Weighted estimates for commutators of multilinear Hausdorff operators ...

WEIGHTED ESTIMATES FOR COMMUTATORS OF MULTILINEAR HAUSDORFF OPERATORS ON VARIABLE EXPONENT MORREY-HERZ TYPE SPACES

arXiv:1710.01299v1 [math.CA] 3 Oct 2017

NGUYEN MINH CHUONG, DAO VAN DUONG, AND KIEU HUU DUNG Abstract. In this paper, we establish the boundedness of the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belong to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previous known results.

1. Introduction Given Φ be a locally integrable function on Rn . The n-dimensional Hausdorff operator HΦ,A [3] is defined by Z Φ(t) f (A(t)x)dt, x ∈ Rn , (1.1) HΦ,A (f )(x) = |t|n Rn

where A(t) is an n × n invertible matrix for almost everywhere t in the support of Φ. It is well known that if the function Φ and the matrix A are taken appropriately, then the Hausdorff operator HΦ,A reduces to many classcial operators in analysis, for example, the Hardy operator, the Cesàro operator, the Hardy-Littlewood average operator and the Riemann-Liouville fractional integral operator. Some of their results have been significantly seen in [3], [8], [9], [17], [30], [35], [36] and references therein. In addition, it is natural to extend the study on the linear operator to multilinear operator, which is actually necessary. Thus, the authors of this paper in [7] have recently investigated the multilinear operators of Hausdorff type HΦ,A~ given as follows: Z m Φ(t) Y ~ HΦ,A~ (f )(x) = fi (Ai (t)x)dt, x ∈ Rn , (1.2) n |t| i=1 Rn

where Φ : Rn → [0, ∞) and Ai (t) (for i = 1, ..., m) are n × n invertible matrices for almost everywhere t in the support of Φ, and f1 , f2 , ..., fm : Rn → C are 2010 Mathematics Subject Classification. Primary 42B30; Secondary 42B20, 47B38. Key words and phrases. Commutator, multilinear Hausdorff operator, Hardy-Cesàro operator, Lipschitz space, central BMO space, Morrey-Herz space, variable exponent. 1

COMMUTATOR OF MULTILINEAR HAUSDORFF OPERATOR

2

~ = (A1 , ..., Am ). It is useful measurable functions and f~ = (f1 , ..., fm ) and A to remark that the weighted multilinear Hardy operators [16] and weighted multilinear Hardy-Cesàro operators [9] are two special cases of the multilinear Hausdorff operators HΦ,A~ . ~ f~ be as above. The Coifman-Rochberg-Weiss type Definition 1.1. Let Φ, A, commutator of multilinear Hausdorff operator is defined by Z m m Y Φ(t) Y ~b ~ HΦ,A~ (f )(x) = fi (Ai (t)x) bi (x) − bi (Ai (t)x) dt, x ∈ Rn , (1.3) n |t| i=1 i=1 Rn

where ~b = (b1 , ..., bm ) and bi are locally integrable functions on Rn for all i = 1, ..., m. Moreover, if we now take m = n ≥ 2, Φ(t) = |t|m .ω(t)χ[0,1]m (t) and Ai (t) = ti .Im (Im is an identity matrix), for t = (t1 , t2 , ..., tm ), where ω : [0, 1]m → ~b [0, ∞) is a measurable function, then HΦ, ~ reduces to the commutator of A weighted multilinear Hardy operator due to Fu et al. [16] defined as the following Z Y m m Y ~b ~ Hω (f )(x) = fi (ti x) bi (x) − bi (ti x) ω(t)dt, x ∈ Rm . (1.4) [0,1]m

i=1

i=1

Also, by 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, it is clear to see that ~b m,n,~b reduces to the commutator of multilinear Hardy-Cesàro operator Uψ,~ HΦ,A s introduced by Hung and Ky [21] as follows Z Y m m Y m,n,~b Uψ,~s (x) = fi (si (t)x) bi (x) − bi (si (t)x) ψ(t)dt, x ∈ Rn . (1.5) [0,1]n

i=1

i=1

In recent years, the theory of function spaces with variable exponents has attracted much more the interest from many mathematicians (see, e.g., [2], [4], [7], [15], [18], [24], [32] and others). It is interesting to see that this theory has had some important applications to the electronic fluid mechanics, elasticity, fluid dynamics, recovery of graphics, harmonic analysis and partial differential equations (see [1], [6], [10], [11], [14], [22], [31]). Let b ∈ BMO(Rn ) and T be a Calderón-Zygmund singular integral operator with rough kernels. From classical result of Coifman, Rochberg, and Weiss [13], Karlovich and Lerner [26] developed the boundedness of commutator [b, T ] to generalized Lp spaces with variable exponent. Also, in order to generalize the result of Chanillo [12], Izuki [23] established the boundedness of the higher order commutator on Herz spaces with variable exponent.


3

More recently, Wu [33] considered the mth-order commutator for the fractional integral as follows m Z f (y) b(x) − b(y) m dy, Iβ,b (f )(x) = |x − y|n−β Rn

where β ∈ (0, n), b ∈ BMO(Rn ), m ∈ N. Then the author established the boundedness for commutators of fractional integrals on Herz-Morrey spaces with variable exponent. Motivated by above mentioned results, the goal of this paper is to establish the boundedness for commutators of multilinear Hausdorff operators on the product of weighted Lebesgue, central Morrey, Herz, and Morrey-Herz spaces with variable exponent with their symbols belong to both Lipschitz spaces and central BMO spaces. Our paper is organized as follows. In Section 2, we give necessary preliminaries on weighted Lebesgue spaces, central Morrey spaces, Herz spaces, Morrey-Herz spaces with variable exponent and Lipschitz spaces, central BMO spaces. In Section 3, our main theorems are given. Finally, the results of this paper are proved in Section 4. 2. Preliminaries In this section, let us recall some basic facts and notations which will be used throughout this paper. 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. For any a ∈ Rn and r > 0, we denote by B(a, r) the ball 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 ≤ Cb. Besides that, we denote that a n k χk = χCk , Ck = Bk \ Bk−1 and Bk = x ∈ R : |x| ≤ 2 , for all k ∈ Z.

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 [11], [14] and [15].

Definition 2.1. 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)| Fp (f /η) := dx + f /η L∞ (Ω∞ ) < ∞, η Rn \Ω∞


4

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

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

1 [ii] If Fp (f ) ≥ C, then f Lp(·) ≥ min C q− , C q+ , for all f ∈ Lp(·) .(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 the set of all complex-valued measurable functions f such that f ω belongs the Lp(·) (Rn ) space and f has norm

f p(·) = f ω p(·) . Lω

L

n Clog 0 (R )

Let denote the set of all log-Hölder continuous functions α(·) satisfying at the origin Cα , for all x ∈ Rn . 0 |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∞ , for all x ∈ Rn , |α(x) − α∞ | ≤ log(e + |x|) . α(·),p

Next, we give the definition of variable exponent weighted Herz spaces K q(·),ω . α(·),λ

and variable exponent weighted Morrey-Herz spaces M K p,q(·),ω (see [29], [32] for more details) .


5

Definition 2.2. 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

k2kα(·) f χk kp q(·) Lω

k=−∞

p1

.

Definition 2.3. 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 ∈Z

−k0 λ

k0 P

k2

kα(·)

f χk k

k=−∞

. α(·),0

. α(·),λ

K p,q(·),ω

p 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(·) = (B(0,R)) Lω 2 ω2 integrable functions.


6

Next, the following theorem is stated as the embedding result on Lebesgue spaces with variable exponent (see [7]). Theorem 2.6. 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(·) . L L L ω

ω

Let us recall to define Lipschitz space and central BMO space (see, for example, [25], [28], [34] for more details). Definition 2.7. Let 0 < β ≤ 1. The Lipschitz space Lipβ (Rn ) is defined as

the set of all functions f : Rn → C satisfying f Lipβ (Rn ) < ∞, where

f β n := Lip (R )

|f (x) − f (y)| . |x − y|β x,y∈Rn ,x6=y sup

Definition 2.8. The space BMO(Rn ) consists of all locally integrable functions f : Rn → C satisfying Z

1

f := sup |f (x) − fQ |dx < ∞, BMO(Rn ) Q |Q| Q

where the supremum is taken over all cubes Q ⊂ Rn with sides parallel to the coordinate axes.

Definition 2.9. Let 1 ≤ q < ∞ .and ω be a weight function. The central q bounded mean oscillation space CMO (ω) is defined as the set of all functions f ∈ Lqloc (Rn ) such that Z 1q

1 q

f . q = sup , |f (x) − f | ω(x)dx ω,B(0,R) CM O (ω) R>0 ω(B(0, R)) B(0,R)

where

ω(B(0, R)) =

Z

B(0,R)

ω(x)dx and fω,B(0,R)

1 = ω(B(0, R))

Z

f (x)ω(x)dx.

B(0,R)

Remark that, Fefferman [34] obtain the famous result that the space BMO(Rn ) is the dual space of Hardy space H 1 (Rn ). When ω = 1, we write simply CMO q (Rn ) := CMO q (ω). The space CMO(Rn ) can be seen as a local version of BMO(Rn ) at the origin. Moreover, BMO(Rn ) $ CMO q (Rn ), where 1 ≤ q < ∞, and the John-Nirenberg inequality is not true in CMO q (Rn ).


7

3. Statement of the results Before stating our main results, we introduce some notations which will be used throughout this section. Let γ1 , ..., γm ∈ R, λ1 , ..., λm ≥ 0, p1 , ..., pm , p ∈ (0, ∞), 0 < β1 , ..., βm ≤ 1, qi ∈ Pb (Rn ), ri ∈ P∞ (Rn ) for i = 1, ..., m and n log n ∗ α1 , ..., αm ∈ L∞ (Rn ) ∩ Clog 0 (R ) ∩ C∞ (R ). The functions α (·), q(·), γ(·) and numbers β, λ are defined as follows β1 + · · · + βm = β, λ1 + λ2 + · · · + λm = λ. γ1 γm γ1 + · · · + γm + +···+ = γ(·), r1 (·) rm (·) 1 1 1 1 1 +···+ + +···+ = , q1 (·) qm (·) r1 (·) rm (·) q(·) γm + n γ1 + n −···− = α∗ (·). α1 (·) + · · · + αm (·) − β1 − · · · − βm − r1 (·) rm (·) 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 that |Ax| ≤ kAk |x| for any vector x ∈ Rn . In particular, if A is invertible, then we have

n (3.2) kAk−n ≤ det(A−1 ) ≤ A−1 . Now, we are ready to state the main results in this paper.

Theorem 3.1. Let ζ > 0, ω1 (x) = |x|γ1 , ..., ωm (x) = |x|γm , ω(x) = |x|γ(x) , βi n log n q(·) ∈ Pb (Rn ), α∗ ∈ L∞ (Rn ) ∩ Clog 0 (R ) ∩ C∞ (R ), bi ∈ Lip , λ1 , ..., λm > 0 and the following conditions are true:

qi (A−1 i (t)·) ≤ ζ.qi (·) and 1 Lϑi (t,·) < ∞, a.e. t ∈ supp(Φ), for all i = 1, ..., m, (3.3) αi (0) − αi∞ ≥ 0, for all i = 1, ..., m, (3.4) either γ1 , ..., γm > −n, r1 (0) = r1+ , r1∞ = r1− , ..., rm (0) = rm+ , rm∞ = rm− or γ1 , ..., γm < −n , r1 (0) = r1− , r1∞ = r1+ , ..., rm (0) = rm− , rm∞ = rm+ , or γ1 = · · · = γm = −n.

(3.5)


8

Then, if C1 =

Z

Rn

where

m

β Φ(t) Y cAi ,qi,γi (t) In − Ai (t) i 1 Lϑi (t,·) φAi ,λi (t)dt < ∞, n |t| i=1

n

λi −αi (0)

λi −αi∞ o

φAi ,λi (t) = max Ai (t) × , Ai (t) ×max

0 n X

2

r(λi −αi (0))

,

r=Θ∗n −1

0 X

r=Θ∗n −1

(3.6)

o 2r(λi −αi∞ ) ,

(3.7)

with 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 i=1,m

−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 (t) i i i 1 1 1 = − , for all i = 1, ..., m, ϑi (t, ·) ζqi (·) qi (A−1 i (t)·) . α1 (·),λ1

~

. αm (·),λm

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

to M K p,ζq(·),ω . Theorem 3.2. Suppose that we have the given supposition of Theorem 3.1. Let 1 ≤ p, p1 , ..., pm < ∞, λi = 0 and αi (0) = αi∞ , for all i = 1, ..., m. At the same time, let 1 1 1 +···+ = , (3.8) p1 pm p Z m Y

∗ m− p1 Φ(t)

In − Ai (t) βi 1 ϑ (t,·) φA ,0 (t)dt < ∞, c (t) C2 = (2 − Θn ) A ,q ,γ i L i |t|n i=1 i i i Rn

Then,

~b HΦ, ~ A

is a bounded operator from

. α1 (·),p1 K ζq1 (·),ω1

. αm (·),pm × · · · × K ζqm(·),ωm

(3.9)

to

. α∗ (·),p K q(·),ω .

By using the ideas in the proof of Theorem 3.1, we give the analogous result for the Lebesgue spaces with variable exponent as follows. Theorem 3.3. Let ζ > 0, γ1 , ..., γm < 0, ω1 (x) = |x|γ1 , ..., ωm (x) = |x|γm , ω(x) = |x|γ(x) , q(·) ∈ Pb (Rn ), b1 , ..., bm ∈ Lipβi , and let the hypothesis (3.3) in Theorem 3.1 hold. Thus, if the following conditions are true:

βi + γi

| · | ri (·) ri (·) < ∞, for all i = 1, ..., m, (3.10) L ωi

C3 =

Z

Rn

Φ(t) |t|n

m Y i=1

β cAi ,qi ,γi (t). In − Ai (t) i 1 Lϑi (t,·) dt < ∞,

(3.11)


9

then we have m Y

~b

H ~ (f~) q(·) . C3 .BLip . kfi kLζqi (·) . Φ,A L ω

ωi

i=1

Next, we consider that all of r1 (·), ..., rm(·) are constant and the following conditions hold: (H1 ) α1 (·) + · · · + αm (·) −

γ1 +n r1

−···−

γm +n rm

= α∗∗ (·),

(H2 ) Ai (t) = si (t).ai (t) for all i = 1, ..., m, where si : supp(Φ) → R is a measurable function such that si (t) 6= 0 for a.e t ∈ supp(Φ) and ai (t) is an n × n rotation matrix for a.e t ∈ supp(Φ). Then, we also obtain the following some interesting results. Theorem 3.4. Let ζ > 0, λ1 , ..., λm > 0, γ1 , ..., γm > −n, ω1 (x) = |x|γ1 , ..., . r1 ωm (x) = |x|γm , ω(x) = |x|γ(x) , q(·) ∈ Pb (Rn ), b1 ∈ CMO (ω1 ),..., bm ∈ . rm CMO (ωm ), the hypothesis (3.3) and (3.4) in Theorem 3.1 hold. Then, if Z m 1 γ +n

Φ(t) Y

1 ϑ (t,·) φAi,λi (t) 1+ψ ri |si (t)| iri +ϕAi (t) dt < ∞, C4 = c (t). A ,q ,γ i i i A ,γ i i L i |t|n i=1 Rn

(3.12)

where

−1 γi

A (t) , Ai (t) −γi , ψAi ,γi (t) = |detA−1 (t)|max i i ϕAi (t) = max log(4|si (t)|), log

2 , |si (t)|

. α1 (·),λ1

~

. αm (·),λm

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

to M K p,ζq(·),ω . Theorem 3.5. Let 1 ≤ p, p1 , ..., pm < ∞, λi = 0, αi (0) = αi∞ , for all i = 1, ..., m. Also, both the assumptions of Theorem 3.4 and the hypothesis (3.8) in Theorem 3.2. In addition, the following condition holds: Z m

1 Φ(t) Y cAi ,qi ,γi (t) 1 Lϑi (t,·) φAi ,0 (t) × C5 = (2 − Θ∗n )m− p n |t| i=1 Rn

1 r

× 1 + ψAii ,γi |si (t)|

γi +n ri

Then, we have

+ ϕAi (t) dt < ∞.

m Y

~b

H ~ (f~) . α∗∗ (·),p . C5

fi . αi (·),pi . Φ,A K q(·),ω

i=1

K ζqi (·),ωi

(3.13)


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Let us now assume that q(·) and qi (·) ∈ P∞ (Rn ), λ, α, γ, β, ri, λi , αi , γi , βi are , 0 , αi , γi ∈ (−n, ∞), βi ∈ (0, 1], real numbers such that ri ∈ (0, ∞), λi ∈ q−1 i∞ i = 1, 2, ..., m and denote

β1 + · · · + βm = β, α1 αm α1 + · · · + αm + +···+ = α, r1 rm 1 1 1 1 1 +···+ + +···+ = . q1 (·) qm (·) r1 rm q(·) We are also interested in the commutators of multilinear Hausdorff operators on the product of weighted λ-central Morrey spaces with variable exponent. More precisely, we have the following useful result. Theorem 3.6. Let ωi (x) = |x|γi , vi (x) = |x|αi , bi ∈ Lipβi for all i = 1, ..., m, ω(x) = |x|γ , v(x) = |x|α and the following conditions are true:

qi (A−1 i (t)·) ≤ qi (·) and 1 Lϑ1i (t,·) < ∞, a.e. t ∈ supp(Φ), for all i = 1, ..., m, (3.14) m X γ γi β+α− + (γi + n)λi − αi + = (γ + n)λ, (3.15) q∞ qi∞ i=1 C6 =

Z

Rn

where

m

Φ(t) Y (n+γi ) q 1 +λi i∞ cAi ,qi ,αi (t) × kAi (t)k n |t| i=1

β × 1 Lϑ1i (t,·) In − Ai (t) i dt < +∞,

(3.16)

1 1 1 − = , for all i = 1, ..., m. −1 ϑ1i (t, ·) qi (Ai (t)·) qi (·) . qm (·),λm

. q1 (·),λ1

~

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

.

. q(·),λ

to B ω,v . ri

Theorem 3.7. Given ωi (x) = |x|γi , vi (x) = |x|αi , bi ∈ CMO (ωi ) for all i = 1, ..., m, ω(x) = |x|γ , v(x) = |x|α , the hypothesis (3.14) in Theorem 3.6 and the condition (H2 ) hold. In addition, the following statements are true: m X γ γi α− + (γi + n)λi − αi + = (γ + n)λ, (3.17) q∞ q i∞ i=1 C7 =

Z

Rn

m

Φ(t) Y (n+γi ) kAi (t)k n |t| i=1

1 r

1 +λi qi∞

× 1 + ψAii ,αi |si (t)| ~

cAi ,qi ,αi (t). 1 Lϑ1i (t,·) ×

αi +n ri

+ ϕAi (t) dt < +∞. (3.18)

. q1 (·),λ1

. qm (·),λm

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

. q(·),λ

to B ω,v .


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4. Proofs of the theorems Fristly, for simplicity of notation, we denote BLip

m Y

bi =

Lipβi

, BCMO,~ω

i=1

m Y

bi =

.

CM O

ri

i=1

m Y

fi and F = (ωi ) M i=1

. αi (·),λi

K pi ,ζqi (·),ωi

.

4.1. Proofs of Theorem 3.1 and Theorem 3.2. By using the versions of the Minkowski inequality for variable Lebesgue spaces from Corollary 2.38 in [11], we have Z m Y

~b Φ(t)

H ~ (f~)χk q(·) .

f (A (t)·) b (·) − b (A (t)·) χk Lωq(·) dt. (4.1) i i i i Φ,A Lω |t|n i=1 Rn

On the other hand, since bi ∈ Lipβi , we get

β |f (Ai (t)x) bi (x)−bi (Ai (t)x) χk (x)| ≤ |f (Ai (t)x)|. bi Lipβi In −Ai (t) i 2βi k χk (x). Thus, by applying the Hölder inequality for variable Lebesgue spaces (see also Corollary 2.30 in [11]), we find m

Y

f (Ai (t)·) bi (·) − bi (Ai (t)·) χk Lq(·) ω

(4.2)

i=1

≤ 2kβ BLip

m m Y γi

Y

r (·) .

fi (Ai (t)·)χk qi (·) | · | ri (·)

In − Ai (t) βi . χ k L i L

We observe that Fri (| · |

γi ri (·)

χk ) =

ωi

i=1

i=1

Z

|x|γi dx =

Ck

Case 1: k < 0. Denote by σi = Case 2: k ≥ 0. Denote by σi =

Z2k Z

r γi +n−1 dσ(x′ )dr . 2k(γi +n) .

2k−1 S n−1

  

  

1 , ri+

if (γi + n) > 0,

1 , ri−

otherwise.

1 , ri−

if (γi + n) > 0,

1 , ri+

otherwise.

From this, by (2.1), we have γi

| · | ri (·) χk Lri (·) . 2k(γi +n)σi ,

(4.3)


12

Therefore, from (4.1)-(4.3), we see that m

P

~b

k(β+ (γi +n)σi )

H ~ (f~)χk q(·) . 2 i=1 BLip × (4.4) Φ,A Lω Z m

Φ(t) Y

In − Ai (t) βi . fi (Ai (t)·)χk qi (·) dt. × n Lω i |t| i=1 n R Let us now fix i ∈ 1, 2, ..., m . Since kAi (t)k = 6 0, there exists an integer ℓi −1 ℓi number ℓi = ℓi (t) such that 2 < kAi (t)k ≤ 2 . By writing ρ∗A~ (t) as

. ρ∗A~ (t) = max Ai (t) . A−1 i (t) i=1,...,m

Hence, by letting y = Ai (t).z with z ∈ Ck , we arrive at ℓi +k−2

−1

∗

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

and

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

(4.5)

Now, we will prove the following inequality

0 X

fi χk+ℓ +r ζqi (·) .

fi (Ai (t)·)χk qi (·) . cA ,q ,γ (t). 1 r (t,·) . i i i i i L L L ωi

ωi

r=Θ∗n −1

(4.6)

Indeed, for η > 0, by (4.5), we get !q (x) Z fi (Ai (t)x)χk (x)ωi (x) i dx η Rn

! −1

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

Z

≤

Ai (t)Ck



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

From this, by the definition of Lebesgue space with variable exponent, we find

fi (Ai (t)·)χk

q (·)

Lωii

0 X

fi χk+ℓ +r ≤ cAi ,qi ,γi (t). i r=Θ∗n −1

q (A−1 (t)·) i

Lωii

.


13

In view of (3.3) and Theorem 2.6, we deduce

f q (A−1 (t)·) . 1 ϑ (t,·) . f ζqi (·) . i L i L i ωi

Lω i

This completes the proof of the inequalities (4.6). Now, combining (4.4) and (4.6), it is easy to see that m P m Z Φ(t) Y

~b k(β+ (γi +n)σi )

H ~ (f~)χk q(·) . 2 i=1 cAi ,qi,γi (t) 1 Lϑi (t,·) × BLip Lω n Φ,A |t| i=1 Rn

m 0 X

β Y

fi χk+ℓ +r ζqi (·) dt . × In − Ai (t) i i L i=1

ωi

r=Θ∗n −1

(4.7)

Thus, by applying Lemma 2.4 in Section 2, we have m Z Φ(t) Y

βi

~b

dt .

H ~ (f~)χk q(·) . BLip .F U(t) c (t) 1 I −A (t) Ai ,qi ,γi n i Lϑi (t,·) Φ,A Lω |t|n i=1 Rn

(4.8)

Here U(t) = 2

k(β+

m P

(γi +n)σi )

i=1

m 0 Y X (k+ℓi )(λi −αi (0)) 2 2r(λi −αi (0)) i=1

r=Θ∗n −1

+2(k+ℓi)(λi −αi∞ )

0 X

r=Θ∗n −1

2r(λi −αi∞ ) .

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

λ −α (0)

λ −α 2ℓi (λi −αi (0)) + 2ℓi (λi −αi∞ ) . max Ai (t) i i , Ai (t) i i∞ . From this, we can estimate U as follows U(t) . 2

k(β+

m P

(γi +n)σi )

i=1

m Y i=1

n

λ −α (0)

λ −α o max Ai (t) i i , Ai (t) i i∞ ×

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

. 2

k(β+

m P

(γi +n)σi )

i=1

m Y i=1

×max

n

0 X

r=Θ∗n −1

2

r=Θ∗n −1

n

λ −α (0)

λ −α o max Ai (t) i i , Ai (t) i i∞ ×

r(λi −αi (0))

,

0 X

r=Θ∗n −1

2r(λi −αi∞ )

on

o 2k(λi −αi (0)) + 2k(λi −αi∞ ) .


14

This implies that U(t) . 2

k(β+

m P

(γi +n)σi )

i=1

m n o Y 2k(λi −αi (0)) + 2k(λi −αi∞ ) φAi ,λ (t). i=1

Thus, by (4.8), it is not difficult to show that m

P m

~b

k(β+ (γi +n)σi ) Y k(λi −αi (0)) k(λi −αi∞ )

H ~ (f~)χk q(·) . C1 .BLip .F .2 i=1 2 + 2 . Φ,A Lω i=1

(4.9)

Next, using Proposition 2.5 in [29], we have

~b

H ~ (f~) Φ,A M

. α∗ (·),λ K p,q(·),ω

. max

where

E1 = 2

−k0 λ

sup k0 0, for all i = 1, ..., m and (3.4), we obtain 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 Consequently, from (4.11), we conclude m Y E1 . C1 .BLip .F . 1 + 2k0

αi (0)−αi∞

i=1

A similar agrument as E1 , we also get

.

(4.13)

E2 . C1 .BLip .F .2−k0 λ .

(4.14)

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

Thus, we see that

E3 . C1 .BLip .F .T∞ ,

(4.15)

where T∞ = 2

−k0 λ

k0 X

2

k(

m P

i=1

αi∞ +

m P

(γi +n)(σi − r 1 )p

i=1

i∞

m Y

2k(λi −αi (0))p + 2k(λi −αi∞ )p

i=1

k=0

p1

.

Remark that, by defining σi and (3.5), we deduce 1 (γi + n)(σi − ) = 0, for all i = 1, ..., m. (4.16) ri∞ Thus, by estimating in the same way as T0 , we also have 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


16

In the case λi + αi∞ − αi (0) = 0, Ti,∞ is dominated by k0 λi p − 1 p1 − 1 1 −k0 λi −k0 λi 2 p + 2λi p − 1 p . . 2 (k + 1) Ti,∞ ≤ 2 + (k + 1) 0 0 2λi .p − 1 Otherwise, we get 2k0 λi p − 1 2k0 (λi +αi∞ −αi (0))p − 1 p1 Ti,∞ ≤ 2−k0 λi + (λi +αi∞ −αi (0))p 2λi .p − 1 2 −1 −1/p . 2k0 (αi∞ −αi (0)) + 2λi p − 1 + 2−k0 λi . which implies T∞ .

m Q

Li . From this, by (4.15), we obtain

i=1

E3 . C1 .BLip .F .

m Y

Li .

(4.17)

i=1

By (4.10), (4.13), (4.14) and (4.17), the proof of Theorem 3.1 is finished. Next, let us give the proof for Theorem 3.2. From Proposition 3.8 in [2], it is easy to see that

~b

H ~ f~ . α∗ (·),p Φ,A

−1 X

.

K q(·),ω

∗ (0)p

2kα

k=−∞

+

∞ X

2

k=0

:= H0 + H1 .

H

~ Φ,A

kα∗∞ p

p p1 f~ χk Lq(·) ω

HΦ,A~

p p1 ~ f χk Lq(·) ω

(4.18)

Next, we need to estimate the upper bound of H0 and H1 . In view of (4.7) and (4.12), by using the Minkowski inequality, we find Z m

Φ(t) Y

1 ϑ (t,·) In − Ai (t) βi × c (t) (4.19) H0 . BLip A ,q ,γ i i i i L |t|n i=1 Rn

m P m −1 0 p o p1 nX αi (0))p Y X k( i=1 × 2 kfi χk+ℓi+r kLζqi (·) dt.

i=1

k=−∞

ωi

r=Θ∗n −1

Using (3.8) and the Hölder inequality, it follows that −1 nX

2

k(

m P

αi (0))p

i=1

i=1

k=−∞

≤

m n X −1 Y i=1

m X 0 Y

k=−∞

kfi χk+ℓi +r kLζqi (·)

r=Θ∗n −1

2kαi(0)pi

0 X

r=Θ∗n −1

ωi

p o 1p

kfi χk+ℓi +r kLζqi (·) ωi

(4.20) 1

pi o pi

.


17

By pi ≥ 1, for all i = 1, ..., m, we have 0 X

kfi χk+ℓi +r kLζqi (·) ωi

r=Θ∗n −1

pi

0 X

≤ (2 − Θ∗n )pi−1

kfi χk+ℓi +r kpiζqi (·) . Lω i

r=Θ∗n −1

Thus, combining (4.19) and (4.20), we deduce Z m Y

β

∗ m− p1 Φ(t) cAi ,qi,γi (t) 1 Lϑi (t,·) In − Ai (t) i H0,i dt. H0 . BLip . (2 − Θn ) n |t| i=1 Rn

Here H0,i =

(4.21)

0 P

r=Θ∗n −1

Hence, we estimate H0,i =

P −1

k=−∞

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

0 X

r=Θ∗n −1

.

0 X

−1+ℓ+r X

1 pi

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

2(t−ℓi −r)αi (0)pi kfi χt kpiζqi (·) Lω i

t=−∞

2−(ℓi +r)αi (0)

∞ X

2tαi (0)pi kfi χt kpiζqi (·) Lω i

t=−∞

r=Θ∗n −1

1 pi

1

pi

.

By αi (0) = αi∞ and Proposition 3.8 in [2], we get H0,i .

0 X

2

−(ℓi +r)αi (0)

kfi k . αi (·),pi = 2 K ζqi (·),ωi

r=Θ∗n −1 ℓi −1

−ℓi αi (0)

.

0 X

2−rαi (0) kfi k . αi (·),pi .

r=Θ∗n −1 −ℓi αi (0)

ℓi

Since 2 < kAi (t)k ≤ 2 , we deduce that 2 . kAi (t)k by (4.22), we have H0,i . φAi ,0 (t).kfi k . αi (·),pi . K ζqi (·),ωi

As above, by (4.21), we make H0 . C2 .BLip .

m Y i=1

By estimating as H0 , we also make H1 . C2 .BLip .

m Y i=1

kfi k . αi (·),pi . K ζqi (·),ωi

kfi k . αi (·),pi . K ζqi (·),ωi

There, by (4.18), we finishes desired conclusion.

K ζqi (·),ωi

−αi (0)

(4.22) . Hence,


18

4.2. Proofs of Theorem 3.4 and Theorem 3.5. Applying the Minkowski inequality and the Hölder inequality for variable Lebesgue spaces, we get Z m

~b Φ(t) Y

fi (Ai (t).)χk qi (·) dt.

bi (·)−bi (Ai (t)·) r

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

By(4.6), we deduce

~b

H ~ (f~)χk q(·) . L Φ,A

Z

ω

Rn

m

Φ(t) Y

bi (·) − bi (Ai (t)·) r × c (t) A ,q ,γ i i i L i (ωi ,Bk ) |t|n i=1

m 0 Y X

fi χk+ℓ +r ζqi (·) dt.

× 1 Lϑi (t,·) i L i=1 r=Θ∗n −1

(4.23)

ωi

On the other hand, we need to prove that 1 k(γi +n) γi +n

ri ri ri

bi (·)−bi (Ai (t)·) r .2 1+ψAi ,γi |si (t)| +ϕAi (t) bi CM. Ori (ωi ) . L i (ωi ,Bk ) (4.24) In fact, we put a1,i (·) = bi (·) − bi,ωi ,Bk , a2,i (·) = bi (Ai (t).) − bi,ωi ,Ai (t)Bk and a3,i (·) = bi,ωi ,Bk − bi,ωi ,Ai (t)Bk . Here Z 1 bi,ωi ,U = bi (x)ωi (x)dx. ωi (U) U

Then, we have

bi (·) − bi (Ai (t)·)

a3,i r

a2,i r

a1,i r . + + ≤ L i (ωi ,Bk ) L i (ωi ,Bk ) L i (ωi ,Bk ) i ,Bk ) (4.25) . ri From defining the space CMO (ωi ), we immediately have k(γi +n)

r1 ri

a1,i r . i . b ri . 2 . bi CM. Ori (ω ) . (4.26) ≤ ω (B ) i i k i L (ωi ,B ) CM O (ω ) Lri (ω

i

k

i

By making the formula for change of variables , we obtain Z

ri

a2,i r = |bi (Ai (t)x) − bi,ωi ,Ai (t)Bk |ri ωi (x)dx L i (ωi ,B ) k

Bk

≤ ψAi ,γi (t).

Z

|bi (z) − bi,ωi ,Ai(t)Bk |ri ωi (z)dx.

Ai (t)Bk

Because of assuming Ai (t) = si (t).ai (t), we deduce 1 γi +n k(γi +n)

ri ri ri

bi . ri .

a2,i r |s (t)| .2 . ψ i Ai ,γi CM O (ωi ) L i (ωi ,B ) k

Next, we observe that

1 ri .

a3,i r b − b ≤ (ω (B )) i,ω ,B i k i,ω ,A (t)B i i i k k L i (ω ,B ) i

k

(4.27)

(4.28)


19

By having si (t) 6= 0, there exists an integer number θi = θi (t) satisfying 2θi −1 < |si (t)| ≤ 2θi . Thus, we define θi − 1, if θi ≥ 1, σ(θi ) = θi , otherwise, and S(θi ) =

  j ∈ Z : 1 ≤ j ≤ θi − 1 , if θi ≥ 1, 

j ∈ Z : θi + 1 ≤ j ≤ 0 , otherwise.

At this point, we give the estimation as below X bi,ω ,2j−1 B −bi,ω ,2j B + b bi,ω ,B −bi,ω ,A (t)B ≤ i i,ωi ,2σ(θi ) Bk −bi,ωi ,Ai (t)Bk . i i k i i k k k j∈S(θi )

(4.29)

When S(θi ) is empty set, we should understand that X bi,ω ,2j−1 B − bi,ω ,2j B := 0. i i k k j∈S(θi )

It is not difficult to show that bi,ω ,2j−1 B − bi,ω ,2j B . bi . ri . i i k k CM O (ωi )

In the case θi ≥ 1, by defining σ, it follows that Z 1 b bi (x) − bi,ω ,A (t)B ωi (x)dt − b ≤ σ(θ ) i,ω ,A (t)B i i i i i i,ωi ,2 Bk k k ωi (2θi −1 Bk ) 2θi −1 Bk

≤

≤

1′ Z ωi (Ai (t)Bk ) ri ωi (2θi −1 Bk )

Ai (t)Bk

1 bi (x) − bi,ω ,A (t)B ri ωi (x)dt ri i i k

ωi (Ai (t)Bk )

bi . ri . CM O (ωi ) ωi (2θi −1 Bk )

Note that Ai (t) = si (t)ai (t). So, we compute n+γi n+γi ωi (Ai (t)Bk ) |si (t)|2k 2 θi 2 k . ≤ θ −1 k n+γ . 1. ωi (2θi −1 Bk ) (2θi −1 2k )n+γi (2 i 2 ) i Consequently, we have b . bi . ri . − b σ(θ ) i,ω ,A (t)B i i i i,ωi ,2 Bk k CM O (ω ) i

(4.30)


20

Otherwise, for θi ≤ 0, by estimating as (4.30), we deduce Z 1 bi (x) − bi,ω ,2θi a (t)B ωi (x)dt b ≤ i,ωi ,2σ(θi ) Bk − bi,ωi ,Ai (t)Bk i i k ωi (Ai (t)Bk ) Ai (t)Bk

ωi (2θi ai (t)Bk ) ωi (Ai (t)Bk )

≤

r1′ i

Z

2θi ai (t)Bk

1 bi (x) − bi,ω ,2θi a (t)B ri ωi (x)dt ri i i k

ωi (2θi ai (t)Bk )

bi . ri . bi CM. Ori (ω ) . (ω ) CM O i i ωi (Ai (t)Bk )

≤

Because of 2θi −1 < |si (t)| ≤ 2θi , we have   log(4|si (t)|), if θi ≥ 0, |θi | + 1 . . ϕAi (t).  log 2 , otherwise. |si (t)|

Therefore, by having (4.29), it follows that

bi,ω ,B − bi,ω ,A (t)B . (|θi | + 1) bi . ri

bi . ri . . ϕ (t). A i i i i k k CM O (ωi ) CM O (ωi )

As above, by (4.28), we get k(n+γi )

ri

bi . ri .

a3,i r ϕ (t). . 2 A i L i (ωi ,B ) CM O (ω ) i

k

From this, by (4.26), (4.27), we finish the proof of the inequality (4.24). Using (4.23) and (4.24), we have m P m γi +n Z

~b

k( ) Φ(t) Y

H ~ (f~)χk q(·) . BCM O,~ω .2 i=1 ri

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

m 0 Y 1 X γi +n

ri ri

fi χk+ℓ +r ζqi (·) dt . × 1 + ψAi ,γi |si (t)| + ϕAi (t) i L ωi

i=1 r=Θ∗n −1

(4.31)

At this point, by making Lemma 2.4 in Section 2 again, we have a similar results to (4.9) as follow m

P γi +n m

~b )Y k( k(λi −αi (0)) k(λi −αi∞ )

H ~ (f~)χk q(·) . C4 .BCM O,~ω .F .2 i=1 ri 2 + 2 . L Φ,A ω

i=1

By using Proposition 2.5 in [29] again, we get

~b

H ~ (f~) . α∗∗ (·),λ . max e1 , sup E Φ,A M K p,q(·),ω

k0 0, we write B := B(0, R) and ∆R as ∆R = ω B

1 q1

∞

~b

H ~ (f~) q(·) . L (B) Φ,A +λ v

By applying the Minkowski inequality for the variable Lebesgue space, we have Z m Y Φ(t) 1

q(·) dt. (4.34) f (A (t)·) b (·) − b (A (t)·) . ∆R . i i i i i 1 Lv (B) n ω(B) q∞ +λ |t| i=1 Rn

By estimating as (4.2) above, we get

m

Y

fi (Ai (t)·) bi (·) − bi (Ai (t)·) Lq(·) v (B) i=1

. Rβ .BLip

m m Y

α

Y

fi (Ai (t)·) qi (·) | · | rii r .

In − Ai (t) βi . L i (B) L (B)

.R

β+

m P

i=1

αi +n ri

vi

i=1

i=1

.BLip

m m Y

Y

In − Ai (t) βi .

fi (Ai (t)·) qi (·) . L (B) i=1

vi

i=1

By (3.14) and the Theorem 2.6, we find

fi (Ai (t)·) qi (·) . cA ,q ,α (t). 1 ϑ (t,·) . fi qi (·) . i i i L 1i L (B) L (B(0,R||A (t)||)) vi

vi

i

(4.35)

(4.36)

By the condition (3.15), we estimate R

β+

m P

i=1

αi +n ri 1

ω(B) q∞ +λ

.

m Y i=1

1

Ai (t) (γi +n)( qi∞ +λi ) 1

ωi (B(0, R||Ai (t)||)) qi∞

+λi

.

(4.37)


23

Thus, by having (4.34)-(4.37) and defining central Morrey spaces with variable exponent, it follows that m Y ∆R . C6 .BLip . kfi k . qi (·),λi B ωi ,vi

i=1

m Q

~b ~ Therefore, we conclude HΦ,A~ (f ) . q(·),λ . C6 .BLip . kfi k . qi (·),λi . B ω,v

B ωi ,vi

i=1

Next, we will prove Theorem 3.7. Indeed, by using the Minkowski inequality and the Hölder inequality for variable Lebesgue spaces again, it is obvious to show that Z m

Φ(t) Y 1

fi (Ai (t).) qi (·) dt.

bi (·) − bi (Ai (t)·) r . ∆R . 1 i (v ,B) L n Lvi (B) i ω(B) q∞ +λ |t| i=1 n R

By (4.24) above, we deduce m Z P αi +n ri i=1 ∆R . R .BCM O,~v Rn

m

1 αi +n Φ(t) Y ri ri . |s (t)| 1 + ψ + ϕ (t) × i A 1 i Ai ,αi n ω(B) q∞ +λ |t| i=1

1

m Y

fi (Ai (t)·) qi (·) dt. × L (B) vi

i=1

For this, by (4.36), we get m P αi +n Z ri BCM O,~v ∆R . Ri=1

m 1 αi +n Φ(t) Y ri ri . |s (t)| 1 + ψ + ϕ (t) × i A 1 i A ,α i i n +λ |t| q∞ ω(B) i=1 Rn

×cAi ,qi,αi (t) 1 Lϑ1i (t,·) . fi Lqi (·) (B(0,R||A (t)||)) dt . (4.38)

1

vi

i

On the other hand, by (3.17), it follows that m P

Ri=1

αi +n ri 1

ω(B) q∞ +λ

.

m Y i=1

1

Ai (t) (γi +n)( qi∞ +λi ) 1

ωi (B(0, R||Ai (t)||)) qi∞

+λi

.

Consequently, by having (4.38), we immediately obtain m

Y

~b ~ kfi k . qi (·),λi ,

HΦ,A~ (f ) . q(·),λ . C7 .BCM O,~v . B ω,v

i=1

B ωi ,vi

which completes the proof.

Acknowledgments. This paper is supported by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2014.51.


24

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