Toán tử tích phân loại hardy và các giao hoán tử của chúng trên một số không gian hàm tt tiếng anh

MINISTY OF EDUCATION AND TRAINING
HANOI NATIONAL UNIVERSITY OF EDUCATION
||||||| * |||||||

NGUYEN THI HONG

THE HARDY TYPE OPERATORS
AND THEIR COMMUTATORS ON FUNCTIONAL SPACES
Speciality: Integral and Differential Equations
Code: 9.46.01.03

SUMMARY OF DOCTORAL THESIS IN MATHEMATICS

Ha Noi - 2019


This thesis has been completed at the Hanoi National University of Education

Scientific Advisor:

1. Prof. PhD. Sci. Nguyen Minh Chuong
2. PhD. Ha Duy Hung

Referee 1: Prof.PhD.Sci. Vu Ngoc Phat, Institute of Mathematics, VAST
Referee 2: Assoc.Prof.PhD. Khuat Van Ninh, Hanoi Padagogical University
Referee 3: Assoc.Prof.PhD. Tran Dinh Ke, Hanoi National University of Education

The thesis shall be defended before the University level Thesis Assessment
Council at ............ on.......

The thesis can be found in the National Library and the Library of Hanoi

National University of Education.


1

dp
d p

INTRODUCTION

1. MOTIVATION AND OUTLINE
One of the core problems in harmonic analysis is to study the
boundedness of an operator T on some functional or distributional spaces.
jjT fjjY

(1)

CjjfjjX ;

where C is a constant, X; Y are functional or distributional spaces with corresponding norms k kX ; k kY . This question arises from natural problems
in investigations about analysis, functional theory, partial di erential
equations. For instance, we consider Riesz operator J de ned by
Z
y)
J (f)(x) =

Rd

f(
x

j j

y

d

dy

(2)


where 1

p<

d

p

then J

and q =

d

q

d

bounded from L (R ) to L (R ).

One important application of this results is the theorem embedding SobolevGagliardo-Nirenberg: W

1;p

d

q

d

1

1 1

(R ) ,! L (R ), with 1 p q p ; p = p d .
One of the main problems in these thesis is study (2) for one particular class
of integrals and their commutators. This operator class includes or closely relates with a lot of classical important operators such as: Hardy operator, maximal
Calderon operators, Riemann-Lioville operators on line, in cases one dimension.
The estimations in form of (1) is called Hardy's inequality. Hardy's integral
inequality and it's discrete version appeared about 1920, related with the
p

continuity of the Hardy operators on L spaces. One of the main motiva-tions
due to these results is began from Hilbert's inequality. The mathemati-cian
Hilbert, while researching for the solutions of some integral equations, due
1
PP

1

ambn

to research the convergence of the double series in form of n=1 m=1 m + n. In
nP P
aman
1
1915, Hardy started that the
1
is convergent i

m+n
1
and

=1 m=1
1
n=1

X

A
a

nn

n

are convergent, where An = a1 +

n=1

X

A
n

n

2
;

+ an. Hence, we can rewrite in the form


2

p

+

p

+

below: if f 2 L (R ), for 1 < p < 1 then Hf 2 L (R ), where
1 Z0 x
Hf(x) = x
f(t)dt:

(3)

In 1920 G. Hardy demonstrated the integral inequality below
1
0x
p
1 p 0 1 fp(x)dx:
x
p
Z

1

Z

f(t)dt

p

dx

(4)

Z

0

where 1 < p < 1, f is a nonnegative measurable function on (0; 1), and the
p

best contant is p 1 .
Hardy operator is one case of the class Hausdor operators, appeared in
the problem for numeral series and exponent series with fundamental in
the research of Siskakis, and Li yand-Moricz in the real eld. Their Hausdor
operators in form of
Z
H ;A(f)(x) =
(u)f(xA(u))du;
(5)
d

R

d

where be measurable function on R and A = A(u) = (aij(u)) be matrix, with
0

order is d d and aij(u) be measurable function of u: In particular, when (u)
= [0;1](u), A(u) = u then H ;A turn in to classical Hardy operators above.
A natural question arises, which spaces replace X; Y spaces and which condition for , matrix A then (1) is true with T = H ;A. Moreover, which is the best
constant C in (1)? The rst question has attracted attention of a lot
mathematician over the world and list some results of K. Andersen, E. Li yand,
F. M•oricz, D.S. Fan. However, the necessary condition about the
boundedness to be given are not su cient conditions and the question about
the best constant in each cases is not easy to answer. The second question
about the best constant in estimation in form of (1) for the class average operator has two directions: The rst is for average operator class on the globular
in form of
Z
d

1
H(f)(x) =
(6)
f(y)dy; x 2 R n f0g:
jj
d

x

d

jyj
p
p

Grafakos and Lacey prove that the norm on L of H equal p

1

.

The second is for average operator class along the parameter curve
given by the form.


3

Z1

U f(x) =

0

f(tx) (t)dt;

(7)

This is operator class has many applications in the operational theory, the
partial di erential equation, because it contains many classical operator
such as Abel, Rieman-Liouville, Hardy operator, the maximum operator. In
q

d

2001, J. Xiao published a important result: U is bounded on L (R ) if only if
R 1 qd (t)dt nite. Moreover,
0 t
jjU jjLq(Rd)!Lq(Rd) =

d

(t)dt:

qd

(8)

0

Similarly, U is bounded on BMO if only if

jjU jj

1t

Z

d

BMO(R )!BMO(R )

R

0

=

1

1(t)dt

nite and then

(t)dt:

Z0

(9)

In 2009, based on the research method for the commutator of CalderonCoifmann-Weiss, the authors Fu, Liu and Lu proved that [M b; U ] is
q

d

d

bounded on L (R ) forall b 2 BMO(R ) if and only if
1

Z

t

d=q

2
(t) log tdt < 1:

0

Where Mb is the multiplication operator M b(f) = bf. This result shows that
the commutators of U is "more singular" than U .
The above two fundamental results are the motivation for many later
studies for the U class. This is also the main research direction that the
author chose in this thesis: studying the norm estimates for multilinear
operators with the following form
Z

;!

Um;n f

s

[0;1]n

!

x

n
Y

()=

!

k=1

f s
k

(

k(

)

tx)

t( )dt:

(10)

In cases m = n = 1, Chuong and Hung nd out the necessary and su cient
conditions( with the appropriate conditions on s(t)) of to ensure the bound1; 1

p

edness of U ;s and their commutators in L and BMO with homogeneous
weighted. The norm of the corresponding operators are nd out. A necessary
p

condition of the weighted function to commutator [M b; U ;s] is bounded on L is
also given. In the cases of Herz spaces, In 2016, Chuong, Hung and Duong give
a necessary for the boundedness of the commutator when b belongs to Lipschitz


4

space. Hung and Ky gave criteria to U

m;n
;!

to L!

p

(R

d

_

) and bounded from B

p ;

p

s

_ p m; m

Rd

1 1

d

bounded from L! 1 (R )
B

1

Rd

However, the criteria for the boundedness, norm of U

m;n

!

pm

d

(R )

m

_ p; R d : Moreto B

cases are nd out.

over the norm of operator in each

L!

together with its

;s

commutator on Herz-type spaces has not been studied before. The Herztype spaces are natural extensions of Lebesgue spaces, which also play
an important role in developing functional theory. For example, the
Taibleson-Weiss molecular functions, which play an important role in the
theory of spaces Hardy, belongs to the space of Herz type. The setting up
the properties that are bounded and norm estimate on the Herz-type

spaces requires changing the approach methods compared to the known
results on the Lebesgue spaces or central Morrey spaces.
From all above reasons, it makes us investigate the boundedness and norm
of operators in (10) from product of spaces of Herz and Morrey-Herz with exponent weighted. We studied commutators of U

m;n

!

on the product of Morrey;s

Herz spaces. The obtained results, we state in the third paper, in the
author's works related to the thesis that has been published, and
presented in chapter 4 in this thesis.
Analysis on p adic numbers or on groups Heisenberg is interested and
strong growth in recent years. In this thesis, we collect investigation some
results of hamornic analysis on local eld, speci c is the boundedness of the
p-adic weighted of Hardy types operator. In 2006, Rim and Lee developed
these results of J. Xiao for weighted Hardy p adic below
Z

p

U f(x) =

where

Zp?

f(tx) (t)dt;

is a nonnegative on ring of p-adic integers.

(11)
In 2014, H.D. Hung

extended the results of Rim and Lee for p adic Hardy-Cesaro operator U
p
where
;s, on the exponent weighted spaces,
?
U

p
;sf(x)

Z Zp

=

f (s(t)x) (t)dt;

(12)

Beside the results about the boundedness, norm of U ;s on weighted p adic
Lebesgue spaces and weighted BMO spaces, the authors shown a
corollary about the convergence of double series in the real eld as below:
!
r
1=r
0

xj+ k yk
1
0
xj 11=r
(13)
r
1

@X X

j2Z k=0

A

X

@

j2Z

1

A

X
k=0

!
yk :

5

where (xj)j2Z and (yk)k 0 are two nonnegative sequences, be natural
number and 1 r < 1: These results give the connection between real
analysis and p-adic analysis, studied p-adic analysis may be become tools
to investigate real analysis. Similarly, the results in real eld, Wu and
Fu(2017) worked out a necessary condition and su cient condition for the
q;
d
boundedness of U on p adic Morrey spaces L (Q p), the p adic central

_q;

d

q;

d

Morrey spaces B
(Q p) and CBM O (Q p). Moreover the corresponding
norm of its operators in each cases work out.
p

Due to this sense, we extended the results of Wu and Fu(2017) for U ;s
on corresponding exponent weighted spaces. Base on the work of Fu at
el(2015), Hung, Ky(2015), Chuong, Duong(2016) we investigated the pp

adic multilinear version of U ;s. We studied the results which Hung, Ky, Fu,

Lu, Gong, Yuan obtained in the real eld, for the p adic eld.
2. PURPOSES, OBJECT AND SCOPE OF THE THESIS
Purposes of the thesis: To study the norm of the weighted Hardy-Cesaro
operators, the weighted multilinear Hardy-Cesaro operators and their commutators on the functional spaces in real case or p-adic case, speci c as

Content 1: Bounds of p-adic weighted Hardy-Cesaro operators and
their commutators on p-adic spaces of Morrey types.
Content 2: Bounds of p-adic weighted multilinear Hardy-Cesaro operators and their commutators in p-adic functional spaces.
Content 3: Estimate the norm of the weighted multilinear Hardy-Cesaro
operators and their commutators on the product of Herz and MorreyHerz spaces .
3. RESEARCH METHODS
To investigate the norm of the weighted Hardy-Cesaro operators, the
weighted multilinear Hardy-Cesaro operators and their commutators on the
functional spaces in real case or p-adic case, we used the known method in
p-adic analy-sis and real analysis, the operational theory, H•older's inequality,
Minkowski's inequality and other inequalities. Moreover to estimate the
boundedness of their commutators, we use the methods of Coifman(1976).
4. STRUCTURE OF THE THESIS
Apart from Introduction, Conclusion, Author's works related to the thesis


6

and References, the thesis includes 4 chapters:
Chapter 1. Preliminaries.
Chapter 2. Bounds of p adic Hardy-Cesaro operators and their commutators on p adic spaces of Morrey types.
Chapter 3. Bounds of p adic multilinear Hardy-Cesaro operators and
their commutators in p adic functional spaces
Chapter 4. Multilinear Hardy-Cesaro Operator and Commutator on the
product of Morrey-Herz types spaces.

7

Chapter 1
PRELIMINARIES

In this chapter, we recall some basic knowledge of p-adic numbers,
the measurement and integral on the p-adic eld, the functional spaces,
some test functions, the H•older's inequality, the Minkowski's inequality,
other inequali-ties and some useful theorems.
1.1. The p-adic

eld

In this section, we recall some concepts of p-adic numbers, the p-adic
norm, the p-adic eld and some special properties.
d

1.2. Measurement and integral on Q p

In this section, we recall the basic knowledge of measurement and
d
integral on Q p.
1.3. The functional spaces

In this section, we recall some concepts of some functional spaces such
as Lebesgue spaces, Herz spaces, Morrey-Herz spaces, BM O spaces,
Morrey spaces and some illustrative examples, the H•older's inequality, the
Minkowski's inequality, other inequalities and some useful theorems.

8

Chapter 2

P
BOUNDS OF -ADIC HARDY-CESARO OPERATORS AND THEIR
COMMUTATORS ON P -ADIC SPACES OF MORREY TYPES

In this chapter we study the norm of the p-adic weighted Hardy-Cesaro
operators on weighted spaces of Morrey types. First, we introduce the motivation due to problem. To solve problem, we used the real variable method
for Hardy integral operator, some results about hamonic analysis on p-adic
eld, combined with constructed the test functions, the Minkowski's inequality, we obtained the norm of the p-adic weighted Hardy-Cesaro operators. In

the last of chapter, due to real variable method of Coifman(1976), p;b
we nd
out a necessary condition and a su cient condition for (t) to U
;s is
q; d
q;
d
bounded on B_! Q p with symbol in CMO ! Q p .
The contents of this chapter is written on the paper 1. in the author's
works related to the thesis that has been published.
2.1. Motivation

Our problem studies the norm of the p-adic weighted Hardy-Cesaro
opera-tors on weighted spaces of Morrey types in p-adic eld.
p

2.2. Bounds of U ;s on weighted spaces of Morrey types
2.2.1.

Some definitions and lemmas

We recall the de nition of the p-adic weighted Hardy-Cesaro operators
which are de ned by Chuong, Hung(2014) as follow:
?
p

De nition 2.1. Let s : Z
d

! Qp and

?
p

:Z

! R+ be measurable functions
d

and ! : Q p ! R+ be a locally integrable function. For a function f on Q p, we
p
d
de ne the p adic weighted Hardy-Cesaro operator U ;s on Q p as
Z
p

U ;sf(x) = Zp?
f (s(t)x) (t)dt:
(2.1)
The weighted functions were introduced by Chuong, Hung which were
ex-tended by exponential weighted functions.


9

De nition 2.2. The class of weights W , which consists of all nonnegative
d
d
locally integrable function ! on Q p so that !(tx) = jtjp !(x) for all x 2 Q p and
t2

?
Q p

and 0 <

R

S0

!(x)dx < 1.
1

Theorem 2.1. Let 1 q < 1; q < 0 be real numbers. Let be a non-negative,
?
measurable function on Z p: Then the following clauses are equivalent

(1) A :=

R

( d+ )

js(t)j p

(t)dt is

nite.

?

Z p

(2) U

p
;s

(3) U

p
;s

is bounded on L

q;

is bounded on B

d
! (Q p).

_ q;

d

!

Q p.

q;

d

Moreover,
Up

;s

L!q;

d

(Qp )!L! (Qp )

= Up

= A:
;s _ q;
q;
B ! (Qpd)!B_! (Qpd)

(2.2)

Remark 2.1. When s(t) = t and ! 1, we obtained the theorem 2.1 and the-orem
2.3 of Fu, Wu(2017). Note that the theorem 2.1 and 2.3 of Fu, Wu(2017) are
only true with 1 < q, while the Theorem 2.1 is even true with q = 1.
p

1;

1;

Corollary 2.1. Operator S is not bounded on L (Qp) and on B (Qp), with
1 < 0.

_ q;

1
q

d

q;

d

Remark 2.2. In this cases =
, B ! Q p spaces and L ! Q p spaces
q
d
become to L ! Q p spaces, although the proof in Theorem 2.1 is not true.
However, due to the result of Theorem 3.1 of Hung(2014), then the
Theorem 2.1 is true if we add the condition js(t)j p jtjp with t 2 Zp (see
theorem 3.1 in chapter 3).
Now, we give the application of this theorem in investigation the solutions
of p adic pseudo-di erential operators.Consider the Cauchy problem

D u + a(jxjp)u = f(jxjp); x 2 Qp
u(0) = 0;
where a; f be continuously functions, the desired function u = u(jxj) is
radius function. To investigate the solvable problem, A. Kochubei(2014)
p

p

considered the solution u in the form u = R (v), where R has the form
p
1
1
Z
f(y)dy
R f(x) = 1
jx yjp
jyjp
1

p

p

1

jyjp jxjp


10

p

with f be the local integrable on Qp. The operator R is right inverse of D on
local constant spaces, plays role the same as Riemann-Lioville operators on
pj 1
the real eld. Let 0(t) = 1
and 1(t) = 0 (1 t) then
t 1
1
1 p

p

x R f(x) = U
j pj

jp

p

0

Corollary 2.2. Assume 0 < < 1 and 1 q <
q;

f(x)

U

1
1; q
q;

p

f(x):

1

< 0. Then Rp
dx; Qp).

< de ned as
q

the operator is bounded from L (Qp) to L (jxjp
Theorem 2.2. Let q; be the real numbers satisfy 1 < q < 1; 0 <
p
q;

d
CBMO
!
U ;s is bounded on
Qp if only if A is nite. Moreover,
kU p

k

;s CBMO!

q;

(Qpd)!CBMO!q; (Qpd)

1
d

then
(2.3)

= A:

2.3. Commutator of the p-adic weighted Hardy-Cesaro operators
2.3.1. Definition of the commutator and lemma
?
p

De nition 2.3. Let s : Z

?
p ! R+ be measurable functions
d
d
Q p, f : Q p ! C be measurable

! Qp and

:Z

and b be a locally integrable function on
functions. Commutator of the p adic weighted Hardy-Cesaro operator de ned
Z
as follow
Z p?

U

p;b
;sf(x)

f(s(t)x)(b(x)

=

b(s(t)x)) (t)dt:
q;

Lemma 2.1. Suppose that b is a function in CBMO !
2

integer numbers. Here
>
n. Then
bB ;! bB 0;! p

=

8

1

Theorem 2.3. Let q; q1; q2
and

and ;

1

and !

2W

0

are

, with

j j maxf! (B ) ; ! (B 0) g c kbkCBMO!q; ; Here and after

p(n+

<(n + ) ln p
:
2.3.2. The main results
1
+q2

1
n

Qp

n+ 0

c

1
q1

1
n,

R so that

(2.4)

1

q1

jp

(n+ )

if

)

j jj

1

if

=0
6= 0:

be real numbers such that 1 < q < q1 < 1,
?

?

1

=

q

< 0. Let s : Zp ! Qp be a measurable function such that
q

d

s(t) 6= 0 almost everywhere t 2 Z p. We assume that b 2 CBMO! 2 Q p . If both A; B
p;b
are nite then the commutator U ;s is determined as a bounded operator from B_!
q; d
q; d
p;b
q; d
1 Q p to B_! 2 Q p . Conversely, if U
;s is bounded B_! 1 Q p to


11

B

_q;
!

2

d

Q p then B? is nite. Here and after,
Z
( d+ )

B=

?

Z p

js(t)j p

j

(t)dt;

(2.5)

(t)dt :

(2.6)

logp s(t)jp

and
Z

B? =

js(t)j(pd+ )

logp js(t)jp

?

Z p

Moreover,
q;
? 1 b
q ;
0
CBMO! 2 2 (Qpd)
kf k
@ 0 B_!q; (Qpd) B A k k
kf k
0 B_!1 (Qpd)

U

p;b k

k

B_! 1 (Qpd)
q;

;s

2A + p
Corollary

2.3. Let q; q1; q2 be

1
1
1
1
= q1 +q2 and q1 < 0.
q
?
such that js(t)jp > 1 a.e t 2 Zp
q

b 2 CBMO! 2 Q

d
p

d+

B_ !

q ;
2

(Qpd)

!

B kbkCBMO!q2 (Qpd):

real numbers such that 1 < q < q1 < 1,
?

Let s : Zp ! Qp be a measurable function

?

or js(t)jp < 1 a.e t 2 Zp . We assume that
p;b
;s is determined as a
_ q; d
B ! Q p if and only if B is nite.

. Then the commutator U

_q;

n

bounded operator from B ! 1 Q p to
Remark 2.3. As we know, commutators of Hardy operators are "more
singu-lar" than corresponding Hardy operators. This problem is not di erent
on in cases the central Morrey spaces. In fact, when js(t)j p < 1 almost
everywhere t 2 Zp then B is nite implies A < 1. In other word, the example
below given that A is nite does imply B < 1.
1
Indeed, choose s(t) = pt, (t) =
, then A < 1; B = 1:
j j

pt p1+(d+

)

(logp pt

j j

1
q

2

p)

1

1

1
q

1

< 1,
= q 1 + q2 ;
< < 0;
q1 <
1
?
1 < 0; 0 < 2 < d and = 1 + 2: Let s : Z p ! Qp be a measur-able function
such that s(t) 6= 0 almost everywhere. If C is nite, then for

q ;
d
p;b
any b 2 CBMO !2 2 (Q p), the corresponding commutator U ;s is bounded
Theorem 2.4. Let 1 < q < q1

_q;
_ q; d
d
from B ! 1 1 (Q p) to B ! (Q p) and we have
U p;b
(2 + pd+ c )
q;
log

k

s(t2) p

;skB_! 1

(t)dt:

1

(Q )!B_!q; (Qpd)
pd

2

Here c is a constant de ned as in Lemma 2.1 ,
j j j
j

C

b

q;

C k kCBMO! 2
=

R
Zp
?

f j

2

(Q )
pd

max 1; s(t)

:

gj

j
(d+ ) 1

jp
s(t)

(d+ ) 1


12

Chapter 3

P
BOUNDS OF -ADIC MULTILINEAR HARDY -CESARO OPERATORS AND
THEIR COMMUTATORS IN P -ADIC FUNCTIONAL SPACES

In this chapter, we study the norm of the p-adic weighted multilinear
Hardy- Cesaro operators on product of Lebesgue spaces and the spaces
of Morrey types. First, we introduce the motivation due to the problem. In
sequel, using the schema proof the results are developed from the schema
in previous chapter, combination with the methods has used in mulltilinear
analysis on the real eld or on the local compact group. The commutator
problem of p-adic Hardy- Cesaro operators has studies in this chapter. The
researching method is the real variable method of Coifman(1976).
Besides, we establish the estimation of di erent of two functions in CBM O
p

space, hence we obtained the estimation on L for the average integral
operators. The di erence is for the singular integral operators, we usually

used John-Nirenberg, but in here, we used immediate estimate by
inequalities such as Minkowski's inequality, H•older inequality.
The contents of this chapter is written on the paper 2. in the author's
works related to the thesis that has been published.
3.1. Motivation

Due to the reasons in the introduce part, we investigate the p-adic weighted
multilinear Hardy- Cesaro operators on some functional spaces in p-adic eld.
3.2. Bounds of the p-adic weighted multilinear Hardy- Cesaro oper-ators

on the product of Lebesgue spaces and spaces of Morrey types
To proof the main results we need some de nitions and lemmas below.
3.2.1. Some definitions and lemmas

We introduce and investigate the p-adic weighted multilinear Hardy- Cesaro operators de ned as follow:


13

m; n

positive integer numbers and

:

De nition 3.1. Let
be ? n
m
p
!

Zp ! Qp be measurable. The adic
[0; +1),
=(1
m) :
multilinear Hardy-Cesaro
;s
!
1
s

s

? n
Z

operator Up;m;n , which de nes on f

we ig hte d

= (f ; : : : ; f

m

!

d

m

!

Qp ! C vector of measurable functions, by
;

U
s

p;m;n
s

?

Z
(Zp

!

(f ; : : : ; f
1

m )(x)

=

!
):

p

; : : :; s

m
Y
k=1

)

f k (s (t)x)

(t)dt;

k

n

(3.1)

s ; : : : ; s m).

!

where = ( 1

Remark 3.1. When m = n = 1; U

p;m;n

!

is reduced to U

p

;s

;s

by Hung(2014).

has been investigated

In this chapter, if not explicitly stated otherwise, q; ; q i; j are real
numbers, 1 q < 1, 1 qj < 1, j > d for each j = 1; : : : ; m so that
1

=

q
and
p

k

q1

q1

=
The weights !k 2 W

1

q

1

+ +

qm

;

qm

+ +

q

1

(3.2)
:

(3.3)

m

; k = 1; : : : ; m, set
m

q

kY

!(x) =

!

=1

q

k

k

(x):

(3.4)

p

W!

It is obvious that ! 2 W .

p

De nition 3.2. We say that (!1; : : : ; !m) satis es the

m

Y

!(S0)

q

!k(S0)

qk

condition if

:

(3.5)

k=1

Example 3.1. For
p example, (!1; : : : ; !m) where !k(x) = jxjp k for k = 1; : : : ; m
satis es the W! condition.
Through out this paper, s1; : : : ; sm are measurable functions from Z

!
Qp and we denote by s the vector (s1; : : : ; sm).
p

Lemma 3.1. Let ! 2 W ;

>

fr; (x) =

r

0

:j j

d

belong to L! (Q ) and jjf jj
p

8

d and

r

d

r; L! (Qp )

=

d+

r

1
2

> 0 then the function
jj

if x p < 1
if x p
1:

p

!(S0)

1 p r=

2

j j1=r
> 0:

? n
p

into


14

3.2.2. The main results
p

Theorem 3.1. Assume that (!1; : : : ; !m) satis es W! condition and there
exists constant

> 0 such that jsk(t1; : : : ; tn)jp

minfjt1jp ; : : : ; jtnjp g holds
? n
p .

for every k = 1; : : : ; m and for almost everywhere (t 1; : : : ; tn) 2 Z
Then there exists a constant C such that the inequality
q

jj U p;m;n(f ; : : : ; f )jj
1

;!

s

m

m

d

Y

C

L!(Qp)

k=1

jjfkjjLq!kk (Qdp)

(3.6)

holds for any measurable f1; : : : ; fm if and only if
m
A

Z

:=

(Zp) k=1

j

?

d+ k
j

Y

n

1

qk

(t)dt < :

sk(t) p

(3.7)

Moreover, A is the best constant C in (3.6).
Remark 3.2. When m = n = 1, we obtained the theorem 3.1 of Hung(2014).
Note that the inequality (13) for two sequences nonnegative real numbers,
is immediate consequence of Theorem 3.1 of Hung(2014).
Theorem 3.2. Let 1

q; qk < 1; ; k;

be as in (3.2), (3.3) such that

k

1
q k

< k < 0 for k = 1; : : : ; m. Assume
condition. We set

=
We assume that

d+

1

1+ +
d +d +

Z

that (!1; ; !m) satis es W!

d+

m

m:

m
Y

k

(d+ )

? n

B=

(Z p )

=1

jsk(t)jp

k

(3.8)

(t)dt < 1;

k

m

and
(!(B0))
d

Here B0 is the ball fx 2 Qp
that the inequality

: jxjp

;

!

1

m

1+ kqk

k

=1

jj U p;m;n(f ; : : : ; f )jj
s

Y

1+ q
q

(!k(B0))

:

qk

(3.9)

1g. Then, there exists a constant C such
q;

p

Q

L! (

m

d

)

C

Y

k=1

jjf jj

q

;

k L !kk

k

(Qdp)

(3.10)

holds for any measurable f1; : : : ; fm. Moreover, the best constant C in
(3.10) equals B.


15

Theorem 3.3. Let q; qk; ; k; k
be as in Theorem 3.2 with q; qk > 1 and
p
conditions (3.2), (3.3) are hold. Assume that (! ;
; ! ) satis es
conW!

dition. Then U
_

q ;

m m

B!m

1

p;m;n
s

_ q; d
(Qp) to B! (Qp). Moreover,

jj

q;

Up;m;n
s

(Qp)

jj

;!

B !1

_1 1

q;

qm; m
B !m

d

_

q ;

1 1

is determined as a bounded operator from B !1

;!

d

m

(Qp) B !

_

d

!

=B :

(Qp)

_

d

(Qp)

d

(3.11)

Remark 3.3. When m = n = 1, from Theorem 3.2 and 3.3, we obtained
Theorem 2.1 in chapter 2 of this thesis.
3.3.

The commutator of weighted bilinear Hardy- Cesaro operators

In p-adic eld, the commutator of operation of Hardy types have researched
by Fu, Lu, Wu, Chuong, Hung,...We have the commutators of the weighted
q

d

bilinear Hardy- Cesaro operators with the symbol in CBM O! (Q p).
3.3.1. Commutator and lemma

We de ned the commutator of weighted bilinear Hardy- Cesaro
operators as follow:
? n
p

De nition 3.3. Let n 2 N;

: Z

? n
p

! [0; 1); s1; s2 : Z

! Qp; b1; b2, be

d
d
p and f1; f2 : Q p ! C be measurable
p;n
The commutator of weighted bilinear Hardy- Cesaro operator U ! is de ned

locally integrable functions on Q

functions.

;s

as:

!

Up;n;

b

;!

s

Z

(

(f ; f )(x) =

1 2
2

Zp? n

)

!

k=1

Y

fk(sk(t)x)
C
Z
(Zp)

We set

(3.12)

k=1

Y

=

!

2

k=1

jj

(t)dt:

(bk(x) bk(sk(t)x))
!

2

D
Z

2

(Z p )

2

=

?

j

n

Y

j

?n

j

2
k=1

Y
sk(t) p

(d+ k) k

Y

Remark 3.4. D2 < 1 does not imply C2 < 1.
Remark 3.5. C2 < 1 does not imply D2 < 1.

logp

(3.13)

(t)dt:

j

sk(t) p

!

k=1

2 =

(d+ k) k

sk(t) p

!

(t)dt:

(3.14)


16

3.3.2. The main results
1

Theorem 3.4. Let 1 < q < qk < 1; 1 < pk < 1; p
+

q q1

and

1

q2

+

+

1

p1

p2
q +

q1

+ 2. Assume that !(x) = !1

q

q

p1

q2

!2

2

1+ q

!(B0)

< k < 0; k = 1; 2 such that

1 1 1 1

=
and =

k

1+ kqk

!k(B0)

q

+

qk

kY

q
p2

q1

; =

q1

+ p1 +

q

1

q2
q

2

+

q2
p

2

1

+

pk

=1

p

CBM O!2

(Qp) then U

(ii) If

(b1; b2)

q ;

d

1 1

_ q1; 1

is bounded from B!1

;!

s

for any b =
_

are nite then for any b = (b1; b2) 2 CBM O!1 (Qp)

p;2;n; b!

d

2

_

q ;

2

CBM O!

p1

_ q;

and =
j

!

s

_

B

1

1 a.e

p

t

q;

1 1

CBM

1

k

!

(Qp)

<

d

2

_

B

q;

t

2 2

( Q p) ?

+

q2

, for each

d

n

!

(Qp) to B!

_ q;

2

d

(Qp):

b

p
d
O! 2 (Qp ),

!

;!

U p;2;n; is
s

is nite.

< k < 0; k = 1; 2 such that

+

p1

1

+

p2

= 1 2. Then

is

+ 2. Furthermore, suppose that jsk(t)jp > 1 a.e t 2

1

!2

d

1 1 1 1

q q1

p

B

(Qp) to B! (Qp) then D2

=
k

_ q2; 2

d

d

Corollary 3.1. Let 1 < q < qk < 1; 1 < pk < 1; p

Z

(Qp)

d
(Q p )
1

d

2 2

bounded from B!1 (Qp) B!2

()

d

p1

(i) If both C2 and D2

Q

p

_ q; (
to B

k

d

;

) if and only if

D2

p;2;n;

U

b

Zp

?

n

or

!

;!

s

is nite.

bounded from


17

Chapter 4
MULTILINEAR HARDY CESARO OPERATOR AND COMMUTATOR ON
THE PRODUCT OF THE SPACES OF HERZ TYPES

In this chapter, we study the boundedness of the weighted multilinear
Hardy-Cesaro operator on the product of Herz and Morrey-Herz spaces. First,
we introduce the motivation due to the problem. In sequel, method of
Xiao(2001) has used in the work of Fu, Wu, Hung, Chuong, Ky, ... , the
technology from multilinear analysis, we obtained the estimate on the
boundedness of this op-erators on the product of Herz and Morrey-Herz
spaces. Finally, due to the real variable method of Coifman(1976), the method
estimates on multilinear analysis and the key lemmas and schema of research
of Fu, Gong, Lu, Yawn,..., and the special case of Hung, Ky, we estimate the
boundedness of their com-mutators from the product of central Morrey spaces
to the central Morrey spaces with symbol in the Lipschitz space.

The contents of this chapter is written on the paper 3. in the author's
works related to the thesis that has been published.
4.1. Motivation

Our problem is investigate the weighted multilinear Hardy- Cesaro
oper-ators on on the product of Herz and Morrey-Herz spaces .
The multilinear version of the weighted multilinear Hardy-Cesaro
operator was considered by Hung, Ky(2015) de ned as

n

De nition 4.1. Let m; n 2 N;

n

: [0; 1] ! [0; 1), s1; : : : ; sm : [0; 1] ! R be

measurable functions. The weighted multilinear Hardy-Cesaro operator U

is de ned by

Z

;!

Um;n

f

!

s

1

where = (f ; : : : ; f

[0;1]n

!

f

m

(

!

x) =

m

f s
k

1

), s = (s ; : : : ; s

!,

;s

!

k=1
Y

m;n

m

):

(

k(

)

tx)

t( )dt;

(4.1)


18

4.2. Boundedness of the weighted multilinear Hardy-Cesaro opera-tor on

the product of Herz and Morrey-Herz spaces
4.2.1. Some definitions and lemmas

We would like to recall the de nition of homogeneous weights introduced
by Chuong, Hung(2014)..
De nition 4.2. Let
be a real number. Let W be the set of all functions

d

d

! on R , which are measurable, !(x) > 0 for almost everywhere x 2 R ,
0 < !(y)d (y) < 1, and are absolutely homogeneous of degree
, that is
R

S

d

d

!(tx) = jtj !(x), for all t 2 R n f0g; x 2 R :
We remark that W =
jxj .

S

W contains strictly the set of power weights !(x) =

For our convenience, we give some common notation throughout this
part Let > 0; ; ; 1; : : : ; m be real numbers, 1; : : : ; m > d, 0 < p < 1,
1 q < 1, 1 pi; qi < 1 with i = 1; : : : ; m and ;

1

1

+

q1

2

+ q2

Sd =

2
(

Y

1

m

= ;

p1

m

+ p2 +

1

+

:::;

1

1

= p ; q1

0 satisfy

m

1

1

pm

+q2 +
= fx 2 R

1

+ qm = q ;
d
+ +
= q ; 1 + 2 + + m = : Sd
: jxj = 1g

qm
d
2
for all i = 1; : : : ; m, and we set
d
) : Functions !i belong to W i

+ +
2

1

1;

2

!(x) =

Lemma 4.1. Let p 1 and (fk)k
n
on [0; 1] . Then
1

X
Lemma 4.2. If f

q

! qi (x):

(4.2)

i

i=1

Obviously, ! 2 W .

k=1

m

fk(t)dt

Z

[0;1]

1

be nonnegative and measurable functions
p

1

0

Zn
n

[0;1]

B
_;
@
MK (!) then f
2
k
k
p;q

k q;!

4.2.2. The main results

p

fk (t)

!1=p

1

p

dt

k=1

X

C

2k(

)

f

A
;

k kMK_p;q

(!) .


n

Theorem 4.1. (i) Let s1(t); : : : ; sm(t) 6= 0 almost everywhere in [0; 1] and
A1 =

Zn

[0;1]

m
i=1

Y

jsi(t)j

i

d

qi

i

+i

+

!

(t)dt < 1:

(4.3)


19

Suppose that 1 p < 1 or 0 < p < 1 and at least one of
Then
m
!

kU

;s

m;n

!

!(

!

A

C;

f )kMK_ ; (!)
p;q

Y

k

fi

1

1;

k

: : : ; m is positive.
i

pi;qi

MK

;

_

(! )

i i

:

(4.4)

i=1

Here
8m

2j k

Q

>
>

if 1 p < 1

+1

kj

<

;

=

>

k=1

m

2

>

p

:(2

1=p

1)

2j k

kj

+ 1 if 0 < p < 1 and > 0:

k=1
QC!!

Conversely, let 0 < p < 1, 0 < i < 1 for i = 1; : : : ; m. Suppose
m
Q
m;n
i
is de ned as a bounded operator from
;!
s
=1

(ii)
that U

Then (4.3) holds and

m;n

kU !k m

;

i; i

;s

1

Q
i=1

MK

_

(!i)!MK

pi;qi

_

p;q

m

!

= i=1(2

!

;

Q

1)

p
(2

1)

1=pi

1 2

1=p

m

i

Q

q(

1=q

)

D

_

i

(!i) to MKp;q (!).

;

;

;!

(4.5)

!

m
ipi

i

pi;qi

A

(!)

where
D

MK

;

_

(q ( ))1=qi

(!(Sd))

i ii

i=1
Q

1=q

qi( i i) 1=qi
(q())

1=q

m

(1 2)

Q

i=1

=
1

1=qi

(!i(Sd))

Theorem 4.2. (i) If 1 p < 1, s1(t); : : : ; sm(t) 6= 0 almost everywhere in
n
!
[0; 1] and
(t)dt < 1;
(4.6)
d i
A2 = Z n
qi i
m

i=1

Y

[0;1]

then

jsi(t)j

m

k

k ;!

U m;n (

A

_
Kq

!)

s

k=1

m

is bounded from
U

Q
i

=1

_ i
K

qi

j

k

min

n

_
;p

i2
m

Q

K

_

i

_

(!i )!K

q

1

n

E =

(mp)
p

i

i

s

(4.8)

qi

;p

2q

1=q
q

;!

A

(!)

Where
1=p

(!i)

; : : : ; t g for i = 1; : : : ; m and U m;n

=
1

1=p

(4.7)

i i

(!). Then (4.6) holds and
E! :

(!i) to Kq
!k

m

fi

+1Y

i
_

;p

m;n

;s

t

f

i

k

!

kk

i=1

Kq

1

;p

j

(!)

(ii) Suppose that j s (t ; : : : ; t )j
i

m

2

2Y

;p

f

i

+

1

m
i=1

Y

1

2qi i

qi

1=qi m
i

i

1=q
(!(Sd))

(! (S ))

1=qi

d

:

:


i
=
1

Q

Q
i=1

20

Remark 4.1. When 1 = = m = 0, we obtain the boundedness and bounds for
multilinear Hardy-Cesaro operator on the product of the Lebesgue spaces.
However, the results are worse than those obtained by Hung, Ky(2015).
m;n

In Theorem 3.1 of Hung, Ky(2015) given that the norm of U
to L is exactly

L!m

!

pm

[0;1]n

R

p

i=1

from L!
;!

jsi(t)j

(t)dt:

i

m

d+ i

Q

p1
1

s

i

q

Remark 4.2. The result of Theorem 4.2 implies that in cases exists positive
constant such that jsi(t1; : : : ; tn)j minft1 ; : : : ; tng with i = 1; : : : ; m (with
the H

m

operators then this condition obviously true), then U
m
;
iQ

an bounded operator from MKpi;qi

_

i

m;n

!

is de ned as

_

i

;s

;

(!i) to MKp;q (!) then the necessary

=1

and su cient condition is A2 nite. The consequence consist of Theorem 5
and Theorem 6 of Gong, Fu and Ma(2014), Moreover, to get necessary
condition, the authors need the condition 1 = = m, p1 = = pm and q1 = = qm.
However, our result does not need this condition. Similarly, for the results
of Morrey-Herz, our results in Theorem 4.1 are better than the results of
Gong, Fu and Ma.

Remark 4.3. In Theorem 4.1 we consider the cases when 0 < p < 1, this
idea derived from the paper of J. Kuang(2008), in his paper, the author
estimates the norm of V on the Herz space. Thus, our results are better
than the cor-responding results of Gong, Fu, Ma(2014) and is extented in
cases multilinear of their work of Kuang, Liu, Fu, Chuong, Duong,....
4.3. Commutator of the weighted multilinear Hardy-Cesaro opera-tor
4.3.1. Some definitions

Commutators of U

m;n

n

!
;s

due to Coifmann-Rochberg-Weiss, de ned as Let
n

m; n 2 N,
: [0; 1] ! [0; 1), s1; : : : ; sm : [0; 1] ! R, b1; : : : ; bm be locally
d
d
integrable functions on R and f1; : : : ; fm : R ! C be measurable functions.
The commutator of weighted multilinear Hardy-Cesaro operator U

ned as

[0;1]n

;!

Um;n;
s

!
b

f

!

x

Z
( ) :=

!

m

k=1 f s t x

( k( ) )

!

is de-

;s
!

m
k=1

k

m;n

b x

( k( )

b s tx
k

( k( ) ))

t dt:

()
(4.9)

According to the idea of Tang, Xue, Zhou(2011), we consider the symbol
belongs to Lipschitz functions, de ned as