Sharp estimates for the p adic Hardy type operators on higher dimensional product spaces

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R E S E A R C H

Open Access

Sharp estimates for the

p

-adic Hardy type

operators on higher-dimensional product

spaces

Ronghui Liu and Jiang Zhou

*

*_{Correspondence:}

[email protected]

College of Mathematics and System Science, Xinjiang University, Urumqi, 830046, People’s Republic of China

Abstract

In this paper, we introduce thep-adic Hardy type operator and obtain its sharp bound on thep-adic Lebesgue product spaces. Meanwhile, an analogous result is computed for thep-adic Lebesgue product spaces with power weights. In addition, we

characterize a suﬃcient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on thep-adic Lebesgue product spaces. Furthermore, thep-adic weighted Hardy-Cesàro operator is also obtained.

MSC: 42B20; 42B25; 42B35

Keywords: Hardy type operator; sharp estimates;p-adic; product Lebesgue spaces

1 Introduction

p-adic numbers were introduced by Hensel at the end of the th century, they constitute an integral part of number theory, algebraic geometry, representation theory and other branches of modern mathematics (see [, ]). However, the geometry of the spaceQpis surprisingly unlike the geometry of the spaceR, in particular the Archimedean axiom is not true inQp. Therefore the ﬁeld ofp-adic numbers has natural hierarchical structures, we refer the reader to [–]. In recent years, theories of functions and operators fromQn p intoRorCplay an important role in thep-adic quantum mechanics, inp-adic analysis. Studies of thep-adic Hardy operators have drawn more and more attention (for example, see [–]).

For a prime numberp, letQp be the field ofp-adic numbers. It is defined as the com-pletion of the field of rational numbersQpwith respect to the non-Archimedeanp-adic norm| · |p. This norm is defined as follows:||p= ; if any non-zero rational numberxis represented asx=pγm

n, whereγ is an integer and the integersm,nare indivisible byp, then|x|p=p–γ. It is easy to see that the norm satisﬁes the following properties:

(i) |x|p≥,∀x∈Qp,|x|p= ⇐⇒x= ; (ii) |xy|p=|x|p|y|p,∀x,y∈Qp;

(iii) |x+y|p≤max{|x|p,|y|p},∀x,y∈Qp, and when|x|p=|y|p, we have

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

It is well known thatQpis a typical model of non-Archimedean local ﬁelds. From the standardp-adic analysis, we know that any non-zero elementxofQp can be uniquely

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represented in the canonical series

x=pγ

∞

j=

ajpj, γ=γ(x)∈Z,

where aj are integrals, ≤aj≤p– ,a= . The series converges in the p-adic norm

because|ajpj|p=p–j. LetZp={x∈Qp:|x|p≤}be the class of allp-adic integrals inQp and denoteZ∗_p=Zp\{}.

The spaceQn

pdenotes a vector space overQpwhich consists of all pointsx= (x,x, . . . , xn), wherexi∈Qp,i= , , . . . ,n. Thep-adic norm onQnpis

|x|p:=max

≤i≤n

|xi|p

, x∈Qn_p.

Denote byBγ(a) ={x∈Qn_p:|x–a|p≤pγ}, the ball with center ata∈Qnpand radiuspγ, and bySγ(a) ={x∈Qnp:|x–a|p=pγ} the sphere with center ata∈Qnp and radiuspγ,

γ ∈Z. It is clear thatSγ(a) =Bγ(a)\Bγ–(a), and we setBγ() =Bγ andSγ() =Sγ.

SinceQn

p is a locally compact commutative group with respect to addition, it follows from the standard analysis that there exists a Harr measuredxonQn

p, which is unique up to a positive constant factor and is translation invariant. We normalize the measuredx

such that

B()

dx=B()_H= ,

where|B|Hdenotes the Harr measure of a measure subset B ofQnp. By simple calculation, we obtain|Bγ(a)|=pγn,|Sγ(a)|=pγn( –p–n).

The most fundamental averaging operator is Hardy operator deﬁned by

Hf(x) :=

x

 f(t)dt,

where the functionf is a nonnegative integrable function onR+_and_x_{> . A celebrated}

integral inequality, due to Hardy [], states that

HfLq_(R+₎≤ q

q– fLq(R+)

holds for  <q<∞, and the constant _qq_–is the best.

For the multidimensional casen≥, generally speaking, there exist two different defi-nitions. One is the rectangle averaging operator defined by

H(f)(x) := 

x· · ·xm

x

 · · ·

xm



f(t, . . . ,tm)dtm· · ·dt,

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The other deﬁnition is then-dimensional spherical averaging operator, which was in-troduced by Christ and Grafakos in [] as follows:

Hf(x) := 

n|x|n

|t|≤|x|f(t)dt, x∈R n_{\ {}__}_,

wherenis the volume of the unit ball inRn. The norm ofHonLq(Rn) was evaluated and found to be equal to that of the one-dimensional averaging operator.HLq_→_Lq, that is to say, does not depend on the dimension of the space.

In , Fuet al.[] deﬁned the followingn-dimensionalp-adic Hardy operator:

Hp_f₍_x_{) :=} 

|B(,|x|p)|H

|t|p≤|x|p

f(t)dt, x∈Qn_p\ {},

wheref is a nonnegative measurable function onQn_p,B(,|x|p) is a ball inQn_pwith center at ∈Qn

pand radius|x|p, and they proved the sharp estimates of thep-adic Hardy operator on Lebesgue spaces with power weights.

In , Luet al.[] gave the deﬁnition of Hardy operator on higher-dimensional prod-uct spaces as follows:

Hm(f)(x) :=

_m

i=



|B(,|xi|)|

|y|<|x| · · ·

|ym|<|xm|

f(y, . . . ,ym)dym· · ·dy,

wheref is a nonnegative measurable function onRn×Rn× · · · ×Rnm_,_m∈N_,_n i∈N,

x= (x,x, . . . ,xm)∈Rn×Rn × · · · ×Rnm,xi∈Rniand

m

i=|xi| = . Furthermore, the corresponding operator norm on the Lebesgue product spaces with power weights was worked out.

Next, we will introduce the deﬁnition of Hardy type operator on the higher-dimensional

p-adic product spaces as follows and discuss the boundedness and best bound on the prod-uct ofp-adic Lebesgue spaces.

Deﬁnition . Letm∈N,ni∈N,xi∈Qnpi,i= , . . . ,m, andfbe a nonnegative measurable function onQn

p ×Qnp× · · · ×Qnpm. Thep-adic Hardy type operator is deﬁned by

Hp m(f)(x) =

_m

i=



|B(,|xi|p)|H

|y|p<|x|p

· · ·

|ym|p<|xm|p

f(y, . . . ,ym)dym· · ·dy,

wherex= (x,x, . . . ,xm)∈Qpn×Qnp× · · · ×Qnpmwith

m

i=|xi|p= .

In , Carton-Lebrun and Fosset [] deﬁned the weighted Hardy operatorHψby

Hψ(f)(x) =





f(tx)ψ(t)dt, x∈Rn,

where ψ : [, ]→[,∞) is a function, and they showed the boundedness of Hψ on

Lebesgue spaces andBMO(Rn_{) spaces. Evidently the operator}_H

ψdeeply depends on the

nonnegative functionψ. For Example, whenn=  andψ(x) =  forx∈[, ], the operator

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In , Rim and Lee [] deﬁned the weightedp-adic Hardy operatorHpψby

Hp

ψ(f)(x) =

Z∗

p

f(tx)ψ(t)dt, x∈Qn_p,

whereψ is a nonnegative function deﬁned onZ∗_p, and they gave the characterization of the functionψ for whichHp_ψ is bounded onLq₍_Qn

p), ≤q≤ ∞, they also got the cor-responding operator norm. Obviously, ifn=  andψ(x) = , thenHpψ just reduces to the

p-adic Hardy operatorHp_on_Q_{p, which is deﬁned by}

Hp_f₍_x_{) :=} 

|x|p

|t|p≤|x|p

f(t)dt, x= .

In , Fuet al.[] introduced the deﬁnition of weighted Hardy operator on higher-dimensional product spaces as follows:

Hm

ϕ(f)(x) =



 · · ·





f(tx, . . . ,tmxm)ϕ(t, . . . ,tm)dt· · ·dtm,

where the nonnegative functionψdeﬁned on [, ]×[, ]× · · · ×[, ]. They obtained a suﬃcient and necessary condition which ensures that the operatorHm

ϕ is bounded on the

Lebesgue product spaces.

Next, we will extend the operatorHm

ϕ to the higher-dimensionalp-adic product spaces.

Deﬁnition . Letm∈N,ni∈N,xi∈Qnpi,i= , . . . ,m, andf be a nonnegative measurable function onQn

p ×Qnp× · · · ×Qnpm. Thep-adic weighted Hardy type operator is deﬁned by

Hp

ϕ,m(f)(x) =

Z∗p

· · ·

Z∗p

f(tx, . . . ,tmxm)ϕ(t, . . . ,tm)dt· · ·dtm,

whereϕis a nonnegative measurable function onZ∗_p×Z∗_p× · · · ×Z∗_p, andx= (x,x, . . . , xm)∈Qnp×Qnp× · · · ×Qnpm.

It follows from the Fubini theorem that we can easily formulate the dual operator of

Hp

ϕ,mand denote it byHpϕ,,∗m.

Deﬁnition . Letm∈N,ni∈N,xi∈Qnpi,i= , . . . ,m, andf be a nonnegative measurable function onQn

p ×Qnp× · · · ×Qnpm. The dual operator of thep-adic weighted Hardy type operator is deﬁned by

Hp,∗

ϕ,m(f)(x) =

Z∗

p

· · ·

Z∗

p

f(x/|t|p, . . . ,xm/|tm|p)ϕ(t, . . . ,tm) |t|n

p· · · |tm|np

dt· · ·dtm,

whereϕis a nonnegative measurable function onZ∗_p×Z∗_p× · · · ×Z∗_p, andx= (x,x, . . . , xm)∈Qnp×Qnp× · · · ×Qnpm.

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Letψ : [, ]→[,∞),s: [, ]→Rbe measurable functions. The weighted Hardy-Cesàro operatorHs

ϕ, associated to parameter curves(x,t) =s(t)x, is deﬁned by

Hs

ϕ(f) =





fs(t)xϕ(t)dt,

for all measurable functionsf onRn_.

In , Hung [] considered the form of Hardy-Cesàro operator inp-adic analysis

Hp,s

ϕ f(x) :=

Z∗

p

fs(t)xϕ(t)dt,

wheres:Z∗_p→Qpandψ:Z∗p→[,∞) are measurable functions. The author investigated the boundedness of thep-adic analog of the weighted Hardy-Cesàro operator on weighted Lebesgue spaces and weighted BMO spaces. In each case, the corresponding operator norms are obtained. In , Chuonget al.[] considered the boundedness of thep-adic weighted Hardy-Cesàro operators and their commutators on weighted functional spaces of Morrey type, and the corresponding operator norms are also computed.

Motivated by these famous results, ﬁrst we will give a higher-dimensional version of the

p-adic weighted Hardy-Cesàro operator.

Deﬁnition . Letm∈N,ni∈N,xi∈Qnpi,i= , . . . ,m, andf be a nonnegative measurable function onQn

p ×Qnp× · · · ×Qnpm. Thep-adic weighted Hardy-Cesàro type operator is deﬁned by

Hp,s

ϕ,m(f)(x) =

Z∗p

· · ·

Z∗p

fs(t)x, . . . ,s(tm)xm

ϕ(t, . . . ,tm)dt· · ·dtm,

wheres:Z∗_p→Qp andϕ:Z∗p×Z∗p× · · · ×Z∗p→[,∞) are measurable functions, and

x= (x,x, . . . ,xm)∈Qpn×Qnp× · · · ×Qnpm.

The paper is organized as follows. Section  is devoted to the sharp estimates ofHpmon thep-adic Lebesgue product spaces with power weights. In Section , we present neces-sary and suﬃcient conditions for the boundedness of the weighted Hardy type operator and its dual operator. Furthermore, thep-adic weighted Hardy-Cesàro type operator also has the corresponding conclusion. In Section , we state explicitly the main conclusions of the research.

2 Sharp estimates for thep-adic Hardy type operator

Theorem . Let <q<∞,m∈N,ni∈N,xi∈Qpni,i= , . . . ,m.If f ∈Lq(Qnp×Qnp×· · ·×

Qnm

p ),then the p-adic Hardy type operatorH p

mis bounded on Lq(Qnp×Qnp× · · · ×Qnpm),

moreover,the norm ofHpmcan be obtained as follows:

_Hp mLq_(Qn

p×Qnp×···×Qnmp )→Lq(Qnp×Qnp×···×Qnmp )= m

i=

 –p–ni

 –pniq–ni .

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Theorem . Let <q<∞,m∈N,ni∈N,xi∈Qpni,i= , . . . ,m.If f ∈Lq(Qnp×Qnp×

· · · ×Qnm

p ,|x|αp),where|x|αp:=|x|

α

p × |x|αp× · · · × |x|pαm andαi< (q– )ni,then the p-adic

Hardy type operatorHpmis bounded on Lq(Qnp×Qnp×· · ·×Qnpm,|x|αp),moreover,the norm

ofHmp can be obtained as follows:

_Hp mLq_(Qn

p×Q n_

p ×···×Qnmp ,|x|αp)→Lq(Q n_ p×Q

n_

p ×···×Qnmp ,|x|αp)= m

i=

 –p–ni

 –pniq+

αi q–ni

.

Whenαi= , the sharp estimate of thep-adic Hardy type operator will be easy to get on thep-adic Lebesgue product spaces, so we only provide the proof of Theorem ..

Proof Without loss of generality, we consider only the situation whenm= . Actually, a similar procedure works for allm∈N. We set

gf(x,x) =

 ( –p–n₎

 ( –p–n₎

|ξ|p=

|ξ|p=

f|x|–p ξ,|x|–p ξ

dξdξ.

It is clear thatgf(x,x) =gf(|x|–p ,|x|–p ), in the following we brieﬂy call this function a radial function on thep-adic Lebesgue product spaces with power weights. We have

Hp

(gf)(x,x) =



|B(,|x|p)|H 

|B(,|x|p)|H

|y|p<|x|p

|y|p<|x|p

gf(y,y)dydy

= 

|B(,|x|p)|H 

|B(,|x|p)|H

|y|p<|x|p

|y|p<|x|p  ( –p–n₎

 ( –p–n₎

×

|ξ|p=

|ξ|p=

f|y|–p ξ,|y|–p ξ

dξdξdydy.

By changing variables,zi=|yi|–p ξi,i= , , we have

Hp

(gf)(x,x) =

 ( –p–n₎

 ( –p–n₎



|B(,|x|p)|H 

|B(,|x|p)|H

×

|y|p<|x|p

|y|p<|x|p

|z|p=|y|p

|z|p=|y|p

f(z,z)

× |y|–n

p |y|–pndzdzdydy.

= 

( –p–n₎

 ( –p–n₎



|B(,|x|p)|H 

|B(,|x|p)|H

×

|z|p<|x|p

|z|p<|x|p

|y|p=|z|p

|y|p=|z|p

|y|–n

p

× |y|–n

p dydyf(z,z)dzdz.

= 

|B(,|x|p)|H 

|B(,|x|p)|H

|z|p<|x|p

|z|p<|x|p

f(z,z)dzdz.

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Using Hölder’s inequality, we get

g_Lq_(Qn

p×Q n_ p ,α) =

Qn

p

Qn

p

_{( –}_p–n₎

 ( –p–n₎

×

|ξ|p=

|ξ|p=

f|x|–_p ξ,|x|–p ξ

dξdξ

q|x|α

p |x|

α

p dxdx

/q

≤

Qn

p

Qn

p  ( –p–n₎q

 ( –p–n₎q

×

|ξ|p=

|ξ|p=

f|x|–_p ξ,|x|–p ξ

q

dξdξ

×

|ξ|p=

|ξ|p= dξdξ

q– |x|α

p |x|

α

p dxdx

/q

=

Qn

p

Qn

p  ( –p–n₎

 ( –p–n₎

×

|ξ|p=

|ξ|p=

f|x|–_p ξ,|x|–p ξ

q

dξdξ

|x|α

p |x|

α

p dxdx

/q

= 

( –p–n₎/q



( –p–n₎/q _Qn_ p

Qn

p |z|p=|x|p

|z|p=|x|p

×f(z,z)q|x|p–n|x|–pndzdz

|x|α

p |x|

α

p dxdx

/q

= 

( –p–n₎/q



( –p–n₎/q _Qn

p

Qn

p |x|p=|z|p

|x|p=|z|p

× |x|α–n

p |x|

α–n

p dxdx

|f(z,z)|qdzdz

/q

=f_Lq_(Qn

p×Qnp,α). Therefore,

Hp

(f)Lq_(Qn

p×Q n_ p ,α)

f_Lq_(Qn

p×Qnp,α)

≤H

p

(gf)_Lq_(Qn

p×Q n_ p ,α)

gf_Lq_(Qn

p×Qnp,α) .

This implies the claim that iff is a radial function, then we havegf =f. This means that the norm of the operatorHp_is equal to the norm that makesHp_ restricted to the set of radial functions. Consequently, without loss of generality, it suﬃces to fulﬁl the proof of the theorem by assumingf is a radial function.

Substituting the variableyi=|xi|–p zi,i= , , we see thatH p

(f)Lq_(Qn

p×Qnp,α)equals

Qn

p

Qn

p

_Hp

(f)(x,x)q|x|αp|x|

α

p dxdx

/q

=

Qn

p

Qn

p

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×

|y|p<|x|p

|y|p<|x|p

f(y,y)dydy

q|x|α

p |x|

α

p dxdx

/q

=

Qn

p

Qn

p

_|

z|p<

|z|p<

f|x|–_p z,|x|–p z

dzdz

q|x|α

p |x|αpdxdx

/q .

Using the generalized Minkowski’s inequality and noting thatf is a radial function, we haveHp_(f)_Lq_(Qn

p×Qnp,α)that is not greater than

Qn

p

Qn

p

_|_z

|p<

|z|p<

f|z|–_p x,|z|–p xdzdz|q|x|αp|x|αpdxdx

/q

≤

|z|p<

|z|p<

Qn

p

Qn

p

f|z|–_p x,|z|–p xq|x|αp|x|

α

p dxdx

/q

dzdz

=

|z|p<

|z|p<

Qn

p

Qn

p

f(y,y)q| y|α

p

|z|α

p

|y|α

p

|z|α

p

dydy

/q

|z|–n/q

p |z|–pn/qdzdz

=

|z|p<

|z|p< |z|

–n–α

q p |z|

–n–α

q

p dzdzf_Lq_(Qn

p×Q n_ p ,α) =



i=

 –p–ni

 –pniq+

α_i

q–ni

f_Lq_(Qn

p×Qnp,α).

Therefore,

_Hp

Lq_(Qn

p×Qnp,|x|αp)→Lq(Qnp×Qnp,|x|αp)≤



i=

 –p–ni

 –pniq+

α_i

q–ni .

Now let us prove that our estimate is sharp. For  <ε<min{, (q– )n/q, (q– )n/q},

we take

fε(x,x) =|x| –n_–α_

q +ε p |x|

–n_–α_

q +ε

p χ{|x|p<}χ{|x|p<}(x,x).

It follows from the elementary calculation thatfε_Lq_(Qn

p×Q n_ p ,α)is

|x|p<

|x|p< |x|

(ε–n+qα)q p |x|

(ε–n+qα)q

p |x|αpdxdx

/q

=

|x|p<

|x|(ε–

n_ q)q p

/q

|x|p<

|x|(ε–

n_ q)q p

/q

=

 –p–n

 –p–qε

/q  –p–n

 –p–qε

/q .

We rewriteHp_(fε)

Hp

(fε)(x,x)

= 

|B(,|x|p)|H 

|B(,|x|p)|H

|y|p<|x|p

|y|p<|x|p

(9)

= 

|B(,|x|p)|H 

|B(,|x|p)|H

|z|p<

|z|p<

fz|x|–n

p ,z|x|–pn

dzdz

= |x|

–n_–α_

q +ε p

|B(,|x|p)|H

|x| –n_–α_

q +ε p

|B(,|x|p)|H

|z|p<

|z|p<

|z| –n_–α

q +ε p |z|

–n_–α

q +ε p dzdz.

Thus, we estimate the norm ofHp_(fε)_Lq_(Qn

p×Qnp,α)as follows:

_Hp

(fε)q

Lq_(Qn

p×Qnp,α) =

Qn

p

Qn

p

_{|

z|p<,|z|p<_|_x |p}

{|z|p<,|z|p<_|_x |p}

|z| –n–α

q +ε p

× |z| –n–α

q +ε p dzdz

q|x|qε–n–α

p |x|pqε–n–α|x|αpdxdx ≥

|x|p<

|x|p<

_|_z

|p<

|z|p<

|z| –n_–α

q +ε p

× |z| –n–α

q +ε p dzdz

q|x|qε–n

p |x|q

ε–n

p |x|

α

pdxdx

= –p

–n

 –p–qε

 –p–n

 –p–qε

|z|p<

|z|p< |z|

–n–α

q +ε p |z|

–n–α

q +ε p dzdz

q

=

_m

i=

 –p–ni

 –pniq+

α_i

q–ni–ε q

fq

Lq_(Qn

p×Qnp,α). Therefore,

_Hp

Lq_(Qn

p×Qnp,|x|αp)→Lq(Qnp×Qnp,|x|αp)≥



i=

 –p–ni

 –pniq+

α_i

q–ni–ε .

Consequently, using the deﬁnition of the norm of the operator and lettingε→, we con-clude that

_Hp

Lq_(Qn

p×Q n_

p ,|x|αp)→Lq(Q n_ p×Q

n_ p ,|x|αp)≥



i=

 –p–ni

 –pniq+

αi q–ni

.

This ﬁnishes the proof of the theorem.

3 Necessary and sufﬁcient conditions for the boundedness of the weighted Hardy type operator

Theorem . Let≤q≤ ∞,m∈N,ni∈N,xi∈Qnpi,i= , . . . ,m.ϕis a nonnegative

mea-surable function onZ∗_p×Z∗_p× · · · ×Z∗_p.If f ∈Lq₍_Qn

p ×Qnp× · · · ×Qnpm),then the p-adic

weighted Hardy type operatorHpϕ,mis bounded on Lq(Qpn×Qnp× · · · ×Qnpm)if and only if

Z∗p

· · ·

Z∗p

|t|–n/q

p · · · |tm|–pnm/qϕ(t, . . . ,tm)dt· · ·dtm<∞.

(10)

We ﬁrst prove the suﬃciency of Theorem .. Assume

Z∗p

· · ·

Z∗p

|t|–n/q

By the generalized Minkowski’s inequality, we getHpϕ,m(f)_Lq_(Qn

p×Q n_

p ×···×Qnmp )that is not greater than

Z∗

p

· · ·

Z∗

p

Qn

p

· · ·

Qnm p

f(tx, . . . ,tmxm) q

dxm· · ·dx

/q

ϕ(t, . . . ,tm)dt· · ·dtm.

Making the change of variablesyi=xiti,i= , . . . ,m, we have

Z∗

p

· · ·

Z∗

p

Qn

p

· · ·

Qnm p

f(y, . . . ,ym) q

dym· · ·dy

/q

× |t|–n/q

p · · · |tm|–pnm/qϕ(t, . . . ,tm)dt· · ·dtm

≤ f_Lq_(Qn

p ×Qnp×···×Qnmp )

Z∗

p

· · ·

Z∗

p

|t|–n/q

p · · · |tm|–pnm/qϕ(t, . . . ,tm)dt· · ·dtm.

Since the inequality

Z∗

p

· · ·

Z∗

p

|t|–n/q

p · · · |tm|–pnm/qϕ(t, . . . ,tm)dt· · ·dtm<∞

holds, this immediately implies that the operatorHpϕ,mis bounded onLq(Qnp×Qnp×· · ·×

Qnm p ), and

_Hp

ϕ,mLq_(Qn

p×Qnp×···×Qnmp )→Lq(Qnp×Qnp···×Qnmp )

≤

Z∗

p

· · ·

Z∗

p

|t|–n/q

p · · · |tm|–pnm/qϕ(t, . . . ,tm)dt· · ·dtm.

This completes the proof of the suﬃciency of Theorem ..

Next we prove the necessary of Theorem ., ifHpϕ,mis bounded onLq(Qnp×Qnp×· · ·×

Qnm

p ), then there exists a constantC>  such that

_Hp

ϕ,m(f)Lq_(Qn

p×Q n_

p ×···×Qnmp )≤CfLq(Q n

p×Qnp×···×Qnmp ).

Now, for any  <ε<  and|ε|p> , we take

f_iε(x) =

⎧ ⎨ ⎩

, |xi|p< ,

|xi|

–ni_q–ε

p , |xi|p≥,

and

fε(x,x, . . . ,xm) =

m

(11)

Then a straightforward computation leads to

fεq

Lq_(Qn

p×Qnp×···×Qnmp )= m

i=

 –p–ni  –p–εq.

Obviously, Hpϕ,m(f)(x,x, . . . ,xm) =  while |xi|p < . Moreover, we also see that, if

|xi|p≥,Hpϕ,m(f)(x,x, . . . ,xm) equals

|x|–

n

q–ε p · · · |xm|

–nmq –ε p

 |x|p≤|t|p≤

· · ·



|xm|p≤|tm|p≤

|t|–

n

q–ε p · · · |tm|

–nmq –ε p

×ϕ(t, . . . ,tm)dtm· · ·dt.

Since the inequality

_Hp

ϕ,m(f)Lq(Qnp×Q n_

p ×···×Qnmp )≤CfLq(Q n

p×Qnp×···×Qnmp )

applies tofε, we have

Cpfq

Lq_(Qn

≥Hp

ϕ,m q Lq_(Qn

≥

|x|p≥

· · ·

|xm|p≥

|x|–

n

q–ε p · · · |xm|

–nm_q –ε

p

q

×

 |x|p≤|t|p≤

· · ·



|xm|p≤|tm|p≤

|t|–

n

q–ε p · · · |tm|

–nmq –ε p

×ϕ(t, . . . ,tm)dtm· · ·dt

q

dxm· · ·dx

≥

|x|p≥|ε|p

· · ·

|xm|p≥|ε|p

|x|–n–εq

p · · · |xm|–pnm–εq

×

 |ε|p≤|t|p≤

· · ·

 |ε|p≤|tm|p≤

|t|–

n

q–ε p · · · |tm|

–nm_q –ε

p

×ϕ(t, . . . ,tm)dtm· · ·dt

q

dxm· · ·dx

=fεq

Lq_(Qn

p×Qnp×···×Qnmp )|ε|

–εmq p

×

 |ε|p≤|t|p≤

· · ·

 |ε|p≤|tm|p≤

|t|–

n

q–ε p · · · |tm|

–nmq –ε

p ϕ(t, . . . ,tm)dtm· · ·dt

q .

Therefore,

 |ε|p≤|t|p≤

· · ·

 |ε|p≤|tm|p≤

|t|–

n

q–ε p · · · |tm|

–nm_q –ε

p ϕ(t, . . . ,tm)dtm· · ·dt≤ C |ε|pεm

(12)

Now takeε=p–k_,_k_{= , , . . . . Then}_|_ε_|

p=pk> . Lettingkapproach∞, thenεapproaches  and|ε|εm

p =p km

pk _{approaches . Then by Fatou’s lemma, we obtain}

Z∗p

· · ·

Z∗p

|t|–n/q

Thus, the proof of Theorem . is completed.

Since the operatorHpϕ,,∗m is the dual operator ofHpϕ,m, we can immediately deduce the following result.

Theorem . Let≤q<∞,m∈N,ni∈N,andϕbe a nonnegative measurable function

onZ∗_p×Z∗_p×· · ·×Z∗_p,i= , . . . ,m.If f ∈Lq₍_Qn

p ×Qnp×· · ·×Qnpm),then the p-adic weighted

Hardy type operatorHpϕ,,∗mis bounded on Lq(Qpn×Qnp× · · · ×Qnpm)if and only if

Z∗

p

· · ·

Z∗

p

|t|– (q–)n

q

p · · · |tm|

–(q–)_qnm

p ϕ(t, . . . ,tm)dt· · ·dtm<∞.

Remark . Using methods similar to Theorem ., we can easily see the proof of The-orem .. So we omit the details of the proof. In particular, ifϕ= , we know that the operatorHpϕ,mis not bounded onL(Qpn×Qnp× · · · ×Qnpm). In the same way, whenϕ= , we can also deduce thatHpϕ,,∗mis not bounded onL∞(Qpn×Qnp× · · · ×Qnpm).

We note that in [], the authors proved the sharp bounds of a weighted multilinear Hardy-Cesàro operator on the product of Lebesgue spaces and central Morrey spaces. They also proved suﬃcient and necessary conditions of the weight functions so that the commutators of weighted multilinear Hardy-Cesàro operator with symbols in central BMO spaces.

Inspired by the paper, we will consider suﬃcient and necessary conditions of the weighted Hardy-Cesàro type operatorHϕp,,smon thep-adic Lebesgue product spaces.

Theorem . Suppose≤q<∞,m∈N,ni∈N.Let s:Z∗p→Qpbe a measurable function

such that|s(ti)|p≥ |ti|βp for some realβ and almost everywhere ti∈Z∗p. If f ∈Lq(Q n

p ×

Qn

p × · · · ×Qnpm),then the p-adic weighted Hardy-Cesàro type operatorH p,s

ϕ,mis bounded

on Lq(Qn

p ×Qnp× · · · ×Qnpm)if and only if

Z∗p

· · ·

Z∗p

s(t) –n/q

p · · ·s(tm)

–nm/q

p ϕ(t, . . . ,tm)dt· · ·dtm<∞.

(13)

4 Conclusions

In the present study, we introduced a class ofp-adic Hardy type operator and considered the problem of ﬁnding a bound for the norm of it fromp-adic Lebesgue product spaces with power weights to itself. Moreover, we characterized a suﬃcient and necessary condi-tion which ensures that the weightedp-adic Hardy type operator is bounded on thep-adic Lebesgue product spaces. In addition, we also extended the results to thep-adic weighted Hardy-Cesàro operator.

Acknowledgements

The authors would like to express their thanks to the referees for valuable advice regarding previous version of this paper. The research was supported by NNSF of China (Grant No. 11661075).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equality and signiﬁcantly in writing this paper. All authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 12 April 2017 Accepted: 29 August 2017

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