Sharp bounds for Hardy type operators on higher dimensional product spaces

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R E S E A R C H

Open Access

Sharp bounds for Hardy type operators on

higher-dimensional product spaces

Shanzhen Lu

1

_{, Dunyan Yan}

2

_{and Fayou Zhao}

3*

*_{Correspondence:}

[email protected]

3_{Department of Mathematics,}

Shanghai University, Shanghai, 200444, P.R. China

Full list of author information is available at the end of the article

Abstract

A new class of Hardy type operator deﬁned on a higher-dimensional product space is discussed. It includes two diﬀerent kinds of the classical Hardy operators. In addition, we also consider the fractional Hardy operatorHβ. The bound of operatorHβfromLp toLq_{is explicitly worked out. Especially, the bound of operator}_H

βfromL1toL n n–β,∞_is sharp.

MSC: 26D10; 26D15; 42B99

Keywords: sharp bound; product space; the fractional Hardy operator

1 Introduction

The most fundamental averaging operator is the Hardy operator deﬁned by

H(f)(x) = 

x x



f(t)dt,

where the functionf is a nonnegative integrable onR+= (,∞) andx> . A classical inequality, due to Hardy [], states that

H(f)_Lp≤ p p– fLp

holds for  <p<∞, and the constant_pp_–is best possible.

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

Rn(f)(x, . . . ,xn) =



x· · ·xn x



· · · xn



f(t, . . . ,tn)dt· · ·dtn, ()

where the functionf is a nonnegative measurable function onG= (,∞)n_and_xi_{> ,} i= , , . . . ,n.

Another deﬁnition is the spherical averaging operator, which was given by Christ and Grafakos in [] as follows:

H(f)(x) = 

|B(,|x|)|

|y|<|x|f(y)dy, x∈R

n_\{__}_, _()

wheref is a nonnegative measurable function onRn_.

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The boundedness of operator Rn are discussed in many papers (cf. [–]). RnLp_→Lp, the norm ofRn, is (_pp_–)nand obviously depends on the dimension of the

space. However,HLp_→Lp, the norm ofH, is still p

p–, and does not depend on the

dimen-sion of the space. The reason of generating the diﬀerence of the two kind of operators is, roughly speaking, that each variable can independently dilate by itself in the operatorRn,

nevertheless, for the operatorH, all variables dilate by the same scale simultaneously. Gen-erally speaking, the spherical averaging operator has better properties than the rectangle averaging operator does, such as the Hardy-Littlewood maximal function. A detailed ac-count of the history of the topic can be found in the book []; see also Kufner and Persson’s book [].

For the operatorRn, we note that every variable is deﬁned on the one-dimensional space.

In this paper, we shall extent the deﬁnition ofRnso that every variable is spherical average

deﬁned on a higher-dimensional space.

Next, we will give the deﬁnition of Hardy type operator on higher-dimensional product spaces as follows and discuss the corresponding properties.

Deﬁnition . Letm∈N,ni∈N,xi∈Rni_{, }_≤_i_≤_m_{, and}_f _{be a nonnegative measurable}

function onRn×Rn× · · · ×Rnm_{. The Hardy type operator is deﬁned by}

Hm(f)(x) = _m

i=



|B(,|xi|)| |y|<|x|

· · ·

|ym|<|xm|f(y, . . . ,ym)dy· · ·dym, ()

wherex= (x,x, . . . ,xm)∈Rn×Rn× · · · ×Rnm with

m

i=|xi| = .

Our ﬁrst aim in this paper is to provide transparent treatments of multivariate inequal-ities of higher-dimensional Hardy type both for the rectangle and for the ball case. Our results subsume those of [] and []. In fact, ifm= , then the operatorHmwill become

Hdeﬁned by (); ifn=n=· · ·=nm= , thenHm will becomeRndeﬁned by ().

Con-sequently, the operatorHmincludes bothRnandH. It is much signiﬁcant to discuss the

properties ofHm.

Our second aim is to consider the fractional Hardy operator on the Lebesgue spaces. Recall that, for a nonnegative measurable functionf onRn_{, the}_n_{-dimensional fractional}

Hardy operatorHβwith spherical mean is deﬁned by

Hβ(f)(x) =



|B(,|x|)|–β_n

|y|<|x|

f(y)dy, x∈Rn\{}, ()

where ≤β<n (cf. []). Clearly,Hβ(|f|)(x)≤CMβ(f)(x), whereMβ is the fractional

Hardy-Littlewood maximal function deﬁned by

Mβ(f)(x) =sup

r>



|B(x,r)|–βn

|y–x|<r

f(y)dy,

wheref is a measurable function.

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(a) Iff ∈Lp₍_Rn₎_{, then} Mβ(f)_Lq≤CfLp;

(b) iff∈L₍_Rn₎_{, then, for any}_λ_{> }_,

x∈Rn:Mβ(f)(x) >λ≤

C λfL

n n–β

.

However the constantC, the bound of operatorMβ, were not given explicit expression of depending on the parametersp,qandβ. In this paper, the bounds of the operatorHβfrom Lp_to_Lq_{and from}_L_to_Ln–nβ,∞_{are explicitly worked out. Furthermore, we will show that}

the constant  is the bound of operatorHβfromLtoL n

n–β,∞_{and is beat possible, that is,} Hβ

L_→Ln–nβ,∞= .

Throughout the paper, we use the following notation. The setB(,|x|) denotes a ball with center at the original point and radius|x|, and|B(,|x|)|denotes the volume of the ballB(,|x|); (B(,|x|))c₌_Rn_\_B_(,_|_x_|_).

2 Sharp bounds for the Hardy type operator on product space

Theorem . Let <p<∞,m∈N,ni∈N,xi∈Rni_,_i_{= , . . . ,}_m_._{If f} _∈_Lp₍_Rn×Rn×

· · · ×Rnm_,_|_x_|α_),_where_|_x_|α_:=_|_x

|α|x|α· · · |xm|αm andαj< (p– )nj,then the Hardy type operatorHm deﬁned in()is bounded on Lp(Rn×Rn× · · · ×Rnm,|x|α),moreover,the norm ofHmcan be obtained as follows:

HmLp₍_|x|α₎_→Lp₍_|x|α₎= m

j=

p p–  –αj

nj

.

Proof of Theorem. We merely give the proof with the casem=  for the sake of clarity in writing, and the same is true for the general casem> . We adapt some ideas and methods used in [].

Set

gf(x,x) =



ωn



ωn

Sn–

Sn–

f|x|ξ,|x|ξ

dσ(ξ)dσ(ξ),

whereωni=π ni



(ni_) andxi∈R

ni_,_i_{= , . Obviously,}_g_{is a nonnegative radial function with}

respect to the variablesxandx, respectively. In the following, we brieﬂy call this function

is a radial function on product space. It follows thatH(gf)(x,x) is equal to



|B(,|x|)|



|B(,|x|)|

B(,|x|)

B(,|x|)



ωnωn

× Sn–

Sn–

f|y|ξ,|y|ξ

dσ(ξ)dσ(ξ)dydy

= 

ωnωn

Sn–

Sn–



|B(,|x|)|



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×

B(,|x|)

B(,|x|)

f|y|ξ,|y|ξ

dydydσ(ξ)dσ(ξ)

=

Sn–

Sn–



|B(,|x|)|



|B(,|x|)|

× |x|



|x|



f(rξ,rξ)rn–r

n–

 drdrdσ(ξ)dσ(ξ)

= 

|B(,|x|)|



|B(,|x|)|

B(,|x|)

B(,|x|)

f(y,y)dydy

=H(f)(x,x).

Using the generalized Minkowski’s inequality and Hölder’s inequality, we conclude that

gf_Lp₍_|x|α₎is equal to

Rn_

Rn_



ωnωn

p

×

Sn–

Sn–

f|x|ξ,|x|ξ

dσ(ξ)dσ(ξ)

p

|x|αdxdx



p

≤ 

ωnωn

Sn–

Sn–

×

Rn

Rn

f(|x|ξ,|x|ξ)

p

|x|α|x|αdxdx



p

dσ(ξ)dσ(ξ)

≤



ωnωn

Sn–

Sn–

Rn_

Rn_

f(|x|ξ,|x|ξ)

p

× |x|α|x|αdxdxdσ(ξ)dσ(ξ)



p

=

Sn–

Sn–

∞



∞



f(rξ,rξ)

p rα+n–

 r

α+n–

 drdrdσ(ξ)dσ(ξ)



p

=f_Lp₍_|x|α₎.

Thus, we conclude that the following inequality:

H(f)_Lp₍_|x|α₎ f_Lp₍_|x|α₎ ≤

H(gf)_Lp₍_|x|α₎ gf_Lp₍_|x|α₎

holds provided thatf_Lp₍_|x|α₎= . In addition, clearly iff is a nonnegative radial function,

then we havegf =f. This means that the norm of the operatorHis equal to the norm thatHrestricts to the set of nonnegative radial functions. Consequently, without loss of generality, it suﬃces to fulﬁl the proof of the theorem by assuming thatf is a nonnegative radial function.

Substituting the variablesz=_|xy__|andz=_|xy__|, we have thatH(f)Lp₍_|x|α₎equals

Rn

Rn

_H(f)(x,x)p|x|α|x|αdxdx



p

=

Rn

Rn



|B(,|x|)||B(,|x|)|

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× |y|<|x|

|y|<|x|

f(y,y)dydy

p



p

= 

νnνn

Rn

Rn

|z|<

|z|<

fz|x|,z|x|

dzdz

p



p

.

Using the generalized Minkowski’s inequality again and noting thatf is a radial func-tion with respect to the ﬁrst variable and the second variable, respectively, we have that

H(f)_Lp₍_|x|α₎is not greater than



νnνn

|z|<

|z|<

Rn

Rn

f(z|x|,z|x|)

p



p dzdz

= 

νnνn

|z|<

|z|<

Rn

Rn

f(x,x)

p

× _|

x|

|z|

α_|_x

|

|z|

α

dxdx



p |z|–

n

p|_z

|–

n

p _dz

dz

= 

νnνn

|z|<

|z|<

|z|–

n_+α_ p |_z

|–

n_+α_ p _dz

dzfLp₍_|x|α₎

=  j= p p–  –αj/nj

f_Lp₍_|x|α₎,

whereνn_i= πni/

(+ni/) is the volume of the unit ball inR

ni_,_i_{= , .}

Therefore, it implies that

HLp₍_|x|α₎_→Lp₍_|x|α₎≤



j=

p p–  –αj/nj

.

Next, we need to prove the converse inequality. For the purpose of getting the sharp bound, we set

 <ε<min

,(p– )n

p

,(p– )n

p

,

and deﬁne

fε(x,x) =|x|–

n_+α_ p +ε|_x

|–

n_+α_ p +ε_χ_{|x

|<,|x|<}(x,x).

It follows from the elementary calculation thatfε_Lp₍_|x|α₎is

|x|<

|x|<

|x|(ε–

n_+α

p )p|_x

|(ε–

n_+α

p )p|_x|α_dx

dx



p

=

|x|<

|x|(–

n

p+ε)p_dx



/p

|x|<

|x|(–

n

p+ε)p_dx





p

=

ωn

pε /p_ω

n

pε 

p

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We rewriteH(fε) as follows:

H(fε)(x,x) =



|B(,|x|)||B(,|x|)|

|y|<|x|

|y|<|x|

fε(y,y)dydy

= 

|B(, )||B(, )|

|z|<

|z|<

fεz|x|,z|x|

dzdz

=|x|

–n+α

p +ε|_x

|–

n_+α

p +ε |B(, )||B(, )|

×

{|z|<,|z|</|x|}

{|z|<,|z|</|x|}

|z|–

n_+α

p +ε|_z

|–

n_+α

p +ε_dz

dz.

Thus, we estimate the norm ofH(fε)Lpas follows: _H(fε)p_Lp₍_|x|α₎ =

 (νnνn)p

Rn

Rn

|

{|z|<,|z|<_|x_|}

{|z|<,|z|<_|x_|}

|z|–

n_+α_ p +ε

× |z|–

n_+α

p +ε_dz

dz|p|x|pε–n–α|x|pε–n–α|x|αdxdx

≥ 

(νnνn)p

|x|<

|x|<

|

|z|<

|z|<

|z|–

n_+α

p +ε

× |z|–

n+α

p +ε_dz

dz|p|x|pε–n|x|pε–ndxdx

=  (νnνn)p

ωn

pε ωn

pε

|z|<

|z|<

|z|–

n_+α

p +ε|_z

|–

n_+α

p +ε_dz

dz

p

=  (νnνn)p

ωn

n( – /p–α/pn+ε/n)

p

×

ωn

n( – /p–α/pn+ε/n)

p fεp_Lp

=

p

p–  –α/n+pε/n

p

p–  –α/n+pε/n

p fεp

Lp₍_|x|α₎.

Therefore, it implies that

H(fε)_Lp₍_|x|α₎ fε_Lp₍_|x|α₎ ≥

p

p–  –α/n+pε/n ·

p

p–  –α/n+pε/n

.

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

HLp₍_|x|α₎_→Lp₍_|x|α₎≥



j=

p p–  –αj/nj

.

This ﬁnishes the proof of the theorem.

3 Explicit bounds for the fractional Hardy operator

Theorem . Suppose that≤β<n,  <p≤_βnand_p–_q=β_n.

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where

p q

/q

p p– 

/q

q q– 

–/q

 –p

q /p–/q

≤C≤

p p– 

p q

.

(ii) Iff ∈L₍_Rn₎_,_{then for any}_λ_{> }_,

x∈Rn:Hβ(f)(x)>λ≤



λfL n

n–β

.

Moreover,

Hβ

L_→Ln–nβ,∞= .

Proof of(i)of Theorem. Setωn= π n

_/ ₍n

) as above. Let

gf(y) = 

ωn

|ξ|=

f|y|ξdξ, y∈Rn.

Clearly,gf is a radial function.

Hβ(gf)(x) =

 (νn|x|n)–

β n

|y|<|x|gf(y)dy

=  (νn|x|n)–

β n

|y|<|x|



ωn

|ξ|=

f|y|ξdξ

dy

=  (νn|x|n)–

β n

|ξ|=



ωn

|y|<|x|f

|y|ξdy

dξ

=  (νn|x|n₎–βn

|y|<|x|f(y)dy

=Hβ(f)(x).

Using the generalized Minkowski inequality and Hölder’s inequality, we have that

gfLp =

Rn

gf(x)pdx 

p

≤  ωn

|ξ|=

Rn

f|x|ξpdx 

p dξ

= ω

/p n ωn

|ξ|=

∞



f(tξ)ptn–dt /p

dξ

≤ω

/p n ωn

|ξ|=

∞



f(tξ)ptn–dt dξ /p

|ξ|=

dξ –/p

=fLp.

Therefore, we have that

Hβ(f)Lq fLp ≤

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As above, this means that the norm of the operatorHβfromLptoLqis equal to the norm

thatHβrestricts to radial functions. Consequently, without loss of generality, it suﬃces to

carry out the proof of the theorem by assuming thatf is a radial function. A simple estimate implies thatHβ(f)Lq equals

Rn



(νn|x|n)– β n

|y|<|x|

f(y)dy q dx  q =ν β n–+ p q n Rn

_|y|_<_|x|f(y)dy q–p

_H(f)(x)p|x|nq(βn–)+pn_dx  q ≤ν β n–+ p q

n f

q–p q p

Rn

νn|x|n

(–_p)(q–p)

H(f)(x)p|x|nq(βn–)+pn_dx 

q

=f q–p

q p

Rn

_H(f)(x)pdx 

q ≤

p p– 

p q

fLp,

where we use the well-known consequence thatHis bounded onLp(Rn_{) with the sharp}

bound_pp_–and the following relationship:

(q–p)

 – 

p

+p+

β n– 

q= .

To obtain a better lower bound ofHLp_→Lq, we can takef_(x) =|x| nC

p _χ_{|x|

<}(x),C> –.

Then

fpLp= νn C+ 

.

We have

_Hβ(f)

q Lq =

|x|>

 (νn|x|n)(–β/n)

|y|<

|y| nC p _dy q dx + |x|≤  (νn|x|n)(–β/n)

|y|<|x||y| nC p _dy q dx = ω q n ν(– β n)q

n (nC/p+n)q

|x|>

|x|(β–n)qdx+

|x|≤

|x|(β+ nC

p )q_dx

= ω

(+q)

n

ν(– β n)q

n (nC/p+n)q

p qn

  +C

+ 

p– 

.

So, we have

Hβ(f)Lq fLp =

ωn(+q)

νn(–β/n)q(nC/p+n)q

p qn

  +C

+ 

p– 



q_C

+  νn  p = p q /q

( +C)/p

 +C/p

  +C

+ 

p– 

 q = p q /q p p– 

/q

( +C)



p–q

( +C/p)–



q

(9)

Letg(t) = ( +t)p–q_{( +}_t_/_p₎–(–q)_,_t_{> –. Taking}_g₍_t∗_{) = , we get}_t∗_{= –}_p_/_q_{, where}_g_is

the derivative of the functiong. It can be easily veriﬁed that the functiong(t) deﬁned over the open interval (–,∞) has a unique global maximum at the pointt∗. TakingC= –p/q

together with inequality (), we get

HβLp_→Lq≥

p q

/q

p p– 

/q

q q– 

–/q

 –p

q /p–/q

. ()

Proof of(ii)of Theorem. Since

Hβ(f)(x) =

 (νn|x|n)–

β n

|y|<|x|

f(y)dy≤ 

(νn|x|n)– β n

fL,

we have that

x∈Rn:  (νn|x|n)–

β n

fL>λ

=

x∈Rn:|x|<

fL

λν– β n n



n–β

=

f_L

λ n

n–β

.

Next, we will show that the constant  is a sharp bound by constructing a suitable func-tion. In fact, set

g(x) =χ|x|<(x).

we have

gL=ν_n.

It follows that

Hβ(g)(x) =

 (νn|x|n)–

β n

|y|<|x|

χ_|y|<(y)dy=|{

y:|y|<min{|x|, }}| (νn|x|n)–

β n

.

We assert thatHβ(g)(x)≤ν β n

n for allx∈Rn. We rewrite x∈Rn:Hβ(g)(x) >ν

β n n

:=A∪B, where

A:=B(, )∩ x∈Rn:Hβ(g)(x) >ν β n n

,

B:=B(, )c∩ x∈Rn:Hβ(g)(x) >ν β n n

(10)

Ifx∈A, then|x|< . It follows that

Hβ(g)(x)≤

|{y:|y|<|x|}|

(νn|x|n₎–βn

=νn|x|n β n _<_ν

β n n.

Ifx∈B, then|x| ≥. It follows that

Hβ(g)(x)≤

|{y:|y|< }| (νn|x|n₎–βn

≤ν β n n.

Consequently, we haveA=B=∅. This implies thatHβ(g)(x)≤ν β n n.

For any  <λ<ν β n

n, we conclude that x∈Rn:Hβ(g)(x)>λ

=B(, )∩ x∈Rn:ν β n

n|x|β>λ+

B(, )c∩

x∈Rn: ν

β n n |x|n–β >λ

=

x∈Rn_: λ/ β

νn/n

<|x|< +

x∈Rn_{: }_{≤ |}_x_|_<

ν β n n λ



n–β

=

νn

λ n

n–β

–λnβ_. _()

If there exists a constantCsuch that

CfL

λ n

n–β

holds for allf ∈L(Rn). Then we can choose that

f(x) =χ|x|<(x).

It follows from equality () that

νn

λ n

n–β

–λβn≤

Cνn λ

n n–β

always holds for every  <λ<ν β n

n. Lettingλ→+, this forces thatC≥. This means that

the constant  is sharp. At the end of this paper, we revisit the lower bound ofHβLp_→Lq. Using L’Hospital’s

rule, we obtain that

lim

q→p+

 –p

q /p–/q

= .

This implies that

(11)

Remark . Ifβ= , the operatorHβ is reduced to the classical Hardy operatorH. In

addition, in order to study the endpoint estimate for Hardy operator, we in [] modiﬁed the deﬁnition of () as follows:

˜

H(f)(x) = 

|B(,|x|)|

|y|<|x|

f(y)dy, x∈Rn\{},

wheref is a measurable function onRn_{. In fact, the operators}_H˜ _and_H_{enjoy the same}

boundedness property forLp_→_Lp_and_L_→_L,∞_{, but they do not have the same property}

involving the Hardy spaceH_{. For instance, the operator}_H˜ _{is bounded from}_H_to_L_{; but}

the operatorHis not sinceHis nonnegative (cf. []). The well-known fact is thatHLp_→Lp= p

p– (cf. []) and ˜HL_→L,∞=HL_→L,∞=  (cf. []). On account of these facts, we guess that the lower bound of (i) in Theorem . is sharp.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Author details

1_{School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, P.R. China.}2_{School of Mathematics Science,}

University of Chinese Academy of Sciences, Beijing, 100190, P.R. China.3_{Department of Mathematics, Shanghai}

University, Shanghai, 200444, P.R. China.

Acknowledgements

The authors would like to express their thanks to the referees for valuable advice regarding a previous version of this paper. This research was supported by NNSF of China (Grant Nos. 11271175, 10931001, 10771221, 11271162 and 11201287), NSF of Beijing (Grant No. 1092004) and NSF of Zhejiang (Grant No. Y6110074). This work was also supported by the FCDU in Shanghai and the KLMCS (Beijing Normal University), Ministry of Education, China.

Received: 11 July 2012 Accepted: 21 March 2013 Published: 3 April 2013

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doi:10.1186/1029-242X-2013-148