Available online www.jsaer.com Journal of Scientific and Engineering Research, 2017, 4(2):127-144 Research Article

ISSN: 2394-2630 CODEN(USA): JSERBR

Adams-Spanne type estimates for the commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups FERİT GÜRBÜZ

Ankara University, Faculty of Science, Department of Mathematics, Tando g an 06100, Ankara, Turkey Abstract In this paper we give BMO (bounded mean oscillation) space estimates for commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are also formulated in terms of Zygmund type integral inequalities. Keywords Heisenberg group; sublinear operator; fractional integral operator; fractional maximal operator; commutator; BMO space; generalized Morrey space 1 . Introduction and Main Results Heisenberg groups play an important role in several branches of mathematics, such as quantum physics, Fourier analysis, several complex variables, geometry and topology; see [23] for more details. It is a remarkable fact that the Heisenberg group, denoted by

H n , arises in two aspects. On the one hand, it can be realized as the

boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental notions of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in the theory of theta functions and related parts of analysis. Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hörmander’s condition. Due to this reason, many interesting works have been devoted to the theory of harmonic analysis on

H n in [6, 8, 9, 19, 20, 23, 26, 27].

We start with some basic knowledge about Heisenberg group in generalized Morrey spaces and refer the reader to [8, 11, 9, 23] and the references therein for more details. The Heisenberg group

H n is a non-commutative

R2 n R and the group structure is given by n x, t x' , t' = x x' , t t' 2x2 j x'2 j 1 x2 j 1 x'2 j . j =1 Using the coordinates g = x, t for points in H n , the left-invariant vector fields for this group structure are nilpotent Lie group, with the underlying manifold

X 2 j 1 =

2 x2 j , x2 j 1 t

j = 1,, n,

X2j =

2 x2 j 1 , x2 j t

j = 1,, n.

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Journal of Scientific and Engineering Research, 2017, 4(2):127-144

These vector fields generate the Lie algebra of satisfy the relation

H n and the commutators of the vector fields X 1 ,, X 2 n

X , X = 4 X n j

j

2 n 1

j = 1,, n,

,

with all other brackets being equal to zero. The inverse element of

g = x, t is g 1 = x,t and we denote the identity (neutral) element of H n as

e = 0,0 R2n1 . The Heisenberg group is a connected, simply connected nilpotent Lie group. Oneparameter Heisenberg dilations

r : Hn Hn

are given by

r x, t = rx, r 2t

for each real number

r > 0 . The Haar measure on H n also coincides with the usual Lebesgue measure on R 2 n1 . These dilations are group automorphisms and Jacobian determinant of

r

where Q = 2n 2 is the homogeneous dimension of

with respect to the Lebesgue measure is equal to

rQ ,

H n . We denote the measure of any measurable set

H n by . Then

r = r Q , The homogeneous norm on

d r x = r Q dx.

H n is defined as follows 1/4

x

Hn

= x1 ,, x2 n , x2 n 1 H

and the Heisenberg distance is given by

n

2 n 2 = x 2j x22n 1 , j =1

d g , h = d g 1h,0 = g 1h .

This distance d is left-invariant in the sense that d g , h = g h remains unchanged when 1

both left-translated by some fixed vector on 320)

H n . Moreover, d satisfies the triangular inequality (see [15], page

d g , h d g , x d x, h,

Using this norm, we define the Heisenberg ball

g , x, h H n .

Bg , r = h H n : g 1h < r

with center by

g and h are

g = x, t and radius r and denote by BC g , r = H n \ Bg , r its complement, and we denote

Br = Be, r = h H n : h < r the open ball centered at e , the identity (neutral) element of H n , with

radius

r . The volume of the ball Bg , r is cQ r Q , where cn is the volume of the unit ball B1 : n 2 cQ = Be,1 = . n 1 n 1n 2 2

n

1 2

For more details about Heisenberg group, one can refer to [8]. In the study of local properties of solutions to of second order elliptic partial differential equations (PDEs), together with weighted Lebesgue spaces, Morrey spaces

Lp , H n play an important role, see [10, 16]. They

were introduced by Morrey in 1938 [18]. For the properties and applications of classical Morrey spaces, see [4, 5, 13] and the references therein. We recall its definition on a Heisenberg group as

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L p , H n = f : f L H = sup r p f p , n gH n ,r > 0 loc where f L p ( H n ) , 0 Q and 1 p < .

Note that

L p ( B ( g ,r ))

Lp ,0 = Lp ( H n ) and Lp,Q = L ( H n ) . If < 0 or > Q , then Lp , = , where is the set

of all functions equivalent to 0 on that

< ,

H n . It is known that Lp , ( H n ) is an expansion of Lp ( H n ) in the sense

Lp ,0 = Lp ( H n ) .

We also denote by

WLp, WLp, ( H n ) the weak Morrey space of all functions f WLloc p ( H n ) for which f

where

WL p ,

f

= sup r

WL p , ( H n )

p

f

gH n ,r > 0

< ,

WL p ( B ( g ,r ))

WLp ( B( g , r )) denotes the weak L p -space of measurable functions f for which f

WL p ( B ( g , r ))

f

B ( g , r ) WL ( H ) p n

= sup h B( g , r ) :| f ( h ) |>

1/p

>0

=

sup 1/p f B ( g , r ) ( ) < .

0 0

In the case of function

= 0 , the fractional

| f (h) | dh.

B ( g ,r )

maximal function

M f coincides with the Hardy-Littlewood maximal

Mf M 0 f (see [8, 23]) and is closely related to the fractional integral T f g =

Hn

The operators

f h

g 1h

Q

dh

0 < < Q.

M and T play an important role in real and harmonic analysis (see [7, 8, 23, 26]).

The classical Riesz potential

I is defined on R n by the formula I f =

where

Q

2

f,

0 < < n,

is the Laplacian operator. It is known that 1 f y I f x = dy T f x , Rn x y n

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2 . The Riesz potential on the Heisenberg group is defined in terms of the sub where = 2 n 2 Laplacian L = H . n 2

n

Definition 1 For 0 < < Q the Riesz potential

I is defined by on the Schwartz space S H n by

the formula

I f g = L

2

1

f g e rL f g r 2 dr , 0

where

e rL f g =

is the semigroups of the operator

1 K r h, g f h d h H n 2

L.

In [26], relations between the Riesz potential and the heat kernel on the Heisenberg group are studied. The following assertion

[26], Theorem 4.2

yields an expression for

I , which allows us to discuss the

boundedness of the Riesz potential. Theorem 1 Let

qs g be the heat kernel on H n . If 0 < Q , then for f S H n 1 1 2 I f g = s qs ds f g . 0 2

The Riesz potential

I satisfies the estimate [26], Theorem 4.2 I f g ≲ T f g

which provides a suitable estimate for the Riesz potential on the Heisenberg group. It is well known that, see [8, 23] for example,

T is bounded from L p H n to Lq H n for all p > 1 and

1 1 = p q

Q (i.e. Hardy-Littlewood Sobolev inequality). > 0 , and T is also of weak type 1, Q Q Spanne (published by Peetre [21]) and Adams [1] have studied boundedness of the fractional integral

operator

T on Lp , R n . This result has been reproved by Chiarenza and Frasca [3], and also studied in [12]. After studying Morrey spaces in detail, researchers have passed to generalized Morrey spaces. Recall

that the concept of the generalized Morrey space

M p, M p, ( H n ) on Heisenberg group has been

introduced in [11].

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Definition 2 [11] Let We denote by

( g , r ) be a positive measurable function on H n (0,) and 1 p < .

M p, M p, ( H n ) the generalized Morrey space, the space of all functions f Lloc p (H n )

with finite quasinorm

f Also by

= sup ( g , r ) | B( g , r ) | 1

M p ,

1 p

gH n ,r >0

f

L p ( B ( g ,r ))

.

WM p, WM p, ( H n ) we denote the weak generalized Morrey space of all functions

f WLloc p ( H n ) for which f

= sup ( g , r ) | B( g , r ) | 1

WM p ,

1 p

f

gH n ,r >0

According to this definition, we recover the Morrey space

WL p ( B ( g ,r ))

< .

L p , and weak Morrey space WLp , under

Q

the choice

(g, r) = r

p

:

Lp , = M p , |

Q

,

WLp , = WM p , |

p

( g ,r )= r

( g ,r )= r

In [11], Guliyev et al. prove the Spanne type boundedness of Riesz potentials

1

the homogeneous dimension of

I , 0, Q from one

2

1 1 = , Q is p q Q

H n and from the space M 1, H n to the weak space WM1, H n , where 1

2

1 = . They also prove the Adams type boundedness of the Riesz potentials I , q Q

0, Q from M p ,

to the weak space

.

p

M p , H n to another M q , H n , where 1 < p < q < ,

generalized Morrey space

1 < q < , 1

Q

1 p

WM 1,

H n to another

M q ,

1 q

1 q

H n for 1 < p < q < and from the space

M1, H n

H n for 1 < q < .

For a locally integrable function b on

H n , suppose that the commutator operator Tb , , 0, Q

represents a linear or a sublinear operator, which satisfies that for any

f L1 ( H n ) with compact support and

x suppf

| Tb, f ( g ) | c0

Hn

where

| f ( h) | dh, 1 Q h|

| b( g ) b( h) | | g

(1.1)

c0 is independent of f and g .

The condition (1.1) is satisfied by many interesting operators in harmonic analysis, such as fractional maximal operator, fractional Marcinkiewicz operator, fractional integral operator and so on (see [17], [22] for details). Let

T be a linear operator. For a locally integrable function b on H n , we define the commutator

[b, T ] by [b,T ] f ( x) = b( x)Tf ( x) T (bf )( x)

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for any suitable function f . Let b be a locally integrable function on

H n , then for 0 < < Q , we define the linear commutator

generated by fractional integral operator and b and the sublinear commutator of the fractional maximal operator as follows, respectively (see also [17]).

[b, T ] f ( g ) b( g )T f ( g ) T (bf )( g ) =

[b( g ) b(h)] | g

Hn

M b, f ( g ) = sup | B( g , r ) | r >0

1

Q

f ( h) dh, h |Q

1

bg bh | f (h) | dh. B ( g ,r )

Now, we will examine some properties related to the space of functions of Bounded Mean Oscillation, BMO , introduced by John and Nirenberg [14] in 1961. This space has become extremely important in various areas of analysis including harmonic analysis, PDEs and function theory. BMO -spaces are also of interest since, in the scale of Lebesgue spaces, they may be considered and appropriate substitute for

L . Appropriate in the sense

that are spaces preserved by a wide class of important operators such as the Hardy-Littlewood maximal function, the Hilbert transform and which can be used as an end point in interpolating Let us recall the defination of the space of

Definition 3 Suppose that

L p spaces.

BMO( H n ) (see, for example, [8, 17, 24]).

b L1loc ( H n ) , let 1 | b(h) bB( g ,r ) | dh < , gH n ,r >0 | B( g , r ) | B ( g ,r )

b = sup

(1.2)

where

bB ( g ,r ) =

1 b(h)dh. | B( g , r ) | B (g ,r )

Define

BMO( H n ) = {b L1loc ( H n ) : b < }. Endowed with the norm given in (1.2), which differ a.e. by constant; clearly,

Remark 1 Note that

BMO( H n ) becomes Banach space provided we identify functions

b = 0 for bh = c a.e. in H n .

L ( H n ) is contained in BMO( H n ) and we have b 2 b .

Moreover, BMO contains unbounded functions, in fact the function log bounded, so

L ( H n ) BMO( H n ) .

Remark 2 (1) all

h on H n , is in BMO but it is not

The John-Nirenberg inequality [14]: there are constants

C1 , C2 > 0 , such that for

b BMO( H n ) and > 0

g B:| b( g ) bB |> C1 | B | e C2/ b ,

B H n .

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(2) The John-Nirenberg inequality implies that 1

p 1 p b sup | b(h) bB( g ,r ) | dh gH n ,r > 0 | B ( g , r ) | B ( g ,r )

(1.3)

for 1 < p < .

(3) Let b BMO( H n ) . Then there is a constant C > 0 such that

bB ( g ,r ) bB ( g , ) C b ln for 0 < 2r < , r where C is independent of b , g , r and .

b BMOH n and Tb , , 0, Q satisfying

Inspired by [11], in this paper, provided that

condition (1.1) is a sublinear operator, we find the sufficient conditions on the pair

(1 , 2 ) which ensures the

Tb , from M p , H n to M q , H n , where

Spanne type boundedness of the commutator operators

1< p < q < , 0 <

2r

>2r

f w dw

f w dw

r sup t Q bB g ,r bB g , f Q

1 = 1 we get p

B g ,

Q q

≲ b r q sup 1 ln t

f

L p B g ,

Bg , B g ,

L p ( B ( g , ))

1

1 p

.

J 2 note that

J 2 = b bB g ,

sup t Q

Lq B g , >2r

f w dw.

B g,

By (1.3), we get Q q

≲ b r sup t Q J 2 >2r

f w dw.

B g ,

Thus, by (2.2) Q q

≲ b r sup t J 2 Summing up

Q q

Lq B

Q q

>2r

J1 and J 2 , for all p 1, we get M b, f 2

≲ b r sup t >2r

Q q

f

L p B g ,

.

1 ln f r

L p B g ,

.

(2.6)

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Finally, we have the following

M b, f 2

Lq B

Q q

≲ b f

≲ b r sup t

Q q

>2r

Q q

b r sup t

L p 2 B

>2r

1 ln f r

L p B g ,

Q q

1 ln f r

L p B g ,

,

which completes the proof. Similarly to Lemma 2 the following lemma can also be proved. Lemma 3 Let 1 < p < ,

b BMOH n and M b is bounded on Lp ( H n ) . Then the inequality

p q ≲ b r 1 ln f sup L p ( B ( g ,r )) r >2r Q

Mb f

Q

L p ( B ( g , ))

f Lloc p (H n ) .

holds for any ball B( g , r ) and for all The following theorem is true.

0

ISSN: 2394-2630 CODEN(USA): JSERBR

Adams-Spanne type estimates for the commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups FERİT GÜRBÜZ

Ankara University, Faculty of Science, Department of Mathematics, Tando g an 06100, Ankara, Turkey Abstract In this paper we give BMO (bounded mean oscillation) space estimates for commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are also formulated in terms of Zygmund type integral inequalities. Keywords Heisenberg group; sublinear operator; fractional integral operator; fractional maximal operator; commutator; BMO space; generalized Morrey space 1 . Introduction and Main Results Heisenberg groups play an important role in several branches of mathematics, such as quantum physics, Fourier analysis, several complex variables, geometry and topology; see [23] for more details. It is a remarkable fact that the Heisenberg group, denoted by

H n , arises in two aspects. On the one hand, it can be realized as the

boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental notions of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in the theory of theta functions and related parts of analysis. Analysis on the groups is also motivated by their role as the simplest and the most important model in the general theory of vector fields satisfying Hörmander’s condition. Due to this reason, many interesting works have been devoted to the theory of harmonic analysis on

H n in [6, 8, 9, 19, 20, 23, 26, 27].

We start with some basic knowledge about Heisenberg group in generalized Morrey spaces and refer the reader to [8, 11, 9, 23] and the references therein for more details. The Heisenberg group

H n is a non-commutative

R2 n R and the group structure is given by n x, t x' , t' = x x' , t t' 2x2 j x'2 j 1 x2 j 1 x'2 j . j =1 Using the coordinates g = x, t for points in H n , the left-invariant vector fields for this group structure are nilpotent Lie group, with the underlying manifold

X 2 j 1 =

2 x2 j , x2 j 1 t

j = 1,, n,

X2j =

2 x2 j 1 , x2 j t

j = 1,, n.

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Journal of Scientific and Engineering Research, 2017, 4(2):127-144

These vector fields generate the Lie algebra of satisfy the relation

H n and the commutators of the vector fields X 1 ,, X 2 n

X , X = 4 X n j

j

2 n 1

j = 1,, n,

,

with all other brackets being equal to zero. The inverse element of

g = x, t is g 1 = x,t and we denote the identity (neutral) element of H n as

e = 0,0 R2n1 . The Heisenberg group is a connected, simply connected nilpotent Lie group. Oneparameter Heisenberg dilations

r : Hn Hn

are given by

r x, t = rx, r 2t

for each real number

r > 0 . The Haar measure on H n also coincides with the usual Lebesgue measure on R 2 n1 . These dilations are group automorphisms and Jacobian determinant of

r

where Q = 2n 2 is the homogeneous dimension of

with respect to the Lebesgue measure is equal to

rQ ,

H n . We denote the measure of any measurable set

H n by . Then

r = r Q , The homogeneous norm on

d r x = r Q dx.

H n is defined as follows 1/4

x

Hn

= x1 ,, x2 n , x2 n 1 H

and the Heisenberg distance is given by

n

2 n 2 = x 2j x22n 1 , j =1

d g , h = d g 1h,0 = g 1h .

This distance d is left-invariant in the sense that d g , h = g h remains unchanged when 1

both left-translated by some fixed vector on 320)

H n . Moreover, d satisfies the triangular inequality (see [15], page

d g , h d g , x d x, h,

Using this norm, we define the Heisenberg ball

g , x, h H n .

Bg , r = h H n : g 1h < r

with center by

g and h are

g = x, t and radius r and denote by BC g , r = H n \ Bg , r its complement, and we denote

Br = Be, r = h H n : h < r the open ball centered at e , the identity (neutral) element of H n , with

radius

r . The volume of the ball Bg , r is cQ r Q , where cn is the volume of the unit ball B1 : n 2 cQ = Be,1 = . n 1 n 1n 2 2

n

1 2

For more details about Heisenberg group, one can refer to [8]. In the study of local properties of solutions to of second order elliptic partial differential equations (PDEs), together with weighted Lebesgue spaces, Morrey spaces

Lp , H n play an important role, see [10, 16]. They

were introduced by Morrey in 1938 [18]. For the properties and applications of classical Morrey spaces, see [4, 5, 13] and the references therein. We recall its definition on a Heisenberg group as

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L p , H n = f : f L H = sup r p f p , n gH n ,r > 0 loc where f L p ( H n ) , 0 Q and 1 p < .

Note that

L p ( B ( g ,r ))

Lp ,0 = Lp ( H n ) and Lp,Q = L ( H n ) . If < 0 or > Q , then Lp , = , where is the set

of all functions equivalent to 0 on that

< ,

H n . It is known that Lp , ( H n ) is an expansion of Lp ( H n ) in the sense

Lp ,0 = Lp ( H n ) .

We also denote by

WLp, WLp, ( H n ) the weak Morrey space of all functions f WLloc p ( H n ) for which f

where

WL p ,

f

= sup r

WL p , ( H n )

p

f

gH n ,r > 0

< ,

WL p ( B ( g ,r ))

WLp ( B( g , r )) denotes the weak L p -space of measurable functions f for which f

WL p ( B ( g , r ))

f

B ( g , r ) WL ( H ) p n

= sup h B( g , r ) :| f ( h ) |>

1/p

>0

=

sup 1/p f B ( g , r ) ( ) < .

0 0

In the case of function

= 0 , the fractional

| f (h) | dh.

B ( g ,r )

maximal function

M f coincides with the Hardy-Littlewood maximal

Mf M 0 f (see [8, 23]) and is closely related to the fractional integral T f g =

Hn

The operators

f h

g 1h

Q

dh

0 < < Q.

M and T play an important role in real and harmonic analysis (see [7, 8, 23, 26]).

The classical Riesz potential

I is defined on R n by the formula I f =

where

Q

2

f,

0 < < n,

is the Laplacian operator. It is known that 1 f y I f x = dy T f x , Rn x y n

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2 . The Riesz potential on the Heisenberg group is defined in terms of the sub where = 2 n 2 Laplacian L = H . n 2

n

Definition 1 For 0 < < Q the Riesz potential

I is defined by on the Schwartz space S H n by

the formula

I f g = L

2

1

f g e rL f g r 2 dr , 0

where

e rL f g =

is the semigroups of the operator

1 K r h, g f h d h H n 2

L.

In [26], relations between the Riesz potential and the heat kernel on the Heisenberg group are studied. The following assertion

[26], Theorem 4.2

yields an expression for

I , which allows us to discuss the

boundedness of the Riesz potential. Theorem 1 Let

qs g be the heat kernel on H n . If 0 < Q , then for f S H n 1 1 2 I f g = s qs ds f g . 0 2

The Riesz potential

I satisfies the estimate [26], Theorem 4.2 I f g ≲ T f g

which provides a suitable estimate for the Riesz potential on the Heisenberg group. It is well known that, see [8, 23] for example,

T is bounded from L p H n to Lq H n for all p > 1 and

1 1 = p q

Q (i.e. Hardy-Littlewood Sobolev inequality). > 0 , and T is also of weak type 1, Q Q Spanne (published by Peetre [21]) and Adams [1] have studied boundedness of the fractional integral

operator

T on Lp , R n . This result has been reproved by Chiarenza and Frasca [3], and also studied in [12]. After studying Morrey spaces in detail, researchers have passed to generalized Morrey spaces. Recall

that the concept of the generalized Morrey space

M p, M p, ( H n ) on Heisenberg group has been

introduced in [11].

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Definition 2 [11] Let We denote by

( g , r ) be a positive measurable function on H n (0,) and 1 p < .

M p, M p, ( H n ) the generalized Morrey space, the space of all functions f Lloc p (H n )

with finite quasinorm

f Also by

= sup ( g , r ) | B( g , r ) | 1

M p ,

1 p

gH n ,r >0

f

L p ( B ( g ,r ))

.

WM p, WM p, ( H n ) we denote the weak generalized Morrey space of all functions

f WLloc p ( H n ) for which f

= sup ( g , r ) | B( g , r ) | 1

WM p ,

1 p

f

gH n ,r >0

According to this definition, we recover the Morrey space

WL p ( B ( g ,r ))

< .

L p , and weak Morrey space WLp , under

Q

the choice

(g, r) = r

p

:

Lp , = M p , |

Q

,

WLp , = WM p , |

p

( g ,r )= r

( g ,r )= r

In [11], Guliyev et al. prove the Spanne type boundedness of Riesz potentials

1

the homogeneous dimension of

I , 0, Q from one

2

1 1 = , Q is p q Q

H n and from the space M 1, H n to the weak space WM1, H n , where 1

2

1 = . They also prove the Adams type boundedness of the Riesz potentials I , q Q

0, Q from M p ,

to the weak space

.

p

M p , H n to another M q , H n , where 1 < p < q < ,

generalized Morrey space

1 < q < , 1

Q

1 p

WM 1,

H n to another

M q ,

1 q

1 q

H n for 1 < p < q < and from the space

M1, H n

H n for 1 < q < .

For a locally integrable function b on

H n , suppose that the commutator operator Tb , , 0, Q

represents a linear or a sublinear operator, which satisfies that for any

f L1 ( H n ) with compact support and

x suppf

| Tb, f ( g ) | c0

Hn

where

| f ( h) | dh, 1 Q h|

| b( g ) b( h) | | g

(1.1)

c0 is independent of f and g .

The condition (1.1) is satisfied by many interesting operators in harmonic analysis, such as fractional maximal operator, fractional Marcinkiewicz operator, fractional integral operator and so on (see [17], [22] for details). Let

T be a linear operator. For a locally integrable function b on H n , we define the commutator

[b, T ] by [b,T ] f ( x) = b( x)Tf ( x) T (bf )( x)

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for any suitable function f . Let b be a locally integrable function on

H n , then for 0 < < Q , we define the linear commutator

generated by fractional integral operator and b and the sublinear commutator of the fractional maximal operator as follows, respectively (see also [17]).

[b, T ] f ( g ) b( g )T f ( g ) T (bf )( g ) =

[b( g ) b(h)] | g

Hn

M b, f ( g ) = sup | B( g , r ) | r >0

1

Q

f ( h) dh, h |Q

1

bg bh | f (h) | dh. B ( g ,r )

Now, we will examine some properties related to the space of functions of Bounded Mean Oscillation, BMO , introduced by John and Nirenberg [14] in 1961. This space has become extremely important in various areas of analysis including harmonic analysis, PDEs and function theory. BMO -spaces are also of interest since, in the scale of Lebesgue spaces, they may be considered and appropriate substitute for

L . Appropriate in the sense

that are spaces preserved by a wide class of important operators such as the Hardy-Littlewood maximal function, the Hilbert transform and which can be used as an end point in interpolating Let us recall the defination of the space of

Definition 3 Suppose that

L p spaces.

BMO( H n ) (see, for example, [8, 17, 24]).

b L1loc ( H n ) , let 1 | b(h) bB( g ,r ) | dh < , gH n ,r >0 | B( g , r ) | B ( g ,r )

b = sup

(1.2)

where

bB ( g ,r ) =

1 b(h)dh. | B( g , r ) | B (g ,r )

Define

BMO( H n ) = {b L1loc ( H n ) : b < }. Endowed with the norm given in (1.2), which differ a.e. by constant; clearly,

Remark 1 Note that

BMO( H n ) becomes Banach space provided we identify functions

b = 0 for bh = c a.e. in H n .

L ( H n ) is contained in BMO( H n ) and we have b 2 b .

Moreover, BMO contains unbounded functions, in fact the function log bounded, so

L ( H n ) BMO( H n ) .

Remark 2 (1) all

h on H n , is in BMO but it is not

The John-Nirenberg inequality [14]: there are constants

C1 , C2 > 0 , such that for

b BMO( H n ) and > 0

g B:| b( g ) bB |> C1 | B | e C2/ b ,

B H n .

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Journal of Scientific and Engineering Research, 2017, 4(2):127-144

(2) The John-Nirenberg inequality implies that 1

p 1 p b sup | b(h) bB( g ,r ) | dh gH n ,r > 0 | B ( g , r ) | B ( g ,r )

(1.3)

for 1 < p < .

(3) Let b BMO( H n ) . Then there is a constant C > 0 such that

bB ( g ,r ) bB ( g , ) C b ln for 0 < 2r < , r where C is independent of b , g , r and .

b BMOH n and Tb , , 0, Q satisfying

Inspired by [11], in this paper, provided that

condition (1.1) is a sublinear operator, we find the sufficient conditions on the pair

(1 , 2 ) which ensures the

Tb , from M p , H n to M q , H n , where

Spanne type boundedness of the commutator operators

1< p < q < , 0 <

2r

>2r

f w dw

f w dw

r sup t Q bB g ,r bB g , f Q

1 = 1 we get p

B g ,

Q q

≲ b r q sup 1 ln t

f

L p B g ,

Bg , B g ,

L p ( B ( g , ))

1

1 p

.

J 2 note that

J 2 = b bB g ,

sup t Q

Lq B g , >2r

f w dw.

B g,

By (1.3), we get Q q

≲ b r sup t Q J 2 >2r

f w dw.

B g ,

Thus, by (2.2) Q q

≲ b r sup t J 2 Summing up

Q q

Lq B

Q q

>2r

J1 and J 2 , for all p 1, we get M b, f 2

≲ b r sup t >2r

Q q

f

L p B g ,

.

1 ln f r

L p B g ,

.

(2.6)

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Journal of Scientific and Engineering Research, 2017, 4(2):127-144

Finally, we have the following

M b, f 2

Lq B

Q q

≲ b f

≲ b r sup t

Q q

>2r

Q q

b r sup t

L p 2 B

>2r

1 ln f r

L p B g ,

Q q

1 ln f r

L p B g ,

,

which completes the proof. Similarly to Lemma 2 the following lemma can also be proved. Lemma 3 Let 1 < p < ,

b BMOH n and M b is bounded on Lp ( H n ) . Then the inequality

p q ≲ b r 1 ln f sup L p ( B ( g ,r )) r >2r Q

Mb f

Q

L p ( B ( g , ))

f Lloc p (H n ) .

holds for any ball B( g , r ) and for all The following theorem is true.

0