New weighted multilinear operators and commutators of hardy-cesàro type

Acta Mathematica Scientia 2015,35B(6):1411–1425 http://actams.wipm.ac.cn

NEW WEIGHTED MULTILINEAR OPERATORS AND ` COMMUTATORS OF HARDY-CESARO TYPE∗ Ha Duy HUNG High School for Gifted Students, Hanoi National University of Education, 136 Xuan Thuy, Hanoi, Vietnam E-mail : [email protected]

Luong Dang KY† Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam E-mail : [email protected] Abstract This paper deals with a general class of weighted multilinear Hardy-Ces` aro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Ces` aro operators (with symbols in central BMO spaces) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators. Key words

Hardy-Ces` aro operators; Hardy’s inequality; multilinear Hardy operator; weighted Hardy-Littlewood averages

2010 MR Subject Classification

1

42B35 (46E30; 42B15; 42B30)

Introduction

The Hardy integral inequality and its variants play an important role in various branches of analysis such as approximation theory, differential equations, theory of function spaces etc (see [4, 10, 18, 20, 29, 30] and references therein). The classical Hardy operator, its variants and extensions were appeared in various papers (refer to [4, 5, 13–15, 20, 21, 29–31] for surveys and historical details about these differerent aspects of the subject). On the other hand, the study of multilinear operators is not motivated by a mere quest to generalize the theory of linear operators but rather by their natural appearance in analysis. Coifman and Meyer in their pioneer work in the 1970s were one of the first to adopt a multilinear point of view in their study of certain singular integral operators, such as the Calder´on commutators, paraproducts, and pseudodifferential operators. ∗ Received

August 7, 2014; revised April 13, 2015. The first author is supported by Vietnam National Foundation for Science and Technology Development (101.02-2014.51). † Corresponding author: Luong Dang KY.

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Let ψ : [0, 1] → [0, ∞) be a measurable function. The weighted Hardy operator Uψ is defined on all complex-valued measurable functions f on Rd as Z 1 Uψ f (x) = f (tx)ψ(t)dt. (1.1) 0

When ψ = 1, this operator is reduced to the usual Hardy operator S defined by Sf (x) =
R 1 x p Rd were first proved by Carton-Lebrun x 0 f (t)dt. Results on the boundedness of Uψ on L and Fosset [5]. Under certain conditions on ψ, the authors [5] found that Uψ is bounded from BMO(Rd ) into itself. Furthermore, Uψ commutes with the Hilbert transform in the case n = 1 and with certain Calder´ on-Zygmund singular integral operators (and thus with the Riesz transform) in the case n ≥ 2. A deeper extension of the results obtained in [5] was due to Jie Xiao [31]. Theorem 1.1 (see [31]) Let 1 < p < ∞ and ψ : [0, 1] → [0, ∞) be a measurable function. Then, Uψ is bounded on Lp (Rd ) if and only if Z 1 t−n/p ψ(t)dt < ∞. (1.2) 0

Furthermore, kUψ kLp (Rd )→Lp (Rd ) =

Z

1

t−n/p ψ(t)dt < ∞.

(1.3)

0

Theorem 1.1 implies immediately the following celebrated integral inequality, due to Hardy [20] kSf kLp(R) ≤

p kf kLp(R) . p−1

(1.4)

For further applications of Theorem 1.1, for examples the sharp bounds of classical RiemannLiouville integral operator, see [13, 14]. Recently, Chuong and Hung [7] introduced a more general form of Uψ operator as follows. Z 1 Uψ,s f (x) = f (s(t)x) ψ(t)dt. (1.5) 0

Here ψ : [0, 1] → [0, ∞), s : [0, 1] → R are measurable functions and f is a measurable complex valued function on Rd . The authors in [7] obtained sharp bounds of Uψ,s on weighted Lebesgue and BMO spaces, where weights are of homogeneous type. They also obtained a characterization on weight functions so that the commutator of Uψ,s , with symbols in BMO, is bounded on Lebesgue spaces. Moreover, the authors in [7] proved the boundedness on Lebesgue spaces of the following operator Z ∞ Hψ,s f (x) = f (s(t)x) ψ(t)dt. 0

1 t,

When d = 1 and s(t) = then Hψ,s reduces to the classical weighted Hausdorff operator (see [25, 26] and references therein). Very recently, the multilinear version of Uψ operators has been introduced by Fu, Gong, Lu and Yuan [14]. They defined the weighted multilinear Hardy operator as Y  Z m Uψm (f1 , · · · , fm ) = fi (ti x) ψ(t1 , · · · , tm )dt1 · · · dtm . (1.6) 0
i=1

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As showed in [14], when d = 1 and ψ(t1 , · · · , tm ) =

1

m−α ,

Γ(α) |(1 − t1 , · · · , 1 − tm )|

where α ∈ (0; m), then Uψm (f1 , · · · , fm )(x) = |x|α Iαm (f1 , · · · , fm )(x). The operator Iαm turns out to be the one-sided analogous to the one-dimensional multilinear Riesz operator Jαm studied by Kenig and Stein [23], where Jαm (f1 , · · ·

, fm )(x) =

m Q

Z

k=1

fk (tk ) m−α dt1

|(x − t1 , · · · , x − tm )|

t1 ,··· ,tm ∈R

· · · dtm .

In [14, 17], the authors obtain the sharp bounds of Uψm on the product of Lebesgue spaces, Herz spaces, Morrey-Herz spaces and central Morrey spaces. They also proved sufficient and necessary conditions of the weight functions so that the commutators of Uψm , with symbols in central BMO spaces, are bounded on the product of central Morrey spaces. Motivated from [7, 13, 14, 17, 21], this paper aims to study the boundedness of a more general multilinear operator of Hardy type as follows. Definition 1.2 Let m, n ∈ N, ψ : [0, 1]n → [0, ∞), s1 , · · · , sm : [0, 1]n → R. Given f1 , · · · , fm : Rd → C be measurable functions, we define the weighted multilinear Hardy-Ces`aro m,n operator Uψ, by → − s m,n Uψ, (f1 , · · · , fm ) (x) := → − s

Z

[0,1]n

Y m

k=1

 fk (sk (t)x) ψ(t)dt,

− where → s = (s1 , · · · , sm ). A mutilinear version of Hψ,s can be defined as  Z Y m m,n Hψ,→ (f1 , · · · , fm ) (x) := fk (sk (t)x) ψ(t)dt, − s Rn +

(1.7)

(1.8)

k=1

where R+ is the set of all positive real numbers. m,n It is obviously that, when n = m and sk (t) = tk , Uψ, is reduced to Uψm as defined in (1.5). → − s m,n m,n The main aim of the paper is to establish the sharp bounds of Uψ, and Hψ, on the product → − → − s s of weighted Lebesgue spaces and weighted central Morrey spaces, with weights of homogeneous types. In addition, we prove sufficient and and necessary conditions of the weight functions so that commutators of such weighted multilinear Hardy-Ces´aro operators (with symbols in m,n is λ-central BMO space) are bounded on the product of central Morrey spaces. Since Uψ, → − s m trivially more general than known operators Uψ , Uψ , our results can be used to recover main results of [7, 13, 14, 31]. Throughout the whole paper, the letter C will indicate an absolute constant, probably different at different occurrences. With χE we will denote the characteristic function of a set E and B(x, r) will be a ball centered at x with radius r. With |A| we will denote the Lebesgue measure of a measurable set E, and E c will be the set Rd \ E. With ω(E) we will denote by R ω(x)dx and Sd will be the unit ball {x ∈ Rd : |x| = 1}. E

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Notations and Definitions

Throughout this paper, ω(x) will be denote a nonnegative measurable function on Rd . Let
us recall that a measurable function f belongs to Lpω Rd if kf kLpω =

Z

Rd

1/p |f (x)|p ω(x)dx < ∞.

(2.1)

The weighted BMO space BMO(ω) is defined as the set of all functions f , which are of bounded mean oscillation with weight ω, that is, Z 1 kf kBMO(ω) = sup |f (x) − fB,ω |ω(x)dx < ∞, (2.2) B ω(B) B where supremum is taken over all the balls B ⊂ Rd . Here we use the standard notation R ω(B) = B ω(x)dx and fB,ω is the mean value of f on B with weight ω: Z 1 fB,ω = f (x)ω(x)dx. ω(B) B The case ω ≡ 1 of (2.2) corresponds to the class of functions of bounded mean oscillation of John and Nirenberg [22]. We obverse that L∞ (Rd ) ⊂ BMO(ω). Next we recall the definition of Morrey spaces. It is well-known that Morrey spaces are useful to study the local behavior of solutions to second-order elliptic partial differential equations and the boundedness of Hardy-Littlewood maximal operator, the fractional integral operators, singular integral operators (see [1, 6, 24]). We notice that the weighted Morrey spaces were first introduced by Komori and Shirai [24], where they used them to study the boundedness of some important classical operators in harmonic analysis like as Hardy-Littlewood maximal operator, Calder´on-Zygmund operators. Definition 2.1 Let λ ∈ R, 1 ≤ p < ∞ and ω be a weight function. The weighted Morrey d space Lp,λ ω (R ) is defined by the set of all locally p-integrable functions f satisfying !1/p Z 1 p kf kLp,λ sup |f (x)| ω(x)dx < ∞. (2.3) d = 1+λp ω (R ) ω (B(a, R)) a∈Rd ,R>0 B(0,R)
The spaces of bounded central mean oscillation CMOq Rd appear naturally when considering the dual spaces of the homogeneous Herz type Hardy spaces and were introduced by Lu and Yang (see [27]). The relationships between central BMO spaces and Morrey spaces were studied by Alvarez, Guzm´ an-Partida and Lakey [2]. Furthermore, they introduced λ-central BMO spaces and central Morrey spaces as follows. Definition 2.2 Let λ ∈ R and 1 < p < ∞. The weighted central Morrey space B˙ ωp,λ (Rd ) is defined by the set of all locally p-integrable functions f satisfying !1/p Z 1 p kf kB˙ ωp,λ(Rd ) = sup |f (x)| ω(x)dx < ∞. (2.4) 1+λp R>0 ω (B(0, R)) B(0,R) Obviously, B˙ ωp,λ (Rd ) is a Banach space and one can easily check that B˙ ωp,λ (Rd ) = {0} if λ < − 1q . Similar to the classical Morrey space, we only consider the case −1/p ≤ λ < 0.

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Definition 2.3 Let λ < 1d and 1 < q < ∞ be two real numbers. The weighted space

CMOq,λ Rd is defined as the set of all function f ∈ Lqω,loc Rd such that ω kf kCMOq,λ d = sup ω (R )

R>0



1

1+λq

ω (B(0, R))

Z

B(0,R)

1/q |f (x) − fB(0,R),ω |q ω(x)dx < ∞.

(2.5)


R x . Let ρ be the measure on (0, ∞) so that ρ(E) = E rd−1 dr and the map Φ(x) = |x|, |x| Then there exists a unique Borel measure σ on Sn such that ρ × σ is the Borel measure induced by Φ from Lebesgue measure on Rd (d > 1). (see [12, page 78] for more details). In one R dimension case, it it conventional that Sd ω(x)dσ(x) refers to 2ω(1).

Definition 2.4 Let α be a real number. Let Wα be the set of all functions ω on Rd , R which are measurable, ω(x) > 0 for almost everywhere x ∈ Rd , 0 < Sd ω(y)dσ(y) < ∞, and are absolutely homogeneous of degree α, that is ω(tx) = |t|α ω(x), for all t ∈ R \ {0}, x ∈ Rd .

Let us describe some typical examples and properties of Wα . Note that, a weight ω ∈ Wα may not need to belong to L1loc (Rd ). In fact, we observe that if ω ∈ Wα , then ω ∈ L1loc (Rd ) if and only if α > −d. If d = 1, then ω(x) = c|x|α , for some positive constant c. For d ≥ 1 and α 6= 0, ω(x) = |x|α belongs to Wα . We refer the readers to [7] for further examples of functions in Wα .

3

m,n Sharp Boundedness of Uψ,~ s on the Product of Weighted Lebesgue Spaces

In this section, we prove sharp bound for Hardy-Ces`aro operators on the product of weighted Lebesgue spaces and weighted central Morrey spaces under certain conditions of the associated weights. Before stating our results we make some conventions on the notation. For m exponents 1 ≤ pj < ∞, j = 1, · · · , m and α1 , · · · , αm > −d we will often write p for the number given by 1 1 1 = + ··· + p p1 pm and α :=

pα1 pαm + ··· + . p1 pm

(3.1)

If ωk ∈ Wαk , k = 1, · · · , m, we will set ω(x) :=

m Y

p/pk

ωk

(x).

(3.2)

k=1 − It is obvious that ω ∈ Wα . We say that (ω1 , · · · , ωm ) satisfies the W→ α condition if

ω(Sd ) ≥

m Y

p

ωk (Sd ) pk .

(3.3)

k=1

Notice that the weights ω as defined in (3.2) were used in [19] to obtain multilinear extrapolation results. On the other hand, Condition (3.3) holds for power weights. In fact, (3.3) becomes to equality for ωk (x) = |x|αk , k = 1, · · · , m. First we will show the following result which can be viewed as an extension of Theorem 1.1 to the multilinear case.

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Theorem 3.1 Let s1 , · · · , sm : [0, 1]n → R be measurable functions such that for every k = 1, · · · , m then |sk (t1 , · · · , tn )| ≥ min{tβ1 , · · · , tβd } almost everywhere (t1 , · · · , tn ) ∈ [0, 1]n , → k = 1, · · · , m, for some β > 0. Let 1 ≤ pk < ∞, k = 1, · · · , m and − s = (s1 , · · · , sm ). Assume

m,n − Rd that (ω1 , · · · , ωm ) satisfies W→ condition. Then U is bounded from Lpω11 Rd ×· · ·×Lpωm → − α m ψ, s
to Lpω Rd if and only if Y  Z m d+α − p k k A= |sk (t)| ψ(t)dt < ∞. (3.4) [0,1]n

Furthermore,



m,n

Uψ,→ − s

k=1

p

p m d d Lω11 (Rd )×···×Lp ωm (R )→Lω (R )

= A.

(3.5)


Proof Suppose that A is finite. Let fk ∈ Lpωkk Rd . Applying Minkowski’s inequality, H¨older’s inequality and change of variable, we have !p !1/p Z Z m

Y

m,n

(f1 , · · · , fm ) p d ≤ |fk (sk (t)x)| ψ(t)dt ω(x)dx

Uψ,→ − s Lω (R )

Rd

≤

Z

Z

Z

m Y

[0,1]n

≤

[0,1]n k=1

!1/p

p

|fk (sk (t)x)| ω(x)dx

Rd k=1

[0,1]n k=1 m Y

= A·

m Y

Z

m Y

|fk (sk (t)x)|

pk

ψ(t)dt !1/pk

ωk (x)dx

Rd k=1

ψ(t)dt

kfk kpk ,ωk .

k=1




m,n The last inequality implies that Uψ, is bounded from Lpω11 Rd × · · · × Lpωm Rd to Lpω Rd → − m s and

m,n ≤ A. (3.6)

Uψ,→ − pm p s p1 d d d Lω1 (R )×···×Lωm (R )→Lω (R )

In order to prove the converse of the theorem, we first need the following lemma. Lemma 3.2 Let w ∈ Wα , α > −d and ε > 0. Then the function  0 if |x| ≤ 1, fp,ε (x) = d+α |x|− p −ε if |x| ≥ 1,


belongs to Lpw Rd and kfp,ε kp,w =

w(Sd )
1/p . pε

Since the proof of the lemma is straightforward, we omit it. Now, let ε be an arbitrary positive number and for each k = 1, · · · , m we set εk = ppεk and  0 if |x| ≤ 1, fpk ,εk (x) = d+αk − −ε k |x| pk if |x| ≥ 1.
Lemma 3.2 implies that fpk ,εk ∈ Lpωkk Rd and 1/pk  1/pk  ωk (Sd ) ωk (Sd ) kfpk ,εk kpk ,ωk = = > 0, pk ε k pε

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for each k = 1, · · · , m. For each x ∈ Rd which |x| ≥ 1, let m \

Sx =

{t ∈ [0, 1]n : |sk (t)x| > 1}.

k=1

From the assumption |sk (t1 , · · · , tn )| ≥ min{tβ1 , · · · , tβn } a.e. t = (t1 , · · · , tn ) ∈ [0, 1]n , there  n exists a null subset E of [0, 1]n so that Sx contains 1/|x|1/β , 1 \ E. From (2.1), we have m,n kUψ, (fp1 ,ε1 , · · · , fpm ,εm ) kpLpω (Rd ) → − s p Y  Z Z m = fk (sk (t)x) ψ(t)dt ω(x)dx Rd [0,1]n k=1 Z p Z Y m m Y d+α d+α − p k −εk − p k −εk k k ω(x) ψ(t)dt dx |sk (t)| = |x| d Sx k=1 R k=1 p Z m Z Y d+α − p k −εk −(d+α+εp) k ≥ |x| ω(x) ψ(t)dt dx |sk (t)| −β n |x|≥ε [ε,1] k=1 p ! Z m Z Y d+α − p k −εk pβε −(d+α+εp) k |sk (t)| =ε |x| ω(x)dx ψ(t)dt [ε,1]n |x|≥1 k=1 p   Z m Y d+αk ω(Sd ) − −εk = εpβε |sk (t)| pk ψ(t)dt [ε,1]n εp k=1 p Z m m Y Y d+αk − −ε k k ≥ εpβε |sk (t)| pk ψ(t)dt . kfpk ,ǫk kp/p pk ,εk [ε,1]n k=1

k=1

This implies that

m,n

Uψ,→ − s

p

p m d d Lω11 (Rd )×···×Lp ωm (R )→Lω (R )

≥ε

Notice that

βε

Z m Y d+α − p k −εk k |sk (t)| ψ(t)dt . [ε,1]n k=1

|sk (t)|−ε ≤ min{t1 , · · · , tn }−εβ ≤ ε−εβ → 1 when ε → 0+ ,

Thus, letting ε → 0+ and by Lebesgue’s dominated convergence theorem, we obtain

m,n ≥ A.

Uψ,→ − pm p s p1 Lω1 (Rd )×···×Lωm (Rd )→Lω (Rd )

Combine (3.6) and (3.7), we obtain the result.

(3.7) 

Analogous to the proof of Theorem 3.1, one can prove the following result. Theorem 3.3 Let s1 , · · · , sm : [0, 1]n → R be measurable functions such that for every k = 1, · · · , m then |sk (t1 , · · · , tn )| ≥ min{tβ1 , · · · , tβd } almost everywhere (t1 , · · · , tn ) ∈ [0, 1]n , → k = 1, · · · , m, for some β > 0. Let 1 ≤ pk < ∞, k = 1, · · · , m and − s = (s1 , · · · , sm ).
m,n − Assume that (ω1 , · · · , ωm ) satisfies W→ is bounded from Lpω11 Rd × − α condition. Then Hψ,→ s

· · · × Lpωm Rd to Lpω Rd if and only if m  Z Y m d+α − p k k A⋆ = |sk (t)| ψ(t)dt < ∞. (3.8) Rn +

Furthermore,



m,n

Hψ,→ − s

k=1

p

p m d d Lω11 (Rd )×···×Lp ωm (R )→Lω (R )

= A⋆ .

(3.9)

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Theorem 3.4 Let 1 ≤ p, pk < ∞, λ, αk , λk be real numbers such that − p1k ≤ λk < 0, d+αm 1 k = 1, · · · , m, 1p = p11 + · · · + p1m , and λ = d+α d+α λ1 + · · · + d+α λm . Let ωk ∈ Wαk for k = 1, · · · , m. (i) If in addition   1+λp  1+λp k pk m  p Y k ω(Sd ) ωk (Sd ) (3.10) ≥ d+α d + αk k=1

and

B=

Z

[0,1]n

Y m

−

|sk (t)|

(d+αk )λk pk

k=1

 ψ(t)dt < ∞,

(3.11)




m,n ,λm Rd to B˙ ωp,λ Rd . Furthermore, the then Uψ, is bounded from B˙ ωp11,λ1 Rd × · · · × B˙ ωpm → − m s m,n operator norm of Uψ, is not greater than B. → − s (ii) Conversely, if  λ λk m  Y ω(Sd ) ω(Sd ) 1/p 1/p (1 + λp) ≤ (1 + λk pk ) k (3.12) d+α d + αk k=1


m,n ,λm and Uψ, is bounded from B˙ ωp11,λ1 Rd × · · · × B˙ ωpm Rd to B˙ ωp,λ Rd , then B is finite. → − m s Furthermore, we have

m,n ≥ B. (3.13)

Uψ,→ − p pm ,λm s p1 ,λ1 B˙ ω1

(Rd )×···×B˙ ωm

(Rd )→Lω (Rd )

Let ωk ≡ 1, we have that αk = 0 thus (3.10) becomes equality and (3.12) holds only when λ1 p1 = · · · = λm pm . Hence, we obtain Theorem 2.1 in [14]. Proof Since p1 = p11 + · · · + p1m , by Minkowski’s inequality and H¨older’s inequality, we see that, for all balls B = B(0, R),  1/p Z p 1 m,n (f , · · · , f ) (x) ω(x)dx U → − 1 m ω(B)1+λp B ψ, s p !1/p Z Z Y m 1 ≤ fk (sk (t)x) ω(x)dx ψ(t)dt ω(B)1+λp B [0,1]n k=1 1/pk Z Z m  Y 1 pk ω (x)dx ψ(t)dt |f (s (t)x)| ≤ k k k 1+λk pk B [0,1]n k=1 ωk (B) 1/pk  Y  Z Z m  m Y 1 pk (d+αk )λk ≤ |fk (y)| ωk (y)dy |sk (t)| ψ(t)dt |sk (t)B|1+λk pk B [0,1]n k=1

≤

m Y

k=1

kfk kB˙ pk ,λk (Rd ) · B.

k=1

This means that

ωk



m,n

Uψ,→ − s

p ,λ pm ,λm p,λ B˙ ω11 1 (Rd )×···×B˙ ω (Rd )→B˙ ω (Rd ) m

≤ B.

(3.14)


m,n ,λm Now we assume that (3.12) holds and Uψ,s is bounded from B˙ ωp11,λ1 Rd × · · · × B˙ ωpm m
p,λ d (d+α )λ k k to B˙ ω R . Let fk (x) = |x| . Then an elementary computation shows that fk ∈
pk ,λk d ˙ Bωk R and λ  d + αk k −1/pk kfk kB˙ pk ,λk (Rd ) = (1 + λk pk ) . ωk ω(Sd )

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Thus, m Y

kfk kB˙ pk ,λk (Rd ) =

k=1

ωk

This leads us to m,n kUψ, (f1 , · · · → − s

Therefore,

λ m  Y d + αk k ω(Sd )

k=1

, fm )kB˙ ωp,λ (Rd ) = B ·

m,n

Uψ,→ − s ˙ p1 ,λ1 Bω1



−1/pk

(1 + λk pk )

d+α ω(Sd )

λ

≤



−1/p

(1 + λp)

d+α ω(Sd )

≥B·

λ

m Y

(1 + λp)−1/p .

kfk kB˙ pk ,λk (Rd ) .

k=1

pm ,λm d (Rd )×···×B˙ ω (Rd )→Lp ω (R ) m

ωk

≥ B. 

From the proof of Theorem 3.4, we also obtain the similar result to

m,n Hψ,s

operator.

Theorem 3.5 Let 1 ≤ p, pk < ∞, λ, αk , λk be real numbers such that − p1k ≤ λk < 0, d+αm 1 k = 1, · · · , m, 1p = p11 + · · · + p1m , and λ = d+α d+α λ1 + · · · + d+α λm . Let ωk ∈ Wαk for k = 1, · · · , m. (i) If in addition  1+λp  1+λp k pk  m  p Y k ωk (Sd ) ω(Sd ) ≥ (3.15) d+α d + αk k=1

and

B⋆ =

Z

Rn +

Y m

−

|sk (t)|

(d+αk )λk pk

k=1

 ψ(t)dt < ∞,

(3.16)




m,n ,λm then Hψ, is bounded from B˙ ωp11,λ1 Rd × · · · × B˙ ωpm Rd to B˙ ωp,λ Rd . Furthermore, the → − m s m,n operator norm of Hψ, is not greater than B⋆ . → − s (ii) Conversely, if  λ λk m  Y ω(Sd ) ω(Sd ) 1/p 1/p (3.17) (1 + λp) ≤ (1 + λk pk ) k d+α d + αk k=1


m,n ,λm and Hψ, is bounded from B˙ ωp11,λ1 Rd × · · · × B˙ ωpm Rd to B˙ ωp,λ Rd , then B⋆ is finite. → − m s Furthermore, we have

m,n ≥ B⋆ . (3.18)

Hψ,→ − p pm ,λm s p1 ,λ1 d d d B˙ ω1

4

(R )×···×B˙ ωm

(R )→Lω (R )

Commutators of Weighted Multilinear Hardy-Ces` aro Operator

We use some analogous tools to study a second set of problems related now to multilinear versions of the commutators of Coifman, Rochberg and Weiss [9]. The boundedness of commutators of weighted Hardy operators on the Lebesgue spaces, with symbols in BMO space, have been studied in [13] by Fu, Liu and Lu. Their main idea is to control the commutators of weighted Hardy operators by the weighted Hardy-Littlewood maximal operators. However this method is not easy to extend to deal with the multilinear case, since the maximal operators which are suitable to control commutator in this case are less known. In [14], the authors give an idea to replace Lebesgue spaces with central Morrey spaces and symbols are taken in central BMO spaces. By this method, we can avoid to use multilinear maximal operators.

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Our purpose of this section is to apply that method in order to characterize weight functions m,n such that the multilinear commutator generated by Uψ, , with symbols in central BMO spaces, → − s are bounded from the product of central Morrey spaces into central Morrey spaces. Definition 4.1 Let m, n ∈ N, ψ : [0, 1]n → [0, ∞), s1 , · · · , sm : [0, 1]n → R, b1 , · · · , bm be locally integrable functions on Rd and f1 , · · · , fm : Rd → C be measurable functions. The m,n commutator of weighted multilinear Hardy-Ces`aro operator Uψ, is defined as → − s Y  Y  Z m m → − m,n, b Uψ,s (f1 , · · · , fm ) (x) := fk (sk (t)x) (bk (x) − bk (sk (t)x)) ψ(t)dt. (4.1) [0,1]n

k=1

k=1

In what follows, we set

C=

Z

[0,1]n

D=

Z

[0,1]n

Y m

Y m

k=1

 |sk (t)|(d+αk )λk ψ(t)dt,  Y m

|sk (t)|(d+αk )λk

k=1

k=1

(4.2)

 |log |sk (t)|| ψ(t)dt.

(4.3)

To state our result, let us introduce some notations. The symbol f . g means that f ≤ Cg. We denote p, p1 , · · · , pm , q1 , · · · , qm , λ1 , · · · , λm and α1 , · · · , αm by real numbers such that 1 < p < pk < ∞, 1 < qk < ∞, − p1k < λk < 0, α1 , · · · , αm > −d and   m m m X X 1 1 X 1 1 d + αk = ,α= pαk + λk . , and λ = p pk pk qk d+α k=1

k=1

k=1

We shall consider weight functions ωk ∈ Wαk , k = 1, · · · , m, such that  m  Y ωk (Sd )

k=1

where ω is determined by ω =

1+λk pk pk

+ q1

k

d + αk

m Q

k=1

p/pk +p/qk

ωk

.



ω(Sd ) d+α

 1+λp p

,

(4.4)

. → −

m,n, b Our main result on the commutator Uψ,s is as follows.

Theorem 4.2 (i) If both C and D are finite then for any b = (b1 , · · · , bm ) ∈ CMOqω11 × → − CMOqm then U m,n, b is bounded from B˙ p1 ,λ1 (Rd ) × · · · × B˙ pm ,λm (Rd ) to B˙ p,λ (Rd ). ωm

ψ,s

ω1

CMOqω11 d

ωm

CMOqωmm ,

× ··· × (ii) If for any b = (b1 , · · · , bm ) ∈ ,λm d ˙ ωp,λ (R ), then D is finite. (R ) to B B˙ ωp11,λ1 (Rd ) × · · · × B˙ ωpm m

ω → − m,n, b Uψ,s is

bounded from

We note here that D is finite is not enough to imply C is finite (see [13, 14]). But it is easy to see that, if we assume in addition that for each k = 1, · · · , m such that |sk (t)| ≥ c > 1 for all t ∈ [0, 1]n or |sk (t)| ≤ c < 1 for all t ∈ [0, 1]n , then C is finite if and only if D is finite. Thus, Theorem 4.2 implies immediately that Corollary 4.3 Suppose that for each k = 1, · · · , m then |sk (t)| ≥ c > 1 for all t ∈ [0, 1]n → −

m,n, b or |sk (t)| ≤ c < 1 for all t ∈ [0, 1]n , for each k = 1, · · · , m. Then, Uψ,s is bounded from p1 ,λ1 d pm ,λm d p,λ d ˙ ˙ ˙ Bω1 (R ) × · · · × Bωm (R ) to Bω (R ) if and only if C is finite.

Now we will give the proof of Theorem 4.2. Proof First we assume that C is finite. We shall give a details analysis for the case m = 2 and by similarity, the general case is proved in the same way. We denote B = B(0, R) for short.

` H.D. Hung & L.D. Ky: MULTILINEAR OPERATORS OF HARDY-CESARO TYPE

No.6

1421

→ − Let b = (b1 , b2 ) ∈ CMOqω11 (Rd ) × CMOqω22 (Rd ). By Minkowski’s inequality, we have  1/p Z p 1 2,n,b Uψ,s (f1 , f2 ) (x) ω(x)dx 1+λp ω (B) B Y   Z Z 2 1 |fk (sk (t)x)| ≤ 1+λp ω (B) B [0,1]n k=1 Y  p 1/p 2 · |bk (x) − bk (sk (t)x)| ψ(t)dt ω(x)dx k=1

≤

Z

[0,1]n

·

2 Y

k=1



1 ω (B)

1+λp

Z Y 2 B

|fk (sk (t)x)|

k=1

p 1/p |bk (x) − bk (sk (t)x)| ω(x) dx ψ(t)dt

=: I.

For any xi , yi , zi , ti ∈ C with i = 1, 2, we have the following elementary inequality 2 Y

|(xi − yi )| ≤

i=1

2 Y

|(xi − zi )| +

i=1

2 Y

|(yi − ti )| +

i=1

2 Y

|(zi − ti )|

i=1

+ (|(x1 − z1 )(z2 − t2 )| + |(x2 − z2 )(z1 − t1 )|) + (|(x1 − z1 )(y2 − t2 )| + |(x2 − z2 )(y1 − t1 )|)

+ (|(z1 − t1 )(y2 − t2 )| + |(z2 − t2 )(y1 − t1 )|) . R 1 bi (x)ω(x)dx for i = 1, 2. Now applying It is convenient to denote by bi,ω,B the integrals B ω(B) the inequality with xi = bi (x), yi = bi (si (t)x), zi = bi,B , ti = bi,si (t)B and using Minkowski’s inequality, we get that I ≤ I1 + I2 + I3 + I4 + I5 + I6 , where, if we set f (x) =

2 Q

|fk (sk (t)x)|, then

k=1

I1 =

Z



Z



Z



[0,1]n

I2 =

[0,1]n

I3 =

[0,1]n

I4 =

Z

[0,1]n



1

1+λp

ω (B)

1 ω (B)1+λp 1 ω (B)1+λp 1 1+λp

ω (B)

p 1/p Z  2 Y f (x) · |bk (x) − bk,ωk ,B | ω(x)dx ψ(t)dt, B

k=1

 1/p Z  2 Y p f (x) · bk (sk (t)x) − bk,ωk ,sk (t)B ω(x)dx ψ(t)dt, B

k=1

 1/p Z  2 Y p f (x) · bk,ωk ,B − bk,ωk ,sk (t)B ω(x)dx ψ(t)dt, B

Z  f (x)

k=1

B

 1/p X
p · (bi (x) − bi,B ) bj,B − bj,ωj ,sj (t)B ω(x)dx ψ(t)dt, i6=j i,j=1,2

I5 =

Z

[0,1]n



1

1+λp

ω (B)

Z  f (x) B

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 1/p X
p (bi (x) − bi,B ) bj (sj (t)x) − bj,ω ,s (t)B ω(x)dx ψ(t)dt, j j

·

i6=j i,j=1,2

I6 =

Z



[0,1]n

1

1+λp

ω (B)

Z  f (x) B

 1/p X
p
bi,B − bi,s (t)B bj (sj (t)x) − bj,ω ,s (t)B ω(x)dx ψ(t)dt. · j j i i6=j i,j=1,2

Choose now p < s1 < ∞ and p < s2 < ∞ such that s11 = p11 + q11 and s12 = that s11 + s12 = p1 . Then by H¨ older’s inequality and (4.4), we deduce that I1 .

Z

2 Y

Z

1

1 p2

+

1 q2 .

!1/pk

pk

|fk (sk (t)x)| ωk (x)dx ωk (B)1+λk pk B 1/qk Z 2  Y 1 qk |bk (x) − bk,ωk ,B | ωk (x) ψ(t)dt × ωk (B) B k=1 ! Z 2 2 2 Y Y Y (d+α )λ k k . kbk kCMOqωk kfk kB˙ pk ,λk × |sk (t)| ψ(t)dt. [0,1]n k=1

k

k=1

[0,1]n

k=1

k=1

Similarly to the estimate of I1 , we have that I2 .

2 Y

2 Y

kbk kCMOqωk

k

k=1

kfk kB˙ pk ,λk ×

k=1

Z

[0,1]n

2 Y

(d+αk )λk

|sk (t)|

k=1

!

ψ(t)dt.

Now we give the estimate for I3 . Applying H¨ older’s inequality, we get that !1/pk Z Z 2 Y 1 pk |fk (y)| ωk (y)dy I3 . 1+p λ ωk (sk (t)B) k k sk (t)B [0,1]n k=1 ×

≤

2 Y

(d+αk )λk

λk

|sk (t)|

k=1 2 Y

ωk (B)

2 Y bk,ω ,B − bk,ω ,s (t)B ψ(t)dt k k k

k=1

kfk kB˙ pk ,λk

k=1

2 Y

Z

|sk (t)|(d+αk )λk

[0,1]n k=1

n

2 Y bk,ω ,B − bk,ω ,s (t)B ψ(t)dt. k k k

k=1

Notice that [0, 1] is the union of pairwise disjoint subsets Sℓ1 ,ℓ2 , where Sℓ1 ,ℓ2 =

2 \ 

i=1

Thus we obtain that I3 .

2 Y

kfk kB˙ pk ,λk

k=1

t ∈ [0, 1]n : 2−ℓi −1 < |si (t)| ≤ 2−ℓi .

∞ X

Z

2 Y

∞ X

Z

2 Y

ℓ1 ,ℓ2 =0

|sk (t)|(d+αk )λk ψ(t)

Sℓ1 ,ℓ2 k=1

  ℓk 2 Y X bk,ω ,2−j−1 B − bk,ω ,2−j B + b  dt  × k,ωk ,2−ℓk −1 B − bk,ωk ,sk (t)B k k k=1

.

2 Y

k=1

j=0

kfk kB˙ pk ,λk

ℓ1 ,ℓ2 =0

Sℓ1 ,ℓ2 k=1

|sk (t)|(d+αk )λk

2 Y

k=1

(ℓk + 2) kbk kCMO ˙ qk ψ(t)dt ωk

Notice

` H.D. Hung & L.D. Ky: MULTILINEAR OPERATORS OF HARDY-CESARO TYPE

No.6

2 Y

.

2 Y

kfk kB˙ pk ,λk

k=1

kbk kCMO ˙ qk

ωk

k=1

2  Y |sk (t)|(d+αk )λk log

Z

[0,1]n k=1

4 |sk (t)|



1 s

=

Now we give the estimate for I4 . Similarly, we choose 1 < s < ∞ such that
bi,j (x) := (bi (x) − bi,B ) bj,B − bj,ωj ,sj (t)B .

1423

ψ(t)dt. 1 q1

+

1 q2 .

Let

Then Minkowski’s inequality and H¨ olders inequality show us that ! !p !1/p Z Z 2 X Y 1 I4 ≤ |fk (sk (t)x)| · bi,j (x) ω(x)dx ψ(t)dt ω (B)1+λp B [0,1]n i6=j k=1 i,j=1,2

.

Z

2  Y

[0,1]n k=1



1 ωk (B)1+λk pk

  X × i6=j i,j=1,2

.

Z

2 Y

1 ωi (B)

Z

B

(d+αk )λk

|sk (t)|

  X × i6=j i,j=1,2

.

2 Y



  X × i6=j i,j=1,2

B

2 Y

1 ωi (B)

Z

B

[0,1]n

1 ωi (B)

Z

|fk (x)|

ωk

ωk (x)dx

|sk (t)|(d+αk )λk ψ(t)

k=1

Z

B

 1/s s bj,B − bj,ω ,s (t)B  |bi (x) − bi,B | ωi (x)dx  dt. j j

k

k=1

!1/pk

sk (t)B

From the estimates of I1 and I3 , we deduce that Z 2 2 2 Y Y Y I4 . kfk kB˙ pk ,λk kbk kCMO |sk (t)|(d+αk )λk ˙ qk ω k=1

pk

 1/s bj,B − bj,ω ,s (t)B  |bi (x) − bi,B |s ωi (x)dx  ψ(t)dt j j

2 Y

Z

1/pk ωk (x)dx

1 ωk (B)1+λk pk

k=1

kfk kB˙ pk ,λk

k=1

pk

|fk (sk (t)x)|

 1/s s bj,B − bj,ω ,s (t)B  |bi (x) − bi,B | ωi (x)dx  j j

[0,1]n k=1



Z

1+

[0,1]n k=1

2 X

2 log |sk (t)|

!

ψ(t)dt.

2 X

2 log |sk (t)|

!

ψ(t)dt.

k=1

It can be deduced from the estimates of I1 , I2 , I3 , I4 that Z 2 2 2 Y Y Y I5 . kfk kB˙ pk ,λk kbk kCMO |sk (t)|(d+αk )λk ψ(t)dt, ˙ qk ω ωk

k=1

I6 .

2 Y

kfk kB˙ pk ,λk

k=1

ωk

k

k=1 2 Y

kbk kCMO ˙ qk

k=1

ωk

[0,1]n k=1

Z

2 Y

(d+αk )λk

|sk (t)|

[0,1]n k=1

1+

k=1

Combining the estimates of I1 , I2 , I3 , I4 , I5 and I6 gives !1/p Z p 1 2,n,b Uψ,s (f1 , f2 ) (x) ω(x)dx 1+λp ω (B) B Z 2 2 2  Y Y Y q . kfk kB˙ pk ,λk kbk kCMO |sk (t)|(d+αk )λk log k ˙ k=1

This proves (i).

k=1

ωk

[0,1]n k=1

4 |sk (t)|



ψ(t)dt.

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ACTA MATHEMATICA SCIENTIA

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2,n,b Now we prove the necessity in (ii). Assume that Uψ,s is bounded from B˙ ωp11,λ1 (Rd ) × ,λm B˙ ωpm (Rd ) to B˙ ωp,λ (Rd ). First, we need the following lemmas. m

Lemma 4.4 ([7, Lemma 2.3]) If ω ∈ Wα and has doubling property, then log |x| ∈ BM O(ω). Lemma 4.5 Let ω ∈ Wα , where α > −d, 1 < p < ∞ and − p1 < λ. Then the function f0 (x) = |x|(d+α)λ belongs to B˙ ωp,λ and  1/p 1 −λ kf kB˙ ωp,λ = (ω(Sd )) . (d + α)(1 + λp) Since the proof of Lemma 4.5 is straightforward, we omit it. Now we set b1 (x) = b2 (x) =

˙ qk Rd . Define log |x|. Lemma 4.4 gives b1 , b2 belong to BMOω Rd , thus b1 , b2 ∈ CMO ωk fk (x) = |x|(d+αk )λk if x ∈ Rd \ {0} and fk (0) = 0. From Lemma 4.5, we get that  1/pk 1 −λk kfk kB˙ pk ,λk = (ωk (Sd )) . ωk (d + αk )(1 + λk pk ) Also we have 2,n,b Uψ,s (f1 , f2 ) (x)

=

2 Y

(d+αk )λk

|x|

k=1

Z

2  Y |sk (t)|(d+αk )λk log

[0,1]n k=1

1 |sk (t)|



dt.

Let B = B(0, R) be any ball of Rd , then  1/p Z p 1 2,n,b (f1 , f2 ) (x) ω(x)dx U ω(B)1+λp B ψ,s !  1/p Z  Z 2  Y 1 1 (d+α)λp ≤ |x| ω(x)dx |sk (t)|(d+αk )λk log ψ(t)dt ω(B)1+λp B |sk (t)| [0,1]n k=1 !  1/p Z  2  Y 1 1 −λ (d+αk )λk = (ω(Sd )) |sk (t)| log ψ(t)dt (d + α)(1 + λp) |sk (t)| [0,1]n k=1 !  Z 2 2  Y Y 1 (d+αk )λk = kfk kB˙ pk ,λk ψ(t)dt . |sk (t)| log ωk |sk (t)| [0,1]n k=1

k=1

Taking the supremum over R > 0, we get Z 2  Y |sk (t)|(d+αk )λk log [0,1]n k=1

1 |sk (t)|

which ends the proof of Theorem 4.2.



ψ(t)dt < ∞, 

Acknowledgements The authors would like to thank Prof. Nguyen Minh Chuong for many helpful suggestions and discussions. References [1] Adams D R. A note on Riesz potentials. Duke Math J, 1975, 42(4): 765–778 [2] Alvarez J, Lakey J, Guzm´ an-Partida M. Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures. Collect Math, 2000, 51(11): 147 [3] Andersen K F. Boundedness of Hausdorff operator on Lp (Rn ), H 1 (Rn ) and BMO(Rn ). Acta Sci Math (Szeged), 2003, 69(1/2): 409–418 [4] Bradley J S. Hardy inequalities with mixed norms. Canad Math Bull, 1978, 21(4): 405–408

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