DUONG, K. H. DUNG
=
\left( \sum
k\geq - \nu \vec{}A
\int
Sk
pk( - n - qip
- r)dx
\right)
1
qi
\simeq
\left( \sum
k\geq - \nu \vec{}A
pk( - n - qip
- r)pkn
\right) 1
qi
=
=
\left( \sum
k\geq - \nu \vec{}A
p - kqip
- r
\right) 1
qi
=
p\nu \vec{}A
.p - r\bigl(
1 - p - qip - r
\bigr) 1
qi
. (3.6)
Next, we define two sets as follows:
Sx =
m\bigcap
i=1
\bigl\{
y \in \BbbQ n
p : | Ai(y)x| p \geq p - \nu \vec{}A
\bigr\}
and
Ur =
\bigl\{
y \in \BbbQ n
p : \| Ai(y)\| p \geq p - r for all i = 1, . . . ,m
\bigr\}
.
From this we derive
Ur \subset Sx for all x \in \BbbQ n
p \setminus Br - 1. (3.7)
In fact, by letting y \in Ur, we get \| Ai(y)\| p| x| p \geq 1 for all x \in \BbbQ n
p \setminus Br - 1. Thus, by applying the
condition (3.1), one has
| Ai(y)x| p \geq
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - 1
p
| x| p =
\| Ai(y)\| p| x| p\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\|
p
\| Ai(y)\| p
\geq p - \nu \vec{}A ,
which confirms the relation (3.7). Now, by taking x \in \BbbQ n
p \setminus Br - 1 and using the relation (3.7), we
obtain
\scrH p
\Phi , \vec{}A
\Bigl(
\vec{}fr
\Bigr)
(x) \geq
\int
Sx
\Phi (y)
| y| np
m\prod
i=1
| Ai(y)x|
- n
qi
- \alpha i
qi
- p - r
p dy \geq
\int
Ur
\Phi (y)
| y| np
m\prod
i=1
| Ai(y)x|
- n
qi
- \alpha i
qi
- p - r
p dy.
From this, by (3.3), one has
\scrH p
\Phi , \vec{}A
\Bigl(
\vec{}fr
\Bigr)
(x) \gtrsim
\left( \int
Ur
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| n+\alpha i
qi
+p - r
p
dy
\right) | x|
- (n+\alpha )
q
- mp - r
p \times
\times \chi \BbbQ n
p\setminus Br - 1
(x) =: prmp - r\scrA rg(x),
where
\scrA r =
\int
Ur
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| n+\alpha i
qi
p
p - rmp - r
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| p - r
p
dy
and
g(x) = | x|
- (n+\alpha )
q
- mp - r
p \chi \BbbQ n
p\setminus Br - 1
(x).
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 987
By estimating as (3.6) above, we also have \| g\| Lq
\omega (\BbbQ n
p)
\simeq p - rmp - r\bigl(
1 - p - qmp - r
\bigr) 1
q
. As a consequence above,
by (3.6), we find that
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}fr
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (\BbbQ n
p)
\gtrsim \scrA r
\prod m
i=1
\| fi,r\| Lqi
\omega i(\BbbQ n
p)\bigl(
1 - p - qmp - r
\bigr) 1
q
\prod m
i=1
p\nu \vec{}A
p - r\bigl(
1 - p - qip - r
\bigr) 1
qi
=:
=: \scrA r\scrT r
m\prod
i=1
\| fi,r\| Lqi
\omega i(\BbbQ n
p)
,
where
\scrT r =
\prod m
i=1
\Bigl(
1 - p - qip
- r
\Bigr) 1
qi\bigl(
1 - p - qmp - r
\bigr) 1
q pm.\nu \vec{}A
.p - r
.
Note that from
1
q1
+ . . . +
1
qm
=
1
q
, it is clear to obtain that \mathrm{l}\mathrm{i}\mathrm{m}r\rightarrow +\infty \scrT r = a > 0. Therefore,
because \scrH p
\Phi , \vec{}A
is bounded from Lq1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times Lqm
\omega m
\bigl(
\BbbQ n
p
\bigr)
to Lq
\omega
\bigl(
\BbbQ n
p
\bigr)
, there exists M > 0 such
that \scrA r \leq M for sufficiently big r. On the other hand, by letting r sufficiently large, y \in Ur and by
(3.1), we get
p - rmp - r
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| p - r
p
\leq
\Biggl(
m\prod
i=1
\| Ai(y)\| p
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\|
p
\Biggr) p - r
\leq p\nu \vec{}A
.m.p - r
\lesssim 1.
Hence, by the dominated convergence theorem of Lebesgue, we obtain\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i+n
qi
p
dy <\infty .
Theorem 3.1 is proved.
Theorem 3.2. Let 1 \leq q\ast , \zeta < \infty and \omega \in A\zeta with the finite critical index r\omega for the reverse
Hölder inequality and \omega (B\gamma ) \lesssim 1 for all \gamma \in \BbbZ . Assume that q > q\ast \zeta r\omega /(r\omega - 1), \delta \in (1, r\omega ) and
the following condition holds:
\scrC 2 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\| Ai(y)\|
\zeta
qi
p \times
\times
\Biggl(
\chi \{ \| Ai(y)\| p\leq 1\} (y)\| Ai(y)\|
- n\zeta
qi
p + \chi \{ \| Ai(y)\| p>1\} (y)\| Ai(y)\|
- n(\delta - 1)
qi\delta
p
\Biggr)
dy <\infty .
Then we have that \scrH p
\Phi , \vec{}A
is bounded from Lq1
\omega
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times Lqm
\omega
\bigl(
\BbbQ n
p
\bigr)
to Lq\ast
\omega
\bigl(
\BbbQ n
p
\bigr)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
988 N. M. CHUONG, D. V. DUONG, K. H. DUNG
Proof. For any R \in \BbbZ , by the Minkowski inequality, we get
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\leq
\int
\BbbQ n
p
\Phi (y)
| y| np
\left( \int
BR
m\prod
i=1
| fi(Ai(y)x)| q
\ast
\omega (x)dx
\right)
1
q\ast
dy.
From the inequality q > q\ast \zeta r\omega /(r\omega - 1), there exists r \in (1, r\omega ) such that q = \zeta q\ast r\prime . Then, by the
Hölder inequality and the reverse Hölder inequality, we obtain\left( \int
BR
m\prod
i=1
| fi(Ai(y)x)| q
\ast
\omega (x)dx
\right)
1
q\ast
\lesssim
\left( \int
BR
m\prod
i=1
| fi(Ai(y)x)|
q
\zeta dx
\right)
\zeta
q
\omega (BR)
1
q\ast | BR| -
\zeta
q .
Next, by using the Hölder inequality and the change of variables formula, and applying Proposi-
tion 2.4, we have\left( \int
BR
m\prod
i=1
| fi(Ai(y)x)|
q
\zeta dx
\right)
\zeta
q
\lesssim
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\bigm| \bigm| \bigm| BR+kAi
\bigm| \bigm| \bigm| \zeta
qi \omega (BR+kAi
)
- 1
qi \| fi\| Lqi
\omega (BR+kAi
),
where kAi(y) = \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p. Thus, by
\bigm| \bigm| BR+kAi
\bigm| \bigm|
| BR|
\simeq p(R+kAi
)n
pRn
= \| Ai(y)\| np , we infer that\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\lesssim
\lesssim \omega (BR)
1
q\ast
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\| Ai(y)\|
\zeta n
qi
p \omega (BR+kAi
)
- 1
qi \| fi\| Lqi
\omega (BR+kAi
)dy. (3.8)
On the other hand, by q > q\ast \geq 1 and \omega (BR) \lesssim 1 for all R \in \BbbZ , we imply that \omega (BR)
1
q\ast \lesssim
\lesssim \omega (BR)
1
q . Hence, by (3.8), we get \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\lesssim
\lesssim
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\| Ai(y)\|
\zeta n
qi
p
\Biggl(
\omega (BR)
\omega (BR+kAi
)
\Biggr) 1
qi
dy
\right) m\prod
i=1
\| fi\| Lqi
\omega (\BbbQ n
p)
.
Next, for i = 1, . . . ,m, by using Proposition 2.3, we have
\Biggl(
\omega (BR)
\omega (BR+kAi
)
\Biggr) 1
qi
\lesssim
\left\{
\Biggl(
| BR|
| BR+kAi
|
\Biggr) \zeta
qi
\lesssim p
(R - R - kAi
)n\zeta
qi = \| Ai(y)\|
- n\zeta
qi
p , \mathrm{i}\mathrm{f} \| Ai(y)\| p \leq 1,\Biggl(
| BR|
| BR+kAi
|
\Biggr) (\delta - 1)
qi\delta
\lesssim p
(R - R - kAi
)n(\delta - 1)
qi\delta = \| Ai(y)\|
- n(\delta - 1)
qi\delta
p , \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}.
(3.9)
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 989
Hence, by letting R \rightarrow +\infty and applying the dominated convergence theorem of Lebesgue, we
obtain \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (\BbbQ n
p)
\lesssim \scrC 2
m\prod
i=1
\| fi\| Lqi
\omega (\BbbQ n
p)
.
Theorem 3.2 is proved.
Theorem 3.3. Let \omega i, \omega be as Theorem 3.1 and \lambda i \in
\biggl(
- 1
qi
, 0
\biggr)
for all i = 1, . . . ,m. Assume
that
(\alpha + n)\lambda = (\alpha 1 + n)\lambda 1 + . . .+ (\alpha m + n)\lambda m. (3.10)
Then \scrH p
\Phi , \vec{}A
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
if and only if
\scrC 3 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)\lambda i
p
dy <\infty .
Furthermore,
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\bigm\| \bigm\| \bigm\| .
B
q1,\lambda 1
\omega 1
(\BbbQ n
p)\times ...\times
.
B
qm,\lambda m
\omega m (\BbbQ n
p)\rightarrow
.
B
q,\lambda
\omega (\BbbQ n
p)
\simeq \scrC 3.
Proof. We will prove the sufficiency of the condition \scrC 3 <\infty . For \gamma \in \BbbZ , by estimating as (3.4)
and (3.5) above, we have\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| (\alpha i+n)
qi
p
m\prod
i=1
\| fi\| Lqi
\omega i
(B\gamma +kAi
)dy.
This implies that
1
\omega (B\gamma )
1
q
+\lambda
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim
\int
\BbbQ n
p
\Phi (y)
| y| np
\Biggl(
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| (\alpha i+n)
qi
p
\Biggr)
\times
\times \scrB i(y)
\left( m\prod
i=1
1
\omega i(B\gamma +kAi
)
1
qi
+\lambda i
\| fi\| Lqi
\omega i
(B\gamma +kAi
)
\right) dy, (3.11)
where
\scrB i(y) =
\prod m
i=1
\omega i(B\gamma +kAi
)
1
qi
+\lambda i
\omega (B\gamma )
1
q
+\lambda
.
On the other hand, by hypothesis (3.10), we immediately get
m\sum
i=1
(\alpha i + n)
\biggl(
1
qi
+ \lambda i
\biggr)
= (\alpha + n)
\biggl(
1
q
+ \lambda
\biggr)
.
Consequently, by the estimation (2.2) and (3.1), we have
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
990 N. M. CHUONG, D. V. DUONG, K. H. DUNG
\scrB i(y) \lesssim
p
\sum m
i=1
(\gamma + kAi)(\alpha i + n)
\biggl(
1
qi
+ \lambda i
\biggr)
p
\gamma (\alpha +n)
\Bigl(
1
q
+\lambda
\Bigr) =
=
p
\sum m
i=1
\gamma (\alpha i + n)
\biggl(
1
qi
+ \lambda i
\biggr)
p
\sum m
i=1
kAi(\alpha i + n)
\biggl(
1
qi
+ \lambda i
\biggr)
p
\gamma (\alpha +n)
\Bigl(
1
q
+\lambda
\Bigr) \lesssim
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr)
p
.
Hence, by (3.11), one has
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q,\lambda
\omega (\BbbQ n
p )
\lesssim \scrC 3
m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
.
Conversely, suppose that \scrH p
\Phi , \vec{}A
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
.
Let us choose the function \vec{}g = (g1, . . . , gm), where
gi(x) = | x| (\alpha i+n)\lambda i
p
for i = 1, . . . ,m. Then, by (2.2), it is not difficult to show that
\| gi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
= \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
1
\omega i(B\gamma )
1
qi
+\lambda i
\left( \int
B\gamma
| x| (\alpha i+n)\lambda iqi+\alpha i
p dx
\right)
1
qi
\simeq
\simeq \mathrm{s}\mathrm{u}\mathrm{p}
\gamma \in \BbbZ
p
\gamma ( (\alpha i+n)\lambda iqi+\alpha i+n) 1
qi
p
\gamma (\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr) = 1,
and, similarly, we also have
\| | \cdot | (\alpha +n)\lambda
p \| .
B
q,\lambda
\omega (\BbbQ n
p )
\simeq 1. (3.12)
Next, by choosing gi’s and using (3.3) and (3.10), we get
\scrH p
\Phi , \vec{}A
(\vec{}g ) (x) \gtrsim
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)\lambda i
p
| x| (\alpha i+n)\lambda i
p dy = \scrC 3| x| (\alpha +n)\lambda
p .
Thus, by (3.12), it follows that
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
(\vec{}g )
\bigm\| \bigm\| \bigm\| .
B
q,\lambda
\omega (\BbbQ n
p)
\gtrsim \scrC 3\| | \cdot | (\alpha +n)\lambda
p \| .
B
q,\lambda
\omega (\BbbQ n
p)
\gtrsim \scrC 3
m\prod
i=1
\| gi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
.
This gives that \scrC 3 <\infty .
Theorem 3.3 is proved.
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 991
Theorem 3.4. Let 1 \leq q\ast , \zeta < \infty , \lambda i \in
\biggl(
- 1
qi
, 0
\biggr)
for all i = 1, . . . ,m and \omega \in A\zeta with
the finite critical index r\omega for the reverse Hölder inequality. Assume that q > q\ast \zeta r\omega /(r\omega - 1),
\delta \in (1, r\omega ) and the following two conditions are true:
\lambda = \lambda 1 + \cdot \cdot \cdot + \lambda m, (3.13)
\scrC 4 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\| Ai(y)\|
\zeta
qi
p \times
\times
\biggl(
\chi \{ \| Ai(y)\| p\leq 1\} (y)\| Ai(y)\| n\zeta \lambda i
p + \chi \{ \| Ai(y)\| p>1\} (y)\| Ai(y)\|
n\lambda i(\delta - 1)
\delta
p
\biggr)
dy <\infty .
Then \scrH p
\Phi , \vec{}A
is bounded from
.
B
q1,\lambda 1
\omega
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
qm,\lambda m
\omega
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q\ast ,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
.
Proof. For \gamma \in \BbbZ , by estimating as (3.8) above and using the relation (3.13), we obtain
1
\omega (B\gamma )
1
q\ast +\lambda
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (B\gamma )
\lesssim
\lesssim
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
qi
p
\| Ai(y)\|
\zeta n
qi
p
\Biggl(
\omega (B\gamma +kAi
)
\omega (B\gamma )
\Biggr) \lambda i
dy
\right) m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega (\BbbQ n
p)
.
In addition, for i = 1, . . . ,m, by using Proposition 2.3 again and \lambda i < 0, we infer
\Biggl(
\omega (B\gamma +kAi
)
\omega (B\gamma )
\Biggr) \lambda i
\lesssim
\left\{
\Biggl(
| B\gamma +kAi
|
| B\gamma |
\Biggr) \zeta \lambda i
\lesssim p(\gamma +kAi
- \gamma )n\zeta \lambda i = \| Ai(y)\| n\zeta \lambda i
p , if \| Ai(y)\| p \leq 1,\Biggl(
| B\gamma +kAi
|
| B\gamma |
\Biggr) \lambda i(\delta - 1)
\delta
\lesssim p
(\gamma +kAi
- \gamma )n\lambda i(\delta - 1)
\delta = \| Ai(y)\|
n\lambda i(\delta - 1)
\delta
p , otherwise.
(3.14)
Thus, we have \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q\ast ,\lambda
\omega (\BbbQ n
p)
\lesssim \scrC 4
m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega (\BbbQ n
p)
.
Theorem 3.4 is proved.
4. Main results about the boundness of \bfscrH \bfitp
\bfPhi , \vec{}\bfitA ,\vec{}\bfitb
. Before stating next results, we introduce
some notations which will be used throughout this section. Let q, qi \in [1,\infty ) and \alpha , \alpha i, ri be real
numbers such that ri \in (1,\infty ), \alpha i \in
\biggl(
- n, nri
r\prime i
\biggr)
, i = 1, 2, . . . ,m. Suppose that
\biggl(
1
q1
+
1
q2
+ . . .+
1
qm
\biggr)
+
\biggl(
1
r1
+
1
r2
+ . . .+
1
rm
\biggr)
=
1
q
,
\biggl(
\alpha 1
q1
+
\alpha 2
q2
+ . . .+
\alpha m
qm
\biggr)
+
\biggl(
\alpha 1
r1
+
\alpha 2
r2
+ . . .+
\alpha m
rm
\biggr)
=
\alpha
q
.
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992 N. M. CHUONG, D. V. DUONG, K. H. DUNG
Lemma 4.1. Let \omega (x) = | x| \alpha p , \omega i(x) = | x| \alpha i
p and bi \in CMOri
\omega i
\bigl(
\BbbQ n
p
\bigr)
for all i = 1, . . . ,m.
Then, for any \gamma \in \BbbZ , we have\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim p
\sum m
i=1
\gamma (n+\alpha i)
ri \scrB \vec{}r,\vec{}\omega
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi i(y)\mu i(y)\| fi\| Lqi
\omega i
(B\gamma +kAi
)dy,
where
\psi i(y) = 1 +
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\}
| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)| p
\Bigr) 1
ri \| Ai(y)\|
(n+\alpha i)
ri
p +
+
\bigm| \bigm| \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p
\bigm| \bigm| + 2
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
,
\mu i(y) =
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} \bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm|
p
\Bigr) 1
qi and \scrB \vec{}r,\vec{}\omega =
m\prod
i=1
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
.
Proof. In what follows, we will write bi,B\gamma instead of (bi)B\gamma for convenience. By the Minkowski
inequality and the Hölder inequality, for any \gamma \in \BbbZ , we get\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega i
(B\gamma )
\| fi(Ai(y)\cdot )\| Lqi
\omega i
(B\gamma )
dy. (4.1)
To prove this lemma, we need to show that the following inequality holds:
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega i
(B\gamma )
\lesssim p
\gamma (\alpha i+n)
ri \psi i(y)\| bi\| CMO
ri
\omega i(\BbbQ n
p)
for all i = 1, . . . ,m. (4.2)
We put I1,i = \| bi(\cdot ) - bi,B\gamma \| Lri
\omega i
(B\gamma )
, I2,i = \| bi(Ai(y)\cdot ) - bi,Ai(y)B\gamma
\| Lri
\omega i
(B\gamma )
and I3,i = \| bi,B\gamma -
- bi,Ai(y)B\gamma
\| Lri
\omega i
(B\gamma )
. It is obvious that
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega i
(B\gamma )
\leq I1,i + I2,i + I3,i for all i = 1, . . . ,m. (4.3)
By the definition of the space CMOri
\omega i
\bigl(
\BbbQ n
p
\bigr)
and the estimation (2.2), we have
I1,i \leq \omega i(B\gamma )
1
ri \| bi\| CMO
ri
\omega i
(\BbbQ n
p )
\lesssim p
\gamma (\alpha i+n)
ri \| bi\| CMO
ri
\omega i(\BbbQ n
p)
. (4.4)
To estimate I2,i, we deduce that
I2,i \leq \omega i(B\gamma )
1
ri
\bigm| \bigm| \bigm| bi,Ai(y)B\gamma
- bi,B\gamma +kAi
\bigm| \bigm| \bigm| +
\left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(x)dx
\right)
1
ri
, (4.5)
where kAi(y) = \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p. Note that, by the formula for change of variables, we get
| Ai(y)B\gamma | =
\int
Ai(y)B\gamma
dx =
\int
B\gamma
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| pdz \simeq | \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| pp\gamma n. (4.6)
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 993
Thus, by using the Hölder inequality and (2.2), it is clear to see that\bigm| \bigm| \bigm| bi,Ai(y)B\gamma
- bi,B\gamma +kAi
\bigm| \bigm| \bigm| \leq
\leq 1
| Ai(y)B\gamma |
\left( \int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(x)dx
\right)
1
ri
\left( \int
B\gamma +kAi
\omega
- r\prime i
ri
i dx
\right)
1
r\prime
i
\lesssim
\lesssim
p
(\gamma +kAi
)(n+\alpha i)
ri p
(\gamma +kAi
)
\biggl(
- \alpha i
ri
+ n
r\prime
i
\biggr)
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| pp\gamma n
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
=
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
. (4.7)
It is easy to see that \left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(x)dx
\right)
1
ri
\leq
\leq
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} \bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm|
p
\omega i(B\gamma +kAi
)
\Bigr) 1
ri \times
\times
\left( 1
\omega i(B\gamma +kAi
)
\int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(z) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(z)dz
\right)
1
ri
. (4.8)
This leads to \left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(x)dx
\right)
1
ri
\lesssim
\lesssim p
\gamma (n+\alpha i)
ri
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} \bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm|
p
\Bigr) 1
ri \| Ai(y)\|
(n+\alpha i)
ri
p \| bi\| CMO
ri
\omega i(\BbbQ n
p)
.
Therefore, by (4.5) and (4.7), we have
I2,i \lesssim
\Biggl(
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
+
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\} \bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm|
p
\Bigr) 1
ri \| Ai(y)\|
(n+\alpha i)
ri
p
\Biggr)
\times
\times p
\gamma (n+\alpha i)
ri \| bi\| CMO
ri
\omega i(\BbbQ n
p)
. (4.9)
Next, we consider the term I3,i. We obtain
I3,i \leq \omega i(B\gamma )
1
ri
\bigm| \bigm| bi,B\gamma - bi,Ai(y)B\gamma
\bigm| \bigm| . (4.10)
Fix y \in \BbbQ n
p . We set
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
994 N. M. CHUONG, D. V. DUONG, K. H. DUNG
SkAi
=
\left\{ \{ j \in \BbbZ : 1 \leq j \leq kAi\} , if kAi \geq 1,
\{ j \in \BbbZ : kAi + 1 \leq j \leq 0\} , otherwise.
As mentioned above, we obtain\bigm| \bigm| bi,B\gamma - bi,Ai(y)B\gamma
\bigm| \bigm| \leq \sum
j\in SkAi
\bigm| \bigm| bi,B\gamma +j - 1 - bi,B\gamma +j
\bigm| \bigm| + \bigm| \bigm| \bigm| bi,B\gamma +kAi
- bi,Ai(y)B\gamma
\bigm| \bigm| \bigm| . (4.11)
Combining the Hölder inequality, the definition of the space CMOri
\omega i
\bigl(
\BbbQ n
p
\bigr)
and (2.2), one has
\bigm| \bigm| bi,B\gamma +j - 1 - bi,B\gamma +j
\bigm| \bigm| \leq \omega i(B\gamma +j)
1
ri
| B\gamma +j |
\left( \int
B\gamma +j
\omega
- r\prime i
ri
i dx
\right)
1
r\prime
i
\times
\times
\left( 1
\omega i(B\gamma +j)
\int
B\gamma +j
\bigm| \bigm| bi(z) - bi,B\gamma +j
\bigm| \bigm| ri \omega i(x)dz
\right)
1
ri
\lesssim
\lesssim
p
(\gamma +j)
(\alpha i+n)
ri
p(\gamma +j)n
p
(\gamma +j)
\biggl(
- \alpha i
ri
+ n
r\prime
i
\biggr)
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
= \| bi\| CMO
ri
\omega i(\BbbQ n
p)
.
Thus, \bigm| \bigm| bi,B\gamma - bi,Ai(y)B\gamma
\bigm| \bigm| \lesssim | kAi | \| bi\| CMO
ri
\omega i(\BbbQ n
p)
+
\bigm| \bigm| \bigm| bi,B\gamma +kAi
- bi,Ai(y)B\gamma
\bigm| \bigm| \bigm| . (4.12)
In addition, by the Hölder inequality again and (4.6), we get\bigm| \bigm| \bigm| bi,B\gamma +kAi
- bi,Ai(y)B\gamma
\bigm| \bigm| \bigm| \leq 1
| Ai(y)B\gamma |
\int
Ai(y)B\gamma
\bigm| \bigm| \bigm| bi(x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| dx \leq
\leq
\omega i(B\gamma +kAi
)
1
ri
| Ai(y)B\gamma |
\left( \int
B\gamma +kAi
\omega
- r\prime i
ri
i dx
\right)
1
r\prime
i
\left( 1
\omega i(B\gamma +kAi
)
\int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega i(x)dx
\right)
1
ri
\lesssim
\lesssim
p
(\gamma +kAi
)
(\alpha i+n)
ri
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| pp\gamma n
p
(\gamma +kAi
)
\biggl(
- \alpha i
ri
+ n
r\prime
i
\biggr)
\| bi
\bigm\| \bigm\|
CMO
ri
\omega i(\BbbQ n
p)
=
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
.
Consequently, by (4.10) – (4.12), it follows that
I3,i \lesssim p
\gamma (n+\alpha i)
ri
\biggl(
| \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| +
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\biggr)
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
.
This together with (4.3), (4.4) and (4.9) follow us to have the proof of the inequality (4.2). Finally,
by estimating as (4.8), we immediately have
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 995
\| fi(Ai(y)\cdot )\| Lqi
\omega i
(B\gamma )
\leq
\Bigl(
\mathrm{m}\mathrm{a}\mathrm{x}
\Bigl\{ \bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| \alpha i
p
, \| Ai(y)\| - \alpha i
p
\Bigr\}
| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)| p
\Bigr) 1
qi \| fi\| Lqi
\omega i
\Bigl(
B\gamma +kAi
\Bigr) =
= \mu i(y)\| fi\| Lqi
\omega
\Bigl(
B\gamma +kAi
\Bigr) .
In view of (4.1) and (4.2), the proof of this lemma is ended.
Lemma 4.2. Let 1 \leq q\ast , r\ast 1, . . . , r
\ast
m, q
\ast
1, . . . , q
\ast
m, \zeta < \infty , \omega \in A\zeta with the finite critical index
r\omega for the reverse Hölder condition, \delta \in (1, r\omega ), \lambda i \in
\biggl(
- 1
q\ast i
, 0
\biggr)
, \zeta \leq r\ast i and bi \in CMO
r\ast i
\omega
\bigl(
\BbbQ n
p
\bigr)
for all i = 1, . . . ,m. Assume that the following condition holds:
1
q\ast
>
\biggl(
1
r\ast 1
+ . . .+
1
r\ast m
+
1
q\ast 1
+ . . .+
1
q\ast m
\biggr)
\zeta
r\omega
r\omega - 1
. (4.13)
Then we have \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (B\gamma )
\lesssim
\lesssim \omega (B\gamma )
1
q\ast \scrB \vec{}r\ast ,\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)
1
\omega (B\gamma +kAi
)
1
q\ast
i
\| fi\|
L
q\ast
i
\omega
\Bigl(
B\gamma +kAi
\Bigr) dy
\right) for all \gamma \in \BbbZ .
Here,
\psi \ast
i (y) = 1 +
2\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
+
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i \| Ai(y)\|
n\zeta
r\ast
i
p + | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| ,
\mu \ast i (y) =
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
q\ast
i
p
\| Ai(y)\|
n\zeta
q\ast
i
p and \scrB \vec{}r\ast ,\omega =
m\prod
i=1
\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
.
Proof. By virtue of the inequality (4.13), there exist r1, . . . , rm, q1, . . . , qm such that
1
qi
>
\zeta
q\ast i
r\omega
r\omega - 1
,
1
ri
>
\zeta
r\ast i
r\omega
r\omega - 1
for all i = 1, . . . ,m, and
1
q1
+ . . .+
1
qm
+
1
r1
+ . . .+
1
rm
=
1
q\ast
.
As mentioned above, for any \gamma \in \BbbZ , by the same argument (4.1), we also get\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (B\gamma )
\lesssim
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega (B\gamma )
\| fi(Ai(y)\cdot )\| Lqi
\omega (B\gamma )
dy. (4.14)
In particular, we need to show the following result:
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega (B\gamma )
\lesssim \omega (B\gamma )
1
ri \psi \ast
i (y)\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
(4.15)
for all i = 1, . . . ,m. Indeed, we see that
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lri
\omega (B\gamma )
\leq
\bigm\| \bigm\| bi(\cdot ) - bi,B\gamma
\bigm\| \bigm\|
L
ri
\omega (B\gamma )
+
\bigm\| \bigm\| bi(Ai(y)\cdot ) - bi,Ai(y)B\gamma
\bigm\| \bigm\|
L
ri
\omega (B\gamma )
+
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996 N. M. CHUONG, D. V. DUONG, K. H. DUNG
+
\bigm\| \bigm\| bi,B\gamma - bi,Ai(y)B\gamma
\bigm\| \bigm\|
L
ri
\omega (B\gamma )
:= J1,i + J2,i + J3,i. (4.16)
By virtue of the inequality r1 < r\ast 1, it is easy to show that
J1,i \leq \omega (B\gamma )
1
ri \| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
. (4.17)
Next, by estimating as (4.5) above, we get
J2,i \leq \omega (B\gamma )
1
ri | bi,Ai(y)B\gamma
- bi,B\gamma +kAi
| +
\left( \int
B\gamma
| bi(Ai(y)x) - bi,B\gamma +kAi
| ri\omega (x)dx
\right)
1
ri
. (4.18)
By having the inequality \zeta \leq r\ast i and applying Proposition 2.4 and (4.6), we infer that\bigm| \bigm| \bigm| bi,B\gamma +kAi
- bi,Ai(y)B\gamma
\bigm| \bigm| \bigm| \leq 1
| Ai(y)B\gamma |
\int
B\gamma +kAi
| bi(x) - bi,B\gamma +kAi
| dx \lesssim
\lesssim
p(\gamma +kAi
)n
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p .p\gamma n
\| bi\| CMO\zeta
\omega (\BbbQ n
p)
\leq
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
. (4.19)
By
1
ri
>
\zeta
r\ast i
r\omega
r\omega - 1
, there exists \beta i \in (1, r\omega ) satisfying
r\ast i
\zeta
= ri\beta
\prime
i. Thus, by combining the Hölder
inequality and the reverse Hölder condition again, we have\left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega (x)dx
\right)
1
ri
\lesssim
\lesssim | B\gamma |
- \zeta
r\ast
i \omega (B\gamma )
1
ri
\left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| r\ast i\zeta dx
\right)
\zeta
r\ast
i
.
According to Proposition 2.4, we get\left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| r\ast i\zeta dx
\right)
\zeta
r\ast
i
\leq
\leq
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i
p
\left( \int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(z) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| r\ast i\zeta dz
\right)
\zeta
r\ast
i
\leq
\leq
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i
p
| B\gamma +kAi
|
\zeta
r\ast
i
\omega (B\gamma +kAi
)
1
r\ast
i
\left( \int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(z) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| r\ast i \omega (z)dz
\right)
1
r\ast
i
.
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 997
In view of (2.2), one has
| B\gamma +kAi
|
| B\gamma |
\simeq \| Ai(y)\| np . From this we give
\left( \int
B\gamma
\bigm| \bigm| \bigm| bi(Ai(y)x) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| ri \omega (x)dx
\right)
1
ri
\lesssim
\lesssim \omega (B\gamma )
1
ri
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i
p
\| Ai(y)\|
n\zeta
r\ast
i
p
\left( 1
\omega (B\gamma +kAi
)
\int
B\gamma +kAi
\bigm| \bigm| \bigm| bi(z) - bi,B\gamma +kAi
\bigm| \bigm| \bigm| r\ast i \omega (z)dz
\right)
1
r\ast
i
\lesssim
\lesssim \omega (B\gamma )
1
ri
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i
p
\| Ai(y)\|
n\zeta
r\ast
i
p \| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
. (4.20)
As a consequence, by (4.18) and (4.19), we infer that
J2,i \lesssim \omega (B\gamma )
1
ri
\Biggl(
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
+
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
r\ast
i
p
\| Ai(y)\|
n\zeta
r\ast
i
p
\Biggr)
.\| b\|
CMO
r\ast
i
\omega (\BbbQ n
p)
. (4.21)
Now, we will estimate J3,i. By a same argument as (4.10), (4.11) and (4.19), we have
J3,i \leq \omega (B\gamma )
1
ri
\left( \sum
j\in SkAi
\bigm| \bigm| bi,B\gamma +j - 1 - bi,B\gamma +j
\bigm| \bigm| + \bigm| \bigm| \bigm| bi,B\gamma +kAi
- bi,Ai(y)B\gamma
\bigm| \bigm| \bigm|
\right) \lesssim
\lesssim \omega (B\gamma )
1
ri
\left( \sum
j\in SkAi
\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
+
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p )
\right) \leq
\leq \omega (B\gamma )
1
ri
\biggl(
| \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| +
\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
\biggr)
\| bi\|
CMO
r\ast
i
\omega (\BbbQ n
p)
.
This together with (4.17) and (4.21) yields that the inequality (4.15) is finished.
In other words, by estimating as (4.20) above, we get
\| fi(Ai(y)\cdot )\| Lqi
\omega (B\gamma )
\lesssim \omega (B\gamma )
1
qi
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| \zeta
q\ast
i
p
\| Ai(y)\|
n\zeta
q\ast
i
p \omega (B\gamma +kAi
)
- 1
q\ast
i \| fi\|
L
q\ast
i
\omega
\Bigl(
B\gamma +kAi
\Bigr) =
= \omega (B\gamma )
1
qi \mu \ast i (y)\omega (B\gamma +kAi
)
- 1
q\ast
i \| fi\|
L
q\ast
i
\omega
\Bigl(
B\gamma +kAi
\Bigr) .
Hence, by (4.14) and (4.15), we conclude that the proof of this lemma is finished.
Theorem 4.1. Let the assumptions of Lemma 4.1 hold and
\scrC 5 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi i(y)\mu i(y)dy <\infty .
Then, for any \gamma \in \BbbZ , we have that \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from Lq1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times Lqm
\omega m
\bigl(
\BbbQ n
p
\bigr)
to Lq
\omega (B\gamma ).
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998 N. M. CHUONG, D. V. DUONG, K. H. DUNG
Proof. For any \gamma \in \BbbZ , by using Lemma 4.1, we infer that
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim p
\sum m
i=1
\gamma (n+ \alpha i)
ri \scrB \vec{}r,\vec{}\omega
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi i(y)\mu i(y)\| fi\| Lqi
\omega i
(B\gamma +kAi
)dy.
Thus, we have
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (B\gamma )
\lesssim \scrC 5\scrB \vec{}r,\vec{}\omega
\prod m
i=1
\| fi\| Lqi
\omega i(\BbbQ n
p)
.
Theorem 4.1 is proved.
Theorem 4.2. Let the assumptions of Lemma 4.2 hold. Suppose that \omega (B\gamma ) \lesssim 1 for all \gamma \in \BbbZ
and
\scrC 6 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)\times
\times
\Biggl(
\chi \{ \| Ai(y)\| p\leq 1\} (y)\| Ai(y)\|
- n\zeta
q\ast
i
p + \chi \{ \| Ai(y)\| p>1\} (y)\| Ai(y)\|
- n(\delta - 1)
q\ast
i
\delta
p
\Biggr)
dy <\infty .
Then we have that \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from L
q\ast 1
\omega
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times L
q\ast m
\omega
\bigl(
\BbbQ n
p
\bigr)
to Lq\ast
\omega
\bigl(
\BbbQ n
p
\bigr)
.
Proof. In view of Lemma 4.2, for any R \in \BbbZ , it is clear to see that\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\lesssim
\lesssim \omega (BR)
1
q\ast \scrB \vec{}r\ast ,\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)
1
\omega
\Bigl(
BR+kAi
\Bigr) 1
q\ast
i
\| fi\|
L
q\ast
i
\omega
\Bigl(
BR+kAi
\Bigr) dy
\right) .
Next, by
1
q\ast
>
1
q\ast 1
+ . . .+
1
q\ast m
and the assumption \omega (BR) \lesssim 1 for any R \in \BbbZ , we have \omega (BR)
1
q\ast \leq
\leq
\prod m
i=1
\omega (BR)
1
q\ast
i . Thus, \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\lesssim
\lesssim \scrB \vec{}r\ast ,\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)
\Biggl(
\omega (BR)
\omega (BR+kAi
)
\Biggr) 1
q\ast
i
dy
\right) m\prod
i=1
\| fi\|
L
q\ast
i
\omega (\BbbQ n
p)
\lesssim
\lesssim \scrC 6\scrB \vec{}r\ast ,\omega
m\prod
i=1
\| fi\|
L
q\ast
i
\omega (\BbbQ n
p)
.
Consequence, by letting R \rightarrow +\infty and applying dominated convergence theorem of Lebesgue,
we have \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\bigl(
\vec{}f
\bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (\BbbQ n
p)
\lesssim \scrC 6\scrB \vec{}r\ast ,\omega
m\prod
i=1
\| fi\|
L
q\ast
i
\omega (\BbbQ n
p)
.
Theorem 4.2 is proved.
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WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 999
Theorem 4.3. Let 1 < \zeta < \infty , 1 \leq q\ast , qi, r
\ast
i < \infty , - n < \alpha i < n(\zeta - 1), \omega (x) = | x| \alpha p ,
\omega i(x) = | x| \alpha i
p for all i = 1, . . . ,m such that
\alpha 1
q1
+ . . .+
\alpha m
qm
=
\zeta \alpha
q\ast
,
1
q1
+ . . .+
1
qm
=
\zeta
q\ast
, (4.22)
1
q1
+ . . .+
1
qm
+
1
r\ast 1
+ . . .+
1
r\ast m
= 1. (4.23)
If bi \in CMOr\ast i
\bigl(
\BbbQ n
p
\bigr)
for all i = 1, . . . ,m and
\scrC 7 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\Gamma i(y)\| Ai(y)\|
- (\zeta +n)
\zeta qi
p dy <\infty ,
where
\Gamma i(y) =
\biggl(
1 + | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| +
2\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
+ \| Ai(y)\|
n
r\ast
i
p
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
r\ast
i
p
\biggr)
\times
\times
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\| Ai(y)\|
n
qi
p , (4.24)
then we have\bigm\| \bigm\| \bigm\| \scrM mod
\Bigl(
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigr) \Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (\BbbQ n
p)
\lesssim \scrC 7
\Biggl(
m\prod
i=1
\| bi\| CMOr\ast
i (\BbbQ n
p)
\Biggr)
m\prod
i=1
\| fi\| L\zeta qi
\omega i (\BbbQ n
p)
.
Proof. For the sake of simplicity, we denote \scrB \vec{}r\ast =
\prod m
i=1
\| bi\| CMOr\ast
i (\BbbQ n
p)
. Now, let x \in \BbbQ n
p
and fix a ball B\gamma such that x \in B\gamma . In view of (4.23), by using the Hölder inequality, we have
1
| B\gamma |
\int
B\gamma
\bigm| \bigm| \bigm| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr)
(z)
\bigm| \bigm| \bigm| dz \leq
\leq 1
| B\gamma |
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\| fi(Ai(y)\cdot )\| Lqi (B\gamma )
m\prod
i=1
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lr\ast
i (B\gamma )
dy.
For i = 1, . . . ,m, by estimating as (4.2) above, we get
\| bi(\cdot ) - bi(Ai(y)\cdot )\| Lr\ast
i (B\gamma )
\lesssim
\lesssim | B\gamma |
1
r\ast
i
\biggl(
1 + | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| +
2\| Ai(y)\| np
| \mathrm{d}\mathrm{e}\mathrm{t}Ai(y)| p
+ \| Ai(y)\|
n
r\ast
i
p
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
r\ast
i
p
\biggr)
\| bi\| CMOr\ast
i (\BbbQ n
p)
.
By x \in B\gamma , we imply that \| Ai(y)\| - 1
p x \in B\gamma +kAi
. Thus, by definition of the Hardy – Littlewood
maximal operator, one has
\| fi(Ai(y)\cdot )\| Lqi (\BbbQ n
p )
=
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\left( \int
Ai(y)B\gamma
| fi(t)| qidt
\right)
1
qi
\leq
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
1000 N. M. CHUONG, D. V. DUONG, K. H. DUNG
\leq
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\left( \int
B\gamma +kAi
| fi(t)| qidt
\right)
1
qi
\lesssim
\lesssim | B\gamma |
1
qi
\bigm| \bigm| \mathrm{d}\mathrm{e}\mathrm{t}A - 1
i (y)
\bigm| \bigm| 1
qi
p
\| Ai(y)\|
n
qi
p
\bigl(
\scrM (| fi| qi)
\bigl(
\| Ai(y)\| - 1
p .x
\bigr) \bigr) 1
qi .
As mentioned above, we give
1
| B\gamma |
\int
B\gamma
\bigm| \bigm| \bigm| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr)
(z)
\bigm| \bigm| \bigm| dz \lesssim \scrB \vec{}r\ast
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\Gamma i(y)
\bigl(
\scrM (| fi| qi)
\bigl(
\| Ai(y)\| - 1
p x
\bigr) \bigr) 1
qi dy.
Hence, we infer that
\scrM mod
\Bigl(
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \Bigr)
(x) \lesssim \scrB \vec{}r\ast
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\Gamma i(y)
\bigl(
\scrM (| fi| qi)
\bigl(
\| Ai(y)\| - 1
p .x
\bigr) \bigr) 1
qi dy.
Thus, by using the assumption (4.22), the Minkowski inequality and the Hölder inequality, we obtain\bigm\| \bigm\| \bigm\| \scrM mod
\Bigl(
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (\BbbQ n
p)
\leq
\leq \scrB \vec{}r\ast
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\Gamma i(y)
m\prod
i=1
\left( \int
\BbbQ n
p
\scrM (| fi| qi)\zeta
\bigl(
\| Ai(y)\| - 1
p x
\bigr)
\omega idx
\right)
1
\zeta qi
dy. (4.25)
For i = 1, . . . ,m, by Proposition 2.2, we have \omega i \in A\zeta . From this, by virtue of the boundedness of
the Hardy – Littlewood maximal operator on the Lebesgue spaces with the Muckenhoupt weights, we
have \left( \int
\BbbQ n
p
M (| fi| qi)\zeta
\bigl(
\| Ai(y)\| - 1
p x
\bigr)
\omega idx
\right)
1
\zeta qi
=
=
\left( \int
\BbbQ n
p
M (| fi| qi)\zeta (z)| \| Ai(y)\| pz| \alpha i
p | \| Ai(y)\| np | pdz
\right)
1
\zeta qi
=
= \| Ai(y)\|
- (\alpha i+n)
\zeta qi
p
\left( \int
\BbbQ n
p
M (| fi| qi)\zeta (z)\omega i(z)dz
\right)
1
\zeta qi
\lesssim \| Ai(y)\|
- (\alpha i+n)
\zeta qi
p \| fi\| L\zeta qi
\omega i (\BbbQ n
p)
.
This together with (4.25) yields that the proof of this theorem is completed.
In what follows, we set
\scrC 8 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)\lambda i
p
| \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| dy.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 1001
Theorem 4.4. Suppose the hypothesis in Lemma 4.1 holds. Let \lambda i \in
\biggl(
- 1
qi
, 0
\biggr)
for all i =
= 1, . . . ,m and conditions (3.1) and (3.10) be true. Assume that
\mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\Phi ) \subset
m
\cap
i=1
\bigl\{
y \in \BbbQ n
p : \| Ai(y)\| p < 1
\bigr\}
. (4.26)
(i) If \scrC 8 <\infty , then \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
.
(ii) If \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . . \times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
for all \vec{}b =
= (b1, . . . , bm) \in CMOr1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times CMOrm
\omega m
\bigl(
\BbbQ n
p
\bigr)
, then \scrC 8 <\infty .
Proof. Firstly, we prove the part (\mathrm{i}) of the theorem. For any R \in \BbbZ , by Lemma 4.1, we get
1
\omega (BR)
1
q
+\lambda
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq
\omega (BR)
\leq
\leq \scrB \vec{}r,\vec{}\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi i(y)\mu i(y)
p
\sum m
i=1
R(n+\alpha i)
ri
\prod m
i=1
\omega i
\Bigl(
BR+kAi
\Bigr) 1
qi
+\lambda i
\omega (BR)
1
q
+\lambda
dy
\right) m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
.
Now, by (2.2) and (3.10), we calculate
p
\sum m
i=1
R(n+\alpha i)
ri
\prod m
i=1
\omega i(BR+kAi
)
1
qi
+\lambda i
\omega (BR)
1
q
+\lambda
\simeq
m\prod
i=1
\| Ai(y)\|
(\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr)
p .
Hence, one has \bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q,\lambda
\omega (\BbbQ n
p)
\lesssim
\lesssim \scrB \vec{}r,\vec{}\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi i(y)\mu i(y)\| Ai(y)\|
(\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr)
p dy
\right) m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
. (4.27)
Note that, by the hypothesis (4.26), we see that | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| \geq 1 for all y \in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p}(\Phi ). As
mentioned above, by (2.1) and (3.1), we make
\psi i(y)\mu i(y)\| Ai(y)\|
(\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr)
p \lesssim
\lesssim
\biggl(
1 +
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| (\alpha i+n)
ri
p
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)
ri
p
+ | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| + 2p\nu \vec{}A
\biggr)
\times
\times
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| (\alpha i+n)
qi
p
\bigm\| \bigm\| A - 1
i (y)
\bigm\| \bigm\| - (\alpha i+n)
\Bigl(
1
qi
+\lambda i
\Bigr)
p
\lesssim | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| \| Ai(y)\| - (\alpha i+n)\lambda i
p .
As an application, by (4.27), we obtain\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q,\lambda
\omega (\BbbQ n
p)
\lesssim \scrC 8\scrB \vec{}r,\vec{}\omega
m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
.
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
1002 N. M. CHUONG, D. V. DUONG, K. H. DUNG
To give the proof for the part (\mathrm{i}\mathrm{i}) of the theorem, for i = 1, . . . ,m, let us choose bi(x) = \mathrm{l}\mathrm{o}\mathrm{g}p | x| p
for all x \in \BbbQ n
p \setminus \{ 0\} , and fi(x) = | x| (n+\alpha i)\lambda i
p for all x \in \BbbQ n
p . Now, we need to prove that
\| bi\| CMO
ri
\omega i(\BbbQ n
p)
<\infty for all i = 1, . . . ,m. (4.28)
In fact, for any R \in \BbbZ , we see that bi,R = p - Rn
\sum
\gamma \leq R
\gamma p\gamma n(1 - p - n) = R - 1
pn - 1
. Thus, we
get
1
\omega i(BR)
\int
BR
| bi(x) - bi,BR
| ri\omega idx = p - R(\alpha i+n)
\sum
\gamma \leq R
\int
S\gamma
\bigm| \bigm| \bigm| \bigm| \gamma -
\biggl(
R - 1
pn - 1
\biggr) \bigm| \bigm| \bigm| \bigm| ri p\gamma \alpha idx \lesssim
\lesssim p - R(\alpha i+n)
\sum
\ell \leq 0
\bigm| \bigm| \bigm| \bigm| \ell + 1
pn - 1
\bigm| \bigm| \bigm| \bigm| ri p(R+\ell )(\alpha i+n) \leq
\sum
\ell \leq 0
\biggl(
| \ell | ri + 1
(pn - 1)ri
\biggr)
p\ell (\alpha i+n) <\infty
uniformly for R \in \BbbZ . As an application, it immediately follows that the inequality (4.28) holds.
By choosing bi and fi, we get
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr)
(x) =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
| Ai(y)x| (\alpha i+n)\lambda i
p
\biggl(
\mathrm{l}\mathrm{o}\mathrm{g}p
| x| p
| Ai(y)x| p
\biggr)
dy.
By the hypothesis (4.26), we have \| Ai(y)\| p < 1 for all y \in \mathrm{s}\mathrm{u}\mathrm{p}\mathrm{p} (\Phi ). Thus, | Ai(y)x| p < | x| p.
This gives that 0 < | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| = \mathrm{l}\mathrm{o}\mathrm{g}p
1
| Ai(y)| p
\leq \mathrm{l}\mathrm{o}\mathrm{g}p
| x| p
| Ai(y)x| p
. Consequently, by (3.1) and
(4.1), we lead to
\scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr)
(x) \gtrsim
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\| Ai(y)\| - (\alpha i+n)\lambda i
p | \mathrm{l}\mathrm{o}\mathrm{g}p \| Ai(y)\| p| dy
\right) | x| (\alpha +n)\lambda
p = \scrC 8| x| (\alpha +n)\lambda
p .
From this, by (3.12) above, we infer that\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q,\lambda
\omega (\BbbQ n
p)
\gtrsim \scrC 8\| | \cdot | (\alpha +n)\lambda
p \| .
B
q,\lambda
\omega (\BbbQ n
p)
\gtrsim \scrC 8
m\prod
i=1
\| fi\| .
B
qi,\lambda i
\omega i
(\BbbQ n
p)
.
Therefore, since \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
, it implies
that \scrC 8 <\infty .
Theorem 4.4 is proved.
Now, we consider Ai(y) = si(y)In for i = 1, . . . ,m. By the similar arguments, we then obtain
the following useful result.
Corollary 4.1. Suppose that the hypothesis in Lemma 4.1 and (3.10) hold. Let
\scrC 9 :=
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
| si(y)| (\alpha i+n)\lambda i
p | \mathrm{l}\mathrm{o}\mathrm{g}p | si(y)| p| dy.
Then the following statements are equivalent:
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
WEIGHTED LEBESGUE AND CENTRAL MORREY ESTIMATES FOR P-ADIC . . . 1003
(i) \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from
.
B
q1,\lambda 1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . . \times
.
B
qm,\lambda m
\omega m
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
for any \vec{}b =
= (b1, . . . , bm) \in CMOr1
\omega 1
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times CMOrm
\omega m
\bigl(
\BbbQ n
p
\bigr)
.
(ii) \scrC 9 <\infty .
Theorem 4.5. Suppose that the hypothesis in Lemma 4.2 holds. Let \lambda i \in
\biggl(
- 1
q\ast i
, 0
\biggr)
for all
i = 1, . . . ,m and condition (3.13) in Theorem 3.4 hold. Then, if
\scrC 10 =
\int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)\times
\times
\biggl(
\chi \{ \| Ai(y)\| p\leq 1\} (y)\| Ai(y)\| n\zeta \lambda i
p + \chi \{ \| Ai(y)\| p>1\} (y)\| Ai(y)\|
n\lambda i(\delta - 1)
\delta
p
\biggr)
dy <\infty ,
we have that \scrH p
\Phi , \vec{}A,\vec{}b
is bounded from
.
B
q\ast 1 ,\lambda 1
\omega
\bigl(
\BbbQ n
p
\bigr)
\times . . .\times
.
B
q\ast m,\lambda m
\omega
\bigl(
\BbbQ n
p
\bigr)
to
.
B
q\ast ,\lambda
\omega
\bigl(
\BbbQ n
p
\bigr)
.
Proof. For any R \in \BbbZ , by Lemma 4.2 and (3.13), we infer
1
\omega (BR)
1
q\ast +\lambda
\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\|
Lq\ast
\omega (BR)
\lesssim
\lesssim \scrB \vec{}r\ast ,\omega
\left( \int
\BbbQ n
p
\Phi (y)
| y| np
m\prod
i=1
\psi \ast
i (y)\mu
\ast
i (y)
\Biggl(
\omega (BR+kAi
)
\omega (BR)
\Biggr) \lambda i
dy
\right) m\prod
i=1
\| fi\| .
B
q\ast
i
,\lambda i
\omega (\BbbQ n
p)
.
From this, by (3.14), we have\bigm\| \bigm\| \bigm\| \scrH p
\Phi , \vec{}A,\vec{}b
\Bigl(
\vec{}f
\Bigr) \bigm\| \bigm\| \bigm\| .
B
q\ast ,\lambda
\omega (\BbbQ n
p)
\lesssim \scrC 10\scrB \vec{}r\ast , \omega
m\prod
i=1
\| fi\| .
B
q\ast
i
,\lambda i
\omega (\BbbQ n
p)
.
Theorem 4.5 is proved.
References
1. S. Albeverio, A. Yu. Khrennikov, V. M. Shelkovich, Harmonic analysis in the p-adic Lizorkin spaces: fractional
operators, pseudo-differential equations, p-wavelets, Tauberian theorems, J. Fourier Anal. and Appl., 12, № 4, 393 –
425 (2006).
2. A. V. Avetisov, A. H. Bikulov, S. V. Kozyrev, V. A. Osipov, p-Adic models of ultrametric diffusion constrained by
hierarchical energy landscapes, J. Phys. A: Math. and Gen., 35, 177 – 189 (2002).
3. A. V. Avetisov, A. H. Bikulov, V. A. Osipov, p-Adic description of characteristic relaxation in complex systems, J.
Phys. A: Math. and Gen., 36, 4239 – 4246 (2003).
4. N. M. Chuong, Yu. V. Egorov, A. Yu. Khrennikov, Y. Meyer, D. Mumford, Harmonic, wavelet and p-adic analysis,
World Sci. (2007).
5. N. M. Chuong, D. V. Duong, Weighted Hardy – Littlewood operators and commutators on p-adic functional spaces,
p-Adic numbers, Ultrametric Anal. and Appl., 5, № 1, 65 – 82 (2013).
6. N. M. Chuong, D. V. Duong, The p-adic weighted Hardy – Cesàro operators on weighted Morrey – Herz space,
p-Adic numbers, Ultrametric Anal. and Appl., 8, № 3, 204 – 216 (2016).
7. N. M. Chuong, D. V. Duong, K. H. Dung, Multilinear Hausdorff operators on some function spaces with variable
exponent (2017), arxiv.org/abs/1709.08185.
8. N. M. Chuong, N. V. Co, The Cauchy problem for a class of pseudo-differential equations over p-adic field, J. Math.
Anal. and Appl., 340, № 1, 629 – 643 (2008).
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
1004 N. M. CHUONG, D. V. DUONG, K. H. DUNG
9. N. M. Chuong, H. D. Hung, Maximal functions and weighted norm inequalities on local fields, Appl. and Comput.
Harmon. Anal., 29, 272 – 286 (2010).
10. B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, I. V. Volovich, On p-adic mathematical physics, p-Adic numbers,
Ultrametric Anal. and Appl., 1, № 1, 1 – 17 (2009).
11. M. Dyachenko, E. Nursultanov, S. Tikhonov, Hardy – Littlewood and Pitt’s inequalities for Hausdorff operators, Bull.
Sci. Math., 147, 40 – 57 (2018).
12. L. Grafakos, Modern Fourier analysis, second ed., Springer (2008).
13. F. Hausdorff, Summation Methoden und Momentfolgen, I. Math. Z., 9, 74 – 109 (1921).
14. H. D. Hung, The p-adic weighted Hardy – Cesàro operator and an application to discrete Hardy inequalities, J. Math.
Anal. and Appl., 409, 868 – 879 (2014).
15. W. A. Hurwitz, L. L. Silverman, The consistency and equivalence of certain definitions of summabilities, Trans.
Amer. Math. Soc., 18, 1 – 20 (1917).
16. T. Hytönen, C. Pérez, E. Rela, Sharp reverse Hölder property for A\infty weights on spaces of homogeneous type, J.
Funct. Anal., 263, 3883 – 3899 (2012).
17. S. Indratno, D. Maldonado, S. Silwal, A visual formalism for weights satisfying reverse inequalities, Expo. Math.,
33, 1 – 29 (2015).
18. S. Haran, Riesz potentials and explicit sums in arithmetic, Invent. Math., 101, 697 – 703 (1990).
19. S. Haran, Analytic potential theory over the p-adics, Ann. Inst. Fourier (Grenoble)., 43, № 4, 905 – 944 (1993).
20. A. Yu. Khrennikov, p-Adic valued distributions in mathematical physics, Kluwer Acad. Publ., Dordrecht etc. (1994).
21. A. N. Kochubei, Pseudo-differential equations and stochastics over non-Archimedean fields, Marcel Dekker, New
York (2001).
22. S. V. Kozyrev, Methods and applications of ultrametric and p-adic analysis: from wavelet theory to biophysics, Proc.
Steklov Inst. Math., 274, 1 – 84 (2011).
23. S. Lu, Y. Ding, D. Yan, Singular integrals and related topics, World Sci. Publ., Co., Singapore (2007).
24. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165, 207 – 226
(1972).
25. J. Ruan, D. Fan, Q. Wu, Weighted Herz space estimates for Hausdorff operators on the Heisenberg group, Banach
J. Math. Anal., 11, 513 – 535 (2017).
26. K. S. Rim, J. Lee, Estimates of weighted Hardy – Littlewood averages on the p-adic vector space, J. Math. Anal. and
Appl., 324, № 2, 1470 – 1477 (2006).
27. E. M. Stein, Harmonic analysis, real-variable methods, orthogonality, and oscillatory integrals, Princeton Univ. Press
(1993).
28. V. S. Varadarajan, Path integrals for a class of p-adic Schrödinger equations, Lett. Math. Phys., 39, 97 – 106 (1997).
29. V. S. Vladimirov, Tables of integrals of complex-valued functions of p-adic arguments, Proc. Steklov Inst. Math.,
284, 1 – 59 (2014).
30. V. S. Vladimirov, I. V. Volovich, p-Adic quantum mechanics, Commum. Math. Phys., 123, 659 – 676 (1989).
31. V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-Adic analysis and mathematical physis, World Sci. (1994).
32. S. S. Volosivets, Multidimensional Hausdorff operator on p-adic field, p-Adic numbers, Ultrametric Anal. and Appl.,
2, 252 – 259 (2010).
33. S. S. Volosivets, Hausdorff operators on p-adic linear spaces and their properties in Hardy, BMO, and Hölder spaces,
Math. Notes, 3, 382 – 391 (2013).
Received 21.10.18,
after revision — 31.03.19
ISSN 1027-3190. Укр. мат. журн., 2021, т. 73, № 7
|
| id | umjimathkievua-article-441 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:02:45Z |
| publishDate | 2021 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/75/85fb757fb539f2e792958b890ea6d775.pdf |
| spelling | umjimathkievua-article-4412025-03-31T08:47:53Z Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Multilinear Hausdorff operator commutator central BMO space Morrey space $A_p$ weight maximal operator $p$-adic analysis Multilinear Hausdorff operator commutator central BMO space Morrey space $A_p$ weight maximal operator $p$-adic analysis UDC 517.9 We establish the sharp boundedness of $p$-adic multilinear Hausdorff operators on the product of Lebesgue and central Morrey spaces associated with both power weights and Muckenhoupt weights.&nbsp;Moreover, the boundedness for the commutators of $p$-adic multilinear Hausdorff operators on the such spaces with symbols in central BMO space is also obtained. УДК 517.9 Зваженi оцiнки для $p$ -адичних багатолiнiйних Хаусдорфових операторiв та їх комутаторiв на просторах Лебега та центральних просторах Моррi Встановлено точну обмеженість $p$-адичних багатолінійних гаусдорфових операторів на добутку просторів Лебега і центральних просторів Моррі, асоційованих як з вагами степенів, так і з вагами Макенхаупта.&nbsp;Також доведено обмеженість комутаторів $p$-адичних багатолінійних гаусдорфових операторів на таких просторах із символами в центральному BMO-просторі. Institute of Mathematics, NAS of Ukraine 2021-07-20 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/441 10.37863/umzh.v73i7.441 Ukrains’kyi Matematychnyi Zhurnal; Vol. 73 No. 7 (2021); 979 - 1004 Український математичний журнал; Том 73 № 7 (2021); 979 - 1004 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/441/9073 Copyright (c) 2021 Nguyen Chuong, Dao Duong, Kieu Dung |
| spellingShingle | Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Chuong, N. M. Duong, D. V. Dung, K. H. Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_alt | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_full | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_fullStr | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_full_unstemmed | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_short | Weighted Lebesgue and central Morrey estimates for $p$-adic multilinear Hausdorff operators and its commutators |
| title_sort | weighted lebesgue and central morrey estimates for $p$-adic multilinear hausdorff operators and its commutators |
| topic_facet | Multilinear Hausdorff operator commutator central BMO space Morrey space $A_p$ weight maximal operator $p$-adic analysis Multilinear Hausdorff operator commutator central BMO space Morrey space $A_p$ weight maximal operator $p$-adic analysis |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/441 |
| work_keys_str_mv | AT chuongnm weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT duongdv weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT dungkh weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT chuongnm weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT duongdv weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT dungkh weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT chuongnm weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT duongdv weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators AT dungkh weightedlebesgueandcentralmorreyestimatesforpadicmultilinearhausdorffoperatorsanditscommutators |