Acta Mathematica Sinica, English Series Aug., 2014, Vol. 30, No. 8, pp. 1407–1421 Published online: July 15, 2014 DOI: 10.1007/s10114-014-3552-2 Http://www.ActaMath.com

Acta Mathematica Sinica, English Series Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2014

Multilinear Fractional Hausdorff Operators Da Shan FAN Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI 53201, USA E-mail : [email protected]

Fa You ZHAO1) Department of Mathematics, Shanghai University, Shanghai 200444, P. R. China E-mail : [email protected] Abstract In this paper, by introducing the space with weak mixed norms, weak type estimates of two kinds of multilinear fractional Hausdorff operators RΦ,β and SΦ,β on Lebesgue spaces are shown. By virtue of Marcinkiewicz interpolation, strong type estimates of these two operators on Lebesgue spaces are also obtained. Our methods shed some new light on dealing with the case of non-radial function Φ. Keywords strong type

Multilinear fractional Hausdorff operators, bilinear fractional Hardy operator, weak type,

MR(2010) Subject Classification

1

42B20, 42B25, 42B35

Introduction

The study of the theory of Hausdorff operator can be dated back almost a century ago. It was started with the method of Hausdorff summability that was initiated by Hurwitz and Silverman in 1917 in connecting with some summabilities of number series [16]. Later, this method was fully developed by Hausdorff [15] in order to solve a famous moment problem in a finite interval. On the other hand, the Hausdorff operator plays important roles not only in real analysis (see [3, 14]), but also in Fourier analysis and complex analysis (see [2, 9–11, 22]). Particularly, the Hausdorff operator and its varieties have attracted a lot of research related to modern harmonic analysis in the last decade (see [1, 17, 18, 21]). For more details we refer to [19] and [20]. We now recall the classical one-dimensional Hausdorff operator    ∞ Φ(t) x hΦ f (x) = f dt, x ∈ R, (1.1) t t 0 where Φ is a suitable function on R+ := (0, ∞). As usually, the above definition is initially defined on the Schwartz functions f ∈ S. From the definition, it is easy to check that the Received September 29, 2013, revised December 4, 2013, accepted December 30, 2013 Supported by National Natural Science Foundation of China (Grant Nos. 11201287 and 11201103) and a grant of the First-class Discipline of Universities in Shanghai 1) Corresponding author

Fan D. S. and Zhao F. Y.

1408

Hausdorff operator takes many classical operators as its special cases if one chooses some different functions Φ. For examples, letting Φ1 (t) = t−1 χ(1,∞) (t), Φ2 (t) = αχ(0,1] (t)(1 − t)α−1 and Φ3 (t) = (1 − 1t )β−1 t−1 χ(1,∞) (t), one respectively obtains the famous Hardy operator [13]  1 x h(f )(x) = f (t)dt, x > 0, x 0 the Ces` aro operator

 Cα (f )(x) = α

0

1

  (1 − t)α−1 x f dt, t t

as well as the Riemann–Liouville fractional derivatives Rβ f :  x 1 Rβ (f )(x) = β (x − t)β−1 f (t)dt, x 0

x > 0,

x > 0.

In the multi-dimensional case the situation becomes more complicated. Andersen [1] studied the n-dimensional Hausdorff operator  Φ( x ) |y| f (y)dy, (1.2) HΦ f (x) = n |y| Rn where Φ is a radial function defined on R+ . A more general Hausdorff operator was studied by Liflyand [18], and Lerner and Liflyand [17]. There are many essential difference between the operator hΦ in the one-dimensional case and the operator HΦ in the n-dimensional case. The Hausdorff operator hΦ in [23] were considered in dimension two only for the so-called product Hardy space H 11 (R × R):    Φ(u) x1 x2 du. HΦ f (x) = , f u1 u2 R2 |u1 u2 | In [26] these and related results were slightly extended. One can find these facts in recent survey papers [4] and [19]. On the other hand, the theory of multilinear analysis is another very active research topic in harmonic analysis (see [7]). Recently, as a coauthor of [8], one of the authors studied multilinear Hardy operator and obtained its sharp boundedness on the Lebesgue space. Inspired by these achievements, in this paper, we will study two multilinear fractional Hausdorff operators. Their definitions are in the following after we first introduce some necessary notation. Let x ∈ Rn , m ∈ N, ni be positive integers and u = (u1 , u2 , . . . , um ) with each ui ∈ Rni , i =  1, 2, . . . , m. Denote |u| = |u1 |2 + |u2 |2 + · · · + |um |2 and du = du1 du2 · · · dum . Analogously,  = (β1 , β2 , . . . , βm ). we denote the constant vector β The first multilinear Hausdorff operator is defined by    Φ( |ux1 | , |ux2 | , . . . , |uxm | ) ··· F (u)du, (1.3) RΦ,β (F )(x) = m ni −βi Rn1 Rn2 Rnm |u | i i=1 where 0 ≤ βi < ni and F (u) is a function on the product space Rn1 × Rn2 × · · · × Rnm for 1 ≤ i ≤ m. The second operator is defined by    Φ( |ux| )  SΦ,β (F )(x) = ··· F (u)du, (1.4) m u| i=1 ni −β Rn1 Rn2 Rnm |

Multilinear Fractional Hausdorff Operators

where 0 ≤ β < 1 ≤ i ≤ m.

m i=1

1409

ni and F (u) is a function on the product space Rn1 × Rn2 × · · · × Rnm for

If we take F (u1 , u2 , . . . , um ) = f1 (u1 )f2 (u2 ) · · · fm (um ), then RΦ,β and SΦ,β can be reduced to the m-linear operators  RΦ,β (f1 , f2 , . . . , fm )(x) =



Rn1

and

 Rn2

··· 

 SΦ,β (f1 , f2 , . . . , fm )(x) =

Rnm

m Φ( |ux1 | , |ux2 | , . . . , |uxm | )  fi (ui )du, m ni −βi i=1 i=1 |ui |

Rn1

Rn2

 ···

Rnm

m  Φ( |ux| ) m fi (ui )du, |u| i=1 ni −β i=1

(1.5)

(1.6)

respectively. We want to make a few remarks on these multilinear operators. First, as a special case, the operators (1.5) and (1.6) will be reduced to the operators studied by Chen et al. [6] when βi = 0 and ni = n for all 1 ≤ i ≤ m. Second, if Φ(t) = tmn χ(1,∞)(t) and βi = 0, then the multilinear Hausdorff operator SΦ,β is the m-linear Hardy operator  1 m f1 (u1 ) · · · fm (um )du H (f1 , f2 , . . . , fm )(x) = |x|mn |u|<|x| which was studied in [8]. Thirdly, if we choose Φ(s1 , s2 , . . . , sm ) =

m  i=1

 s−n i χ m

i=1

(s1 , s2 , . . . , sm ), s−2 i <1

then the operator RΦ,β (f1 , f2 , . . . , fm ) also becomes the m-linear Hardy operator Hm (f1 , f2 , . . . , fm ). The aim of this paper is to obtain some Lp → Lq weak and strong type estimates for these multilinear Hausdorff operators. It is worth pointing out that the function Φ in the operators defined in (1.3) and (1.4) is not necessarily radial as in [6] or [5]. Thus, the method that will be used is different from the ones in [8] and [6], and itself seems to be of independent interest. Our strategy is to first establish some weak type estimates, then use the Marcinkiewicz interpolation to achieve the strong boundedness. 2

Notation and Auxiliary Results

In this section, we present some notation and preliminary lemmas that will be used in the proofs of the main results. Let us begin with the notation of mixed-norm Lp spaces. Given a measurable function f on Rn1 × Rn2 × · · · × Rnm and an index vector p = (p1 , p2 , . . . , pm ) where 0 < pi ≤ ∞ for 1 ≤ i ≤ n, one can calculate the number f Lp by calculating first the Lp1 -norm of f (x1 , x2 , . . . , xm ) with respect to the variable x1 , and then the Lp2 -norm of the result with respect to the variable x2 , and so on, finishing with the Lpm -norm with respect to xm : f Lp =  · · · f Lp1 (Rn1 ,

dx1 ) Lp2 (Rn2 , dx2 )

· · · Lpm (Rnm ,

dxm ) ,

Fan D. S. and Zhao F. Y.

1410

where f Lpi (Rni ,

⎧  ⎪ ⎨ dxi )

=

⎪ ⎩

Rni

pi

|f (. . . , xi , . . .)| dxi

 p1

i

,

ess supxi ∈Rni |f (. . . , xi , . . .)| ,

if 0 < pi < ∞, if pi = ∞.

It is easy to verify the properties of the mixed norm below. Proposition 2.1 ([25, p. 50]) (i) If each pi is equal to p for 1 ≤ i ≤ m, then  p1    ··· |F (u1 , u2 , . . . , um )|p du1 du2 · · · dum ; F Lp = Rnm

Rn2

Rn1

(ii) if F (u1 , u2 , . . . , um ) = f1 (u1 )f2 (u2 ) · · · fm (um ), then F Lp =

m 

fi Lpi (Rni ) .

i=1

The following lemma is H¨ older’s inequality for mixed norms. Lemma 2.2 ([25, p. 50]) Let 0 < pi ≤ ∞ and let 0 < qi ≤ ∞ for 1 ≤ i ≤ m. If F ∈ Lp and G ∈ Lq , then F G ∈ Lr where 1 1 1 = + , ri pi qi

1 ≤ i ≤ m,

and we have H¨ older’s inequality : F GLr ≤ F Lp GLq . Mixed-norm Lp spaces play a key role in partial differential equations, we refer the reader to [25]. We now consider the measure d(r) = r d−1 dr on the positive line R+ . For an f ∈ Lq (Rd ), 0 < q < ∞, using the spherical coordinates, we see  ∞    q1  q  |f (rx )| dσ(x ) d(r) . f Lq (Rd ) = 0

S d−1

Thus we can write it as the mixed norm f Lq (Rd ) = f (r·)Lq (S d−1 ,dσ) Lq (R+ ,d) . In general, for 0 < q ≤ ∞, the weak Lq (Rd ) space is defined as the set of all measurable functions f such that  1 f Lq,∞ (Rd ) = sup t {x ∈ Rd : |f (x)| > t} q . (2.1) t>0

This observation motivates us to give a weak mixed norm as follows. Definition 2.3 Let 0 < q ≤ ∞ and the measure d(r) = r d−1 dr for r > 0. The weak mixed norm space Lq,∞ (Lq (S d−1 , dσ), (R+ , d)) is defined as the set of all measurable functions f such that    1q f (r·)Lq (S d−1 ,dσ) Lq,∞ (R+ ,d) = sup t  r > 0 : f (r·)Lq (S d−1 ,dσ) > t . (2.2) t>0

Multilinear Fractional Hausdorff Operators

1411

It is easy to verify that (2.1) and (2.2) are equivalent when f is radial. We need to introduce some more notation. Let 1 ≤ p ≤ ∞ and denote by p the expo1 nent conjugate to p; that is, let p1 + p1 = 1 with the agreement that ∞ := 0. Denote by d−1 d d−1 S the unit sphere in R with Lebesgue measure |S |. Let n = (n1 , n2 , . . . , nm ) with each ni be a positive integer and s = (s1 , s2 , . . . , sm ) with each si ∈ R+ , i = 1, 2, . . . , m and  |s| = |s1 |2 + |s2 |2 + · · · + |sm |2 . For simplicity, let εi = (0, . . . , 0, ε, 0, . . . , 0) and p ± εi = (p1 , p2 , . . . , pi−1 , pi ± ε, pi+1 , . . . , pm ) for 1 ≤ i ≤ m. The symbol A  B means that there exists a constant C > 0 independent of all essential variables such that A ≤ CB. 3

Weak and Strong Type Estimates for RΦ,β

Our results can be stated as follows: Theorem 3.1

Let pi ≥ 1, 0 ≤ βi < ni for 1 ≤ i ≤ m and q > 0 satisfying  m   ni N − βi = . p q i i=1

For x = 0, write x = 

C1,Φ,β,  p (x ) =

x |x|

and

 0

∞

 ···

∞

0

(n2 −β2 )p2

s · 2 n2 +1 ds2 s2



 p3 p2

∞

0





|Φ(s1 x , s2 x , . . . , sm x (n −β )p

sm m m m ··· dsm sm nm +1



1 pm



(n1 −β1 )p1 p1 s1 )| s1 n1 +1

 p2 ds1

p1

.

 ∞ N −1 , dσ), then we have If C1,Φ,β,  p (x ) ∈ L (S

 RΦ,β (F )Lq,∞ (RN ) ≤

|S N −1 | N

 q1

Cn C1,Φ,β,  (Rn1 ×Rn2 ×···×Rnm ) .  p L∞ (S N −1 ) F Lp

Here and in what follows Cn =

m 

1 

|S ni −1 | pi .

i=1

To prove Theorem 3.1, we need the following lemma. For simplicity, define A by     x   p2  p3  1   pm p1 p2  Φ |u1 | , |ux2 | , . . . , |uxm | p1    ··· du du · · · du . A(x) = 1 2 m m   ni −βi Rnm Rn2 Rn1 i=1 |ui | Lemma 3.2

 Let pi , βi , q, Cn and C1,Φ,β,  p (x ) be as in Theorem 3.1. Then 

A(x) = Cn C1,Φ,β,  p (x )

m 

n

− p i +βi

|x|

i

.

i=1

Proof Since the case m ≥ 3 presents only notational differences and does not require any new ideas; for simplicity we only provide the proof in the case m = 2. Using the spherical coordinates, we have        p2  1  p p  Φ |ux1 | , |ux2 | p1 1 2   du1 du2 a(x) =   n −β n −β 1 1 2 2 |u2 | Rn2 Rn1 |u1 |

Fan D. S. and Zhao F. Y.

1412

=

2 

|S ni −1 |

1 p i



∞ 0

i=1

r2n2 −1



∞ 0

  |x|  |x|   p1  p2  1 p p  Φ r1 x , r2 x   dr1 1 dr2 2  r1n1 −β1 r2n2 −β2

r1n1 −1 

:= Cn1 ,n2 a0 (x). Changing variables, s1 = then r1 =

|x| , s1

r2 =

|x| , s2

|x| , r1

s2 =

|x| , r2

dr1 = −|x|s−2 1 ds1 ,

dr2 = −|x|s−2 2 ds2 .

Thus, it is easy to see that a0 (x) is equal to  n1 −β1 n2 −β2    p2  1   p1 n1 p1 |x|n2 p2  s1 s Φ(s x , s x ) |x| 1 2 2   ds ds 2 n2 +1   sn1 +1 1 n1 −β1 +n2 −β2 s2 |x| 0 0 1  ∞  ∞  p2 (n1 −β2 )p2  1 (n1 −β1 )p1 n1 n2 p1 s p2  s + −n −n +β +β 1 2 1 2     p 1 2 = |x| p1 p2 |Φ(s1 x , s2 x )| 1 ds ds 1 2 n1 +1 n2 +1 s1 s2 0 0  p  ∞ ∞  2 (n1 −β2 )p2  1 (n1 −β1 )p1 n n p1 s p2  s − 1 − 2 +β +β 2 = |x| p1 p2 1 2 |Φ(s1 x , s2 x )|p1 1 n1 +1 ds1 ds . 2 n2 +1 s1 s2 0 0



∞



∞



The lemma is proved. Proof of Theorem 3.1

By H¨older’s inequality on the mix norm

 · · · F (u1 , u2 , . . . , um )Lp1 (Rn1 ,du1 ) Lp2 (Rn2 ,du2 ) · · · Lpm (Rnm ,dum ) , we have |RΦ,β (F )(x)| ≤ A(x)F p .  ∞ N −1 Since C1,Φ,β, ), then by Lemma 3.2, we deduce that  p (x ) ∈ L (S

|RΦ,β (F )(x)| ≤ |x|− q Cn C1,Φ,β, .  p L∞ (S N −1 ) F p N

For any λ > 0, |{x ∈ RN : |RΦ,β (F )(x)| > λ}| ≤ |{x ∈ RN : |x|− q Cn C1,Φ,β,  p L∞ (S N −1 ) F p  > λ}|   q   Cn C1,Φ,β,   N   p L∞ (S N −1 ) F p N   ≤  x ∈ R : |x| <  λ   q  p L∞ (S N −1 ) F p  |S N −1 | Cn C1,Φ,β, = . N λ N

This shows

 RΦ,β (F )Lq,∞ (RN ) ≤

|S N −1 | N

Therefore, we finish the proof of Theorem 3.1.

 q1

Cn C1,Φ,β,  p L∞ (S N −1 ) F p . 

We also obtain another weak type estimate for the operator RΦ,β by virtue of Definition 2.3.

Multilinear Fractional Hausdorff Operators

Theorem 3.3

1413

Let pi ≥ 1, 0 ≤ βi < ni for 1 ≤ i ≤ m and q > 0 satisfying  m   ni N − βi = . p q i i=1

For x = 0, write x = 

x |x|

and



C1,Φ,β,  p (x ) =

0

·

∞

 ···

(n −β )p s2 2 2 2 s2 n2 +1

∞

0



 p3

ds2

p2

∞

0





|Φ(s1 x , s2 x , . . . , sm x 

(n −β )p

sm m m m ··· dsm sm nm +1

1  pm



(n1 −β1 )p1  s )|p1 1 n1 +1 s1

.

 q N −1 If C1,Φ,β, , dσ), then we have  p (x ) ∈ L (S

RΦ,β (F )Lq (S N −1 ,dσ) Lq,∞ (R+ ,d)   1q 1 ≤ Cn C1,Φ,β,  (Rn1 ×Rn2 ×···×Rnm ) .  p Lq (S N −1 ) F Lp N Proof of Theorem 3.3

By the proof of Theorem 3.1, we have |RΦ,β (F )(x)| ≤ A(x)F p .

For x = 0, we write r = |x| and x =

x |x| .

So we can write

 A(x) = Cn C1,Φ,β,  p (x )

m 

n

r

− p i +βi i

.

i=1

Thus we have RΦ,β (F )(·)Lq (S N −1 ) ≤ Cn

m 

n

r

− p i +βi i

C1,Φ,β, .  p Lq (S N −1 ) F Lp

i=1

Let d(r) = r N −1 dr. Furthermore, since

m

ni i=1 ( pi

− βi ) =

N q ,

we obtain

{r > 0 : RΦ,β (F )(r·)Lq (S N −1 ) > t}   m  n − p i +βi ≤ r>0: r i Cn C1,Φ,β,  > t  p Lq (S N −1 ) F Lp i=1

= {r > 0 : r < C1 (t)}, where

 C1 (t) =

Cn C1,Φ,β,   p Lq (S N −1 ) F Lp t

 Nq .

Therefore, {r > 0 : RΦ,β (F )(r·)Lq (S N −1 ) > t} q q  C1 (t) Cnq 1 ,n2 ,...,nm C1,Φ,β,  p Lq (S N −1 ) F Lp  N −1 r dr = . ≤ q Nt 0 By Definition 2.3, it shows that RΦ,β (F )Lq (S N −1 ,dσ) Lq,∞ (R+ ,d)

 p2 ds1

p 1

Fan D. S. and Zhao F. Y.

1414

 ≤

1 N

 1q

Cn C1,Φ,β, .  p Lq (S N −1 ) F Lp



 When F (u1 , u2 , . . . , um ) = m i=1 fi (ui ), applying Theorem 3.1 and the Marcinkiewicz interpolation, we have the following result. Theorem 3.4 Let N, q, β, C3,Φ,β,p (x ), pi and ni for 1 ≤ i ≤ m be as in Theorem 3.1. If  ∞ N −1 C1,Φ,β, , dσ), then we have  p (x ) ∈ L (S  RΦ,β (f1 , f2 , . . . , fm )Lq,∞ (RN ) ≤

|S N −1 | N

 q1

m 

Cn C1,Φ,β,  p L∞ (S N −1 )

fi Lpi (Rni ) .

(3.1)

i=1

 ∞ N −1 , dσ) for some i ∈ {1, 2, . . . , m}, If there exists an ε > 0 such that C1,Φ,β,  p± εi (x ) ∈ L (S then m  RΦ,β (f1 , f2 , . . . , fm )Lq (RN )  fi Lpi (Rni ) . (3.2) i=1

Proof Theorem 3.1 and Property (ii) of Proposition 2.1 imply that (3.1) holds. For some i ∈ {1, 2, . . . , m}, fix fj ∈ Lpj (Rnj ) with 1 ≤ j ≤ m and j = i. There exists an ε such that q1 and q2 satisfy   m  ni nj N − βi + − βj = , pi − ε pj q1 j=1,j=i   m  ni nj N − βi + − βj = , pi + ε pj q2 j=1,j=i

and q1 < q < q2 . According to the above weak type estimate, we have RΦ,β (f1 , f2 , . . . , fm )Lq1 ,∞ (RN )  N −1  q1 | 1 |S ≤ Cn C1,Φ,β,  p− εi L∞ (S N −1 ) fi Lpi −εi (Rni ) N

m 

fj Lpj (Rnj ) ,

j=1,j=i

and RΦ,β (f1 , f2 , . . . , fm )Lq2 ,∞ (RN )  N −1  q1 | 2 |S p +ε n ≤ Cn C1,Φ,β,  p+ εi L∞ (S N −1 ) fi L i i (R i ) N

m 

fj Lpj (Rnj ) .

j=1,j=i

By the Marcinkiewicz interpolation [12, p. 56], we get RΦ,β (f1 , f2 , . . . , fm )Lq (RN ) 

m 

fi Lpi (Rni ) ,

i=1

which proves the theorem. If the function Φ is radial, we have the following statements. Theorem 3.5

Let pi ≥ 1, 0 ≤ βi < ni for 1 ≤ i ≤ m and q > 0 satisfying  m   ni N − βi = . pi q i=1



Multilinear Fractional Hausdorff Operators

1415

Assume that Φ is a radial function. If  C1,Φ,β,  p (1) =

0

∞

 ···

∞

0

 p3

(n2 −β2 )p2

s · 2 n2 +1 ds2 s2 then

 RΦ,β (F )Lq,∞ (RN ) ≤

Proof of Theorem 3.5



p2

|S N −1 | N

∞

0

(n1 −β1 )p1 p1 s1 |Φ(s1 , s2 , . . . , sm )| s1 n1 +1 (n −β )p

sm m m m ··· dsm sm nm +1  1q



1 pm

 p2 ds1

p1

< ∞,

Cn C1,Φ,β,  (Rn1 ×Rn2 ×···×Rnm ) .  p (1)F Lp

By H¨older’s inequality on the mix norm

 · · · F (u1 , u2 , . . . , um )Lp1 (Rn1 ) Lp2 (Rn2 ) · · · Lpm (Rnm ) , we have |RΦ,β (F )(x)| ≤ A(x)F p .   Since Φ is a radial function, C1,Φ,β,  p (x ) = C1,Φ,β,  p (|x |) = C1,Φ,β,  p (1). By Lemma 3.2, we have, for any λ > 0,

|{x ∈ RN : |RΦ,β (F )(x)| > λ}| −q ≤ |{x ∈ RN : Cn C1,Φ,β, F p > λ}|  p (1)|x|   q   Cn C1,Φ,β,   p (1)F p  N  N   ≤  x ∈ R : |x| <  λ   q  p (1)F p  |S N −1 | Cn C1,Φ,β, = . N λ N

This shows

 RΦ,β (F )Lq,∞ (RN ) ≤

|S N −1 | N

 1q

Cn C1,Φ,β,  p (1)F p . 

Therefore, we finish the proof of Theorem 3.5. Since Φ is a radial function, C1,Φ,β,  p Lq (S N −1 ) = C1,Φ,β,  p (1)|S from Theorem 3.3 that Theorem 3.5 holds. Using Property (i) of Proposition 2.1, we obtain Corollary 3.6

N −1

1 q

| . We can also conclude

Let p ≥ 1, q > 0 and 0 ≤ βi < ni for 1 ≤ i ≤ m satisfying  m   ni N − βi = . p q i=1

Assume that Φ is a radial function. If  ∞  (1) = · · · C2,Φ,β,p  ·

0 (n2 −β2 )p s2 s2 n2 +1

then

 RΦ,β (F )Lq,∞ (RN ) ≤

∞

0

ds2





∞

0

(n1 −β1 )p p s1 |Φ(s1 , s2 , . . . , sm )| s1 n1 +1

(n −β )p

sm m m ··· dsm sm nm +1

|S N −1 | N

 1q

 1 p

 ds1

< ∞,

Cn C2,Φ,β,p  (Rn1 ×Rn2 ×···×Rnm ) .  (1)F Lp

Fan D. S. and Zhao F. Y.

1416

The next corollary follows by the same argument as in Theorem 3.4. Corollary 3.7 Let N , q and C1,Φ,β,  p (1), pi and βi for 1 ≤ i ≤ m be as in Theorem 3.5. Assume that Φ is a radial function. If C1,Φ,β,  p (1) < ∞, then

 RΦ,β (f1 , f2 , . . . , fm )Lq,∞ (RN ) ≤

|S N −1 | N

 1q Cn C1,Φ,β,  p (1)

m 

fi Lpi (Rni ) .

i=1

If there exists an ε > 0 such that C1,Φ,β,  p± εi (1) < ∞ for some i ∈ {1, 2, . . . , m}, then RΦ,β (f1 , f2 , . . . , fm )Lq (RN ) 

m 

fi Lpi (Rni ) .

i=1

4

Weak and Strong Type Estimates for SΦ,β

Our results can be stated as follows: Theorem 4.1

 Let pi ≥ 1 for 1 ≤ i ≤ m, 0 ≤ β < m i=1 ni and q > 0 satisfying   m  ni N −β = . p q i i=1

For x = 0, write x = 

x |x|

C3,Φ,β,p (x ) =

and 

∞ 0

 ···

·s2n2 −1 ds2

∞

0

 p3 p2



∞ 0

 x      p2 p1  Φ | p1 n −1 s | 1  m  ds1  |s| i=1 ni −β  s1

nm −1 · · · sm dsm



1  pm

.

If C3,Φ,β,p (x ) ∈ L∞ (S N −1 , dσ), then we have  N −1  q1 | |S SΦ,β (F )Lq,∞ (RN ) ≤ Cn C3,Φ,β,p L∞ (S N −1 ) F Lp (Rn1 ×Rn2 ×···×Rnm ) . N To prove Theorem 4.1, we need the following lemma. Write x       p2  p3  1  pm p1 p2  Φ |u| p1  m  du1 B(x) = ··· du · · · du . 2 m   n −β i Rnm Rn2 Rn1 |u| i=1 Lemma 4.2

Let pi , β, q and C3,Φ,β,p (x ) be as in Theorem 4.1. Then 

B(x) = Cn C3,Φ,β,p (x )

m 

n

− p i +β

|x|

i

.

i=1

Proof We only provide the proof in the case m = 2. Using the spherical coordinates, we have x      p2  1  p1 p2  Φ |u| p1   b(x) = du1 du2   n +n −β 1 2 Rn2 Rn1 |u|     Φ √|x|x  ∞  ∞ 2 1  2 +r 2  r  1 2 = |S ni −1 | pi r2n2 −1 r1n1 −1  n +n −β 2 + r2 ) 1 22 0 0 (r i=1 1 2

p1  p2  1 p p   dr1 1 dr2 2 

Multilinear Fractional Hausdorff Operators

1417

:= Cn1 ,n2 b0 (x). Changing variables, s1 =

r1 , |x|

s2 =

r2 , |x|

then r1 = |x|s1 ,

r2 = |x|s2 ,

dr1 = |x|ds1 ,

dr2 = |x|ds2 .

Thus, it is easy to see that b0 (x) is equal to     p1  ∞  ∞  p2  1 Φ √ x2 2 p1 p2   s +s 1 2 n2 −1 n1 −1  n1 n2  s2 s1 |x| ds |x| ds 1 2 n1 +n2 −β  2  n +n −β 2 0 0 (s1 + s22 ) |x| 1 2    Φ √ x p1  ∞  ∞  p2  1 2 2 n1 n2 p1 p2   s1 +s2 − p − p +β n −1 n −1 2 1   = |x| 1 2 s2 s1 ds ds . 1 2 n1 +n2 −β   2 2 0 0 (s1 + s22 ) 

The lemma is proved. Proof of Theorem 4.5

By H¨older’s inequality on the mix norm

 · · · F (u1 , u2 , . . . , um )Lp1 (Rn1 ) Lp2 (Rn2 ) · · · Lpm (Rnm ) , we have |SΦ,β (F )(x)| ≤ B(x)F p . Under the assumption that C3,Φ,β,p (x ) ∈ L∞ (S N −1 ), by Lemma 4.2, we see that, for any λ > 0,    x ∈ RN : |SΦ,β (F )(x)| > λ  ≤ |t{x ∈ RN : |x|− q Cn C3,Φ,β,p L∞ (S N −1 ) F p > λ}|   q    Cn C3,Φ,β,p L∞ (S N −1 ) F p N  N   ≤  x ∈ R : |x| <  λ   q |S N −1 | Cn C3,Φ,β,p L∞ (S N −1 ) F p = . N λ N

This shows

 SΦ,β (F )Lq,∞ (RN ) ≤

|S N −1 | N

 q1

Cn C3,Φ,β,p L∞ (S N −1 ) F p , 

which is the desired conclusion. The theorem below is an analogue of Theorem 3.3.

Theorem 4.3 Let N, β, q, C3,Φ,β,p (x ), pi and ni for 1 ≤ i ≤ m be as in Theorem 4.1. If C3,Φ,β,p (x ) ∈ Lq (S N −1 , dσ), then we have   q1 1 SΦ,β (F )Lq (S N −1 ,dσ) Lq,∞ (R+ ,d) ≤ Cn C3,Φ,β,p Lq (S N −1 ) F Lp (Rn1 ×···×Rnm ) . N Proof of Theorem 4.3

By the proof of Theorem 4.1, we have |SΦ,β (F )(x)| ≤ B(x)F p .

For x = 0, we write r = |x| and x =

x |x| .

So we can write

B(x) = Cn C3,Φ,β,p (x )

m  i=1

n

r

− p i +β i

.

Fan D. S. and Zhao F. Y.

1418

Thus we have SΦ,β (F )(r·)Lq (S N −1 ) ≤ Cn

m 

n

r

− p i +β i

C3,Φ,β,p Lq (S N −1 ) F Lp .

i=1

 m  Let d(r) = r N −1 dr. Furthermore, since i=1 npii − β = Nq , we obtain    r > 0 : SΦ,β (F )(r·)Lq (S N −1 ) > t   m  n − p i +β ≤ r>0: r i Cn C3,Φ,β,p Lq (S N −1 ) F Lp > t i=1

=  {r > 0 : r < C2 (t)} , where

 C2 (t) =

Cn C3,Φ,β,p Lq (S N −1 ) F Lp t

 Nq .

Therefore,    r > 0 : SΦ,β (F )(r·)Lq (S N −1 ) > t  C2 (t) Cnq 1 ,n2 ,...,nm C3,Φ,β,p qLq (S N −1 ) F qLp r N −1 dr = . ≤ N tq 0 By Definition 2.3, it shows that SΦ,β (F )Lq (S N −1 ,dσ) Lq,∞ (R+ ,d)   1q 1 ≤ Cn C3,Φ,β,p Lq (S N −1 ) F Lp . N The next theorem specializes to the result of Theorem 4.1 if F (u1 , u2 , . . . , um ) =

 m i=1

fi (ui ).

Theorem 4.4 Let N, β, q, C3,Φ,β,p (x ), pi and ni for 1 ≤ i ≤ m be as in Theorem 4.1. If C3,Φ,β,p (x ) ∈ L∞ (S N −1 , dσ), then we have  N −1  q1 m  | |S Cn C3,Φ,β,p L∞ (S N −1 ) fi Lpi (Rni ) . (4.1) SΦ,β (f1 , f2 , . . . , fm )Lq,∞ (RN ) ≤ N i=1 If there exists an ε > 0 such that C3,Φ,β,p±εi (x ) ∈ L∞ (S N −1 , dσ), i ∈ {1, 2, . . . , m}, then SΦ,β (f1 , f2 , . . . , fm )Lq (RN ) 

m 

fi Lpi (Rni ) .

(4.2)

i=1

Proof By virtue of Theorem 4.1 and Property (ii) of Proposition 2.1, (4.1) can be obtained. As in the proof of (3.2), for some i ∈ {1, 2, . . . , m}, fix fj ∈ Lpj (Rnj ) with 1 ≤ j ≤ m and j = i. There exists an ε such that q1 and q2 satisfy that   m  ni nj N −β+ −β = , pi − ε pj q1 j=1,j=i   m  nj ni N −β+ −β = , pi + ε pj q2 j=1,j=i

and q1 < q < q2 .

Multilinear Fractional Hausdorff Operators

1419

Now we have SΦ,β (f1 , f2 , . . . , fm )Lq1 ,∞ (RN )  N −1  q1 | 1 |S ≤ Cn C3,Φ,β,p−εi L∞ (S N −1 ) fi Lpi −εi (Rni ) N

m 

fj Lpj (Rnj ) ,

j=1,j=i

and SΦ,β (f1 , f2 , . . . , fm )Lq2 ,∞ (RN )  N −1  q1 | 2 |S ≤ Cn C3,Φ,β,p+εi L∞ (S N −1 ) fi Lpi +εi (Rni ) N

m 

fj Lpj (yRnj ) .

j=1,j=i

By the Marcinkiewicz interpolation, we get SΦ,β (f1 , f2 , . . . , fm )Lq (RN ) 

m 

fi Lpi (Rni ) ,

i=1



which proves the theorem. If the function Φ is radial, we have the following statements.  Theorem 4.5 Let pi ≥ 1 for 1 ≤ i ≤ m, 0 ≤ β < m i=1 ni and q > 0 satisfying   m  ni N −β = . p q i i=1 Assume that Φ is a radial function. If  ∞  ··· C3,Φ,β,p (1) = 0

0

·s2n2 −1 ds2 then

∞

 p3 p2



∞

0

      p2 p1  Φ |1s| p1 n −1 1  m  ds1  |s| i=1 ni −β  s1

nm −1 · · · sm dsm



1 pm

< ∞,

1 |S N −1 | q Cn C3,Φ,β,p (1)F Lp (Rn1 ×Rn2 ×···×Rnm ) . N Analysis similar to that in the proof of Theorem 3.5 shows that 

SΦ,β (F )Lq,∞ (RN ) ≤ Proof of Theorem 4.5

|SΦ,β (F )(x)| ≤ B(x)F p . The function Φ is radial, and consequently C3,Φ,β,p (x ) = C3,Φ,β (|x |) = C3,Φ,β,p (1). By Lemma 4.2, we have, for any λ > 0,    x ∈ RN : |SΦ,β (F )(x)| > λ  ≤ |{x ∈ RN : Cn C3,Φ,β,p (1)|x|− q F p > λ}|   Nq     C (1)F  C  n 3,Φ,β, p p  N  ≤  x ∈ R : |x| <  λ   q |S N −1 | Cn C3,Φ,β,p (1)F p = . N λ N

This shows

 SΦ,β Lq,∞ (RN ) ≤

|S N −1 | N

 q1

Cn C3,Φ,β,p (1)F p ,

Fan D. S. and Zhao F. Y.

1420



which completes the proof of Theorem 4.5. The following corollaries follow by the same method as in Corollaries 3.6 and 3.7.  Corollary 4.6 Let p ≥ 1, q > 0 and 0 ≤ β < m i=1 ni satisfying   m  N ni −β = . p q i=1 Assume that Φ is a radial function. If  ∞  ∞ C4,Φ,β,p (1) = ··· 0

· · · snmm −1 dsm

0

 1 p

then

0

        Φ |1s| p n −1  m  s 1 ds1 sn2 −1 ds2 2  |s| i=1 ni −β  1

< ∞, 

SΦ,β (F )Lq,∞ (RN ) ≤ Corollary 4.7

∞

|S N −1 | N

 q1

Cn C4,Φ,β,p (1)F Lp .

Let N , β, q, C3,Φ,β,p (1) and pi for 1 ≤ i ≤ m be as in Theorem 4.5. If C3,Φ,β,p (1) < ∞,

then

 SΦ,β (f1 , f2 , . . . , fm )Lq,∞ (RN ) ≤

|S N −1 | N

 1q Cn C3,Φ,β,p (1)

m 

fi Lpi (Rni ) .

i=1

If there exists an ε > 0 such that C3,Φ,β,p±εi (1) < ∞ for some i ∈ {1, 2, . . . , m}, then SΦ,β (f1 , f2 , . . . , fm )Lq (RN ) 

m 

fi Lpi (Rni ) .

i=1

Remark 4.8 Assume that Φ is radial. Let B(0, |x|) denote the ball of a radius |x| centered n−1 β at the origin. Take m = 1, n1 = n and Φ(t) = χ(1,∞) t−n+β ( |S n | ) n −1 . Then SΦ,β is the fractional Hardy operator Hβ defined by  1 f (y)dy, x = 0, Hβ (f )(x) = |B(0, |x|)|n−β B(0,|x|)  n−1  1 if 0 < n < β. It is easy to check the constant |S n | q Cn C4,Φ,β,p (1) is sharp which was shown in [24]. Acknowledgements The authors are indebted to the anonymous referees for valuable comments that improved the paper significantly. References [1] Andersen, K. F.: Boundedness of Hausdorff operators on Lp (Rn ), H 1 (Rn ) and BMO(Rn ). Acta Sci. Math. (Szeged), 69, 409–418 (2003) [2] Aizenberg, L., Liflyand, E.: Hardy spaces in Reinhardt domains, and Hausdorff operators. Illinois J. Math., 53, 1033–1049 (2009) [3] Bennett, G.: An inequality for Hausdorff means. Houston J. Math., 25, 709–744 (1999) [4] Chen, J., Fan, D., Wang, S.: Hausdorff operators on Eulidean spaces. Appl. Math. J. Chinese Univ., 28, 548–564 (2013)

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[5] Chen, J., Wu, X.: Best constant for Hausdorff operators on n-dimensional product spaces. Sci. China Math., 57, 569–578 (2014) [6] Chen, J., Fan, D., Zhang, C.: Multilinear Hausdorff operators and their best constants. Acta Math. Sin., Engl. Series, 28, 1521–1530 (2012) [7] Coifman, R. R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Amer. Math. Soc., 212, 315–331 (1975) [8] Fu, Z., Grafakos, L., Lu, S., et al.: Sharp bounds for m-linear Hardy and Hilbert operators. Houston J. Math., 38, 225–243 (2012) [9] Galanopoulos, P., Papadimitrakis, M.: Hausdorff and quasi-Hausdorff matrices on spaces of analytic functions. Canad. J. Math., 58, 548–579 (2006) [10] Galanopoulos, P., Siskakis, A. G.: Hausdorff matrices and composition operators. Illinois J. Math., 45, 757–773 (2001) [11] Georgakis, C.: The Hausdorff mean of a Fourier–Stieltjes transform. Proc. Amer. Math. Soc., 116, 465–471 (1992) [12] Grafakos, L.: Classical Fourier Analysis, Second edition, Graduate Texts in Math., no. 249, Springer, New York, 2008 [13] Hardy, G. H.: Note on a theorem of Hilbert. Math. Z., 6, 314–317 (1920) [14] Hardy, G. H.: An inequality for Hausdorff means. J. London Math. Soc., 18, 46–50 (1943) [15] Hausdorff, F.: Summation methoden und Momentfolgen, I. Math. Z., 9, 74–109 (1921) [16] Hurwitz, W. A., Silverman, L. L.: On the consistency and equivalence of certain definitions of summability. Trans. Amer. Math. Soc., 18, 1–20 (1917) [17] Lerner, A., Liflyand, E.: Multidimensional Hausdor operators on the real Hardy spaces. J. Austr. Math. Soc., 83, 79–86 (2007) [18] Liflyand, E.: Boundedness of multidimensional Hausdorff operators on H 1 (Rn ). Acta Sci. Math. (Szeged), 74, 845–851 (2008) [19] Liflyand, E.: Complex and Real Hausdorff Operators. CRM, preprint 1046, 2011, 45p. Or http://www. crm.es/Publications/11/Pr1046.pdf [20] Liflyand, E.: Open problems on Hausdorff operators. In: Complex Analysis and Potential Theory (Aliyev Azeroglu, T. and Tamrazov, P.M. eds.), Proc. Conf. Satellite to ICM 2006, Gebze, Turkey, Sept. 2006, 8–14; World Sci., 2007, 280–285 [21] Liflyand, E., Miyachi, A.: Boundedness of the Hausdorff operators in H p spaces, 0 < p < 1. Studia Math., 194, 279–292 (2009) [22] Liflyand, E., M´ oricz, F.: The Hausdorff operator is bounded on the real Hardy space H 1 (R). Proc. Amer. Math. Soc., 128, 1391–1396 (2000) [23] Liflyand, E., M´ oricz, F.: The multi-parameter Hausdorff operator is bounded on the product Hardy space H 11 (R × R). Analysis (Munich), 21, 107–118 (2001) [24] Lu, S., Yan, D., Zhao, F.: Sharp bounds for Hardy type operators on higher-dimensional product spaces. J. Inequal. Appl., 2013, 1–11 (2013) [25] Robert, A., John, J. F.: Sobolev spaces. In: Pure and Applied Mathematics, 2nd ed., Vol. 140, Elsevier/Academic Press, Amsterdam, 2003 [26] Weisz, F.: The boundedness of the Hausdorff operator on multi-dimensional Hardy spaces. Analysis (Munich), 24, 183–195 (2004)