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Boundedness for commutators of fractional p-adic Hardy operators

Abstract

In this paper we prove that the commutators generated by the fractional p-adic Hardy operators and the central BMO function are bounded on weighted homogeneous Herz spaces.

MSC:11E95, 11K70, 42B99.

1 Introduction

In recent years, p-adic fields have been introduced into some aspects of mathematical physics. There are a lot of articles where different applications of the p-adic analysis in the string theory, quantum mechanics, stochastics, the theory of dynamical systems, cognitive sciences, and psychology are studied [19] (see also the references therein). As a consequence, new mathematical problems have emerged, among them, the study of harmonic analysis on a p-adic field has been drawing more and more concern (cf. [1014] and the references therein).

For a prime number p, let Q p n be the field of p-adic numbers. It is defined as the completion of the field of rational numbers with respect to the non-Archimedean p-adic norm | | p . This norm is defined as follows: | 0 | p =0. If any non-zero rational number x is represented as x= p γ m n , where m and n are integers which are not divisible by p, and γ is an integer, then | x | p = p γ . It is not difficult to show that the norm satisfies the following properties:

| x y | p = | x | p | y | p , | x + y | p max { | x | p , | y | p } .

It follows from the second property that when | x | p | y | p , then | x + y | p =max{ | x | p , | y | p }. From the standard p-adic analysis [7], we see that any non-zero p-adic number x Q p can be uniquely represented in the canonical series

x= p γ j = 0 a j p j ,γ=γ(x)Z,
(1.1)

where a j are integers, 0 a j p1, a 0 0. The series (1.1) converges in the p-adic norm because | a j p j | p = p j . Set Q p = Q p {0}.

The space Q p n consists of points x=( x 1 , x 2 ,, x n ), where x j Q p , j=1,2,,n. The p-adic norm on Q p n is

| x | p := max 1 j n | x j | p ,x Q p n .
(1.2)

Denote by

B γ (a)= { x Q p n : | x a | p p γ } ,

the ball with center at a Q p n and radius p γ , and

S γ (a)= { x Q p n : | x a | p = p γ } = B γ (a) B γ 1 (a).

Since Q p n is a locally compact commutative group under addition, it follows from the standard analysis that there exists a Haar measure dx on Q p n , which is unique up to a positive constant multiple and is translation invariant. We normalize the measure dx by the equality

B 0 ( 0 ) dx= | B 0 ( 0 ) | H =1,

where | E | H denotes the Haar measure of a measurable subset E of Q p n . By simple calculation, we can obtain that

| B γ ( a ) | H = p γ n , | S γ ( a ) | H = p γ n ( 1 p n ) ,

for any a Q p n . For a more complete introduction to the p-adic field, see [15] or [7].

The classical Hardy operators are defined by

Hf(x):= 1 x 0 x f(t)dt, H f(x):= x f ( t ) t dt,x>0,

for a non-negative integrable function f on R + . Obviously, and H satisfy

R n g(x)Hf(x)dx= R n f(x) H g(x)dx.

The well-known Hardy integral inequality [16] tells us that for 1<q<,

H f L q ( R + ) q q 1 f L q ( R + ) ,

where the constant q q 1 is the best possible. The generalized result [17] is that

H f L q ( R + ) q q 1 f L q ( R + ) ,

and

H L q ( R + ) L q ( R + ) = q q 1 ,

where 1 q + 1 q =1.

The Hardy integral inequalities have received considerable attention due to their usefulness in analysis and their applications. There are numerous papers dealing with their various generalizations, variants and applications (cf. [1820] and the references cited therein). We have obtained the Hardy integral inequalities for p-adic Hardy operators and their commutators [21]. The boundedness of commutators is an active topic in harmonic analysis because of its important applications; for example, it can be applied to characterizing some function spaces. There are a lot of works about the boundedness of commutators of various Hardy-type operators on Euclidean spaces (cf. [22, 23], etc.). In this paper, we will establish the Hardy integral inequalities for commutators generated by fractional p-adic Hardy operators and CMO functions.

Definition 1.1 For a function f on Q p n , we define the p-adic Hardy operators as follows:

H p f ( x ) = 1 | x | p n B ( 0 , | x | p ) f ( t ) d t , H p , f ( x ) = Q p n B ( 0 , | x | p ) f ( t ) | t | p n d t , x Q p n { 0 } ,
(1.3)

where B(0, | x | p ) is a ball in Q p n with center at 0 Q p n and radius | x | p .

Definition 1.2 Let f L loc ( Q p n ), 0β<n. The fractional p-adic Hardy operators are defined by

H β p f ( x ) = 1 | x | p n β B ( 0 , | x | p ) f ( t ) d t , H β p , f ( x ) = Q p n B ( 0 , | x | p ) f ( t ) | t | p n β d t , x Q p n { 0 } ,
(1.4)

where B(0, | x | p ) is the ball as in Definition 1.1.

It is clear that when β=0, then H β p becomes H p .

Definition 1.3 Let b L loc ( Q p n ), 0β<n. The commutators of fractional p-adic Hardy operators are defined by

H β , b p f=b H β p f H β p (bf), H β , b p , f=b H β p , f H β p , (bf).
(1.5)

In [2426], the CMO spaces (central BMO spaces) on R n have been introduced and studied. CMO spaces bear a simple relationship with BMO: gBMO precisely when g and all of its translates belong to BMO spaces uniformly a.e. Many precise analogies exist between CMO spaces and BMO spaces from the point of view of real Hardy spaces. Similarly, we define the CMO q spaces on Q p n .

Definition 1.4 Let 1q<, a function f L loc q ( Q p n ) is said to be in CMO q ( Q p n ) if

f CMO q ( Q p n ) := sup γ Z ( 1 | B γ ( 0 ) | H B γ ( 0 ) | f ( x ) f B γ ( 0 ) | q d x ) 1 q <,

where

f B γ ( 0 ) = 1 | B γ ( 0 ) | H B γ ( 0 ) f(x)dx.

Remark 1.1 It is obvious that L ( Q p n )BMO( Q p n ) CMO q ( Q p n ).

Let B k = B k (0)={x Q p n : | x | p p k }, S k = B k B k 1 and χ k be the characteristic function of the set S k .

Definition 1.5 [27]

Suppose that αR, 0<q< and 0<r<. The homogeneous p-adic Herz space K r α , q ( Q p n ) is defined by

K r α , q ( Q p n ) = { f L loc r ( Q p n ) : f K r α , q ( Q p n ) < } ,

where

f K r α , q ( Q p n ) = ( k = + p k α q f χ k L r ( Q p n ) q ) 1 q ,

with the usual modifications made when q= or r=.

Remark 1.2 K r α , q ( Q p n ) is the generalization of L q ( | x | p α dx), and K q 0 , q ( Q p n )= L q ( Q p n ), K q α q , q ( Q p n )= L q ( | x | p α dx) for all 0<q and αR.

Motivated by [22], we get the following operator boundedness results. Throughout this paper, we use C to denote different positive constants which are independent of the essential variables.

Theorem 1.1 Suppose that β0, 0< q 1 q 2 <, 1 r 1 1 r 2 = β n , 1< r 1 <, 1 r 1 + 1 r 1 =1, b CMO max { r 1 , r 2 } ( Q p n ). Then

  1. (1)

    If α< n r 1 , then

    H β , b p f K r 2 α , q 2 ( Q p n ) C b CMO max { r 1 , r 2 } ( Q p n ) f K r 1 α , q 1 ( Q p n ) .
    (1.6)
  2. (2)

    If α> n r 2 , then

    H β , b p , f K r 2 α , q 2 ( Q p n ) C b CMO max { r 1 , r 2 } ( Q p n ) f K r 1 α , q 1 ( Q p n ) .
    (1.7)

When α=0, q j = r j , j=1,2, we can get the following result.

Corollary 1.1 Suppose that β0, 0< q 1 q 2 <, 1 q 1 1 q 2 = β n , 1< q 1 <, 1 q 1 + 1 q 1 =1, b CMO max { q 1 , q 2 } ( Q p n ). Then

H β , b p f L q 2 ( Q p n ) C b CMO max { q 1 , q 2 } ( Q p n ) f L q 1 ( Q p n ) ,
(1.8)

and

H β , b p , f L q 2 ( Q p n ) C b CMO max { q 1 , q 2 } ( Q p n ) f L q 1 ( Q p n ) .
(1.9)

When β=0, we can get the boundedness of a p-adic Hardy operator in [21].

Corollary 1.2 Let 0< q 1 q 2 <, 1<r<, b CMO max { r , r } ( Q p n ). Then

  1. (1)

    If α< n r , then

    H b p f K r α , q 2 ( Q p n ) C b CMO max { r , r } ( Q p n ) f K r α , q 1 ( Q p n ) .
    (1.10)
  2. (2)

    If α> n r , then

    H b p , f K r 2 α , q 2 ( Q p n ) C b CMO max { r , r } ( Q p n ) f K r 1 α , q 1 ( Q p n ) .
    (1.11)

By the similar proof of Theorem 1.1, we can obtain the following result.

Corollary 1.3 Suppose that β0, 0< q 1 q 2 <, 1 r 1 1 r 2 = β n , 1< r 1 <, 1 r 1 + 1 r 1 =1. Then

  1. (1)

    If α< n r 1 , then

    H β p f K r 2 α , q 2 ( Q p n ) C f K r 1 α , q 1 ( Q p n ) .
    (1.12)
  2. (2)

    If α> n r 2 , then

    H β p , f K r 2 α , q 2 ( Q p n ) C f K r 1 α , q 1 ( Q p n ) .
    (1.13)

2 Boundedness of commutators of fractional p-adic Hardy operator

In order to prove Theorem 1.1, we firstly give the following lemmas.

Lemma 2.1 Suppose that b is a CMO function and 1q<r<, then CMO r ( Q p n ) CMO q ( Q p n ) and b CMO q b CMO r .

Proof For any b CMO r ( Q p n ), by Hölder’s inequality, we have

Therefore, b CMO q ( Q p n ) and b CMO q b CMO r . This completes the proof. □

Lemma 2.2 Suppose that b is a CMO function, j,kZ, then

| b ( t ) b B k | | b ( t ) b B j | + p n |jk| b CMO 1 ( Q p n ) .
(2.1)

Proof For iZ, recall that b B i = 1 | B i | H B i b(x)dx, we have

| b B i b B i + 1 | 1 | B i | H B i | b ( t ) b B i + 1 | d t p n | B i + 1 | H B i + 1 | b ( t ) b B i + 1 | d t p n b CMO 1 ( Q p n ) .
(2.2)

For j,kZ, without loss of generality, we can assume that jk, by (2.2), we get

| b ( t ) b B k | | b ( t ) b B j | + i = k j 1 | b B i b B i + 1 | | b ( t ) b B j | + p n | j k | b CMO 1 ( Q p n ) .
(2.3)

The lemma is proved. □

Proof of Theorem 1.1 Denote f(x) χ i (x)= f i (x).

  1. (1)

    By definition,

    ( H β , b p f ) χ k L r 2 ( Q p n ) r 2 = S k | x | p r 2 ( n β ) | B ( 0 , | x | p ) f ( t ) ( b ( x ) b ( t ) ) d t | r 2 d x S k p k r 2 ( n β ) ( B ( 0 , p k ) | f ( t ) ( b ( x ) b ( t ) ) | d t ) r 2 d x = p k r 2 ( n β ) S k ( j = k S j | f ( t ) ( b ( x ) b ( t ) ) | d t ) r 2 d x C p k r 2 ( n β ) S k ( j = k S j | f ( t ) ( b ( x ) b B k ) | d t ) r 2 d x + C p k r 2 ( n β ) S k ( j = k S j | f ( t ) ( b ( t ) b B k ) | d t ) r 2 d x : = I + II .

Now let us estimate I and II, respectively. For I, by Hölder’s inequality ( 1 r 1 + 1 r 1 =1), we have

I = C p k r 2 ( n β ) ( S k | b ( x ) b B k | r 2 d x ) ( j = k S j | f ( t ) | d t ) r 2 C p k r 2 n r 1 ( 1 | B k | H B k | b ( x ) b B k | r 2 d x ) × { j = k ( S j | f ( t ) | r 1 d t ) 1 r 1 ( S j d t ) 1 r 1 } r 2 C b CMO r 2 ( Q p n ) r 2 { j = k p ( j k ) n r 1 f j L r 1 ( Q p n ) } r 2 .

For II, by Lemma 2.2, we get

II = C p k r 2 ( n β ) S k ( j = k S j | f ( t ) ( b ( t ) b B k ) | d t ) r 2 d x = C p k r 2 ( n β ) p k n ( 1 p n ) ( j = k S j | f ( t ) ( b ( t ) b B k ) | d t ) r 2 C p k r 2 n r 1 ( j = k S j | f ( t ) ( b ( t ) b B j ) | d t ) r 2 + C p k r 2 n r 1 b CMO 1 ( Q p n ) r 2 ( j = k ( k j ) S j | f ( t ) | d t ) r 2 = II 1 + II 2 .

For II 1 and II 2 , by Hölder’s inequality, we obtain

II 1 C p k r 2 n r 1 { ( j = k S j | f ( t ) | r 1 d t ) 1 r 1 ( S j | b ( t ) b B j | r 1 d t ) 1 r 1 } r 2 C p k r 2 n r 1 { j = k f j L r 1 ( Q p n ) p j n r 1 ( 1 | B j | H B j | b ( t ) b B j | r 1 d t ) 1 r 1 } r 2 C b CMO r 1 ( Q p n ) r 2 { j = k p ( j k ) n r 1 f j L r 1 ( Q p n ) } r 2

and

II 2 C p k r 2 n r 1 b CMO 1 ( Q p n ) r 2 { j = k ( k j ) ( S j | f ( t ) | r 1 d t ) 1 r 1 ( S j d t ) 1 r 1 } r 2 C b CMO 1 ( Q p n ) r 2 { j = k ( k j ) p ( j k ) n r 1 f j L r 1 ( Q p n ) } r 2 .

Then the above inequalities together with Lemma 2.1 imply that

H β , b p f K r 2 α , q 2 ( Q p n ) = ( k = + p k α q 2 ( H β , b p f ) χ k L r 2 ( Q p n ) q 2 ) 1 q 2 ( k = + p k α q 1 ( H β , b p f ) χ k L r 2 ( Q p n ) q 1 ) 1 q 1 C { k = + p k α q 1 b CMO r 2 ( Q p n ) q 1 ( j = k p ( j k ) n r 1 f j L r 1 ( Q p n ) ) q 1 } 1 q 1 + C { k = + p k α q 1 b CMO r 1 ( Q p n ) q 1 ( j = k p ( j k ) n r 1 f j L r 1 ( Q p n ) ) q 1 } 1 q 1 + C { k = + p k α q 1 b CMO 1 ( Q p n ) q 1 ( j = k ( k j ) p ( j k ) n r 1 f j L r 1 ( Q p n ) ) q 1 } 1 q 1 C b CMO max { r 1 , r 2 } ( Q p n ) { k = + p k α q 1 ( j = k ( k j ) p ( j k ) n r 1 f j L r 1 ( Q p n ) ) q 1 } 1 q 1 = J .
(2.4)

For the case 0< q 1 1, since α< n r 1 , we have

J q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + p k α q 1 ( j = k ( k j ) p ( j k ) n r 1 f j L r 1 ( Q p n ) ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + ( j = k p j α f j L r 1 ( Q p n ) ( k j ) p ( j k ) ( n r 1 α ) ) q 1 C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + j = k p j α q 1 f j L r 1 ( Q p n ) q 1 ( k j ) q 1 p ( j k ) ( n r 1 α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 j = + p j α q 1 f j L r 1 ( Q p n ) q 1 k = j + ( k j ) q 1 p ( j k ) ( n r 1 α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 f K r 1 α , r 1 ( Q p n ) q 1 .
(2.5)

For the case q 1 >1, by Hölder’s inequality, we have

J q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + ( j = k p j α f j L r 1 ( Q p n ) ( k j ) p ( j k ) ( n r 1 α ) ) q 1 C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + ( j = k p j α q 1 f j L r 1 ( Q p n ) q 1 p ( j k ) 2 ( n r 1 α ) q 1 ) × ( j = k ( k j ) q 1 p ( j k ) 2 ( n r 1 α ) q 1 ) q 1 q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 j = + p j α q 1 f j L r 1 ( Q p n ) q 1 k = j + p ( j k ) 2 ( n r 1 α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 f K r 1 α , r 1 ( Q p n ) q 1 .
(2.6)

Then (1.6) follows from (2.4)-(2.6).

  1. (2)

    By definition,

    ( H β , b p , f ) χ k L r 2 ( Q p n ) r 2 = S k | Q p n B ( 0 , | x | p ) | t | p β n f ( t ) ( b ( x ) b ( t ) ) d t | r 2 d x S k ( Q p n B ( 0 , p k ) | t | p β n | f ( t ) ( b ( x ) b ( t ) ) | d t ) r 2 d x = S k ( j = k S j p j ( β n ) | f ( t ) ( b ( x ) b ( t ) ) | d t ) r 2 d x C S k ( j = k S j p j ( β n ) | f ( t ) ( b ( x ) b B k ) | d t ) r 2 d x + C S k ( j = k S j p j ( β n ) | f ( t ) ( b ( t ) b B k ) | d t ) r 2 d x : = K + L .

By Hölder’s inequality, we get

K = C ( S k | b ( x ) b B k | r 2 d x ) ( j = k S j p j ( β n ) | f ( t ) | d t ) r 2 C p k n ( 1 | B k | H B k | b ( x ) b B k | r 2 d x ) × { j = k ( S j | f ( t ) | r 1 d t ) 1 r 1 p j ( β n r 1 ) } r 2 C b CMO r 2 ( Q p n ) r 2 { j = k p ( k j ) n r 2 f j L r 1 ( Q p n ) } r 2 .
(2.7)

By Lemma 2.2, we have

L = C S k ( j = k S j p j ( β n ) | f ( t ) ( b ( t ) b B k ) | d t ) r 2 d x = C p k n ( 1 p n ) ( j = k S j p j ( β n ) | f ( t ) ( b ( t ) b B k ) | d t ) r 2 C p k n ( j = k S j p j ( β n ) | f ( t ) ( b ( t ) b B j ) | d t ) r 2 + C p k n b CMO 1 ( Q p n ) r 2 ( j = k ( j k ) p j ( β n ) S j | f ( t ) | d t ) r 2 = L 1 + L 2 .

For L 1 and L 2 , by Hölder’s inequality, we obtain

L 1 C p k n { j = k p j ( β n ) ( S j | b ( t ) b B j | r 1 d t ) 1 r 1 ( S j | f ( t ) | r 1 d t ) 1 r 1 } r 2 C p k n { j = k p j n r 1 + j ( β n ) f j L r 1 ( Q p n ) ( 1 | B j | H B j | b ( t ) b B j | r 1 d t ) 1 r 1 } r 2 C b CMO r 1 ( Q p n ) r 2 { j = k p ( k j ) n r 2 f j L r 1 ( Q p n ) } r 2

and

L 2 C b CMO 1 ( Q p n ) r 2 p k n { j = k ( j k ) p j ( β n ) ( S j | f ( t ) | r 1 d t ) 1 r 1 ( S j d t ) 1 r 1 } r 2 C b CMO 1 ( Q p n ) r 2 { j = k ( j k ) p ( k j ) n r 2 f j L r 1 ( Q p n ) } r 2 .
(2.8)

Then (2.7)-(2.8) together with Lemma 2.1 imply that

H β , b p , f K r 2 α , q 2 ( Q p n ) = ( k = + p k α q 2 ( H β , b p , f ) χ k L r 2 ( Q p n ) q 2 ) 1 q 2 ( k = + p k α q 1 ( H β , b p , f ) χ k L r 2 ( Q p n ) q 1 ) 1 q 1 C ( k = + p k α q 1 b CMO r 2 ( Q p n ) q 1 ( j = k p ( k j ) n r 2 f j L r 1 ( Q p n ) ) q 1 ) 1 q 1 + C ( k = + p k α q 1 b CMO r 1 ( Q p n ) q 1 ( j = k p ( k j ) n r 2 f j L r 1 ( Q p n ) ) q 1 ) 1 q 1 + C ( k = + p k α q 1 b CMO 1 ( Q p n ) q 1 ( j = k ( k j ) p ( j k ) n r 2 f j L r 1 ( Q p n ) ) q 1 ) 1 q 1 C b CMO max { r 1 , r 2 } ( Q p n ) ( k = + p k α q 1 ( j = k ( j k ) p ( k j ) n r 2 f j L r 1 ( Q p n ) ) q 1 ) 1 q 1 = S .

For the case 0< q 1 1, since α> n r 2 , we have

S q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + p k α q 1 ( j = k ( j k ) p ( k j ) n r 2 f j L r 1 ( Q p n ) ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + ( j = k p j α f j L r 1 ( Q p n ) ( j k ) p ( k j ) ( n r 2 + α ) ) q 1 C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + j = k p j α q 1 f j L r 1 ( Q p n ) q 1 ( j k ) q 1 p ( k j ) ( n r 2 + α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 j = + p j α q 1 f j L r 1 ( Q p n ) q 1 k = j ( j k ) q 1 p ( k j ) ( n r 2 + α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 f K r 1 α , q 1 ( Q p n ) q 1 .

For the case q 1 >1, by Hölder’s inequality, we have

S q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + p k α q 1 ( j = k ( j k ) p ( k j ) n r 2 f j L r 1 ( Q p n ) ) q 1 C b CMO max { r 1 , r 2 } ( Q p n ) q 1 k = + ( j = k p j α q 1 f j L r 1 ( Q p n ) q 1 p ( k j ) 2 ( n r 2 + α ) q 1 ) × ( j = k ( j k ) q 1 p ( k j ) 2 ( n r 2 + α ) q 1 ) q 1 q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 j = + p j α q 1 f j L r 1 ( Q p n ) q 1 k = j p ( k j ) 2 ( n r 2 + α ) q 1 = C b CMO max { r 1 , r 2 } ( Q p n ) q 1 f K r 1 α , q 1 ( Q p n ) q 1 .

Then the above inequalities imply (1.7). Theorem 1.1 is proved. □

References

  1. Albeverio S, Karwoski W: A random walk on p -adics: the generator and its spectrum. Stoch. Process. Appl. 1994, 53: 1–22. 10.1016/0304-4149(94)90054-X

    Article  MathSciNet  Google Scholar 

  2. Avetisov AV, Bikulov AH, Kozyrev SV, Osipov VA: p -adic models of ultrametric diffusion constrained by hierarchical energy landscapes. J. Phys. A, Math. Gen. 2002, 35: 177–189. 10.1088/0305-4470/35/2/301

    MathSciNet  Article  MATH  Google Scholar 

  3. Khrennikov A: p-Adic Valued Distributions in Mathematical Physics. Kluwer, Dordrecht; 1994.

    Book  MATH  Google Scholar 

  4. Khrennikov A: Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Kluwer, Dordrecht; 1997.

    Book  MATH  Google Scholar 

  5. Varadarajan VS: Path integrals for a class of p -adic Schrödinger equations. Lett. Math. Phys. 1997, 39: 97–106. 10.1023/A:1007364631796

    MathSciNet  Article  MATH  Google Scholar 

  6. Vladimirov VS, Volovich IV: p -adic quantum mechanics. Commun. Math. Phys. 1989, 123: 659–676. 10.1007/BF01218590

    MathSciNet  Article  MATH  Google Scholar 

  7. Vladimirov VS, Volovich IV, Zelenov EI Series on Soviet and East European Mathematics I. In p-Adic Analysis and Mathematical Physics. World Scientific, Singapore; 1992.

    Google Scholar 

  8. Volovich IV: p -adic space-time and the string theory. Theor. Math. Phys. 1987, 71: 337–340.

    MathSciNet  Article  MATH  Google Scholar 

  9. Volovich IV: p -adic string. Class. Quantum Gravity 1987, 4: 83–87. 10.1088/0264-9381/4/4/003

    MathSciNet  Article  MATH  Google Scholar 

  10. Kim YC: Carleson measures and the BMO space on the p -adic vector space. Math. Nachr. 2009, 282: 1278–1304. 10.1002/mana.200610806

    MathSciNet  Article  MATH  Google Scholar 

  11. Kim YC: Weak type estimates of square functions associated with quasiradial Bochner-Riesz means on certain Hardy spaces. J. Math. Anal. Appl. 2008, 339: 266–280. 10.1016/j.jmaa.2007.06.050

    MathSciNet  Article  MATH  Google Scholar 

  12. Rim KS, Lee J: Estimates of weighted Hardy-Littlewood averages on the p -adic vector space. J. Math. Anal. Appl. 2006, 324: 1470–1477. 10.1016/j.jmaa.2006.01.038

    MathSciNet  Article  MATH  Google Scholar 

  13. Rogers KM: A van der Corput lemma for the p -adic numbers. Proc. Am. Math. Soc. 2005, 133: 3525–3534. 10.1090/S0002-9939-05-07919-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Rogers KM: Maximal averages along curves over the p -adic numbers. Bull. Aust. Math. Soc. 2004, 70: 357–375. 10.1017/S0004972700034602

    Article  MathSciNet  MATH  Google Scholar 

  15. Taibleson MH: Fourier Analysis on Local Fields. Princeton University Press, Princeton; 1975.

    MATH  Google Scholar 

  16. Hardy GH: Note on a theorem of Hilbert. Math. Z. 1920, 6: 314–317. 10.1007/BF01199965

    MathSciNet  Article  MATH  Google Scholar 

  17. Haran S: Riesz potentials and explicit sums in arithmetic. Invent. Math. 1990, 101: 697–703. 10.1007/BF01231521

    MathSciNet  Article  MATH  Google Scholar 

  18. Faris W: Weak Lebesgue spaces and quantum mechanical binding. Duke Math. J. 1976, 43: 365–373. 10.1215/S0012-7094-76-04332-5

    MathSciNet  Article  MATH  Google Scholar 

  19. Fu ZW, Grafakos L, Lu SZ, Zhao FY: Sharp bounds for m-linear Hardy and Hilbert operators. Houst. J. Math. 2012, 38: 225–244.

    MathSciNet  MATH  Google Scholar 

  20. Long SC, Wang J: Commutators of Hardy operators. J. Math. Anal. Appl. 2002, 274: 626–644. 10.1016/S0022-247X(02)00321-9

    MathSciNet  Article  MATH  Google Scholar 

  21. Fu, ZW, Wu, QY, Lu, SZ: Sharp estimates for p-adic Hardy, Hardy-Littlewood-Polya operators and commutators. Acta Math. Sin. (Engl. Ser.). Preprint

  22. Fu ZW, Liu ZG, Lu SZ, Wang HB: Characterization for commutators of n -dimensional fractional Hardy operators. Sci. China Ser. A 2007, 50: 1418–1426. 10.1007/s11425-007-0094-4

    MathSciNet  Article  MATH  Google Scholar 

  23. Fu ZW, Lu SZ: Commutators of generalized Hardy operators. Math. Nachr. 2009, 282: 832–845. 10.1002/mana.200610775

    MathSciNet  Article  MATH  Google Scholar 

  24. Chen YZ, Lau KS: Some new classes of Hardy spaces. J. Funct. Anal. 1989, 84: 255–278. 10.1016/0022-1236(89)90097-9

    MathSciNet  Article  MATH  Google Scholar 

  25. Garcia-Cuerva J: Hardy spaces and Beurling algebras. J. Lond. Math. Soc. 1989, 39: 499–513. 10.1112/jlms/s2-39.3.499

    MathSciNet  Article  MATH  Google Scholar 

  26. Lu SZ, Yang DC: The central BMO spaces and Littlewood-Paley operators. Approx. Theory Appl. 1995, 11: 72–94.

    MathSciNet  MATH  Google Scholar 

  27. Zhu YP, Zheng WX: Besov spaces and Herz spaces on local fields. Sci. China Ser. A 1998, 41: 1051–1060. 10.1007/BF02871839

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The author sincerely thanks Professor Zunwei Fu for his useful discussions. This work was supported by NSF of China (Grant No. 11126203), NSF of Shandong Province (Grant Nos. ZR2010AL006).

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Correspondence to Qing Yan Wu.

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Wu, Q.Y. Boundedness for commutators of fractional p-adic Hardy operators. J Inequal Appl 2012, 293 (2012). https://doi.org/10.1186/1029-242X-2012-293

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Keywords

  • fractional p-adic Hardy operator
  • central BMO function
  • weighted homogeneous Herz space