Sublinear operators with rough kernel generated by fractional ...

Available online www.jsaer.com Journal of Scientific and Engineering Research, 2017, 4(2):145-163 ISSN: 2394-2630 CODEN(USA): JSERBR

Research Article

Sublinear operators with rough kernel generated by fractional integrals and their commutators on generalized Morrey spaces FERİT GÜRBÜZ



Ankara University, Faculty of Science, Department of Mathematics, Tando g an 06100, Ankara, Turkey Abstract In this paper some results for the boundedness of certain sublinear operators, including fractional integral operators, with rough kernels on generalized Morrey spaces are given. Moreover, the corresponding results of the commutators with rough kernels are discussed. Also, Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example. Keywords Sublinear operator, fractional integral operator, rough kernel, generalized Morrey space, commutator, BMO 1. Introduction and Main Results The classical Morrey spaces

M p , have been introduced by Morrey in [26] to study the local behavior of

solutions of second order elliptic partial differential equations (PDEs). Later, there are many applications of Morrey space to the Navier-Stokes equations (see [24]), the Schrödinger equations (see [32]) and the elliptic problems with discontinuous coefficients (see [2, 29]). We recall the definition of classical Morrey spaces

M p , as

    M p , R n =  f : f M  Rn  = sup r p f p ,    xR n ,r > 0  loc n where f  L p (R ) , 0    n and 1  p <  .

 

Note that

L p ( B ( x ,r ))

  < ,  

M p,0 = Lp (Rn ) and M p,n = L (Rn ) . If  < 0 or  > n , then M p , =  , where  Rn . n  WM p, (Rn ) the weak Morrey space of all functions f WLloc p (R )

is the set of all functions equivalent to 0 on We also denote by

WM p ,

for which

f where

WM p ,

 f

 WM p , (Rn )

= sup r xRn ,r > 0

 p

f

WL p ( B ( x ,r ))

< ,

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

Journal of Scientific and Engineering Research 145

GÜRBÜZ F

Journal of Scientific and Engineering Research, 2017, 4(2):145-163

f

WL p ( B ( x ,r ))

 f

B ( x ,r ) WL (R n ) p

= sup t y  B( x, r ) :| f ( y ) |> t

1/p

t >0

0 0 and also let B( x, r ) denotes the open ball n

x of radius r , BC ( x, r ) denotes its complement and | B( x, r ) | is the Lebesgue measure of the

centered at

ball B( x, r ) and

| B( x, r ) |= vn r n , where vn =| B(0,1) | .

For the boundedness of the Hardy–Littlewood maximal operator, the fractional integral operator and the Calderón–Zygmund singular integral operator on these spaces, we refer the readers to [1, 5, 31]. For further properties and applications of classical Morrey spaces, see [6, 7, 14, 17] and references therein. After studying Morrey spaces in detail, researchers have passed to generalized Morrey spaces. Mizuhara [25] has given generalized Morrey spaces

M p , considering  r  instead of r  in the above

definition of the Morrey space. Later, Guliyev [12], Guliyev et al. [13] and Karaman [22] have defined the

M p , with normalized norm as follows:

generalized Morrey spaces

Definition 1 (Generalized Morrey space) Let

 ( x, r ) be a positive measurable function on

Rn  (0,) and 1  p <  . We denote by M p,  M p, (Rn ) the generalized Morrey space, the space of all functions

n f  Lloc p (R ) with finite quasinorm

f Also by

= sup  ( x, r ) | B( x, r ) | 1

M p ,



1 p

f

xRn ,r > 0

L p ( B ( x ,r ))

< .

WM p ,  WM p, (Rn ) we denote the weak generalized Morrey space of all functions

n f WLloc p (R ) for which

f

= sup  ( x, r ) | B( x, r ) | 1

WM p ,



xRn ,r > 0

According to this definition, we recover the Morrey space

1 p

f

WL p ( B ( x ,r ))

< .

M p , and weak Morrey space WM p ,

 n

under the choice

 ( x, r ) = r

p

:

M p , = M p , |  ( x ,r )= r

 n

,

p

WM p , = WM p , |  ( x ,r )= r

 n

.

p

During the last decades various classical operators, such as maximal, singular and potential operators have been widely investigated in generalized Morrey spaces (see [8, 12, 13, 18, 22, 28, 34] for details). n 1

R n (n  2) equipped with the normalized Lebesgue measure s for any d . Let   Ls ( S n1 ) with 1 < s   be homogeneous of degree zero. We define s' = s 1 Suppose that S

is the unit sphere on


GÜRBÜZ F


s > 1 . Suppose that T, ,   0, n represents a linear or a sublinear operator, which satisfies that for any

f  L1 (Rn ) with compact support and x  suppf | T, f ( x) | c0 where

| ( x  y ) | | f ( y ) |dy, | x  y |n  Rn



(1.1)

c0 is independent of f and x . R n , suppose that the commutator operator T,b, ,   0, n

For a locally integrable function b on

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

f  L1 (Rn ) with compact support and

x  suppf

| T,b, f ( x) | c0

| ( x  y) |

 | b( x )  b( y ) | | x  y |

n 

| f ( y) |dy,

(1.2)

Rn

where

c0 is independent of f and x .

We point out that the condition (1.1) in the case of   1 ,  = 0 has been introduced by Soria and Weiss in [35]. The conditions (1.1) and (1.2) are satisfied by many interesting operators in harmonic analysis, such as fractional Marcinkiewicz operator, fractional maximal operator, fractional integral operator (Riesz potential) and so on (see [23], [35] for details). In 1971, Muckenhoupt and Wheeden [27] defined the fractional integral operator with rough kernel

T , by T , f ( x) =

( x  y)

 | x y|

n 

f ( y )dy

0 0

where



x  y  | f ( y ) | dy



n

M , is given by 0 <  < n,

B ( x ,t )

  Ls (S n1 ) with 1 < s   is homogeneous of degree zero on R n and also T , and M ,

satisfy condition (1.1). If

 = 0 , then M ,0  M 

is the Hardy-Littlewood maximal operator with rough kernel and

also becomes a Calderón-Zygmund singular integral operator with rough kernel. It is obvious that when

T ,

 1,

M1,  M  and T 1,  T  are the fractional maximal operator and the fractional integral operator, respectively. In recent years, the mapping properties of

T , on some kinds of function spaces have been studied in

many papers (see [4, 9, 10, 27] for details). In particular, the boundedness of obtained. Lemma 1 [4, 9, 27] Let 0 <  < n , 1 < p
, q p n n 

then we have


GÜRBÜZ F


T , f

Lq

C f

Lp

.

Corollary 1 Under the assumptions of Lemma 1, the operator

M , is bounded from L p (R n ) to

Lq (R n ) . Moreover, we have M , f

Lq

C f

Lp

.

Proof. Set ∼ 𝑇

 ,

 f ( x) = 

Rn

where

( x  y )

f ( y ) dy

| x  y |n

0 <  < n,

  Ls (S n1 )s > 1 is homogeneous of degree zero on R n . It is easy to see that, for 𝑇

∼  ,

, Lemma 1

is also hold. On the other hand, for any t > 0 , we have ∼ 𝑇

 ,

 f ( x)   ( x  y)  | x  y |

f ( y ) dy

n

B x ,t



1 t

n 



( x  y ) f ( y ) dy.

B  x ,t 

Taking the supremum for t > 0 on the inequality above, we get

M , f x   Cn,1 𝑇

∼

For

  f ( x)

,

Cn, = B0,1

n  n

.

b  L1loc (Rn ) , the commutator [b, T  ] of fractional integral operator (also known as the Riesz

potential) is defined by

[b, T  ] f ( x) = b( x)T  f ( x)  T  (bf )( x) =

b( x )  b( y ) f ( y)dy n  | x  y | n R



for any suitable function f . The function b is also called the symbol function of boundedness of the commutator

[b, T  ] . The characterization of Lp , Lq  -

[b, T  ] of fractional integral operator has been given by Chanillo [3]. A well

known result of Chanillo [3] states that the commutator

1< p < q < ,

0 0 | B( x, r ) | B ( x ,r )

b  = sup where

bB ( x ,r ) =

1 b( y )dy. | B( x, r ) | B (x ,r )

Define

BMO(R n ) = {b  L1loc (R n ) : b  < }. If one regards two functions whose difference is a constant as one, then the space space with respect to norm

BMO(Rn ) is a Banach

 .

Remark 1 [22] (1) The John-Nirenberg inequality [21]: there are constants for all

C1 , C2 > 0 , such that

b  BMO(Rn ) and  > 0

x  B:| b( x)  bB |>    C1 | B | e C2/ b  ,

B  R n .

(2) The John-Nirenberg inequality implies that

  1 b   sup  | b( y )  bB ( x ,r ) | p dy     xR n ,r > 0  | B ( x, r ) | B ( x ,r ) 

1 p

(1.3)

for 1 < p <  .

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

t bB ( x ,r )  bB ( x ,t )  C b  ln for 0 < 2r < t , r where C is independent of b , x , r and t .

Remark 2 [22] Note that

L (Rn ) is contained in BMO(Rn ) and we have b

BMO

 2 b .

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

(1.4)

x on R n , is in BMO but it is not

L (Rn )  BMO(Rn ) .

R n , then for 0 <  < n and f is a suitable function, we define the commutators generated by fractional integral and maximal operators with rough kernel and b as Let b be a locally integrable function on

follows, respectively:

[b, T , ] f ( x)  b( x)T , f ( x)  T , (bf )( x) =  [b( x)  b( y)] Rn

( x  y) f ( y)dy, | x  y |n


GÜRBÜZ F


M ,b,  f ( x) = sup | B( x, t ) |

1

 n

t >0

 bx   b y  x  y  | f ( y) | dy B ( x ,t )

satisfy condition (1.2). The proof of boundedness of

[b, T , ] in Lebesgue spaces can be found in [9] (by

taking w = 1 there). Theorem 1 [9] Suppose that mean value zero on S

n 1

  Ls (S n1 ) , 1 < s   , is homogeneous of degree zero and has

. Let 0 <  < n , 1  p
1 and q < s , then the inequality

T, f holds for any ball

n n  n n   1 s s q

≲rq Lq B x0 , r 

Bx0 , r  and for all f  L R  . loc p

t

f

2r

   dt

L p B x0 ,t

n

Moreover, for p = 1 < q < s the inequality

T, f holds for any ball

≲r 

Bx0 , r  and for all f  L1loc Rn  .

Lemma 4 Suppose that

0