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Boundedness of sublinear operators and their commutators on generalized central Morrey spaces

Abstract

In this paper, we introduce the generalized central Morrey spaces B ˙ p , φ ( R n ) and get the boundedness of a large class of rough operators on them. We also consider the CBMO estimates of their commutators on generalized central Morrey spaces. As applications, we obtain the boundedness characterizations of rough Hardy-Littlewood maximal function, rough Calderón-Zygmund singular integral, rough fractional integral, etc. on generalized central Morrey spaces.

MSC:42B20, 42B25.

1 Introduction

Let Ω L s ( S n 1 ) be homogeneous of degree zero on R n , where S n 1 denotes the unit sphere of R n and s>1. We define s =s/(s1) for any s>1. Suppose that T Ω represents a sublinear operator, which satisfies that for any f L 1 ( R n ) with compact support and xsuppf,

| T Ω f ( x ) | C R n | Ω ( x y ) | | x y | n | f ( y ) | dy,
(1.1)

where C>0 is an absolute constant. Similarly, for any 0<α<n, we assume that T Ω , α represents a sublinear operator, which satisfies that

| T Ω , α f ( x ) | C R n | Ω ( x y ) | | x y | n α | f ( y ) | dy
(1.2)

for any f L 1 ( R n ) with compact support and xsuppf.

Let T be a linear operator. For a locally integrable function b on R n , we define the commutator [T,b] by

[T,b]f(x)=b(x)Tf(x)T(bf)(x)
(1.3)

for any suitable function f.

To study the local behavior of solutions to second-order elliptic partial differential equations, Morrey [1] introduced the classical Morrey spaces L p , λ ( R n ). The readers can find more details in [2].

Let 1p< and λ0. B( x 0 ,t) denotes a ball centered at x 0 of radius t. Morrey spaces L p , λ ( R n ) are defined by

L p , λ ( R n ) = { f L loc p ( R n ) : f L p , λ ( R n ) < } ,

where

f L p , λ ( R n ) = sup x 0 R n , t > 0 ( 1 t λ B ( x 0 , t ) | f ( x ) | p d x ) 1 p .

When 1p<, L p , 0 ( R n )= L p ( R n ) and L p , n ( R n )= L ( R n ); when λ>n, L p , λ ( R n )={0}.

Many authors have studied the mapping properties of many operators on Morrey spaces; see [35] and [6]. Alvarez et al. [7], in order to study the relationship between central BMO spaces and Morrey spaces, introduced λ-central bounded mean oscillation spaces and central Morrey spaces.

Let λ< 1 n and 1<p<. A function f L loc p ( R n ) belongs to the λ-central bounded mean oscillation spaces CBMO p , λ ( R n ) if

f CBMO p , λ = sup r > 0 ( 1 | B ( 0 , r ) | 1 + λ p B ( 0 , r ) | f ( x ) f B ( 0 , r ) | p ) 1 p <,

where f B ( 0 , r ) = 1 | B ( 0 , r ) | B ( 0 , r ) f(y)dy. If two functions which differ by a constant are regarded as functions in the spaces CBMO p , λ ( R n ), then CBMO p , λ ( R n ) spaces become Banach spaces. CBMO p , λ ( R n ) spaces become the spaces of constants when λ< 1 p and they coincide with L p ( R n ) modulo constants when λ= 1 p .

Let λR and 1<p<. The central Morrey spaces B ˙ p , λ ( R n ) are defined by

f B ˙ p , λ = sup r > 0 ( 1 | B ( 0 , r ) | 1 + λ p B ( 0 , r ) | f ( x ) | p ) 1 p <.

It follows that B ˙ p , λ ( R n ) spaces are Banach spaces continuously included in CBMO p , λ ( R n ) spaces. B ˙ p , λ ( R n ) spaces reduce to {0} when λ< 1 p , and it is true that B ˙ p , 1 p ( R n )= L p ( R n ), B ˙ p , 0 ( R n )= B ˙ p ( R n ).

Recently, Guliyev [8] introduced the generalized Morrey spaces L p , φ ( R n ), where φ(x,r) is a positive measurable function on R n ×(0,) and 1p<. For all functions f L loc p ( R n ), the generalized Morrey spaces L p , φ ( R n ) are defined by

f L p , φ ( R n ) = sup x R n , r > 0 φ ( x , r ) 1 | B ( x , r ) | 1 p f L p ( B ( x , r ) ) <.

Obviously, if φ(x,r)= r λ n p , L p , λ ( R n )= L p , φ ( R n ).

When Ω1, Guliyev obtained the sufficient condition on φ 1 and φ 2

r ess inf t < τ < φ 1 ( τ ) τ n p t n p + 1 dtC φ 2 (r)

for the boundedness of T Ω satisfying (1.1) from L p , φ 1 ( R n ) to L p , φ 2 ( R n ) in [9] and gave the condition on the pair of ( φ 1 , φ 2 )

r ess inf t < τ < φ 1 ( τ ) τ n p t n q + 1 dtC φ 2 (r)

for the boundedness of T Ω , α satisfying (1.2) from L p , φ 1 ( R n ) to L q , φ 2 ( R n ) in [10], where 1 q = 1 p α n .

Inspired by the above, we consider the boundedness of sublinear operators on the following generalized central Morrey spaces and give the λ-central bounded mean oscillation estimates for linear operator commutators.

Definition 1.1 Let φ(r) be a positive measurable function on R + and 1<p<. We denote by B ˙ p , φ ( R n ) the generalized central Morrey spaces, the spaces of all f L loc p ( R n ) with finite quasinorm

f B ˙ p , φ ( R n ) = sup r > 0 φ ( r ) 1 | B ( 0 , r ) | 1 p f L p ( B ( 0 , r ) ) .

We can recover the spaces B ˙ p , λ ( R n ) under the choice φ(r)= r n λ .

Recall that in 1994 the doctoral thesis [11] by Guliyev (see also [1215]) introduced the local Morrey-type space L M p θ , ω given by

f L M p θ , ω = ω ( r ) f L p ( B ( 0 , r ) ) L θ ( 0 , ) <,

where ω is a positive measurable function defined on (0,). The main purpose of [11] (also of [1215]) is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type space L M p θ , ω . In a series of papers by Burenkov, H Guliyev and V Guliyev, etc. (see [1621]), some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators and singular integral operators in local Morrey-type spaces L M p θ , ω were given.

Particularly, if θ=, L M p θ , ω =L M p , ω , then the generalized central Morrey spaces B ˙ p , φ ( R n ) are the same spaces as the local Morrey spaces L M p , ω with ω(r)=φ ( r ) 1 r n p .

The following statements were proved in [11] (see also [14]).

Theorem A Let 1<p< and ( φ 1 , φ 2 ) satisfy the condition

r φ 1 (t) d t t C φ 2 (r),

where C does not depend on r. Then the Calderón-Zygmund operator T is bounded from B ˙ p , φ 1 to B ˙ p , φ 2 .

Theorem B Let 1<p<, 0<α< n p , 1 q = 1 p α n and ( φ 1 , φ 2 ) satisfy the condition

r t α φ 1 (t) d t t C φ 2 (r),

where C does not depend on r. Then the Riesz potential I α is bounded from B ˙ p , φ 1 to B ˙ q , φ 2 .

From Lemmas 4.4 and 5.3 in [9] we get the following for the generalized central (local) Morrey spaces B ˙ p , φ .

Theorem C Let 1<p<, T be a sublinear operator satisfying that for any f L 1 ( R n ) with compact support and xsuppf,

| T f ( x ) | C R n | f ( y ) | | x y | n dy,

and bounded on f L p ( R n ). Let also the pair ( φ 1 , φ 2 ) satisfy the condition

r ess inf t < τ < φ 1 ( τ ) τ n p t n p + 1 dtC φ 2 (r),

where C does not depend on r. Then the operator T is bounded from B ˙ p , φ 1 to B ˙ p , φ 2 .

Theorem D Let 1<p<, 0<α< n p , 1 q = 1 p α n , T α be a sublinear operator satisfying that for any f L 1 ( R n ) with compact support and xsuppf,

| T α f ( x ) | C R n | f ( y ) | | x y | n α dy,

and bounded from f L p ( R n ) to f L q ( R n ). Let also the pair ( φ 1 , φ 2 ) satisfy the condition

r ess inf t < τ < φ 1 ( τ ) τ n p t n q + 1 dtC φ 2 (r),

where C does not depend on r. Then the operator T α is bounded from B ˙ p , φ 1 to B ˙ q , φ 2 .

2 Sublinear operator with rough kernel

Theorem E Let ω be a positive weight function on (0,). The inequality

ess sup t > 0 ν 2 (t) H ω g(t)cess sup t > 0 ν 1 (t)g(t)

holds for all non-negative and non-increasing g on (0,) if and only if

A:= sup t > 0 ν 2 ( t ) t 0 t ω ( r ) d r ess sup 0 < τ < r ν 1 ( τ ) <,

and cA, where the H ω is the weighted Hardy operator

H ω g(t):= 1 t 0 t g(r)ω(r)dr,0<t<.

Note that Theorem E can be proved analogously to Theorem 1 in [22]; particularly, when ω1, it was proved in [23].

In this section we are going to discuss the boundedness of T Ω and T Ω , α on generalized central Morrey spaces.

Lemma 2.1 Let 1<p<, T Ω be a sublinear operator and satisfy (1.1) with Ω L s ( S n 1 ).

When s p and T Ω is bounded on L p ( R n ) for 1<p<, then the inequality

T Ω f L p ( B ( 0 , r ) ) C r n p 2 r t n p 1 f L p ( B ( 0 , t ) ) dt

holds for any ball B(0,r) and for all f L loc p ( R n ); or p<s and T Ω is bounded on L p ( R n ) for 1<p<, then the inequality

T Ω f L p ( B ( 0 , r ) ) C r n p n s 2 r t n p + n s 1 f L p ( B ( 0 , t ) ) dt

holds for any ball B(0,r) and for all f L loc p ( R n ).

Proof Let 1<p<. For any r>0, set B=B(0,r) and 2B=B(0,2r). We write

f(x)=f(x) χ 2 B (x)+f(x) χ ( 2 B ) c (x):= f 1 (x)+ f 2 (x)

and have

T Ω f L p ( B ) T Ω f 1 L p ( B ) + T Ω f 2 L p ( B ) .

Since T Ω f 1 is bounded on L p ( R n ), it follows that

T Ω f 1 L p ( B ) T Ω f 1 L p ( R n ) C f 1 L p ( R n ) =C f L p ( 2 B ) ,

where the constant C>0 is independent of f.

It is known that xB, y ( 2 B ) c , which implies 1 2 |y||xy| 3 2 |y|. Thus

| T Ω f 2 ( x ) | C ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n .
  1. (i)

    When s p and by Fubini’s theorem, we have

    ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n = C ( 2 B ) c | f ( y ) | | Ω ( x y ) | | y | d t t n + 1 d y C 2 r 2 r | y | < t | f ( y ) | | Ω ( x y ) | d y d t t n + 1 C 2 r f L p ( B ( 0 , t ) ) ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 s d t t n + 1 C 2 r f L p ( B ( 0 , t ) ) d t t n p + 1 .

Hence, for all p(0,), the inequality

T Ω f 2 L p ( B ) C r n p 2 r f L p ( B ( 0 , t ) ) d t t n p + 1

holds.

  1. (ii)

    When p<s, by Fubini’s theorem and the Minkowski inequality, we get

    T Ω f 2 L p ( B ) ( B | 2 r B ( 0 , t ) | f ( y ) | | Ω ( x y ) | d y d t t n + 1 | p d x ) 1 p 2 r B ( 0 , t ) | f ( y ) | ( B | Ω ( x y ) | p d x ) 1 p d y d t t n + 1 C 2 r B ( 0 , t ) | f ( y ) | ( B ( 0 , t ) | Ω ( x y ) | s d x ) 1 s | B | 1 p 1 s d y d t t n + 1 C r n p n s 2 r B ( 0 , t ) | f ( y ) | d y d t t n n s + 1 C r n p n s 2 r f L p ( B ( 0 , t ) ) d t t n p n s + 1 .

On the other hand, for any q>0, we have

f L p ( 2 B ) =C r n q f L p ( 2 B ) 2 r d t t n q + 1 C r n q 2 r f L p ( B ( 0 , t ) ) d t t n q + 1 .

Combining the above estimates, we complete the proof of Lemma 2.1. □

Theorem 2.2 Let 1<p< and Ω L s ( S n 1 ). Let T Ω be a sublinear operator satisfying (1.1) and bounded on L p ( R n ) for p>1. If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n p + 1 dtC φ 2 (r),
  2. (ii)

    when p<s, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n p n s + 1 dtC φ 2 (r) r n s

is satisfied, then the operator T Ω is bounded from B ˙ p , φ 1 to B ˙ p , φ 2 .

Proof When s p, by Lemma 2.1 and Theorem E, for ω1, p>1, we have

T Ω f B ˙ p , φ 2 C sup r > 0 φ 2 ( r ) 1 r f L p ( B ( 0 , t ) ) d t t n p + 1 = C sup r > 0 φ 2 ( r ) 1 0 r n p f L p ( B ( 0 , t p n ) ) d t = C sup r > 0 φ 2 ( r p n ) 1 0 r f L p ( B ( 0 , t p n ) ) d t C sup r > 0 φ 1 ( r p n ) 1 r f L p ( B ( 0 , r p n ) ) = C f B ˙ p , φ 1 .

For the case of p<s, we can use the same method to prove the desirable conclusion. □

The Calderón-Zygmund operator with rough kernel T ˜ Ω has the following integral expression:

T ˜ Ω f(x)= R n K(x,y)f(y)Ω(y)dy

for any test function f and xsuppf. The kernel is a locally integral function defined away from the diagonal satisfying the size condition

K(x,y)C 1 | x y | n

for all x,y R n and xy.

f L loc 1 , the rough Hardy-Littlewood maximal function M Ω is defined by

M Ω f(x)= sup t > 0 1 | B ( x , t ) | B ( x , t ) | Ω ( y ) | | f ( y ) | dy.

Then we can get the following corollary.

Corollary 2.3 Let 1<p< and Ω L s ( S n 1 ). If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n p + 1 C φ 2 (r),
  2. (ii)

    when p<s, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n p n s + 1 C φ 2 (r) r n s

is satisfied, then M Ω and T ˜ Ω are both bounded from B ˙ p , φ 1 to B ˙ p , φ 2 .

In the following statements, the boundedness of T Ω , α satisfying (1.2) in generalized central Morrey spaces is proved.

Lemma 2.4 Let 0<α<n, 1<p< n α and 1 q = 1 p α n , T Ω , α be a sublinear operator and satisfy (1.2) with Ω L s ( S n 1 ).

When s p and T Ω , α is bounded from L p ( R n ) to L q ( R n ), then the inequality

T Ω , α f L q ( B ( 0 , r ) ) C r n q 2 r t n q 1 f L p ( B ( 0 , t ) ) dt

holds for any ball B(0,r) and for all f L loc p ( R n ); or q<s and T Ω , α is bounded from L p ( R n ) to L q ( R n ), then the inequality

T Ω , α f L q ( B ( 0 , r ) ) C r n q n s 2 r t n q + n s 1 f L p ( B ( 0 , t ) ) dt

holds for any ball B(0,r) and for all f L loc p ( R n ).

Proof Let 0<α<n, 1<p< n α and 1 q = 1 p α n . For any r>0, set B=B(0,r) and 2B=B(0,2r). We write

f(x)=f(x) χ 2 B (x)+f(x) χ ( 2 B ) c (x):= f 1 (x)+ f 2 (x)

and have

T Ω , α f L q ( B ) T Ω , α f 1 L q ( B ) + T Ω , α f 2 L q ( B ) .

Since T Ω , α f 1 is bounded from L p ( R n ) to L q ( R n ), it follows that

T Ω , α f 1 L q ( B ) T Ω , α f 1 L q ( R n ) C f 1 L p ( R n ) =C f L p ( 2 B ) ,

where the constant C>0 is independent of f.

Since xB, y ( 2 B ) c , thus

| T Ω , α f 2 ( x ) | C ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n α .
  1. (i)

    When s p and by Fubini’s theorem, we have

    ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n α = C ( 2 B ) c | f ( y ) | | Ω ( x y ) | | y | d t t n + 1 α d y C 2 r 2 r | y | < t | f ( y ) | | Ω ( x y ) | d y d t t n + 1 α C 2 r f L p ( B ( 0 , t ) ) ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 s d t t n + 1 α C 2 r f L p ( B ( 0 , t ) ) d t t n q + 1 .

Hence, for all p(0,), the inequality

T Ω , α f 2 L q ( B ) C r n q 2 r f L p ( B ( 0 , t ) ) d t t n q + 1

holds.

  1. (ii)

    When q<s, by Fubini’s theorem and the Minkowski inequality, we get

    T Ω , α f 2 L p ( B ) ( B | 2 r B ( 0 , t ) | f ( y ) | | Ω ( x y ) | d y d t t n + 1 α | q d x ) 1 q 2 r B ( 0 , t ) | f ( y ) | ( B | Ω ( x y ) | q d x ) 1 q d y d t t n + 1 α C 2 r B ( 0 , t ) | f ( y ) | ( B ( 0 , t ) | Ω ( x y ) | s d x ) 1 s | B | 1 q 1 s d y d t t n + 1 α C r n q n s 2 r B ( 0 , t ) | f ( y ) | d y d t t n n s + 1 α C r n q n s 2 r f L p ( B ( 0 , t ) ) d t t n q n s + 1 .

Similarly, combining the above estimates, we finish this proof. □

Theorem 2.5 Let 0<α<n, 1<p< n α , 1 q = 1 p α n and Ω L s ( S n 1 ). Let T Ω , α be a sublinear operator T Ω satisfying (1.2) and bounded from L p ( R n ) to L q ( R n ). If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n q + 1 dtC φ 2 (r),
  2. (ii)

    when q<s, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n q n s + 1 dtC φ 2 (r) r n s

is satisfied, then the operator T Ω , α is bounded from B ˙ p , φ 1 to B ˙ q , φ 2 .

Proof When s p, by Lemma 2.1 and Theorem E, for 1<p< n α and 1 q = 1 p α n , we have

T Ω , α f B ˙ q , φ 2 C sup r > 0 φ 2 ( r ) 1 r f L p ( B ( 0 , t ) ) d t t n q + 1 = C sup r > 0 φ 2 ( r ) 1 0 r n q f L p ( B ( 0 , t q n ) ) d t = C sup r > 0 φ 2 ( r q n ) 1 0 r f L p ( B ( 0 , t q n ) ) d t C sup r > 0 φ 1 ( r q n ) 1 r p q f L p ( B ( 0 , r q n ) ) = C f B ˙ p , φ 1 .

For the case of q<s, we can also use the same method to prove the desirable conclusion. □

f L loc 1 , the rough fractional maximal function M Ω , α and the rough fractional integral I Ω , α are defined by

M Ω , α f ( x ) = sup t > 0 1 | B ( x , t ) | 1 + α B ( x , t ) | Ω ( y ) | | f ( y ) | d y , I Ω , α f ( x ) = sup t > 0 R n Ω ( y ) f ( y ) | x y | n α d y

for 0<α<n.

Corollary 2.6 Let 0<α<n, 1<p< n α , 1 q = 1 p α n and Ω L s ( S n 1 ). If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n q + 1 dtC φ 2 (r),
  2. (ii)

    when q<s, ( φ 1 , φ 2 ) satisfies the condition

    r ess inf t < τ < φ 1 ( τ ) τ n p t n q n s + 1 dtC φ 2 (r) r n s

is satisfied, then M Ω , α and I Ω , α are both bounded from B ˙ p , φ 1 to B ˙ q , φ 2 .

Remark 1 When s=, the comments in Theorem 2.2 and in Theorem 2.5 can be obtained from Lemmas 4.4 and 5.3 in [9].

3 The commutators of a linear operator with rough kernel

Let T ˜ be a Calderón-Zygmund singular integral operator and bBMO( R n ). The commutator operator [ T ˜ ,b] is defined by

[ T ˜ ,b]f(x)=b(x) T ˜ f(x) T ˜ (bf)(x).

A well-known result of Coifman et al. [24] states that the commutator [ T ˜ ,b] is bounded on L p ( R n ) for 1<p< if and only if bBMO( R n ).

Since BMO p > 1 CBMO p , λ when λ=0, if we only assume b CBMO p , λ , then [ T ˜ ,b] may not be a bounded operator on L p ( R n ). However, it has some boundedness properties on other spaces. As a matter of fact, in [25] and [26], they considered the commutators with b CBMO p , λ . Here we also obtain some boundedness of the commutators with b CBMO p , λ on generalized central Morrey spaces.

We need the following statement on the boundedness of the Hardy-type operator

H 1 g(t):= 1 t 0 t ln ( 1 + t r ) g(r)dr,0<t<.

Theorem F [27]

The inequality

ess sup t > 0 ν 2 (t) H 1 g(t)cess sup t > 0 ν 1 (t)g(t)

holds for all non-negative and non-increasing g on (0,) if and only if

A:= sup t > 0 ν 2 ( t ) t 0 t ln ( 1 + t r ) d r ess sup 0 < τ < r ν 1 ( τ ) <,

and cA.

Lemma 3.1 Let 1<p<, b CBMO p 2 , λ , 0<λ< 1 n and 1 p = 1 p 1 + 1 p 2 , T Ω is a sublinear operator and satisfies (1.1) with Ω L s ( S n 1 ).

When s p and T Ω is bounded on L p ( R n ) for 1<p<, then the inequality

[ T Ω , b ] f L p ( B ( 0 , r ) ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1

holds for any ball B(0,r) and for all f L loc p 1 ( R n ); or p 1 <s and T Ω is bounded on L p ( R n ) for 1<p<, then the inequality

[ T Ω , b ] f L p ( B ( 0 , r ) ) C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1

holds for any ball B(0,r) and for all f L loc p 1 ( R n ).

Proof Let 1<p<, b CBMO p 2 , λ and 1 p = 1 p 1 + 1 p 2 . For any r>0, set B=B(0,r) and 2B=B(0,2r). We can write

f(x)=f(x) χ 2 B (x)+f(x) χ ( 2 B ) c (x):= f 1 (x)+ f 2 (x)

and

[ T Ω , b ] f ( x ) = ( b ( x ) b B ) T Ω f 1 ( x ) T Ω ( ( b ( ) b B ) f 1 ) ( x ) + ( b ( x ) b B ) T Ω f 2 ( x ) T Ω ( ( b ( ) b B ) f 2 ) ( x ) = : I 1 + I 2 + I 3 + I 4 .

Hence, we have

[ T Ω , b ] f L p ( B ) I 1 L p ( B ) + I 2 L p ( B ) + I 3 L p ( B ) + I 4 L p ( B ) .

Since T Ω is bounded on L p ( R n ), it follows that

I 1 L p ( B ) = ( B | b ( x ) b B | p | T Ω f 1 ( x ) | p d x ) 1 p ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | T Ω f 1 ( x ) | p 1 d x ) 1 p 1 C r n p 2 + n λ b CBMO p 2 , λ f L p 1 ( 2 B ) ,

where the constant C>0 is independent of f.

For I 2 , we have

I 2 L p ( B ) = ( B | T Ω ( ( b ( ) b B ) f 1 ) ( x ) | p d x ) 1 p C ( R n | b ( x ) b B | p | f 1 ( x ) | p d x ) 1 p C ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | f ( x ) | p 1 d x ) 1 p 1 C r n p 2 + n λ b CBMO p 2 , λ f L p 1 ( 2 B ) .

For I 3 , it is known that xB, y ( 2 B ) c , which implies 1 2 |y||xy| 3 2 |y|.

  1. (i)

    When s p and by Fubini’s theorem, we have

    | T Ω f 2 ( x ) | C ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n = C ( 2 B ) c | f ( y ) | | Ω ( x y ) | | y | d t t n + 1 d y = C 2 r 2 r | y | < t | f ( y ) | | Ω ( x y ) | d y d t t n + 1 C 2 r f L p 1 ( B ( 0 , t ) ) ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 1 s d t t n + 1 C 2 r f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 ,

thus

I 3 L p ( B ) C ( B | b ( x ) b B | p d x ) 1 p 2 r f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 C r n p + n λ 2 r f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 .
  1. (ii)

    When p 1 <s, by Fubini’s theorem and the Minkowski inequality, we get

    I 3 L p ( B ) ( B | 2 r B ( 0 , t ) | b ( x ) b B | | f ( y ) | | Ω ( x y ) | d y d t t n + 1 | p d x ) 1 p 2 r B ( 0 , t ) | f ( y ) | ( B | b ( x ) b B | p | Ω ( x y ) | p d x ) 1 p d y d t t n + 1 2 r B ( 0 , t ) | f ( y ) | ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | Ω ( x y ) | p 1 d x ) 1 p 1 d y d t t n + 1 C r n p 2 + n λ 2 r B ( 0 , t ) | f ( y ) | ( B ( 0 , t ) | Ω ( x y ) | s d x ) 1 s | B | 1 p 1 1 s d y d t t n + 1 C r n p n s + n λ 2 r B ( 0 , t ) | f ( y ) | d y d t t n n s + 1 C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1 .

On the other hand, for I 4 , by Fubini’s theorem, we have

| [ T Ω , b ] f 2 ( x ) | C ( 2 B ) c | b ( y ) b B | | f ( y ) | | Ω ( x y ) | d y | y | n C 2 r B ( 0 , t ) | b ( y ) b B | | f ( y ) | | Ω ( x y ) | d y d t t n + 1 C 2 r B ( 0 , t ) | b ( y ) b B ( 0 , t ) | | f ( y ) | | Ω ( x y ) | d y d t t n + 1 + C 2 r B ( 0 , t ) | b B ( 0 , r ) b B ( 0 , t ) | | f ( y ) | | Ω ( x y ) | d y d t t n + 1 = : I 41 + I 42 .
  1. (i)

    When s p, we obtain

    I 41 C 2 r ( B ( 0 , t ) | b ( y ) b B ( 0 , t ) | p | f ( y ) | p d y ) 1 p ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 s d t t n + 1 C 2 r ( B ( 0 , t ) | b ( y ) b B ( 0 , t ) | p 2 d y ) 1 p 2 f L p 1 ( B ( 0 , t ) ) d t t n p + 1 C 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 ,

then

I 41 L p ( B ) C r n p 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 .

Moreover,

I 42 C 2 r | b B ( 0 , r ) b B ( 0 , t ) | B ( 0 , t ) | f ( y ) | | Ω ( x y ) | d y d t t n + 1 C 2 r | b B ( 0 , r ) b B ( 0 , t ) | ( B ( 0 , t ) | f ( y ) | p 1 d y ) 1 p 1 ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 1 s d t t n + 1 C 2 r ( 1 + ln t r ) t n λ f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 ,

then

I 42 L p ( B ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 .

By estimating I 41 and I 42 , we obtain

I 4 L p ( B ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 + 1 .
  1. (ii)

    When p 1 <s, by the Minkowski inequality, we get

    I 41 L p ( B ) C 2 r B ( 0 , t ) | b ( y ) b B ( 0 , t ) | | f ( y ) | ( B | Ω ( x y ) | p d x ) 1 p d y d t t n + 1 C r n p n s 2 r B ( 0 , t ) | b ( y ) b B ( 0 , t ) | | f ( y ) | d y d t t n n s + 1 C r n p n s 2 r ( B ( 0 , t ) | b ( y ) b B ( 0 , t ) | p 2 d y ) 1 p 2 f L p 1 ( B ( 0 , t ) ) t n n p d t t n n s + 1 C r n p n s 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1

and

I 42 L p ( B ) C 2 r | b B ( 0 , r ) b B ( 0 , t ) | B ( 0 , t ) | f ( y ) | ( B | Ω ( x y ) | p d x ) 1 p d y d t t n + 1 C r n p n s 2 r | b B ( 0 , r ) b B ( 0 , t ) | B ( 0 , t ) | f ( y ) | d y d t t n n s + 1 C r n p n s 2 r | b B ( 0 , r ) b B ( 0 , t ) | f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1 C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1 .

Hence, we have

I 4 L p ( B ) C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n p 1 n s + 1 .

Moreover, for any q>0, we have

r n p 2 + n λ f L p 1 ( 2 B ) = C r n p 2 + n λ + n q f L p 1 ( 2 B ) 2 r d t t n q + 1 C r n p 2 + n q 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n q + 1 C r n p 2 + n q 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1 .

Now combining all the above estimates, we end the proof. □

Then we have the following conclusions.

Theorem 3.2 Let 1<p<, b CBMO p 2 , λ , 0<λ< 1 n , 1 p = 1 p 1 + 1 p 2 and Ω L s ( S n 1 ). Let T Ω be a linear operator satisfying (1.1) and bounded on L p ( R n ) for p>1. If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ( 1 + ln t r ) t n λ ess inf t < τ < φ 1 ( τ ) τ n p 1 t n p 1 + 1 dtC φ 2 (r),
  2. (ii)

    when p 1 <s, ( φ 1 , φ 2 ) satisfies the condition

    r ( 1 + ln t r ) t n λ ess inf t < τ < φ 1 ( τ ) τ n p 1 t n p 1 n s + 1 dtC φ 2 (r) r n s

is satisfied, then the operator [ T Ω ,b] is bounded from B ˙ p 1 , φ 1 to B ˙ p , φ 2 .

Corollary 3.3 Let 1<p<, b CBMO p 2 , λ , 0<λ< 1 n , 1 p = 1 p 1 + 1 p 2 and Ω L s ( S n 1 ). If either of the two conditions

  1. (i)

    when s p, ( φ 1 , φ 2 ) satisfies the condition

    r ( 1 + ln t r ) t n λ ess inf t < τ < φ 1 ( τ ) τ n p 1 t n p 1 + 1 dtC φ 2 (r),
  2. (ii)

    when p 1 <s, ( φ 1 , φ 2 ) satisfies the condition

    r ( 1 + ln t r ) t n λ ess inf t < τ < φ 1 ( τ ) τ n p 1 t n p 1 n s + 1 dtC φ 2 (r) r n s

is satisfied, then the operator [ T ˜ Ω ,b] is bounded from B ˙ p 1 , φ 1 to B ˙ p , φ 2 .

About the commutator of linear operator T Ω , α satisfying (1.2), we get the following corresponding results.

Lemma 3.4 Let 0<α<n, 1< p 1 < n α , b CBMO p 2 , λ , 0<λ< 1 n , p 1 < p 2 < and let

1 p = 1 p 1 + 1 p 2 α n , 1 q = 1 p 1 α n , 1 h = 1 p 1 + 1 p 2 .

T Ω α is a sublinear operator and satisfies (1.2) with Ω L s ( S n 1 ).

When s h, T Ω , α is bounded from L t ( R n ) to L m ( R n ) for any 1<t< n α and 1 m = 1 t α n , then the inequality

[ T Ω , α , b ] f L p ( B ( 0 , r ) ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1

holds for any ball B(0,r) and all f L loc p 1 ( R n ); or q<s, T Ω , α is bounded from L t ( R n ) to L m ( R n ) for any 1<t< n α and 1 m = 1 t α n , then the inequality

[ T Ω , α , b ] f L p ( B ( 0 , r ) ) C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q n s + 1

holds for any ball B(0,r) and all f L loc p 1 ( R n ).

Proof Let 0<α<n, 1< p 1 < n α , b CBMO p 2 , λ , 0<λ< 1 n . For any r>0, set B=B(0,r) and 2B=B(0,2r). We also write

f(x)=f(x) χ 2 B (x)+f(x) χ ( 2 B ) c (x):= f 1 (x)+ f 2 (x)

and

[ T Ω , α , b ] f ( x ) = ( b ( x ) b B ) T Ω , α f 1 ( x ) T Ω , α ( ( b ( ) b B ) f 1 ) ( x ) + ( b ( x ) b B ) T Ω , α f 2 ( x ) T Ω , α ( ( b ( ) b B ) f 2 ) ( x ) = : II 1 + II 2 + II 3 + II 4 .

Hence, we have

[ T Ω , α , b ] f L p ( B ) II 1 L p ( B ) + II 2 L p ( B ) + II 3 L p ( B ) + II 4 L p ( B ) .

Since T Ω , α is bounded from L p 1 ( R n ) to L q ( R n ) and 1 q = 1 p 1 α n , it follows that

II 1 L p ( B ) = ( B | b ( x ) b B | p | T Ω , α f 1 ( x ) | p d x ) 1 p ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | T Ω , α f 1 ( x ) | q d x ) 1 q C r n p 2 + n λ b CBMO p 2 , λ f L p 1 ( 2 B ) ,

where the constant C>0 is independent of f.

For II 2 , 1 p = 1 h α n and 1 h = 1 p 1 + 1 p 2 ,

II 2 L p ( B ) = ( B | T Ω , α ( ( b ( ) b B ) f 1 ) ( x ) | p d x ) 1 p ( R n | b ( x ) b B | h | f 1 ( x ) | h d x ) 1 h ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | f ( x ) | p 1 d x ) 1 p 1 C r n p 2 + n λ b CBMO p 2 , λ f L p 1 ( 2 B ) .

For I 3 ,

  1. (i)

    when s h, by Fubini’s theorem, since xB, we have

    | T Ω , α f 2 ( x ) | C ( 2 B ) c | f ( y ) | | Ω ( x y ) | d y | y | n α = C ( 2 B ) c | f ( y ) | | Ω ( x y ) | | y | d t t n α + 1 d y = C 2 r 2 r | y | < t | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 C 2 r f L p 1 ( B ( 0 , t ) ) ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 1 s d t t n α + 1 C 2 r f L p 1 ( B ( 0 , t ) ) d t t n q + 1 ,

thus

I 3 L p ( B ) C ( B | b ( x ) b B | p d x ) 1 p 2 r f L p 1 ( B ( 0 , t ) ) d t t n q + 1 C ( B | b ( x ) b B | p 2 d x ) 1 p 2 r n q 2 r f L p 1 ( B ( 0 , t ) ) d t t n q + 1 C r n p + n λ 2 r ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1 C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1 ;
  1. (ii)

    when q<s, by Fubini’s theorem and the Minkowski inequality, we get

    I 3 L p ( B ) ( B | 2 r B ( 0 , t ) | b ( x ) b B | | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 | p d x ) 1 p 2 r B ( 0 , t ) | f ( y ) | ( B | b ( x ) b B | p | Ω ( x y ) | p d x ) 1 p d y d t t n α + 1 2 r B ( 0 , t ) | f ( y ) | ( B | b ( x ) b B | p 2 d x ) 1 p 2 ( B | Ω ( x y ) | q d x ) 1 q d y d t t n α + 1 C r n p 2 + n λ 2 r B ( 0 , t ) | f ( y ) | ( B ( 0 , t ) | Ω ( x y ) | s d x ) 1 s | B | 1 q 1 s d y d t t n α + 1 C r n p n s + n λ 2 r B ( 0 , t ) | f ( y ) | d y d t t n n s α + 1 C r n p n s 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q n s + 1 .

On the other hand, for I 4 , by Fubini’s theorem, we have

| [ T Ω , α , b ] f 2 ( x ) | C ( 2 B ) c | b ( y ) b B | | f ( y ) | | Ω ( x y ) | d y | y | n α C 2 r B ( 0 , t ) | b ( y ) b B | | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 C 2 r B ( 0 , t ) | b ( y ) b B ( 0 , t ) | | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 + C 2 r B ( 0 , t ) | b B ( 0 , r ) b B ( 0 , t ) | | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 = : I 41 + I 42 .
  1. (i)

    When s h, we obtain

    I 41 C 2 r ( B ( 0 , t ) | b ( y ) b B ( 0 , t ) | h | f ( y ) | h d y ) 1 h ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 h 1 s d t t n α + 1 C 2 r ( B ( 0 , t ) | b ( y ) b B ( 0 , t ) | p 2 d y ) 1 p 2 f L p 1 ( B ( 0 , t ) ) t n n h d t t n α + 1 C 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n q + 1 ,

then

I 41 L p ( B ) C r n p 2 r t n λ f L p 1 ( B ( 0 , t ) ) d t t n q + 1 .

For I 42 , we have

I 42 C 2 r | b B ( 0 , r ) b B ( 0 , t ) | B ( 0 , t ) | f ( y ) | | Ω ( x y ) | d y d t t n α + 1 C 2 r | b B ( 0 , r ) b B ( 0 , t ) | ( B ( 0 , t ) | f ( y ) | p 1 d y ) 1 p 1 ( B ( 0 , t ) | Ω ( x y ) | s d y ) 1 s | B ( 0 , t ) | 1 1 p 1 1 s d t t n α + 1 C 2 r ( 1 + ln t r ) t n λ f L p 1 ( B ( 0 , t ) ) d t t n q + 1 ,

then

I 42 L p ( B ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1 .

Then, by estimating I 41 and I 42 , we obtain

I 4 L p ( B ) C r n p 2 r t n λ ( 1 + ln t r ) f L p 1 ( B ( 0 , t ) ) d t t n q + 1 .
  1. (ii)

    When q<s, by the Minkowski inequality, we get

    I 41 L p ( B ) C 2 r B ( 0 , t ) | b ( y ) b B ( 0 , t ) | | f ( y ) | ( B | Ω ( x y ) | p d x ) 1 p d y d t t n α + 1 C r n p