Multilinear Hausdorff operators on some function spaces with variable

MULTILINEAR HAUSDORFF OPERATORS ON SOME FUNCTION SPACES WITH VARIABLE EXPONENT

arXiv:1709.08185v1 [math.CA] 24 Sep 2017

NGUYEN MINH CHUONG, DAO VAN DUONG, AND KIEU HUU DUNG Abstract. The aim of the present paper is to give necessary and sufficient conditions for the boundedness of a general class of multilinear Hausdorff operators that acts on the product of some weighted function spaces with variable exponent such as the weighted Lebesgue, Herz, central Morrey and Morrey-Herz type spaces with variable exponent. Our results improve and generalize some previous known results.

1. Introduction The one dimensional Hausdorff operator is defined by   Z∞ x Φ(y) dy, f HΦ (f )(x) = y y 0

where Φ is an integrable function on the positive half-line. The Hausdorff operator may be originated by Hurwitz and Silverman [25] in order to study summability of number series (see also [24]). It is well known that the Hausdorff operator is one of important operators in harmonic analysis, and it is used to solve certain classical problems in analysis. It is worth pointing out that if the kernel function Φ is taken appropriately, then the Hausdorff operator reduces to many classcial operators in analysis such as the Hardy operator, the Ces`aro operator, the Riemann-Liouville fractional integral operator and the Hardy-Littlewood average operator (see, e.g., [2], [16], [21], [34] and references therein). In 2002, Brown and M´oricz [5] extended the study of Hausdorff operator to the high dimensional space. Given Φ be a locally integrable function on Rn , the n-dimensional Hausdorff operator HΦ,A associated to the kernel function Φ is then defined in terms of the integral form as follows Z Φ(t) HΦ,A (f )(x) = f (A(t)x)dt, x ∈ Rn , (1.1) |t|n Rn

2010 Mathematics Subject Classification. Primary 42B30; Secondary 42B20, 47B38. Key words and phrases. Multilinear operator, Hausdorff operator, weighted HardyLittlewood operator, Lebesgue space, Morrey-Herz space, variable exponent. 1

MULTILINEAR HAUSDORFF OPERATOR

2

where A(t) is an n × n invertible matrix for almost everywhere t in the support of Φ. It should be pointed out that if we take Φ(t) = |t|n ψ(t1 )χ[0,1]n (t) and A(t) = t1 .In (In is an identity matrix), for t = (t1 , t2 , ..., tn ), where ψ : [0, 1] → [0, ∞) is a measurable function, HΦ,A then reduces to the weighted HardyLittlewood average operator due to Carton-Lebrun and Fosset [8] defined as the following Z1 Uψ (f )(x) = f (tx)ψ(t)dt, x ∈ Rn . (1.2) 0

n

Similarly, by taking Φ(t) = |t| ψ(t1 )χ[0,1]n (t) and A(t) = s(t1 ).In , with s : [0, 1] → R being a measurable function, it is easy to see that HΦ,A reduces to the weighted Hardy-Ces`aro operator Uψ,s defined by Chuong and Hung [12] as follows Z1 Uψ,s (f )(x) = f (s(t)x)ψ(t)dt, x ∈ Rn . (1.3) 0

In recent years, the theory of weighted Hardy-Littlewood average operators, Hardy-Ces`aro operators and Hausdorff operators have been significantly developed into different contexts (for more details see [5], [12], [13], [14], [35], [41] and references therein). Also, remark that Coifman and Meyer [9], [10] discovered a multilinear point of view in their study of certain singular integral operators. Thus, the research of the theory of multilinear operators is not only attracted by a pure question to generalize the theory of linear ones but also by their deep applications in harmonic analysis. In 2015, Hung and Ky [26] introduced the weighted multilinear Hardy-Ces`aro type operators, which are generalized of weighted multilinear Hardy operators in [21], defined as follows: Z Y m  m,n Uψ,~s (f1 , ..., fm )(x) = fi (si (t)x) ψ(t)dt, x ∈ Rn , (1.4) [0,1]n

i=1

where ψ : [0, 1]n → [0, ∞), and s1 , ..., sm : [0, 1]n → R are measurable functions. By the relation between Hausdorff operator and Hardy-Ces`aro operator as mentioned above, we shall introduce in this paper a more general multilinear operator of Hausdorff type defined as follows. Definition 1.1. Let Φ : Rn → [0, ∞) and Ai (t), for i = 1, ..., m, be n × n invertible matrices for almost everywhere t in the support of Φ. Given f1 , f2 , ..., fm : Rn → C be measurable functions, the multilinear Hausdorff operator HΦ,A~ is defined by Z m Φ(t) Y ~ HΦ,A~ (f )(x) = fi (Ai (t)x)dt, x ∈ Rn , (1.5) n |t| i=1 Rn

MULTILINEAR HAUSDORFF OPERATOR

3

~ = (A1 , ..., Am ). for f~ = (f1 , ..., fm ) and A It is obvious that when Φ(t) = |t|n .ψ(t)χ[0,1]n (t) and Ai (t) = si (t).In , where ψ : [0, 1]n → [0, ∞), s1, ..., sm : [0, 1]n → R are measurable functions, then the multilinear Hausdorff operator HΦ,A~ reduces to the weighted multilinear m,n Hardy-Ces`aro operator Uψ,~ s above. It is also interesting that the theory of function spaces with variable exponents has attracted much more attention because of the necessary in the field of electronic fluid mechanics and its important applications to the elasticity, fluid dynamics, recovery of graphics, and differential equations (see [3], [11], [15], [19], [27], [36], [18]). The foundational results and powerful applications of some function spaces with variable exponents in harmonic analysis and partial differential equations are given in the books [18], [20] and the references therein. It is well-known that the Calder´on-Zygmund singular operators, the Hardy-type operators and their commutators have been extensively investigated on the Lebesgue, Herz, Morrey, and Morrey-Herz spaces with variable exponent (see, e.g., [6], [7], [17], [22], [32], [33], [31], [37], [38], and others). Motivated by above mentioned results, the goal of this paper is to establish the necessary and sufficient conditions for the boundedness of multilinear Hausdorff operators on the product of weighted Lebesgue, central Morrey, Herz, and Morrey-Herz spaces with variable exponent. In each case, the estimates for operator norms are worked out. It should be pointed out that all results in this paper are new even in the case of linear Hausdorff operators. Our paper is organized as follows. In Section 2, we give necessary preliminaries on weighted Lebesgue spaces, central Morrey spaces, Herz spaces and Morrey-Herz spaces with variable exponent. Our main theorems are given and proved in Section 3. 2. Preliminaries Before stating our results in the next section, let us give some basic facts and notations which will be used throughout this paper. By kT kX→Y , we denote the norm of T between two normed vector spaces X and Y . The letter C denotes a positive constant which is independent of the main parameters, but may be different from line to line. Given a measurable set Ω, let us denote by χΩ its characteristic function, by |Ω| its Lebesgue measure, and by ω(Ω) the R integral ω(x)dx. For any a ∈ Rn and r > 0, we denote by B(a, r) the ball Ω

centered at a with radius r.

Next, we write a . b to mean that there is a positive constant C, independent of the main parameters, such that a ≤ Cb. The symbol f ≃ g means that f is equivalent to g (i.e. C −1 f ≤ g ≤ Cf ).

MULTILINEAR HAUSDORFF OPERATOR

4

 In what follows, we denote χ = χ , C = B \ B and B = x ∈ Rn : k C k k k−1 k k |x| ≤ 2k , for all k ∈ Z. Now, we are in a position to give some notations and definitions of Lebesgue, Herz, Morrey and Morrey-Herz spaces with constant parameters. For further information on these spaces as well as their deep applications in analysis, the interested readers may refer to the work [1] and to the monograph [30]. In this paper, as usual, we will denote by ω(·) a non-negative weighted function on Rn . Definition 2.1. Let 1 ≤ p < ∞, we define the weighted Lebesgue space Lp (ω) of a measurable function f by   p1 Z



f p =  |f (x)|p ω(x)dx < ∞, L (ω) Rn

and for p = ∞ by

 


f ∞ = inf M > 0 : ω x ∈ Rn : |f (x)| > M = 0 < ∞. L (ω)

Definition 2.2. Let α ∈ R, 0. < q < ∞, and 0 < p < ∞. The weighted α,p homogeneous Herz-type space K q (ω) is defined by n o . α,p q n . α,p (ω) = f ∈ L (R \ {0}, ω) : kf k < ∞ , Kq loc K q (ω) where kf k

. α,p K q (ω)

=



∞ P

2

k=−∞

kαp

kf χk kpLq (ω)

 1p

.

Definition 2.3. Let λ ∈ R and 1 ≤ p < ∞. The weighted central Mor. p,λ rey space B (ω) is defined by the set of all locally p-integrable functions f satisfying   p1 Z 1   kf k . p,λ (ω) = sup  |f (x)|p ω(x)dx < ∞. 1+λp B ω(B(0, R)) R >0 B(0,R)

Definition 2.4. Let α ∈ R, 0 < p < ∞, 0 < q < ∞, λ ≥ 0. The homogeneous . α,λ weighted Morrey-Herz type space M K p,q (ω) is defined by n o . α,λ M K p,q (ω) = f ∈ Lqloc (Rn \ {0}, ω) : kf kM . α,λ (ω) < ∞ , K p,q

where kf kM

. α,λ

K p,q (ω)

= sup 2 k0 ∈Z

−k0 λ



k0 P

k=−∞

2

kαp

kf χk kpLq (ω)

 p1

.

MULTILINEAR HAUSDORFF OPERATOR

5

. 0,p

Remark 1. It is useful to note that K p (Rn ) = Lp (Rn ) for 0 < p < ∞; . α/p,p

. α,0

K p (Rn ) = Lp (|x|α dx) for all 0 < p < ∞ and α ∈ R. Since M K p,q (Rn ) . α,p =K q (Rn ), it follows that the Herz space is a special case of Morrey-Herz space. Therefore, the Herz spaces are natural generalisations of the Lebesgue spaces with power weights. Now, we present the definition of the Lebesgue space with variable exponent. For further readings on its deep applications in harmonic analysis, the interested reader may find in the works [18], [19] and [20].

Definition 2.5. Let P(Rn ) be the set of all measurable functions p(·) : R → [1, ∞]. For p(·) ∈ P(Rn ), the variable exponent Lebesgue space Lp(·) (Rn ) is the set of all complex-valued measurable functions f defined on Rn such that there exists constant η > 0 satisfying p(x) Z 

|f (x)| dx + f /η L∞ (Ω∞ ) < ∞, Fp (f /η) := η Rn \Ω∞

where Ω∞

 = x ∈ Rn : p(x) = ∞ . When |Ω∞ | = 0, it is straightforward p(x) Z  |f (x)| dx < ∞. Fp (f /η) = η Rn

The variable exponent Lebesgue space Lp(·) (Rn ) then becomes a norm space equipped with a norm given by     f kf kLp(·) = inf η > 0 : Fp ≤1 . η

Let us denote by Pb (Rn ) the class of exponents q(·) ∈ P(Rn ) such that 1 < q− ≤ q(x) ≤ q+ < ∞, for all x ∈ Rn ,

where q− = ess infx∈Rn q(x) and q+ = ess supx∈Rn q(x). For p ∈ Pb (Rn ), it is useful to remark that we have the following inequalities which are usually used in the sequel. 1

 1 [i] If Fp (f ) ≤ C, then f Lp(·) ≤ max C q− , C q+ , for all f ∈ Lp(·) (Rn ), 1

 1 [ii] If Fp (f ) ≥ C, then f Lp(·) ≥ min C q− , C q+ , for all f ∈ Lp(·) (Rn ).(2.1) The space P∞ (Rn ) is defined by the set of all measurable functions q(·) ∈ P(Rn ) and there exists a constant q∞ such that q∞ = lim q(x). |x|→∞

p(·)

For p(·) ∈ P(Rn ), the weighted variable exponent Lebesgue space Lω (Rn ) is

MULTILINEAR HAUSDORFF OPERATOR

6

the set of all complex-valued measurable functions f such that f ω belongs to p(·) the Lp(·) (Rn ) space, and the norm of f in Lω (Rn ) is given by





f p(·) = f ω p(·) . Lω

L

n Let Clog older continuous functions α(·) satis0 (R ) denote the set of all log-H¨ fying at the origin Cα  0  , for all x ∈ Rn . |α(x) − α(0)| ≤ 1 log e + |x|

n Denote by Clog older continuous functions α(·) satisfying ∞ (R ) the set of all log-H¨ at infinity α C∞ |α(x) − α∞ | ≤ , for all x ∈ Rn , log(e + |x|)

Next, we would like to give the definition of variable exponent weighted Herz . α(·),p spaces K q(·),ω and the definition of variable exponent weighted Morrey-Herz . α(·),λ

spaces M K p,q(·),ω (see [31], [38] for more details). Definition 2.6. Let 0 < p < ∞, q(·) ∈ Pb (Rn ) and α(·) : Rn → R with . α(·),p

α(·) ∈ L∞ (Rn ). The variable exponent weighted Herz space K q(·),ω is defined by   . α(·),p q(·) n K q(·),ω = f ∈ Lloc (R \ {0}) : kf k . α(·),p < ∞ , K q(·),ω

where kf k . α(·),p = K q(·),ω



∞ P

k=−∞

k2

kα(·)

f χk k

p q(·)

Lω

 p1

.

Definition 2.7. Assume that 0 ≤ λ < ∞, 0 < p < ∞, q(·) ∈ Pb (Rn ) and α(·) : Rn → R with α(·) ∈ L∞ (Rn ). The variable exponent weighted Morrey. α(·),λ

Herz space M K p,q(·),ω is defined by  . α(·),λ q(·) M K p,q(·),ω = f ∈ Lloc (Rn \ {0}) : kf kM where kf k

. α(·),λ

M K p,q(·),ω

= sup 2−k0 λ k0 ∈Z



k0 P

k=−∞

. α(·),λ

K p,q(·),ω

k2kα(·) f χk kp q(·) Lω

 1p



0 ω B(0, R) λ+ p∞ 1



f χB(0,R) p(·) and ω1 , ω2 are non-negative and local where f Lp(·) = Lω ω (B(0,R)) 2

2

1

integrable functions. Moreover, as p(·) is constant and ω1 = ω and ω2 = ω p , . p(·),λ

. p,λ

it is natural to see that B ω,ω1/p = B

(ω).

Later, the next theorem is stated as an embedding result for the Lebesgue spaces with variable exponent (see, for example, Theorem 2 in [6], Theorem 2.45 in [18], Lemma 3.3.1 in [20]). Theorem 2.10. Let p(·), q(·) ∈ P(Rn ) and q(x) ≤ p(x) almost everywhere x ∈ Rn , and

1 1 1 := − and 1 Lr(·) < ∞. r(·) q(·) p(·) Then there exists a constant K such that







f q(·) ≤ K 1 r(·) f p(·) . Lω

L

Lω

3. Main results and their proofs

Before stating the next main results, we introduce some notations which will be used throughout this section. Let γ1 , ..., γm ∈ R, λ1 , ..., λm ≥ 0, p, pi ∈ n (0, ∞), qi ∈ Pb (Rn ) for i = 1, ..., m and α1 , ..., αm ∈ L∞ (Rn ) ∩ Clog 0 (R ) ∩ n Clog ∞ (R ). The functions α(·), q(·) and numbers γ, λ are defined by α1 (·) + · · · + αm (·) = α(·),

1 1 1 +···+ = , q1 (·) qm (·) q(·) γ1 + · · · + γm = γ, λ1 + λ2 + · · · + λm = λ.

MULTILINEAR HAUSDORFF OPERATOR

8

Thus, it is clear to see that the function α also belongs to the L∞ (Rn ) ∩ n log n Clog 0 (R ) ∩ C∞ (R ) space. For a matrix A = (aij )n×n , we define the norm of A as follow !1/2 n X kAk = |aij |2 .

(3.1)

i,j=1

As above we conclude |Ax| ≤ kAk |x| for any vector x ∈ Rn . In particular, if A is invertible, then we find

n (3.2) kAk−n ≤ det(A−1 ) ≤ A−1 . In this section, we will investigate the boundedness of multilinear Hausdorff operators on variable exponent Herz, central Morrey and Morrey-Herz spaces with power weights associated to the case of matrices having the important property as follows: there exists ρA~ ≥ 1 such that

≤ ρ ~ , for all i = 1, ..., m, (3.3) (t) kAi (t)k . A−1 i A

for almost everywhere t ∈ Rn . Thus, by the property of invertible matrice, it is easy to show that

−σ

, for all σ ∈ R, kAi (t)kσ . A−1 (t) (3.4) i and

−σ

.|x|σ , for all σ ∈ R, x ∈ Rn . |Ai (t)x|σ & A−1 i (t)

(3.5)

Remark 2. If A(t) = (aij (t))n×n is a real orthogonal matrix for almost everywhere t in Rn , then A(t) satisfies the property (3.3). Indeed, we know that the definition of the matrix norm (3.1) is the special case of Frobenius matrix norm. We recall the property of the above norm as follows p

√ p ρ(B ∗ .B) ≤ B ≤ n. ρ(B ∗ .B), for all B ∈ Mn (C), where B ∗ is the conjugate matrix of B and ρ(B) is the largest modulus of the eigenvalues of B. Thus, since A−1 (t) is also a real orthogonal matrix, we get

√

√ kA(t)k ≤ n and A−1 (t) ≤ n,

which immediately obtain the desired result. Now, we are ready to state our first main result in this paper. Theorem 3.1. Let ω1 (x) = |x|γ1 , ..., ωm (x) = |x|γm , ω(x) = |x|γ , q(·) ∈ Pb (Rn ), ζ > 0 and the following conditions are true:



(3.6) qi (A−1 i (t)·) ≤ ζ.qi (·) and 1 Lri (t,·) < ∞, a.e. t ∈ supp(Φ), Z m

Φ(t) Y

1 r (t,·) dt < ∞, c (t). (3.7) C1 = A ,q ,γ i i i L i |t|n i=1 Rn

MULTILINEAR HAUSDORFF OPERATOR

9

where

−γ

γi 1 1  

max det A−1 (t) qi+ , det A−1 (t) qi− , cAi ,qi,γi (t) = max Ai (t) i , A−1 i (t) i i 1 1 1 = − , for all i = 1, ..., m. −1 ri (t, ·) qi (Ai (t)·) ζqi (·) ζq (·)

Then, HΦ,A~ is a bounded operator from Lω11

ζq (·)

× · · · × Lωmm

q(·)

to Lω .

Proof. By using the versions of the Minkowski inequality for variable Lebesgue spaces from Corollary 2.38 in [18], we have Z m Y

Φ(t)

H ~ (f~) q(·) .

q(·) dt.

(3.8) f (A (t).) i Φ,A Lω Lω |t|n i=1 Rn

By assuming

m P

i=1

1 qi (·)

1 q(·)

=

and applying the H¨older inequality for variable

Lebesgue spaces (see also Corollary 2.28 in [18]), we imply that m

Y

fi (Ai (t).)

q(·)

Lω

i=1

Consequently, we obtain

H

~) q(·) . ( f ~ Φ,A Lω

Z

Rn

For η > 0, we see that ! Z fi (Ai (t).x) ωi (x) qi (x) dx η

m Y

fi (Ai (t).) qi (·) . . L ωi

i=1

m

Φ(t) Y

fi (Ai (t).) qi (·) dt. Lω i |t|n i=1

(3.9)

Rn

=

Z

Rn

!q (A−1 (t).z) fi (z) A−1 (t)z γi i i i det A−1 (t) dz i η

≤ det A−1 i (t) .

Z

Rn

≤

Z

Rn

! −1

γi 

, kAi (t)k−γi fi (z) ωi (z) qi (Ai (t).z) max A−1 (t) i dz η

!qi (A−1 i (t).z) cAi ,qi,γi (t). fi (z) ωi (z) dz. η

Hence, it follows from the definition of the Lebesgue space with variable exponent that



fi (Ai (t).) qi (·) ≤ cAi ,qi,γi (t). fi q (A−1 (t)·) . (3.10) i L i ωi

Lω i

MULTILINEAR HAUSDORFF OPERATOR

In view of (3.6) and Theorem 2.10, we deduce







f q (A−1 (t)·) . 1 r (t,·) . f Lωii

L

i

i

ζq (·)

Lωii

10

,

(3.11)

Therefore, by (3.9)-(3.11), we obtain

m Y



H ~ (f~) q(·) . C1 .

fi ζqi (·) , Φ,A Lω L ωi

i=1

which finishes the proof of this theorem.



In particular, when ζ ≤ 1, we have the following important result to the case of matrices having property (3.3) above. Theorem 3.2. Let us have the given supposition of Theorem 3.1 and assume that



qi (A−1 (3.12) i (t)·) ≤ qi (·), 1 Lr1i (t,·) < ∞, 1 1 1 where = , a.e. t ∈ supp(Φ), for all i = 1, ..., m. − −1 r1i (t, ·) qi (Ai (t)·) qi (·) (a) If Z m n o

n

n

Φ(t) Y

A−1 (t) qi+ +γi , A−1 (t) qi− +γi 1 r (t,·) dt < ∞, C2 = max i i L 1i |t|n i=1 Rn

then

m Y



fi qi (·) .

H ~ (f~) q(·) . C2 Φ,A L L ω

ωi

i=1

(b) Let C2∗

=

Z

Rn

m o n

n

n Φ(t) Y

A−1 (t) qi+ +γi , A−1 (t) qi− +γi dt. min i i |t|n i=1 q (·)

q (·)

Assume that HΦ,A~ is a bounded operator from Lω11 × · · · × Lωmm the following condition is satisfied: 1 1 1 1 + +···+ = . q1− q2− qm− q+ Then, C2∗ is finite. Moreover,

H ~

q(·) (·) Φ,A Lqω1 (·) ×···×Lqωm m →Lω 1

q(·)

to Lω

and

(3.13)

& C2∗ .

Proof. We begin with the proof for the case (a). From (3.12), by using the Theorem 2.10 again, we have −1







f q (A−1 (t).) . 1 r (t,·) . f qi (.) , for all f ∈ Lqωii(Ai (t).) . (3.14) 1i i L L i Lω i

ωi

MULTILINEAR HAUSDORFF OPERATOR

11

On the other hand, by (3.2) and (3.4), for i = 1, 2, ..., m, we find n

n +γi −1 n +γi o −1

. cAi ,qi,γi (t) . max Ai (t) qi+ , Ai (t) qi−

(3.15)

By the similar arguments as Theorem 3.1, by (3.14) and (3.15), we also obtain m Y



H ~ (f~) q(·) . C2

fi qi (·) . Φ,A Lω L i=1

ωi

Now, for the case (b), we make the functions fi for all i = 1, ...m as follows: ( 0, if |x| < ρ−1 ~ , A n fi (x) = − q (x) −γi −ε , otherwise. |x| i

This immediately deduces that fi Lqi (·) > 0. Besides that, we also compute ωi

Z

Fqi (fi ωi ) =

|x|−n−εqi(x) dx =

A

=

′

r −εqi(rx )−1 dσ(x′ )dr

S n−1 ρ−1 ~

|x|≥ρ−1 ~

Z1

Z+∞ Z A

Z

r

−εqi (rx′ )−1

′

dσ(x )dr +

Z+∞ Z

′

r −εqi(rx )−1 dσ(x′ )dr.

1 S n−1

S n−1 ρ−1 ~ A

Thus, Fqi (fi ωi ) is dominated by Z1 Z

r

−1−εqi+

′

dσ(x )dr +

Z+∞ Z

1 S n−1

S n−1 ρ−1 ~ A

  εq r −1−εqi− dσ(x′ )dr . (ρA~ i+ −1)qi− +qi+ ε−1 .

From the above estimation, by (2.1), we get   1 −1



fi qi (·) . (ρεqi+ − 1)qi− + qi+ qi− ε qi− . ~ L A ωi

(3.16)

Next, let us denote two useful sets as follows m \  Sx = t ∈ Rn : |Ai (t)x| ≥ ρ−1 , ~ A i=1

and

 U = t ∈ Rn : Ai (t) ≥ ε, for all i = 1, ..., m . Then, we claim that

U ⊂ Sx , for all x ∈ Rn \ B(0, ε−1). (3.17)

Indeed, let t ∈ U. This leads that Ai (t) .|x| ≥ 1, for all x ∈ Rn \ B(0, ε−1 ). Therefore, it follows from applying the condition (3.3) that

−1

|x| ≥ ρ−1 , |Ai (t)x| ≥ A−1 (t) i ~ A

MULTILINEAR HAUSDORFF OPERATOR

12

which finishes the proof of the relation (3.17). Now, by letting x ∈ Rn \ B(0, ε−1 ) and using (3.17), we see that Z Z m m Φ(t) Y Φ(t) Y − q n(x) −γi −ε − n −γ −ε ~ |Ai (t)x| i dt ≥ |Ai (t)x| qi (x) i dt. HΦ,A~ (f )(x) ≥ n n |t| i=1 |t| i=1 U

Sx

Thus, by (3.5), we find m  Z Φ(t) Y

−1 n +γi +ε  − n −γ−mε

A (t) qi (x) HΦ,A~ (f~)(x) & dt |x| q(x) χRn \B(0,ε−1 ) (x). n i |t| i=1 U

(3.18) For convenience, we denote Z m m oY n

n

−1 ε mε

n Φ(t) Y

A (t) ε dt.

A−1 (t) qi+ +γi , A−1 (t) qi− +γi Γε = min i i i |t|n i=1 i=1 U

Hence, by (3.18), we arrive at

− n −γ−mε

−mε

H ~ (f~) q(·)

| · | q(·) n \B(0,ε−1 ) q(·) χ & ε Γ . ε R Φ,A Lω Lω

−mε

=: ε Γε . h q(·) , Lω

(3.19)

n

where h(x) = |x|− q(x) −γ−mε χRn \B(0,ε−1 ) . Next, we will prove the following result q+ −1



h q(·) & εmε q− .ε q+ . (3.20) Lω Indeed, for ε sufficiently small such that ε−1 > 1, we compute Z Z+∞ Z ′ Fq (hω) = |x|−n−mεq(x) dx = r −1−mεq(rx ) dσ(x′ )dr ε−1 S n−1

|x|≥ε−1

≥

Z+∞ Z

r −1−mεq+ dσ(x′ )dr & εmεq+ .ε−1 .

(3.21)

ε−1 S n−1

From this, by the inequality (2.1), we immediately obtain the inequality (3.20). By writing ϑ(ε) as q




ϑ ε =

ε m  Q

i=1

εq (ρA~ i+

mε q+ −

−1

.ε q+

− 1)qi− + qi+

then, by (3.19) and (3.20), we estimate

 q1

i−

ε

−1 qi−

,

m Y




H ~ f~ q(·) & ε−mε ϑ.Γε .

fi qi (·) . Φ,A Lω L i=1

ωi

(3.22)

MULTILINEAR HAUSDORFF OPERATOR

13

Note that, by lettting ε sufficiently small and t ∈ U, we find m Y

−1 ε mε

A (t) .ε ≤ ρε~ . 1. i A

(3.23)

i=1

By the relation (3.13), we get the limit of function ε−mε ϑ is a positive number when ε tends to zero. Thus, by (3.22), (3.23) and the dominated convergence theorem of Lebesgue, we also have Z m n

n +γi −1 n +γi o Φ(t) Y −1

min Ai (t) qi+ , Ai (t) qi− dt < ∞. |t|n i=1 Rn

which completes the proof of the theorem.



Next, we discuss the boundedness of the multilinear Hausdorff operators on the product of weighted Morrey-Herz spaces with variable exponent. Theorem 3.3. Let ω1 (x) = |x|γ1 ,..., ωm (x) = |x|γm , ω(x) = |x|γ , q(·) ∈ Pb (Rn ), λ1 , ..., λm , ζ > 0, and the hypothesis (3.6) in Theorem 3.1 hold. Suppose that for all i = 1, ..., m, we have αi (0) − αi∞ ≥ 0.

(3.24)

At the same time, let Z m n

λi −αi∞ o

λi −αi (0)

Φ(t) Y



× , Ai (t) C3 = cA ,q ,γ (t) 1 Lri (t,·) .max Ai (t) |t|n i=1 i i i Rn

×max

0 n X

2

r(λi −αi (0))

r=Θ∗n −1

,

0 X

r=Θ∗n −1

o 2r(λi −αi∞ ) dt < ∞,

where Θ∗n = Θ∗n (t) is the greatest integer number satisfying  −Θ∗n max kAi (t)k.kA−1 , for a.e. t ∈ Rn . i (t)k < 2

(3.25)

i=1,...,m

. α1 (·),λ1

. αm (·),λm

Then, HΦ,A~ is a bounded operator from M K p1 ,ζq1(·),ω1 × · · · × M K pm ,ζqm (·),ωm . α(·),λ

to M K p,q(·),ω . Proof. By estimating as (3.9) above, we have Z m

Φ(t) Y

fi (Ai (t).)χk qi (·) dt.

H ~ (f~)χk q(·) . Φ,A Lω Lω i |t|n i=1

(3.26)

Rn

 Let us now fix i ∈ 1, 2, ..., m . Since kAi (t)k = 6 0, there exists an integer number ℓi = ℓi (t) such that 2ℓi −1 < kAi (t)k ≤ 2ℓi . For simplicity of notation,

MULTILINEAR HAUSDORFF OPERATOR

we write ρ∗A~ (t) = max

i=1,...,m

14





Ai (t) . A−1 (t) . Then, by letting y = Ai (t).z i

with z ∈ Ck , it follows that ℓi +k−2

−1 ∗

|z| ≥ 2 > 2k+ℓi−2+Θn , |y| ≥ A−1 (t) i ∗ ρA~

and

|y| ≤ kAi (t)k . |z| ≤ 2ℓi+k . These estimations can be used to get  ∗ Ai (t).Ck ⊂ z ∈ Rn : 2k+ℓi−2+Θn < |z| ≤ 2k+ℓi .

(3.27)

Now, we need to prove that

fi (Ai (t).)χk

q (·) Lωii

0 X



fi χk+ℓ +r ζqi (·) . . cAi ,qi ,γi (t) 1 Lri (t,·) . i L ωi

r=Θ∗n −1

(3.28)

Indeed, for η > 0, by (3.27), we find ! Z fi (Ai (t)x) χk (x)ωi (x) qi (x) dx η Rn

=

Z

Ai (t)Ck

!q (A−1 (t)z) fi (z) A−1 (t)z γi i i i detA−1 (t) dz i η

≤ detA−1 i (t)

!q (A−1 (t).z)

γi



, Ai (t) −γi fi (z) ωi (z) i i max A−1 (t) i dz. η

Z

Ai (t)Ck

So, we have that !q (x) Z fi (Ai (t)x)χk (x) ωi (x) i dx η Rn



Z   ≤   Rn

cA ,q ,γ (t) i

i

i

qi (A−1 i (t).z) fi (z)χk+ℓi +r (z) ωi (z)  r=Θ∗n −1  dz.  η  0 P

Therefore, by the definition of Lebesgue space with variable exponent, it is easy to get that 0 X



fi χk+ℓ +r

fi (Ai (t).)χk qi (·) ≤ cA ,q ,γ (t). i i i i L ωi

r=Θ∗n −1

q (A−1 (t)·) i

Lωii

,

MULTILINEAR HAUSDORFF OPERATOR

15

which completes the proof of the inequalities (3.28), by (3.11). Now, it immediately follows from (3.26) and (3.28) that Z m



Φ(t) Y

1 r (t,·) ×

H ~ (f~)χk q(·) . c (t) A ,q ,γ i i i Φ,A L i Lω |t|n i=1 Rn

m  X 0 Y

f χk+ℓ +r ×

 ζq (·) dt. L i

i

i=1

r=Θ∗n −1

ωi

(3.29)

Consequently, by applying Lemma 2.8 in Section 2, we get Z m Y

Φ(t)

H ~ (f~)χk q(·) .

1 r (t,·) × L(t). c (t) A ,q ,γ i i i Φ,A Lω L i |t|n i=1 Rn

× where L(t) =

m  Y

2

0 X

(k+ℓi )(λi −αi (0))

i=1

2

r(λi −αi (0))

m Y



fi i=1

+2

. α (·),λi i i (·),ωi

M K p i,ζq

(k+ℓi )(λi −αi∞ )

r=Θ∗n −1

dt,

0 X

2

(3.30)

r(λi −αi∞ )

r=Θ∗n −1

 .

By having 2ℓi −1 < Ai (t) ≤ 2ℓi , for all i = 1, ..., m, it implies that

λ −α

λ −α (0)  2ℓi (λi −αi (0)) + 2ℓi (λi −αi∞ ) . max Ai (t) i i , Ai (t) i i∞ . Thus, we can estimate L as follows m n Y

λ −α (0)

λ −α o max Ai (t) i i , Ai (t) i i∞ × L(t) . i=1

0 0 o n X X r(λi −αi (0)) k(λi −αi∞ ) k(λi −αi (0)) 2r(λi −αi∞ ) 2 +2 × 2 r=Θ∗n −1

r=Θ∗n −1

.

m Y i=1

n

λi −αi (0)

λi −αi∞ o

× max Ai (t) , Ai (t)

×max

0 n X

2

r(λi −αi (0))

r=Θ∗n −1

,

0 X

2

r(λi −αi∞ )

r=Θ∗n −1

on o k(λi −αi (0)) k(λi −αi∞ ) 2 +2 .

From this, by (3.30), it is not difficult to show that m m Y


Y k(λi −αi (0)) k(λi −αi∞ )

H ~ (f~)χk q(·) . C3

fi 2 +2 . Φ,A Lω M i=1

i=1

. αi (·),λi

K pi ,ζqi (·),ωi

.

(3.31)

MULTILINEAR HAUSDORFF OPERATOR

16

On the other hand, using Proposition 2.5 in [31], we get



H ~ (f~) . α(·),λ . max sup E sup (E 1, 2 + E3 ) , Φ,A M K p,q(·),ω

k0 0, for all i = 1, ..., m and (3.24), we see at once that m Y n 2k0 λi p 2k0 (λi −αi∞ +αi (0))p o p1 −k0 λi p T0 . 2 + 1 − 2−λi p 1 − 2−(λi −αi∞ +αi (0))p i=1
 m m n Y 2k0 (−αi∞ +αi (0)) o Y  1 k0 αi (0)−αi∞ . . 1+2 + . 1 − 2−λi p 1 − 2−(λi −αi∞ +αi (0))p i=1 i=1

MULTILINEAR HAUSDORFF OPERATOR

Then, from (3.33), we have m  Y E1 . C3 1 + 2k0

αi (0)−αi∞

i=1


 Y m



fi . i=1

. α (·),λi i i (·),ωi

M K p i,ζq

By estimating in the same way as E1 , we also get m Y



fi . αi (·),λi E2 . C3 .2−k0 λ M

K pi ,ζqi (·),ωi

i=1

17

.

.

(3.34)

(3.35)

For i = 1, ..., m, we denote  1  2k0 (αi∞ −αi (0)) + 2λi p − 1 − p + 2−k0 λi , if λi + αi∞ − αi (0) 6= 0, Ki =  2−k0 λi (k + 1) p1 + 2λi p − 1 − p1 , otherwise. 0

Then, we may show that

E3 . C3

m Y i=1

m  Y



fi Ki . M i=1

. αi (·),λi

K pi ,ζqi (·),ωi

.

(3.36)

The proof of inequality (3.36) is not difficult, but for convenience to the reader, we briefly give here. By employing (3.31) again, we make m Y



fi E3 . C3 .T∞ . M i=1

where T∞ = 2

−k0 λ

P k0

2

kα∞ p

m Q

2

k(λi −αi (0))p

i=1

k=0

. αi (·),λi

K pi ,ζqi (·),ωi

+2

,

k(λi −αi∞ )p

(3.37)
 p1

. By a similar

argument as T0 , we also get m m k0 k0 X 1 Y Y X −k0 λi kλi p k(λi +αi∞ −αi (0))p p ≡ Ti,∞ . T∞ . 2 2 + 2 i=1

k=0

k=0

i=1

In the case λi + αi∞ − αi (0) 6= 0, we deduce that Ti,∞ is dominated by  k0 λi p − 1 2k0 (λi +αi∞ −αi (0))p − 1  p1 −k0 λi 2 Ti,∞ ≤ 2 + (λi +αi∞ −αi (0))p 2λi .p − 1 2 −1 λp −1/p k0 (αi∞ −αi (0)) −k λ . 2 + 2 i − 1 + 2 0 i.

Otherwise, we have  2k0 λi p − 1  p1 − 1 1 −k0 λi p + 2λi p − 1 p . Ti,∞ ≤ . 2 (k + 1) + (k + 1) 0 0 2λi .p − 1 m Q This leads that T∞ . Ki . Hence, by (3.37), we obtain the inequality (3.36). i=1

From (3.32) and (3.34)-(3.36), we conclude that the proof of Theorem 3.3 is finished. 

MULTILINEAR HAUSDORFF OPERATOR

18

Next, we will discuss the interesting case when λ1 = · · · = λm = 0. Remark that these special cases of variable exponent Morrey-Herz spaces are variable exponent Herz spaces. Hence, we also have the boundedness for the multilinear Hausdorff operators on the product of weighted Herz spaces with variable exponent as follows. Theorem 3.4. Suppose that we have the given supposition of Theorem 3.3 and αi (0) = αi∞ , for all i = 1, ..., m. Let 1 ≤ p, pi < ∞ such that 1 1 1 +···+ = . (3.38) p1 pm p At the same time, let Z m

1 Φ(t) Y C4 = (2 − Θ∗n )m− p n cAi ,qi ,γi (t) 1 Lri (t,·) × |t| i=1 Rn

−αi (0)

×kAi (t)k

0 X

r=Θ∗n −1


2−rαi (0) dt < ∞.

. α1 (·),p1

(3.39)

. αm (·),pm

. α(·),p

Then, HΦ,A~ is a bounded operator from K ζq1 (·),ω1 × · · · × K ζqm (·),ωm to K q(·),ω .

Proof. It follows from Proposition 3.8 in [4] that −1  X



p  p1 kα(0)p ~

H ~ f~ . α(·),p . 2 H f χk Lq(·) ~ Φ,A Φ,A K q(·),ω

ω

k=−∞

+

∞ X k=0

2


p  p1 ~ HΦ,A~ f χk Lq(·) .

kα∞ p

ω

From this, by α(0) = α∞ , we conclude that ∞  X


p  p1
kα(0)p ~

H ~ f~ . α(·),p . H f χk Lq(·) . 2 ~ Φ,A Φ,A K q(·),ω

ω

(3.40)

k=−∞

For convenience, let us denote by ∞  X


p  p1 kα(0)p ~ H= 2 HΦ,A~ f χk Lq(·) . ω

k=−∞

Next, we need to estimate the upper bound of H. By (3.29) and using the Minkowski inequality, we get Z m

Φ(t) Y

1 r (t,·) × H ≤ c (t) (3.41) A ,q ,γ L i |t|n i=1 i i i Rn

m  X ∞ 0 nX p o p1 Y kα(0)p × 2 kfi χk+ℓi +r kLζqi (·) dt. k=−∞

i=1

r=Θ∗n −1

ωi

MULTILINEAR HAUSDORFF OPERATOR

19

By (3.38) and the H¨older inequality, it follows that m  X ∞ 0 p o p1 nX Y kα(0)p 2 kfi χk+ℓi +r kLζqi (·) i=1

k=−∞

m n X ∞ Y

≤

i=1

2kαi(0)pi

k=−∞

(3.42)

ωi

r=Θ∗n −1

0  X

r=Θ∗n −1

1

kfi χk+ℓi +r kLζqi (·) ωi

pi o pi

.

On the other hand, by pi ≥ 1, we have 0 0  X pi X kfi χk+ℓi +r kLζqi (·) ≤ (2 − Θ∗n )pi−1 kfi χk+ℓi +r kpiζqi (·) . ωi

r=Θ∗n −1

Lω i

r=Θ∗n −1

Hence, combining (3.41) and (3.42), we obtain Z m m Y Y

∗ m− p1 Φ(t)

cA ,q ,γ (t) 1 Lri (t,·) . Hi dt, H ≤ (2 − Θn ) |t|n i=1 i i i i=1

(3.43)

Rn

1

where Hi =

0 P

r=Θ∗n −1

Then, we find

 P ∞

k=−∞

2kαi (0)pi kfi χk+ℓi +r kpiζqi (·) Lω i

 pi

for all i = 1, 2, ..., m. 1

Hi = =

0 X

r=Θ∗n −1 0 X

∞ X

t=−∞

2(t−ℓi −r)αi (0)pi kfi χt kpiζqi (·)

2−(ℓi +r)αi (0)

≤

r=Θ∗n −1

∞ X

t=−∞

r=Θ∗n −1 0 X

Lω i

2tαi (0)pi kfi χt kpiζqi (·) Lω i


−ℓ α (0) 2−rαi (0) .2 i i kfi k . αi (·),pi . K ζqi (·),ωi

Since 2ℓi−1 < kAi (t)k ≤ 2ℓi , we imply that 2 (3.43) and (3.44), we get

−ℓi αi (0)

which finishes our desired conclusion.

i=1

1

 pi (3.44)

. kAi (t)k−αi (0) . Thus, by

m Y




H ~ f~ . α(·),p . C4 . kfi k . αi (·),pi , Φ,A K q(·),ω

 pi

K ζqi (·),ωi



Remark 3. We would like to give several comments on Theorem 3.3 and Theorem 3.4. If we suppose that

ess sup Ai (t) < ∞, for all i = 1, ..., m, t∈supp(Φ)

MULTILINEAR HAUSDORFF OPERATOR

20

then we do not need to assume the conditions αi (0) − αi∞ ≥ 0 in Theorem 3.3 and αi (0) = αi∞ in Theorem 3.4. Indeed, by putting β=

ess sup

ℓi (t),

t∈supp(Φ) and i=1,...,m

and applying Lemma 2.8 in Section 2, we refine the estimation as follow: In the case k < β, we get



fi χk+ℓ +r ζqi (·) . 2(k+ℓi +r)(λi −αi (0)) fi . αi (·),λi . i L M K pi ,ζqi (·),ωi

ωi

Θ∗n

In the case k ≥ β − + 1, we have



fi χk+ℓ +r ζqi (·) . 2(k+ℓi +r)(λi −αi∞ ) fi i

. α (·),λi i i (·),ωi

M K p i,ζq

Lω i

Otherwise, then we obtain




fi χk+ℓi +r ζqi (·) . 2(k+ℓi +r)(λi −αi (0)) + 2(k+ℓi+r)(λi −αi∞ ) fi

.

. α (·),λi i i (·),ωi

M K p i,ζq

Lω i

.

Also, the other estimations can be done by similar arguments as two theorems above. From this we omit details, and their proof are left to reader. Theorem 3.5. Suppose that the given supposition of Theorem 3.3 and the hypothesis (3.12) in Theorem 3.2 are true. (a) If Z m n o

n

n

Φ(t) Y

A−1 (t) qi+ +γi , A−1 (t) qi− +γi A−1 (t) −λi × C5 = max i i i |t|n i=1 Rn n

αi (0) −1 αi∞ o −1

1 r (t,·) dt < ∞,

× max Ai (t) , Ai (t) L 1i then



H ~ (f~) Φ,A

. α(·),λ

M K p,q(·),ω

m Y



fi . C5 i=1

. α (·),λi i i (·),ωi

M K p i,q

.

(b) Denote by Z m o n

n

n Φ(t) Y ∗

A−1 (t) qi+ +γi , A−1 (t) qi− +γi A−1 (t) −λi × C5 = min i i i |t|n i=1 Rn n

αi (0)+C0αi −1 αi (0)−C0αi o

dt. × min A−1 (t) , Ai (t) i . α1 (·),λ1

. αm (·),λm

Suppose that HΦ,A~ is a bounded operator from M K p1 ,q1 (·),ω1 ×· · ·×M K pm ,qm (·),ωm . α(·),λ

to M K p,q(·),ω and one of the following conditions holds: αi (b1) qi+ = qi− , C0αi and C∞ ≤ αi (0) − αi∞ , for all i = 1, ..., m; αi αi (b2) qi+ 6= qi− , C0 = C∞ = 0, λi = αi (0) = αi∞ , for all i = 1, ..., m; αi αi (b3) qi+ 6= qi− , both C0αi and C∞ are less than αi (0)−αi∞ , C∞ +C0αi ≤ C αi , 0 1 0 1 λi ∈ [ηi , ηi ] ∩ [ζi , ζi ], for all i = 1, ..., m.

MULTILINEAR HAUSDORFF OPERATOR

Here C αi = ηi0 ζi0

qi− (αi (0) − αi∞ )(1 +

qi+ ) qi−

qi+

=

− αi (0) qqi− + αi∞ C0αi qqi− i+ i+

=

i+ αi qi+ − αi (0) + αi∞ qqi− C∞ qi−

1− qi+ qi−

qi− qi+

−1

21

and ηi0 , ηi1 , ζi0, ζi1 are defined by , ηi1 , ζi1

=

i+ i+ − αi (0) qqi− + αi∞ C0αi qqi−

,

=

αi qi− C∞ − αi (0) + αi∞ qqi− qi+ i+

.

1− qi− qi+

Then, we have that C5∗ is finite. Furthermore,

H ~ . α1 (·),λ1 . αm (·),λm

qi+ qi−

−1

& C5∗ .

. α(·),λ Φ,A M K p1 ,q1 (·),ω1 ×···×M K pm ,qm (·),ωm →M K p,q(·),ω

Proof. Firstly, we prove for the case (a). From (3.3), we call Θn is the greatest integer number such that ρA~ < 2−Θn . Now, we replace Θ∗n by Θn in the proof of Theorem 3.3, and the other results are estimated in the same way. Then, by (3.14), we get m  Z Φ(t) Y



H ~ (f~) . α(·),λ .

c (3.45) Ai ,qi ,γi (t) 1 Lr1i (t,·) × Φ,A n M K p,q(·),ω |t| i=1 Rn

m n

λi −αi (0)

λi −αi∞ o  Y





fi × max Ai (t) dt , Ai (t) M i=1

. αi (·),λi

K pi ,qi (·),ωi

.

By the inequality (3.4), we have n

λ −α (0)

λ −α o max Ai (t) i i , Ai (t) i i∞ n

αi (0) −1 αi∞ o

−1 −λi −1

. , Ai (t) max Ai (t) . Ai (t)

Thus, by (3.15), (3.45) and C3 < ∞, we finish the proof for this case. Next, we will prove for case (b). By choosing

it is evident that fi

fi (x) = |x|

show that



fi M

Indeed, we find Fqi (fi ωi .χk ) =

Z

Ck

. α (·),λi i i (·),ωi

M K p i,q

. αi (·),λi

K pi ,qi (·),ωi

n −γi +λi i (x)

−αi (x)− q

,

> 0, for all i = 1, ..., m. Now, we need to < ∞, for all i = 1, ..., m.

|x|(λi −αi (x))qi (x)−n dx =

Z2k Z

′

(3.46)

′

r (λi −αi (r.x ))qi (r.x )−1 dσ(x′ )dr.

2k−1 S n−1

n Case 1 : k ≤ 0. Since αi ∈ Clog ∞ (R ), it follows that

αi αi −C∞ + αi∞ ≤ αi (x) ≤ αi∞ + C∞ .

MULTILINEAR HAUSDORFF OPERATOR

22

As a consequence, we get Z2k Z

Fqi (fi ωi .χk ) ≤

αi

′

r (λi −αi∞ −C∞ )qi (r.x )−1 dσ(x′ )dr

2k−1 S n−1

 αi αi . max 2k(λi −αi∞ −C∞ )qi− , 2k(λi −αi∞ −C∞ )qi+ .

Thus, by (2.1), we obtain αi qi− αi qi+



fi χk qi (·) . max 2k(λi −αi∞ −C∞ ) qi+ , 2k(λi −αi∞ −C∞ ) qi− L ωi

αi

= 2k(λi −αi∞ −C∞ )βi∞ ,

where βi∞ =

 qi+ αi  , if λi − αi∞ − C∞ < 0,   qi−    qi− , otherwise. qi+

n Case 2 : k > 0. Since αi ∈ Clog 0 (R ), we have

Denote

− C0αi + αi (0) ≤ αi (x) ≤ αi (0) + C0αi .

βi0 =

(3.47)

(3.48)

 qi+  , if λi − αi (0) + C0αi ≥ 0,   qi−

   qi− , otherwise. qi+ By having (3.48) and estimating in the same way as the case 1, we deduce

α

fi χk qi (·) . 2k(λi −αi (0)+C0 i )βi0 . (3.49) L ωi

Next, it follows from Proposition 2.5 in [31] that



fi . αi (·),λ . max sup Ei,1 , sup M K pi ,qi (·),ωi

k0 log(2)A~ . Estimating as (3.18), we have m  Z Φ(t) Y

−1 αi (0)+ n +γi +ε  −α(0)− n −γ−mε qi ~

A (t) q dt |x| χRn \B(0,ε−1 ) (x). HΦ,A~ (f )(x) & i |t|n i=1 U

(3.56)

MULTILINEAR HAUSDORFF OPERATOR

26

Let k0 be the smallest integer number such that 2k0 −1 ≥ ε−1 . Using Proposition 3.8 in [4] again, αi (0) = αi∞ and (3.56), we obtain Z ∞   pq X


−εmq−α(0)q−n kα(0)p

H ~ f~ p. α(·),p & 2 |x| dx × Φ,A K q,ω

k=k0

2k−1 1, we get m ( kαi kL∞ + εm)q+ i=1

i=1

Fq (gω.χk ) =

Z

|x|

−(

m P

i=1

kαi kL∞ +mε)q(x)−n

dx

Ck

≥

Z2k Z

r

−(

m P

i=1

kαi kL∞ +mε)q+ −1

′

dσ(x )dr & η+ .2

−k(

m P

i=1

kαi kL∞ +mε)q+

.

2k−1 S n−1 m

P q 1

−k( kαi kL∞ +mε) q+ q − . This finishes the Thus, by (2.1), we have gχk Lq(·) & η++ .2 i=1 ω proof of the estimation (3.65).

MULTILINEAR HAUSDORFF OPERATOR

29

Now, we define 1 q+

η+ 2

k0 (α(0)−(

m P

i=1

q m kαi kL∞ +εm) q+ ) Q −

i=1

ϑ∗∗ (ε) =

m Q

i=1



q

1−2

pi (αi (0)−(kαi kL∞ +ε) qi− ) i+

m P q 1  α(0)−( kαi kL∞ +εm) q+  p − i=1 Ii (ε). 1 − 2

 p1

i

.

By (3.63)-(3.65), we estimate

H ~ (f~) . α(·),p Φ,A

&ε

K q(·),ω

m o  Z Φ(t) Y n

n

n

A−1 (t) qi+ +γi , A−1 (t) qi− +γi A−1 (t) kαi kL∞ ϑ . min i i i |t|n i=1

−mε ∗∗

U

m m Y



−1 ε mε  Y

fi . αi (·),pi .

Ai (t) ε dt . × i=1

i=1

(3.66)

K qi (·),ωi

Because of assuming αi (0) < kαi kL∞ qqi− , we have α(0) < i+

m P

αi

L∞

i=1

. From this,

the limit of function ε−mε ϑ∗∗ is a positive number when ε tends to zero. Therefore, by (3.23), (3.66) and the dominated convergence theorem of Lebesgue, we obtain m Y



H ~ (f~) . α(·),p & C6∗ .

fi . αi (·),pi , Φ,A

K q(·),ω

i=1

K qi (·),ωi

which ends the proof for this case.



When all of α1 (·),..., αm (·) and q1 (·),..., qm (·) are constant, we obtain the following useful result which is seen as an extension of Theorem 3.1 and Theorem 3.2 in the work [14] to the case of matrices having property (3.3) as mentioned above. Theorem 3.7. Let ω(x) = |x|γ , γ1 , ..., γm ∈ R, λ1 , ..., λm ∈ R+ , α1 , ..., αm ∈ R, 1 ≤ qi , q < ∞, 0 < pi , p < ∞ and ωi (x) = |x|γi for all i = 1, ..., m. Simultaneously, let γ γm γ1 +···+ . (3.67) = q q1 qm . α1 ,λ1

. αm ,λm

Then HΦ,A~ is a bounded operator from M K p1 ,q1 (ω1 ) × · · · × M K pm ,qm (ωm ) to . α,λ

M K p,q (ω) if and only if Z m

−λi +αi + n+γi Φ(t) Y −1 qi

dt < +∞, C7 = A (t) i |t|n i=1 Rn

Moreover,



H ~ Φ,A M

. α ,λ

. αm ,λm

. α,λ

1 1 K p1 ,q1 (ω1 )×···×M K pm ,qm (ωm )→M K p,q (ω)

≃ C7 .

MULTILINEAR HAUSDORFF OPERATOR

30

Proof. It is clear to see that the resuts of Theorem 3.7 can be viewed as consequence of Theorem 3.5. Indeed, we put γ ∗ = γq ,γi∗ = γqii for i = 1, ..., m and ∗ ∗ ω ∗ = |x|γ , ωi∗ = |x|γi for i = 1, ..., m. By having (3.67) and assuming that α1 (·),..., αm (·) and q1 (·),..., qm (·) are constant, we have Z m

−λi +αi + n +γi∗ Φ(t) Y −1 ∗ qi

A (t) C5 = C5 = dt = C7 . i |t|n i=1 Rn

Thus, combining the case (a) and case (b1) of Theore 3.5, we deduce

H ~ . α1 ,λ1 . αm ,λm . α,λ ≃ C7 . Φ,A M ×···×M ∗ →M ∗ K p1 ,q1 ,ω ∗ 1

K pm ,qm ,ωm

K p,q,ω

At this point, by relation (2.2), we immediately get the desired result.



As a consequence of Theorem 3.6, we also obtain the analogous result for the constant parameters case as follows. Theorem 3.8. Let 1 ≤ p, p1 , ..., pm < ∞, the assumptions of Theorem 3.7 and the hypothesis (3.38) in Theorem 3.4 hold. We have that HΦ,A~ is a bounded . α1 ,p1

. αm ,pm

. α,p

(ω1 ) × · · · × K qm (ωm ) to K q (ω) if and only if Z m

αi + n+γi Φ(t) Y −1 qi

dt < +∞. C8 = A (t) i |t|n i=1

operator from K q1

Rn

Furthermore,



H ~ . α1 ,p1 . αm ,pm . α,p ≃ C8 . Φ,A K (ω1 )×···×K (ωm )→K (ω) q1

qm

q

∗ ∗ Proof. By putting γ ∗ , γ1∗ , ..., γm and ω ∗ , ω1∗, ..., ωm above, it is not hard to see ∗ that C6 = C6 = C8 . Therefore, by using case a, case b1 of Theorem 3.6 and the relation (2.2), we finish the proof of this theorem. 

Now, let us take measurable functions s1 (t), ..., sm (t) 6= 0 almost everywhere in Rn . We consider a special case that the matrices Ai (t) = diag[si1 (t), ..., sin (t)] with |si1 | = · · · = |sin | = |si |, for almost everywhere t ∈ Rn , for all i = 1, ..., m. It is obvious that the matrices Ai ’s satisfy the condition (3.3). Therefore, since the Lebesgue space with power weights is a special case of the Herz space, we also obtain the following corollary. Corollary 3.9. Let 1 ≤ p, p1 , ..., pm < ∞, α1 , ..., αm ∈ R, and the hypothesis (3.38) in Theorem 3.4 is true. Then HΦ,A~ is a bounded operator from Lp1 (|x|α1 p1 dx) × · · · × Lpn (|x|αn pn dx) to Lp (|x|αp dx) if and only if Z m Φ(t) Y −α − n C9 = |si (t)| i pi dt < +∞. n |t| i=1 Rn

MULTILINEAR HAUSDORFF OPERATOR

31

Furthermore,

H ~ p α p Φ,A L 1 (|x| 1 1 dx)×···×Lpn (|x|αn pn dx)→Lp (|x|αp dx) = C9 .

Proof. By the assumption of the matrices Ai ’s, it is easy to see that |Ai (t)x|α = |si (t)|α .|x|α , for all α ∈ R, i = 1, ..., m.

Hence, we immedialately obtain the desired result.



By the relation between the Hausdorff operators and the Hardy-Ces`aro operators as mentioned in Section 1, we see that Corollary 3.9 extends and strengthens the results of Theorem 3.1 in [26] with power weights. Let us now assume that q(·)
and qi (·) ∈ P∞ (Rn ), λ, α, γ, αi , λi , γi are real numbers such that λi ∈ q−1 , 0 , γi ∈ (−n, ∞), i = 1, 2, ..., m and i∞

1 1 1 1 + +···+ = , q1 (·) q2 (·) qm (·) q(·) γ1 γ2 γm γ + +···+ = , q1∞ q2∞ qm∞ q∞ n + γ2 n + γm n + γ1 λ1 + λ2 + · · · + λm = λ, n+γ n+γ n+γ α1 + · · · + αm = α. We are also interested in the multilinear Hausdorff operators on the product of weighted λ-central Morrey spaces with variable exponent. We have the following interesting result.

Theorem 3.10. Let ω1 (x) = |x|γ1 , ..., ωm (x) = |x|γm , ω(x) = |x|γ and v1 (x) = |x|α1 , ..., vm (x) = |x|αm , v(x) = |x|α . In addition, the hypothesis (3.12) in Theorem 3.2 holds and the following condition is true: Z   m

Φ(t) Y (n+γi ) q 1 +λi i∞ C10 = kAi (t)k cAi ,qi,αi (t) 1 Lr1i (t,·) dt < +∞, (3.68) n |t| i=1 Rn

. q1 (·),λ1

. qm (·),λm

Then, we have HΦ,A~ is bounded from B ω1 ,v1 × · · · × B ωm ,vm

. q(·),λ

to B ω,v .

Proof. For R > 0, we denote ∆R =

1 ω B(0, R)


q1

∞



H ~ (f~) q(·) . Φ, A Lv (B(0,R)) +λ

It follows from using the Minkowski inequality for the variable Lebesgue space that Z m Y

1 Φ(t)

q(·) ∆R . f (A (t).) dt. (3.69) . i i 1 Lv (B(0,R)) +λ |t|n q ∞ ω(B(0, R)) i=1 n R

MULTILINEAR HAUSDORFF OPERATOR

32

On the other hand, we apply the H¨older inequality for the variable Lebesgue space to obtain m m Y



Y

fi (Ai (t).) qi (·)

. (3.70) fi (Ai (t).) Lq(·) (B(0,R)) . L (B(0,R)) v

vi

i=1

i=1

By estimating as (3.10) and (3.14), we have





fi (Ai (t).) qi (·)

1 r (t,·) . fi qi (·) . c (t). . (3.71) A ,q ,α i i i 1i L L (B(0,R)) L (B(0,R||A (t)||)) vi

In view of

n+γ1 λ n+γ 1

i

vi

+

n+γ2 λ n+γ 2

+···+

1

n+γm λ n+γ m

= λ, we estimate 1

Ai (t) (γi +n)( qi∞ +λi )

. 1 +λ ωi (B(0, RkAi (t)k)) qi∞ i m Q Thus, by (3.69) and (3.71), it follows that ∆R . C10 kfi k . qi (·),λi . B ωi ,vi i=1

m Q

kfi k . qi (·),λi . Consequently, it is straightforward HΦ,A~ (f~) . q(·),λ . C10 1

ω(B(0, R)) q∞ +λ

.

B ω,v

i=1

B ωi ,vi



As a consequence of Theorem 3.10, by the reason (3.2) and (3.4), we also have the analogous result for the q, q1 , ..., qm -constant case as follows. Corollary 3.11. Let ωi , vi , ω, v be as Theorem 3.10. In addition, the following condition holds: Z m

γi Φ(t) Y

A−1 (t) αi − qi −λi (n+γi ) dt < +∞. (3.72) C11 = n i |t| i=1 Rn

Then, we have

m

Y

kfi k . qi ,λi .

HΦ,A~ (f~) . q,λ . C11 . B ω,v

i=1

B ωi ,vi

Proof. By (3.2) and (3.4), it is clear to see that  

αi − γi −λi (n+γi ) (n+γi ) q 1 +λi

qi i∞ cAi ,qi ,αi (t) . A−1 kAi (t)k . i (t)

Hence, by Theorem (3.10), the proof is finished.



Moreover, we also obtain the above operator norm on the product of weighted λ-central Morrey spaces as follows. Theorem 3.12. Let ω(x) = |x|γ and ωi (x) = |x|γi for i = 1, ..., m. Then, we . q1 ,λ1 . qm ,λm . q,λ have that HΦ,A~ is bounded from B (ω1 ) × · · ·× B (ωm ) to B (ω) if and only if Z m

Φ(t) Y

A−1 (t) −(n+γi )λi dt < +∞. C12 = n i |t| i=1 Rn

MULTILINEAR HAUSDORFF OPERATOR

Furthermore, we obtain

H ~ . q1 ,λ1 Φ,A B

. qm ,λm

(ω1 )×···×B

. q,λ

(ωm )→B

(ω )

33

≃ C12 .

Proof. We first note that the sufficient condition of the theorem is derived from Corollary 3.11. In more details, by letting αi = γqii for i = 1, ..., m, we have C11 = C12 < ∞. Hence, by Corollary 3.11, we find m

Y

~ kfi k . qi ,λi . . C . H ( f )

Φ,A~ . q,λ 12 B B ω,ω 1/q

. q,λ

ωi ,ωi 1/qi

i=1

. q,λ

. qi ,λi

. qi ,λi

From this, by B ω,ω1/q = B (ω) and B ωi ,ωi 1/qi = B (ωi ) for i = 1, ..., m, the proof of sufficient condition of this theorem is ended. To give the proof for the necessary condition, let us now choose fi (x) = |x|(n+γi )λi . Then, it is not hard to show that kfi k . qi ,λi (ω ) = B

i



n + γi |Sn−1 |

λi

1 1

.

(1 + qi λi ) qi

Thus, we have λ  m Y

−1 γ + n

fi . qi ,λi q . (1 + λq) . (ωi ) B |S | n−1 i=1

By choosing fi ’s, we also have

 

HΦ,A~ f~ . q,λ B

= sup R>0



(3.73)

(ω)

1

ω(B(0, R))1+qλ

Z

m q Z Φ(t) Y  q1 (n+γi )λi |Ai (t)x| dt ω(x)dx . |t|n i=1

B(0,R) Rn

−(n+γi )λi (n+γi )λi

By (3.5), we get |Ai (t)x|(n+γi )λi & A−1 .|x| . Therefore, we i (t) imply that m

Z Φ(t) Y

−1 −(n+γi )λi 


~

A (t) dt × &

HΦ,A~ f . q,λ i |t|n i=1 B (ω) Rn

× sup R>0



1

ω(B(0, R))1+qλ

Z

B(0,R)

m Y i=1

  1q |x|(n+γi )λi q |x|γ dx

m Z Φ(t) Y

−1 −(n+γi )λi  γ + n λ −1

A (t) dt (1 + λq) q . = n i |t| i=1 |Sn−1 | Rn

MULTILINEAR HAUSDORFF OPERATOR

34

Hence, it follows from (3.73) that m m

Z Φ(t) Y

−1 −(n+γi )λi  Y


~

dt . kfi k . qi ,λi (ω ) . Ai (t)

HΦ,A~ f . q,λ & B i |t|n i=1 B (ω) i=1 Rn

. q1 ,λ1

Because of assuming that HΦ,A~ is bounded from B . q,λ

. qm ,λm

(ω1 ) × · · · × B

(ωm )

to B (ω), it immediately deduces that C12 < ∞, and hence, the proof of the theorem is completed.  Acknowledgments. This paper is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.51. References 1. J. Alvarez, J. Lakey, M. Guzm´ an-Partida, Spaces of bounded λ-central mean oscillation, Morrey spaces, and λ-central Carleson measures, Collect. Math. 51 (2000), 1-47. 2. K. Andersen and E. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), 547-558. 3. A. Almeida, and P. H¨ ast¨ o, Besov spaces with variable smoothness and integrability, J. Funct. Anal. 258(5) (2010), 1628-1655. 4. A. Almeida, D. Drihem, Maximal, potential and singular type operators on Herz spaces with variable exponents, J. Math. Anal. Appl. 394 (2012), 781-795. 5. G. Brown, F. M´ oricz, Multivariate Hausdorff operators on the spaces Lp (Rn ), J. Math. Anal. Appl. 271 (2002), 443-454. 6. R.A. Bandaliev, The boundedness of multidimensional hardy operators in weighted variable Lebesgue spaces, Lith. Math. J. 50 (2010), 249-259. 7. C. Capone, D. Cruz-Uribe, and A. Fiorenza, The fractional maximal operator and fractional integrals on variable Lp spaces, Rev. Mat. Iberoam. 23 (3) (2007), 743-770. 8. C. Carton-Lebrun and M. Fosset, Moyennes et quotients de Taylor dans BMO, Bull. Soc. Roy. Sci. Li´ege 53, No. 2 (1984), 85-87. 9. R. R. Coifman, Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331. 10. R. R. Coifman, Y. Meyer, Au del` a des op´erateurs pseudo-diff´erentiels, Ast´erisque, 57 (1978). 11. N. M. Chuong, Degenerate parabolic pseudodifferential operators of variable order, Dokl. Akad. Nauk SSSR 268 (1983), 1055-1058. 12. N. M. Chuong, H. D. Hung, Bounds of weighted Hardy-Ces` aro operators on weighted Lebesgue and BMO spaces, Integr. Transforms and Special Funct. 25 (2014), 697-710. 13. N. M. Chuong, D. V. Duong, H. D. Hung, Bounds for the weighted Hardy-Ces` aro operator and its commutator on weighted Morrey-Herz type spaces, Z. Anal. Anwend. 35 (2016) 489-504. 14. N. M. Chuong, N. T. Hong, H. D. Hung, Multilinear Hardy-Cesaro operator and commutator on the product of Morrey-Herz spaces, Analysis Math. (To appear). 15. N. M. Chuong, Pseudodifferential operators and wavelets over real and p-adic fields, Springer (Submitted to the Editor of Springer). 16. M. Christ and L. Grafakos, Best constants for two non-convolution inequalities, Proc. Amer. Math. Soc. 123 (1995), 1687-1693.

MULTILINEAR HAUSDORFF OPERATOR

35

17. D. Cruz-Uribe, A. Fiorenza, J. M. Martell, and C. Perez, The boundedness of classical operators on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 31(1) (2006), 239-264. 18. D. Cruz-Uribe, A. Fiorenza, Variable Lebesgue Spaces: Foundations and Harmonic Analysis, Springer-Basel, 2013. 19. L. Diening, M. Ruˇziˇcka, Calder´ on-Zygmund operators on generalized Lebesgue spaces Lp(x) and problems related to uid dynamics, J. Reine Angew. Math. 563 (2003), 197220. 20. L. Diening, P. Harjulehto, P. H¨ ast¨ o, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer-Verlag, (2011). 21. Z. W. Fu, S. L. Gong, S. Z. Lu and W. Yuan, Weighted multilinear Hardy operators and commutators, Forum Math. 27 (2015), 2825-2851. 22. V. S. Guliyev, J. Hasanov, and S. Samko, Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces, Math. Scand., 107 (2010), 285-304. 23. C. Georgakis, The Hausdorff mean of a Fourier-Stieltjes transform, Proc. Amer. Math. Soc. 116 (1992), 465 - 471. 24. F. Hausdorff, Summation methoden und Momentfolgen, I, Math. Z. 9 (1921), 74-109. 25. W. A. Hurwitz, L. L. Silverman, The consistency and equivalence of certain definitions of summabilities, Trans. Amer. Math. Soc. 18 (1917), 1-20. 26. H. D. Hung, L. D. Ky, New weighted multilinear operators and commutators of HardyCes` aro type, Acta Math. Sci. Ser. B Engl. Ed. 35 (2015)(6), 1411-1425. 27. W. Hoh, Pseudodifferential operators with negative definite symbols of varable order, Revista Mat. Iberoamer. 18, No.2 (2000), 219-241. 28. O. Kov´aˇcik, J. R´ akosn´ık, On spaces Lp(x) and W k,p(x) , Czechoslovak Math. J. 41(116) (1991), 592-618. 29. A. Lerner and E. Liflyand, Multidimensional Hausdorff operators on real Hardy spaces, J. Austr. Math. Soc. 83 (2007), 79-86. 30. S. Z. Lu, D. C. Yang, G. E. Hu, Herz type spaces and their applications, Beijing Sci. Press (2008). 31. Y. Lu, Y. P. Zhu, Boundedness of multilinear Calder´ on-Zygmund singular operators on Morrey-Herz spaces with variable exponents, Acta Math. Sin.(Engl. Ser.) 30 (2014), 1180-1194. 32. F. I. Mamedov, A. Harman, On a Hardy type general weighted inequality in spaces Lp(·) , Integr. Equations Oper. Theor. 66 (2010), 565-592. 33. R. Mashiyev, B. C ¸ eki¸c, F. I. Mamedov, S. Ogras, Hardy’s inequality in power-type weighted Lp(·) (0, ∞), J. Math. Anal. Appl. 334(1) (2007), 289-298. 34. A. Miyachi, Boundedness of the Ces` aro operator in Hardy space, J. Fourier Anal. Appl. 10 (2004), 83-92. 35. F. M´ oricz, Multivariate Hausdorff operators on the spaces H 1 (Rn ) and BM O(Rn ), Analysis Math. 31 (2005), 31-41. 36. N. Jacob, H. G. Leopold, Pseudodifferential operators with variable order of differentiation generating Feller semigroups, Integr. Equations Oper. Theor. 17 (1993), 544-553. 37. H. Rafeiro, S. Samko, Hardy type inequality in variable Lebesgue spaces, Ann. Acad. Sci. Fenn., Ser. A 1 Math. 34(1) (2009), 279-289 . 38. J. L. Wu, W. J. Zhao, Boundedness for fractional Hardy-type operator on variableexponent Herz-Morrey spaces, Kyoto J. Math. Vol. 56, No. 4 (2016), 831-845. 39. X. Wu, Necessary and sufficient conditions for generalized Hausdorff operators and commutators, Ann. Funct. Anal. Vol. 6, No. 3 (2015), 60-72.

MULTILINEAR HAUSDORFF OPERATOR

36

40. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton University Press, (1993). 41. J. Xiao, Lp and BM O bounds of weighted Hardy-Littlewood averages, J. Math. Anal. Appl. 262 (2001), 660-666. Institute of mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam. E-mail address: [email protected] Shool of Mathematics, Mientrung University of Civil Engineering, Phu Yen, Vietnam. E-mail address: [email protected] Shool of Mathematics, University of Transport and Communications- Campus in Ho Chi Minh City, Vietnam. E-mail address: [email protected]