Boundedness of intrinsic square functions and their commutators on

arXiv:1406.4957v1 [math.FA] 19 Jun 2014

Boundedness of intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces Vagif Guliyeva,b,1 , Mehriban Omarovab,c , Yoshihiro Sawanod Abstract. We shall investigate the boundedness of the intrinsic square functions and their commutators on generalized weighted Orlicz-Morrey spaces MwΦ,ϕ (Rn ). In all the cases, the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights ϕ without assuming any monotonicity property of ϕ(x, ·) with x fixed.

1

Introduction

In the present paper, we are concerned with the intrinsic square functions, which Wilson introduced initially [36, 37]. For 0 < α ≤ 1, let Cα be the family of Lipschitz functions φ : Rn → R of order α with the homogeneous norm 1 such in the closed ball {x : |x| ≤ 1}, and that Rthat the support of φ is contained n+1 φ(x)dx = 0. For (y, t) ∈ R and f ∈ L1,loc (Rn ), set n + R Aα f (t, y) ≡ sup |f ∗ φt (y)|, φ∈Cα


where φt ≡ t−n φ t· . Let β be an auxiliary parameter. Then we define the varying-aperture intrinsic square (intrinsic Lusin) function of f by the formula; Gα,β (f )(x) ≡

ZZ

Γβ (x)

dydt (Aα f (t, y))2 n+1 t

! 12

,

where Γβ (x) ≡ {(y, t) ∈ Rn+1 : |x − y| < βt}. Write Gα (f ) = Gα,1 (f ) . + Everywhere in the sequel, B(x, r) stands for the ball in Rn of radius r centered at x and we let |B(x, r)| be the Lebesgue measure of the ball B(x, r); |B(x, r)| = vn rn , where vn is the volume of the unit ball in Rn . We recall generalized weighted Orlicz-Morrey spaces, on which we work in the present paper. Definition 1.1 (Generalized weighted Orlicz-Morrey Space). Let ϕ be a positive measurable function on Rn × (0, ∞), let w be non-negative measurable function on Rn and Φ any Young function. Denote by MwΦ,ϕ (Rn ) the generalized weighted Orlicz-Morrey space, the space of all functions f ∈ LΦ,loc (Rn ) w 1 Corresponding author. 2010 Mathematics Subject Classification. Primary 42B25; Secondary 42B20, 42B35, 46E30. Key words and phrases. Generalized weighted Orlicz-Morrey space, intrinsic square functions, commutator, BMO.

1

such that kf kMwΦ,ϕ ≡

sup x∈Rn ,r>0


ϕ(x, r)−1 Φ−1 w(B(x, r))−1 kf kLΦw (B(x,r)) ,

 n  o R where kf kLΦw (B(x,r)) ≡ inf λ > 0 : B(x,r) Φ |f (x)| w(x) dx ≤ 1 . λ

According to this definition, we recover the generalized weighted Morrey space Mwp,ϕ (Rn ) by the choice Φ(r) = rp , 1 ≤ p < ∞. If Φ(r) = rp , 1 ≤ p < ∞ λ and ϕ(x, r) = r− p , 0 ≤ λ ≤ n, then MwΦ,ϕ (Rn ) coincides with the weighted Morrey space Mwp,ϕ (Rn ) and if ϕ(x, r) = Φ−1 (w(B(x, r)−1 )), then MwΦ,ϕ (Rn ) n Φ n coincides with the weighted Orlicz space LΦ w (R ). When w = 1, then Lw (R ) Φ n Φ n is abbreviated to L (R ). The space L (R ) is the classical Orlicz space. Our first theorem of the present paper is the following one: Theorem 1.2. Let α ∈ (0, 1] and 1 < p0 ≤ p1 < ∞. Let Φ be a Young function which is lower type p0 and upper type p1 . Namely, Φ(st1 ) ≤ Ct1 p1 Φ(s)

Φ(st0 ) ≤ Ct0 p0 Φ(s),

for all s > 0 and 0 < t0 ≤ 1 ≤ t1 < ∞. Assume that w ∈ Ap0 and that the measurable functions ϕ1 , ϕ2 : Rn × (0, ∞) → (0, ∞) and Φ satisfy the condition; Z ∞
dt ϕ1 (x, s)
Φ−1 w(B(x0 , t))−1 ess inf −1 ≤ C ϕ2 (x, r), (1.1) −1 t 0 : λ Rn

See [30, Section 3, Theorem 10] for example. In particular, we have Z  |f (x)|  Φ w(x)dx ≤ 1. kf kLΦw Rn p n If Φ(r) = rp , 1 ≤ p < ∞, then LΦ w = Lw (R ) with norm coincidence. If ∞ n Φ(r) = 0, (0 ≤ r ≤ 1) and Φ(r) = ∞, (r > 1), then LΦ w = Lw (R ). For a Young function Φ and 0 ≤ s ≤ +∞, let

Φ−1 (s) ≡ inf{r ≥ 0 : Φ(r) > s}

(inf ∅ = +∞).

If Φ ∈ Y, then Φ−1 is the usual inverse function of Φ. We also note that Φ(Φ−1 (r)) ≤ r ≤ Φ−1 (Φ(r))

for 0 ≤ r < +∞.

(2.2)

A Young function Φ is said to satisfy the ∆2 -condition, denoted by Φ ∈ ∆2 , if Φ(2r) ≤ kΦ(r) for r > 0 for some k > 1. If Φ ∈ ∆2 , then Φ ∈ Y. A Young function Φ is said to satisfy the ∇2 -condition, denoted also by Φ ∈ ∇2 , if Φ(r) ≤

1 Φ(kr), 2k

r ≥ 0,

for some k > 1. The function Φ(r) = r satisfies the ∆2 -condition and it fails the ∇2 -condition. If 1 < p < ∞, then Φ(r) = rp satisfies both the conditions. The function Φ(r) = er −r−1 satisfies the ∇2 -condition but it fails the ∆2 -condition. Definition 2.3. A Young function Φ is said to be of upper type p (resp. lower type p) for some p ∈ [0, ∞), if there exists a positive constant C such that, for all t ∈ [1, ∞) (resp. t ∈ [0, 1] ) and s ∈ [0, ∞), Φ(st) ≤ Ctp Φ(s). 6

Remark 2.4. If Φ is lower type p0 and upper type p1 with 1 < p0 ≤ p1 < ∞, then Φ ∈ ∆2 ∩ ∇2 . Conversely if Φ ∈ ∆2 ∩ ∇2 , then Φ is lower type p0 and upper type p1 with 1 < p0 ≤ p1 < ∞; see [17] for example. About the norm kf kMwΦ,ϕ , we have the following equivalent expression: If Φ satisfies the ∆2 -condition, then the norm kf kMwΦ,ϕ is equivalent to the norm n kf kM Φ,ϕ (w) ≡ inf λ > 0 :

ϕ(x, r)−1 Φ−1 w(B(x, r))−1

sup x∈Rn ,r>0

×

Z

B(x,r)

Φ

 |f (x)|  λ

o w(x)dx ≤ 1 .




See [22, p. 416]. The latter was used in [22, 25, 26, 31], see also references e we have the following estimate, whose proof is similar to therein. For Φ and Φ, [21, Lemmas 4.2]. So, we omit the details.

e be a positive constant. Suppose Lemma 2.5. Let 0 < p0 ≤ p1 < ∞ and let C that we are given a non-negative measurable function w on Rn and a Young function Φ which is lower type p0 and upper type p1 . Then there exists a positive constant C such that for any ball B of Rn and µ ∈ (0, ∞)   Z |f (x)| e w(x)dx ≤ C Φ µ B implies that kf kLΦw (B) ≤ Cµ.

e For a Young function Φ, the complementary function Φ(r) is defined by  sup{rs − Φ(s) : s ∈ [0, ∞)} if r ∈ [0, ∞), e Φ(r) ≡ (2.3) +∞ if r = +∞.

e e is also a Young function and it satisfies Φ e The complementary function Φ = Φ. Here we recall three examples. Example.

e e 1. If Φ(r) = r, then Φ(r) = 0 for 0 ≤ r ≤ 1 and Φ(r) = +∞ for r > 1.

′ e 2. If 1 < p < ∞, 1/p + 1/p′ = 1 and Φ(r) = rp /p, then Φ(r) = rp /p′ .

e 3. If Φ(r) = er − r − 1, then a calculation shows Φ(r) = (1 + r) log(1 + r) − r. e ∈ ∆2 . It is also known that Note that Φ ∈ ∇2 if and only if Φ e −1 (r) ≤ 2r r ≤ Φ−1 (r)Φ

for r ≥ 0.

(2.4)

Note that Young functions satisfy the properties; Φ(αt) ≤ αΦ(t) 7

(2.5)

for all 0 ≤ α ≤ 1 and 0 ≤ t < ∞, and Φ(βt) ≥ βΦ(t)

(2.6)

for all β > 1 and 0 ≤ t < ∞. The following analogue of the H¨ older inequality is known, see [35]. Theorem 2.6. [35] For a non-negative measurable function w on Rn , a Young e the following inequality is valid function Φ and its complementary function Φ, for all measurable functions f and g: kf gkL1(Rn ) ≤ 2kf kLΦw kw−1 gkLΦe . w

An analogy of Theorem 2.6 for weak type spaces is available. If we define kf kW LΦw ≡ sup λkχ{|f |>λ} kLΦw , λ>0

we can prove the following by a direct calculation: Corollary 2.7. Let Φ be a Young function and let B be a measurable set in 1 Rn . Then kχB kW LΦw = kχB kLΦw = Φ−1 (w(B) −1 ) . In the next sections where we prove our main estimates, we need the following lemma, which follows from Theorem 2.6. Corollary 2.8. For a non-negative measurable function w on Rn , a Young function Φ and a ball B = B(x, r), the following inequality is valid:

1

kf kL1(B) ≤ 2 Φe kf kLΦw (B) . w Lw (B)

Lemma 2.9. Let α ∈ (0, 1] and 1 < p0 ≤ p1 < ∞. Let also Φ be a Young function which is lower type p0 and upper type p1 . Assume in addition w ∈ Ap0 . For a ball B = B(x, r), the following inequality is valid:
kf kL1(B) . |B|Φ−1 w(B)−1 kf kLΦw (B) . Proof. We know that M is bounded on LΦ w (B); see [20]. Thus,

kf kL1(B) kχB kLΦw (B) ≤ kM f kLΦw (B) . kf kLΦw (B) . |B| So, Lemma 2.9 is proved.

2.2

Weighted Hardy operator

We will use the following statement on the boundedness of the weighted Hardy operator Z ∞ g(s)w(s)ds, 0 < t < ∞, Hw∗ g(t) := t

where w is a weight. The following theorem was proved in [9]. In (2.7) and (2.8) below, it will be 1 = 0 and 0 · ∞ = 0. understood that ∞ 8

Theorem 2.10. Let v1 , v2 and w be weights on (0, ∞). Assume that v1 is bounded outside a neighborhood of the origin. Then the inequality sup v2 (t)Hw∗ g(t) ≤ C sup v1 (t)g(t) t>0

(2.7)

t>0

holds for some C > 0 for all non-negative and non-decreasing g on (0, ∞) if and only if Z ∞ w(s)ds < ∞. (2.8) B := sup v2 (t) sup t>0 t s β}| ≤ C1 |B|e−C2 β/kbk∗ , for all β > 0. (2) The following norm equivalence holds: kbk∗ ≈

sup x∈Rn ,r>0

1 |B(x, r)|

Z

p

|b(y) − bB(x,r)| dy B(x,r)

! p1

(2.9)

for 1 < p < ∞. (3) There exists a constant C > 0 such that bB(x,r) − bB(x,t) ≤ Ckbk∗ ln t for 0 < 2r < t, r

(2.10)

where C is independent of b, x, r and t.

3

Intrinsic square functions in MwΦ,ϕ (Rn )

The following lemma generalizes Guliyev’s lemma [4, 5, 6] for Orlicz spaces: Lemma 3.1. Let α ∈ (0, 1] and 1 < p0 ≤ p1 < ∞. Let Φ be a Young function which is lower type p0 and upper type p1 . Assume that the weight belongs to the class w ∈ Ap0 . Then for the operator Gα the following inequality is valid:
Z ∞ Φ−1 w(B(x0 , t))−1 dt
kGα f kLΦw (B) . kf kLΦ(B(x0 ,t)) −1 (3.1) w(B(x0 , r))−1 t Φ 2r for all f ∈ LΦ,loc (Rn ), B = B(x0 , r), x0 ∈ Rn and r > 0. w 9

Proof. With the notation 2B = B(x0 , 2r), we decompose f as f1 (y) ≡ f (y)χ2B (y),

f = f1 + f2 ,

f2 (y) ≡ f (y)χ ∁(2B) (y).

We have kGα f kLΦw (B) ≤ kGα f1 kLΦw (B) + kGα f2 kLΦw (B) n by the triangle inequality. Since f1 ∈ LΦ w (R ), it follows from Theorem 1.3 that

kGα f1 kLΦw (B) ≤ kGα f1 kLΦw (Rn ) . kf1 kLΦw (Rn ) = kf kLΦw (2B) .

(3.2)

So, we can control f1 . Now let us estimate kGα f2 kLΦw (B) . Let x ∈ B = B(x0 , r) and write out Gα f2 (x) in full: 

Gα (f )(x) ≡ 

ZZ

sup |f2 ∗ φt (y)|

Γ(x)

φ∈Cα

!2

 21 dydt  . tn+1

(3.3)

Let (y, t) ∈ Γ(x). We next write the convolution f2 ∗ φt (y) out in full:   Z Z y−z 1 −n |f2 ∗ φt (y)| = t φ f2 (z)dz . n |f2 (z)|dz. (3.4) t |y−z|≤t t |y−z|≤t ∁

Recall that the suppost of f is contained in (2B). Keeping this in mind, let ∁ z ∈ B(y, t) ∩ (2B). Since (y, t) ∈ Γ(x), we have |z − x| ≤ |z − y| + |y − x| ≤ 2t.

(3.5)

Another geometric observation shows r = 2r − r ≤ |z − x0 | − |x0 − x| ≤ |x − z|. Thus, we obtain 2t ≥ r

(3.6)

from (3.5). So, putting together (3.3)–(3.6), we obtain

Gα f2 (x)



. 



≤  

. 

1 2 Z dydt 2 −n |f2 (z)|dz n+1  t t Γ(x) |y−z|≤t

Z Z Z

t>r/2

Z

t>r/2

Z

|x−y| |z−x| 2

t2n+1

! 21

|f (z)|dz.

Thanks to (3.7), we have Gα f2 (x)

Z

.

|z−x0 |>2r

Z

|f (z)| dz |z − x|n

|f (z)| dz n |z−x0 |>2r |z − x0 | Z Z +∞ |f (z)|

. =

|z−x0 |>2r

Z

=

∞

2r

Z

|z−x0 |

|f (z)|dz

B(x0 ,t)

dt

!

tn+1

!

dz

dt . tn+1

If we invoke Lemma 2.9, then we obtain Z ∞
dt . Gα f2 (x) . kf kLΦw (B(x0 ,t)) Φ−1 w(B(x0 , t))−1 t 2r

(3.8)

Moreover,

kGα f2 kLΦw (B) .

Z

∞

2r


Φ−1 w(B(x0 , t))−1 dt
. kf kLΦw (B(x0 ,t)) −1 Φ w(B(x0 , r))−1 t

(3.9)

Thus, it follows from (3.2) and (3.8) that
Z ∞ Φ−1 w(B(x0 , t))−1 dt
. (3.10) kGα f kLΦw (B) . kf kLΦw (2B) + kf kLΦw (B(x0 ,t)) −1 w(B(x0 , r))−1 t Φ 2r On the other hand, by (2.4) we get

and hence

Z ∞

dt Φ−1 w(B(x0 , r))−1 ≈ Φ−1 w(B(x0 , r))−1 rn n+1 t 2r Z ∞
−1 dt −1 w(B(x0 , t)) Φ . t 2r

kf kLΦw (2B) .

Z

∞

2r


Φ−1 w(B(x0 , t))−1 dt
. kf kLΦw (B(x0 ,t)) −1 Φ w(B(x0 , r))−1 t 11

(3.11)

Thus, it follows from (3.10) and (3.11) that kGα f kLΦw (B) .

Z

∞

2r

So, we are done.


Φ−1 w(B(x0 , t))−1 dt
. kf kLΦw (B(x0 ,t)) −1 Φ w(B(x0 , r))−1 t

With this preparation, we can prove Theorem 1.2 Proof. Fix x ∈ Rn . Write v1 (r) ≡ ϕ1 (x, r)−1 ,

v2 (r) ≡

g(r) ≡ kf kLΦw (B(x0 ,r)) ,

1 ϕ2 (x, r)Φ−1 (w(B(x0 , r))−1 ) Φ−1 (w(B(x0 , r))−1 )

ω(r) ≡

,

.

r

We omit a routine procude of truncation to justify the application of Theorem 2.10. By Lemma 3.1 and Theorem 2.10, we have kGα f kMwΦ,ϕ2 (Rn )

Z ∞
dt 1 kf kLΦw (B(x0 ,t)) Φ−1 w(B(x0 , t))−1 . sup n ϕ (x, r) t x∈R , r>0 2 r
1 . sup Φ−1 w(B(x0 , r))−1 kf kLΦw (B(x0 ,r)) n ϕ (x, r) x∈R , r>0 1 = kf kM Φ,ϕ1 .

So we are done. The following lemma is an easy consequence of the monotonicity of the norm k · kLΦw and Wilson’s estimate; Gα,β (f )(x) ≤ β

3n 2 +α

Gα (f )(x)

(x ∈ Rn ),

which was proved in [36]. Lemma 3.2. For j ∈ Z+ , denote Gα,2j (f )(x) ≡

Z

0

∞

Z

|x−y|≤2j t

dydt (Aα f (t, y))2 n+1 t

Let Φ be a Young function and 0 < α ≤ 1. Then we have kGα,2j (f )kLΦw . 2j( n for all f ∈ LΦ w (R ).

12

3n 2 +α)

kGα (f )kLΦw

! 21

∗ Now we can prove Thoerem 1.4. We write gλ,α (f )(x) out in full:

∗ [gλ,α (f )(x)]2 =

ZZ

ZZ

+

Γ(x)

∁

Γ(x)



t t + |x − y|

nλ

(Aα f (t, y))2

dydt := I + II. tn+1

As for I, a crude estimate suffices; ZZ dydt I≤ (Aα f (t, y))2 n+1 ≤ (Gα f (x))2 . t Γ(x)

(3.12)

Thus, the heart of the matters is to control II. We decompose the ambient space Rn : II

≤ .

∞ Z X

j=1 0 ∞ Z ∞ X j=1

.

∞

0

∞ ZZ X j=1

Z

2j−1 t≤|x−y|≤2j t

Z



t t + |x − y|

nλ

(Aα f (t, y))2

2−jnλ (Aα f (t, y))2

2j−1 t≤|x−y|≤2j t

dydt tn+1

dydt tn+1

∞

Γ2j (x)

X (Gα,2j (f )(x))2 (Aα f (t, y))2 dydt := . 2jnλ tn+1 2jnλ j=1

(3.13)

Thus, putting together (3.12) and (3.13), we obtain ∗ kgλ,α (f )kM Φ,ϕ2 . kGα f kM Φ,ϕ2 + w

w

∞ X

2−

jnλ 2

kGα,2j (f )kM Φ,ϕ2 . w

(3.14)

j=1

By Theorem 1.2, we have kGα f kMwΦ,ϕ2 (Rn ) . kf kMwΦ,ϕ1 (Rn ) .

(3.15)

In the sequel, we will estimate kGα,2j (f )kMwΦ,ϕ2 . We divide kGα,2j (f )kLΦw (B) into two parts: kGα,2j (f )kLΦw (B) ≤ kGα,2j (f1 )kLΦw (B) + kGα,2j (f2 )kLΦw (B) ,

(3.16)

where f1 (y) ≡ f (y)χ2B (y) and f2 (y) ≡ f (y) − f1 (y). For kGα,2j (f1 )kLΦw (B) , by Lemma 3.2 and (3.11), we have (see also, [8, p. 47, (5.4)]) kGα,2j (f1 )kLΦw (B) . 2j(

3n 2 +α)

kGα (f1 )kLΦw (Rn )

j( 3n 2 +α)

kf kLΦw (2B) (3.17)
Z ∞ −1 −1 Φ w(B(x0 , t)) dt 3n
. kf kLΦw (B(x0 ,t)) −1 . 2j( 2 +α) −1 t Φ w(B(x0 , r)) 2r (3.18)

.2

13

For kGα,2j (f2 )kLΦw (B) , we first write the quantity out in full: Gα,2j (f2 )(x)

ZZ

=

Γ2j (x)

=

dydt (Aα f (t, y))2 n+1 t

 ZZ 

! 12

sup |f ∗ φt (y)|

φ∈Cα

Γ2j (x)

 21

!2

dydt  . tn+1

A geometric observation shows that

Gα,2j (f2 )(x)

 

.

ZZ

Z

Γ2j (x)

|f2 (z)|dz

|z−y|≤t

!2

Since |z − x| ≤ |z − y| + |y − x| ≤ 2j+1 t, we get

Gα,2j (f2 )(x)

 ZZ 

.

Z

Γ2j (x)

 Z 

.

Z

∞

2

.

Z

|f2 (z)|dz

|z−x|≤2j+1 t

0

jn 2

|f2 (z)|dz |z−x|≤2j+1 t

Rn

Z

∞ |z−x| 2j+1

|f2 (z)|2 dt t2n+1

! 21

!2

!2

 21 dydt  . t3n+1

 21 dydt  t3n+1  21

jn

2 dt  t2n+1

dz . 2

3jn 2

Z

∁

B(x0 ,2r)

|f (z)|dz . |z − x|n

A geometric observation shows 1 1 |z − x| ≥ |z − x0 | − |x0 − x| ≥ |z − x0 | − |z − x0 | = |z − x0 |. 2 2 Thus, we have Gα,2j (f2 )(x) . 2

Z

3jn 2

|z−x0 |>2r

|f (z)| dz. |z − x0 |n

By Fubini’s theorem and Lemma 2.9, we obtain Gα,2j (f2 )(x)

.

2

3jn 2

Z

|f (z)|

|z−x0 |>2r

. .

2 2

3jn 2

3jn 2

Z

∞

2r Z ∞ 2r

Z

Z

∞

|z−x0 |

|f (z)|

|z−x0 |0


Φ−1 w(B(x0 , t))−1 dt
. Φ−1 w(B(x0 , r))−1 t

(3.19)


Φ−1 w(B(x0 , t))−1 dt
. kf kLΦw (B(x0 ,t)) −1 w(B(x0 , r))−1 t Φ Z

∞

Φ−1 w(B(x0 , t))−1

r


kf kLΦw (B(x,t)) dt . ϕ2 (x0 , r) t

Thus by Theorem 2.10 we have kGα,2j f kM Φ,ϕ2 (Rn ) . 2

j( 3n 2 +α)

=2

j( 3n 2 +α)

w


Φ−1 w(B(x0 , r))−1 sup kf kLΦw (B(x,r)) ϕ1 (x0 , r) x0 ∈Rn r>0

kf kMwΦ,ϕ1 (Rn ) .

(3.20)

2α , by (3.14), (3.15) and (3.20), we can conclude the proof of the Since λ > 3 + n theorem.

4

Commutators of the intrinsic square functions in MwΦ,ϕ (Rn)

We start with a characterization of the BMO norm. Lemma 4.1. Let 0 < p0 ≤ p1 < ∞. Let b ∈ BMO(Rn ) and Φ be a Young function which is lower type p0 and upper type p1 . Then


kbk∗ ≈ sup Φ−1 w(B(x, r))−1 b − bB(x,r) LΦ (B(x,r)) . w

x∈Rn ,r>0

Proof. By H¨ older’s inequality, we have kbk∗ .

sup x∈Rn ,r>0

Now we show that sup x∈Rn ,r>0


Φ−1 w(B(x, r))−1 b − bB(x,r) LΦ (B(x,r)) . w


Φ−1 w(B(x, r))−1 b − bB(x,r) LΦ (B(x,r)) . kbk∗ . w

15

Without loss of generality, we may assume that kbk∗ = 1; otherwise, we replace b by b/kbk∗. By the fact that Φ is lower type p0 and upper type p1 and (2.2) it follows that
! Z |b(y) − bB(x,r) |Φ−1 |B(x, r)|−1 Φ dy kbk∗ B(x,r) Z

Φ |b(y) − bB(x,r) |Φ−1 |B(x, r)|−1 dy = B(x,r) Z   1 . |b(y) − bB(x,r)|p0 + |b(y) − bB(x,r) |p1 dy . 1. |B(x, r)| B(x,r) By Lemma 2.5 we get the desired result.

Remark 4.2. Note that a counterpart to Lemma 4.1 for the variable exponent Lebesgue space Lp(·) case was obtained in [13]. Lemma 4.3. Let α ∈ (0, 1], 1 < p0 ≤ p1 < ∞ and b ∈ BMO(Rn ). Let Φ be a Young function which is lower type p0 and upper type p1 . Then the inequality k[b, Gα ]f kLΦw (B(x0 ,r)) .

Φ−1

kbk∗
w(B(x0 , r))−1

Z

∞

2r



1 + ln


dt t kf kLΦw (B(x0 ,t)) Φ−1 w(B(x0 , t))−1 r t

holds for any ball B(x0 , r) and for any f ∈ LΦ,loc (Rn ). w

Proof. For an arbitrary x0 ∈ Rn , set B ≡ B(x0 , r) for the ball centered at x0 and of radius r. Write f = f1 + f2 with f1 ≡ f χ2B and f2 ≡ f χ ∁ . (2B)

We have k[b, Gα ]f kLΦw (B) ≤ k[b, Gα ]f1 kLΦw (B) + k[b, Gα ]f2 kLΦw (B) by the triangle n inequality. From Theorem 1.6, the boundedness of [b, Gα ] in LΦ w (R ) it follows that k[b, Gα ]f1 kLΦw (B) ≤ k[b, Gα ]f1 kLΦw (Rn ) . kbk∗ kf1 kLΦw (Rn ) = kbk∗ kf kLΦw (2B) . For k[b, Gα ]f2 kLΦw (B) , we write it out in full [b, Gα ]f2 (x) =

!1 2 dydt 2 . [b(y) − b(z)]φt (y − z)f2 (z)dz n+1 t Rn

Z sup

ZZ

Γ(x) φ∈Cα

We then divide it into two parts: [b, Gα ]f2 (x)

≤

:=

!1 2 dydt 2 [b(y) − bB ]φt (y − z)f2 (z)dz n+1 t Γ(x) φ∈Cα Rn !1 Z 2 ZZ dydt 2 + sup [bB − b(z)]φt (y − z)f2 (z)dz n+1 t Γ(x) φ∈Cα Rn ZZ

Z sup

A + B.

16

First, for the quantity A, we proceed as follows: ZZ

A =

Γ(x)∩Rn ×[r,∞)

.

 ZZ 

|b(y) − bB |2

Γ(x)∩Rn ×[r,∞)

Note that

Z

B(x,t)

!1 2 dydt 2 φt (y − z)f2 (z)dz n+1 t Rn α 1  !2 2 Z dydt  1 . |f (z)| dz tn B(x,t) tn+1

Z |b(y) − bB | sup φ∈C 2

|f (z)| dz . |B(x, t)|Φ−1 (w(B(x, t)−1 )kf kLΦw (B(x,t)) .

Thus, by virtue of the embedding ℓ2 (N) ֒→ ℓ1 (N), we obtain ZZ

dydt A. |b(y) − bB |2 Φ−1 (w(B(x, t)−1 )2 kf kLΦw (B(x,t)) 2 n+1 t n Γ(x)∩R ×[r,∞) ! 21  2 Z ∞ t dt . Φ−1 (w(B(x, t)−1 )2 log 2 + kf kLΦw (B(x,t)) 2 r t r   12 ∞ X
2 . Φ−1 (w(B(x, 2j r)−1 )2 log 2 + 2j kf kLΦw (B(x,2j r)) 2 

! 12

j=1

.

∞ X


Φ−1 (w(B(x, 2j r)−1 ) log 2 + 2j kf kLΦw (B(x,2j r))

j=1 ∞

.

Z

r

  t dt Φ−1 (w(B(x, t)−1 ) log 2 + kf kLΦw (B(x,t)) . r t

For the quantity B, since |y − x| < t, we have |x − z| < 2t. Thus, by Minkowski’s inequality, we have a pointwise estimate: 1  Z 2 ZZ dydt 2 B ≤  |bB − b(z)||f2 (z)|dz 3n+1  t Γ(x) B(x,2t) . .

  Z

Z

0

∁

∞

 21 Z 2 dt |bB − b(z)||f2 (z)|dz 2n+1  B(x,2t) t

B(x0 ,2r)

|bB − b(z)||f (z)| dz. |x − z|n

Thus, we have

Z

kBkLΦw (B) .

∁

(2B)

|b(z) − bB |

|f (z)|dz

Φ . |x0 − z|n Lw (B) 17

1 |z − x0 |, we obtain 2 Z |b(z) − bB | 1
|f (z)|dz n ∁ Φ−1 w(B(x0 , r))−1 (2B) |x0 − z| Z ∞ Z 1 dt
|b(z) − bB ||f (z)| dz n+1 ∁ t Φ−1 w(B(x0 , r))−1 (2B) |x0 −z| ! Z ∞ Z dt 1
|b(z) − bB ||f (z)|dz n+1 t Φ−1 w(B(x0 , r))−1 2r 2r≤|x0 −z|≤t ! Z ∞ Z 1 dt
|b(z) − bB ||f (z)|dz n+1 . −1 −1 t w(B(x0 , r)) Φ 2r B(x0 ,t)

Since |z − x| ≥ kBkLΦw (B) . ≈

≈

.

We decompose the matters by using the trinangle inequality: ! Z ∞ Z 1 dt
kBkLΦw (B) . −1 |b(z) − bB(x0 ,t) ||f (z)|dz n+1 −1 t w(B(x0 , r)) Φ 2r B(x0 ,t) ! Z Z ∞ |bB − bB(x0 ,t) | dt
|f (z)|dz n+1 + −1 −1 t w(B(x , r)) Φ 0 B(x0 ,t) 2r

Applying H¨ older’s inequality, by Lemma 4.1 and (2.10) we get Z ∞

kf kLΦw (B(x0 ,t)) dt

|b − bB(x ,t) |w(·)−1 Φe
kBkLΦw (B) . 0 Lw (B) tn+1 Φ−1 w(B(x , r))−1 0 2r
Z ∞ Φ−1 w(B(x0 , t))−1 dt
+ |bB − bB(x0 ,t) |kf kLΦw (B(x0 ,t)) −1 w(B(x0 , r))−1 t Φ 2r
Z ∞ Φ−1 w(B(x0 , t))−1 dt t
. . kbk∗ 1 + ln kf kLΦw (B(x0 ,t)) −1 r w(B(x0 , r))−1 t Φ 2r

Summing kAkLΦw (B) and kBkLΦw (B) , we obtain k[b, Gα ]f2 kLΦw (B) .

Φ−1

Finally,

kbk∗
w(B(x0 , r))−1

Z

∞

2r


dt t 1 + ln kf kLΦw (B(x0 ,t)) Φ−1 w(B(x0 , t))−1 . r t

k[b, Gα ]f kLΦw (B) . kbk∗ kf kLΦw (2B) Z ∞
dt t kbk∗
kf kLΦw (B(x0 ,t)) Φ−1 w(B(x0 , t))−1 1 + ln , + −1 −1 r t w(B(x0 , r)) Φ 2r

and the statement of Lemma 4.3 follows by (3.11).

Finally, Theorem 1.7 follows by Lemma 4.3 and Theorem 2.10 in the same manner as in the proof of Theorem 1.2.

18

5

Acknowledgements

The research of V. Guliyev and F. Deringoz was partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003) and (PYO.FEN.4003-2.13.007). We thank the referee for he/her valuable comments to the paper.

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[27] F. John, L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. ¨ [28] W. Orlicz, Uber eine gewisse Klasse von R¨ aumen vom Typus B, Bull. Acad. Polon. A (1932), 207–220; reprinted in: Collected Papers, PWN, Warszawa (1988), 217–230. ¨ [29] W. Orlicz, Uber R¨ aume (LM ), Bull. Acad. Polon. A (1936), 93–107, reprinted in: Collected Papers, PWN, Warszawa (1988), 345–359. [30] M.M. Rao, Z.D. Ren, Theory of Orlicz Spaces, M. Dekker, Inc., New York, 1991. [31] Y. Sawano, S. Sugano, H. Tanaka, Orlicz-Morrey spaces and fractional operators, Potential Anal. 36 (2012), no. 4, 517–556. [32] H. Wang, Intrinsic square functions on the weighted Morrey spaces, J. Math. Anal. Appl. 396 (2012), 302–314. [33] H. Wang, Boundedness of intrinsic square functions on the weighted weak Hardy spaces, Integr. Equ. Oper. Theory 75 (2013), 135–149. [34] H. Wang, H.P. Liu, Weak type estimates of intrinsic square functions on the weighted Hardy spaces, Arch. Math. 97 (2011), 49–59. [35] G. Weiss, A note on Orlicz spaces, Portugal Math. 15 (1956), 35–47. [36] M. Wilson, The intrinsic square function, Rev. Mat. Iberoam. 23 (2007), 771–791. [37] M.Wilson, Weighted Littlewood-Paley theory and Exponential-square integrability, Lecture Notes in Math. vol. 1924, Springer-Verlag, 2007. [38] X. Wu, Commutators of Intrinsic Square Functions on Generalized Morrey Spaces, Journal of Inequalities and Applications, 2014:128 (2014)

a

Department of Mathematics, Ahi Evran University, Kirsehir, Turkey Institute of Mathematics and Mechanics, Baku, Azerbaijan E-mail address: [email protected] b

c

Baku State University, Baku, AZ 1148, Azerbaijan E-mail address: mehriban [email protected] d

Department of Mathematics and Information Science, Tokyo Metropolitan University E-mail address: [email protected]

21