WEIGHTED INEQUALITIES FOR COMMUTATORS OF POTENTIAL ...

J. Korean Math. Soc. 44 (2007), No. 6, pp. 1233–1241

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** **OF**

**POTENTIAL** TYPE OPERATORS

Wenming Li

Reprinted from the

Journal of the Korean Mathematical Society

Vol. 44, No. 6, November 2007

c○2007 The Korean Mathematical Society

J. Korean Math. Soc. 44 (2007), No. 6, pp. 1233–1241

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** **OF**

**POTENTIAL** TYPE OPERATORS

Wenming Li

Abstract. We derive a kind of weighted norm inequalities which relate

the commutators of potential type operators to the corresponding maximal

operators.

1. Introduction

For a nonnegative, locally integrable function Φ on R n , define the potential

type operator T Φ by

∫

T Φ f(x) = Φ(x − y)f(y)dy.

R n

The basic example is provided by the Riesz potentials or fractional integrals

I α , defined by the kernel Φ(x) = |x| α−n , 0 < α < n.

Now assume that the kernel Φ satisfies the following weak growth condition:

there are constants δ, c > 0, 0 ≤ ε < 1 with the property that for all k ∈ Z,

(1) sup Φ(x) ≤ c ∫

2 k

1234 WENMING LI

Let b ∈ BMO, and Φ satisfy (1), the higher order commutators of T Φ are

defined by

T b,m

Φ f(x) = ∫R n (b(x) − b(y)) m Φ(x − y)f(y)dy,

where m = 0, 1, 2, . . ..

In this paper, using the technique developed in [4], [5] and [7], we derive

weighted norm inequalities which relate the commutators of potential type

operators to the corresponding maximal operators.

2. Preliminaries and main results

We will need the following facts about Orlicz spaces, for further information

see [1]. A function B : [0, ∞) → [0, ∞) is a Young function if it is continuous,

convex and increasing, and if B(0) = 0 and B(t) → ∞ as t → ∞. A Young

function B is doubling if there exists a positive constant C such that B(2t) ≤

CB(t) for all t > 0. If A, B are two Young functions, we write A(t) ≈ B(t) if

there are constants c, c 1 , c 2 > 0 with c 1 A(t) ≤ B(t) ≤ c 2 A(t) for t > c. Given

a Young function B, define the mean Luxemburg norm of f on a cube Q by

∫ ( |f(y)|

}

‖f‖ B,Q = inf

{

λ > 0 :

1

|Q|

Q

B

λ

)

dy ≤ 1

For a given Young function B, there exists a complementary Young function

B such that

t ≤ B −1 (t)B −1 (t) ≤ 2t.

There is another characterization of the Luxemburg norm, which we will

need:

(2) ‖f‖ B,Q ≤ inf

s>0

{

s +

s ∫

B

|Q| Q

( |f(x)|

s

) }

dx ≤ 2‖f‖ B,Q .

Given three Young functions A, B and C such that for all t > 0,

A −1 (t)C −1 (t) ≤ B −1 (t),

then we have the following generalized Hölder inequality due to O’Neil [3]: for

all functions f and g and for any cube Q,

(3) ‖fg‖ B,Q ≤ 2‖f‖ A,Q ‖g‖ C,Q .

In particular, given any Young function B,

∫

1

(4)

|f(x)g(x)|dx ≤ 2‖f‖ B,Q ‖g‖

|Q|

B,Q

.

Q

Given a Young function B, define the associated Orlicz maximal operator

by

M B f(x) = sup ‖f‖ B,Q .

Q∋x

The dyadic maximal operator MB d is defined similarly, except the supremum is

restricted to dyadic cubes containing x.

.

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** 1235

For an increasing function φ : [0, ∞) → [0, ∞) and a Young function B, we

define the maximal operator associated to φ and B by

M B,φ f(x) = sup φ(|Q| 1/n )‖f‖ B,Q .

Q∋x

For the case B(t) = t, we write M B,φ simply as M φ . For α > 0, φ(t) = t α , the

maximal operator associated to φ and B is denoted by M B,α . When B(t) = t,

we denote M B and M B,α simply as M and M α respectively. Clearly M is

the well-known Hardy-Littlewood maximal operator and M α is the fractional

maximal operator. It is easy to prove that M B,α f(y) ≤ M α M B f(y) for any

Young function B.

The main example that we are going to be using is Ψ m (t) = t log(e + t) m ,

m = 1, 2, . . .. For the Young function Ψ m , we denote the mean Luxemburg

norm of f on a cube Q by ‖f‖ L(log L) m ,Q, the maximal function by M L(log L) mf.

For this maximal function, Pérez [6] obtains that M L(log L) mf(y) ≈ M m+1 f(y)

whenever m = 0, 1, 2, . . . , where M k denotes the Hardy-Littlewood maximal

operator M iterated k times. The complementary Young function of Ψ m is

given by Ψ m (t) ≈ e t1/m , with the corresponding mean Luxemburg norm and

the maximal function denoted by ‖f‖ exp L

1

m ,Q

and M exp L

1

m

f respectively. Otherwise,

we need the following lemmas about the maximal functions.

Lemma 2.1 ([2]). Let 0 < α < n and Mf, M α f be locally integrable. Then

there are constants C > 0, C 1 > 0 and C 2 > 0, independent of f and y, such

that

(i) MM α f(y) ≤ CM α Mf(y),

(ii) C 1 M α M m f(y) ≤ M L(log L) m ,αf(y) ≤ C 2 M α M m f(y), m = 0, 1, 2, . . . .

Let 1 < p < ∞. We say that a doubling Young function B satisfies the

B p -condition if there is a positive constant c such that

∫ ∞

c

B(t) dt

t p t < ∞.

Lemma 2.2 ([6]). For 1 < p < ∞, let B be a doubling Young function. Then

the following are equivalent:

(i) B ∈ B p ;

(ii) there is a constant C such that

∫

∫

M B f(y) p dy ≤ C f(y) p dy

R n R n

for all nonnegative functions f;

(iii) there is a constant C such that

∫

∫

M B f(y) p w(y)dy ≤ C f(y) p Mw(y)dy

R n R n

for all nonnegative functions f and w.

1236 WENMING LI

Our main results are the following Theorem and Corollary. Here and below,

by weights we mean nonnegative locally integrable functions.

Theorem 2.3. Given m ≥ 0 an integer, Ψ m (t) = t log(e+t) m . Let T b,m

Φ

be the

commutator of potential type operator T Φ with Φ satisfies (1) and b ∈ BMO.

(i) There is a constant C such that for any weight w and all f,

∫

(5) |T b,m

Φ f(y)|w(y)dy ≤ C M L(log L) mf(y)M L(log L)

R

∫R m ,˜Φ w(y)dy.

n (ii) For 1 < p < ∞, there is a constant C such that for any weight w and

all f,

∫

(6) |T b,m

Φ

f(y)|p w(y)dy ≤ C

R n ∫

|f(y)| p M(M L(log L) [(m+1)p] ,˜Φ

)w(y)dy.

p

R n

For 0 < α < n the case Φ(x) = |x| α−n corresponds to the Riesz potential of

order α. In this case ˜Φ(t) ∼ = t α . By Lemma 2.1 and Theorem 2.3, we have the

following corollary:

Corollary 2.4. Given m ≥ 0 an integer, 0 < α < n and Ψ m (t) = t log(e+t) m .

Let

Iα b,m

(b(x) − b(y))

f(x) =

∫R m

f(y)dy

|x − y|

n−α n

be the commutator of order m for I α and b ∈ BMO.

(i) There is a constant C such that for any weight w and all f,

∫

∫

|Iα

b,m f(y)|w(y)dy ≤ C

R n M L(log L) mf(y)M L(log L) m ,αw(y)dy.

R n

(ii) If 1 < p < ∞, there is a constant C such that for any weight w and all

f,

∫

|Iα

b,m f(y)| p w(y)dy ≤ C

R n ∫

|f(y)| p M L(log L) ,pαw(y)dy.

[(m+1)p]+1

R n

3. The proof of Theorem 2.3

In order to prove Theorem 2.3, we need the following generalized Calderón-

Zygmund decomposition.

Lemma 3.1 ([6]). Given a doubling Young function B, suppose f is a nonnegative

function such that ‖f‖ B,Q tends to zero as l(Q) tends to infinity. Then

for each t > 0 there exists a disjoint collection of maximal dyadic cubes {Q t,j }

such that for each j, t < ‖f‖ B,Qt,j ≤ 2 n t, and

{x ∈ R n : M d Bf(x) > t} = ⋃ j

Q t,j ,

{x ∈ R n : M B f(x) > 4 n t} ⊂ ⋃ j

3Q t,j .

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** 1237

The collection of dyadic cubes {Q t,j } is referred to as the Calderón-Zygmund

decomposition of f with respect to B at height t.

We fix a constant a > 2 n , and for each integer k we let

Ω k = {x ∈ R n : M B f(x) > 4 n a k },

D k = {x ∈ R n : M d Bf(x) > a k }.

Hence, by Lemma 3.1 with t = a k there is a family of maximal non-overlapping

dyadic cubes {Q k,j } for which D k = ∪ j Q k,j , Ω k ⊂ ∪ j 3Q k,j , and a k <

‖f‖ B,Qk,j ≤ 2 n a k .

Lemma 3.2 ([6]). Suppose a > 2 n . For all integers k, j we let E k,j = Q k,j \

(Q k,j ∩ D k+1 ). Then {E k,j } is a disjoint family of sets which satisfies

|Q k,j | ≤

1

1 − 2 n /a |E k,j|.

Proof of Theorem 2.3. We set for each t > 0,

and

Φ(t) =

sup Φ(y),

t

1238 WENMING LI

Now we prove (i). By a limiting argument we may assume that w is bounded

with compact support. Then, by the Hölder inequality (3) and the John-

Nirenberg theorem, we have

∫

|T b,m

Φ

f(y)|w(y)dy

R n

≤ ∑ m∑

( ) ∫ m

|b(y) − b Q | m−l w(y)dy · Φ( l(Q)

l

Q∈∆ l=0

Q

2

∫3Q

) |b(z) − b Q | l |f(z)|dz

≤ ∑ m∑

( ) m

|Q|‖(b − b Q ) m−l ‖ 1

l ‖w‖ exp L (m−l) L(log L)

,Q m−l ,Q

Q∈∆

l=0

· Φ( l(Q)

2 )|3Q|‖(b − b Q) l ‖ 1 exp L l

‖f‖ ,3Q L(log L) l ,3Q

≤ ∑ m∑

( ) m

|Q|‖b − b Q ‖ m−l

exp L,Q

l ‖w‖ L(log L) m−l ,Q

Q∈∆

l=0

· Φ( l(Q)

2 )|3Q|‖b − b Q‖ l exp L,3Q‖f‖ L(log L) l ,3Q

∑ m∑

≤ C‖b‖ m BMO ‖w‖ L(log L) ,Q|Q| · Φ( l(Q)

m−l 2 )|3Q|‖f‖ L(log L) l ,3Q

≤ C‖b‖ m BMO

Q∈∆ l=0

∑

Q∈∆

‖w‖ L(log L) m ,Q|Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q.

Since w is bounded with compact support, for the Young function Ψ m (t) =

t log(e + t) m , we have ‖w‖ Ψm ,Q → 0 as l(Q) → ∞. For a constant a > 2 n

and an integer k, by Lemma 3.1 with t = a k there is a family of maximal

non-overlapping dyadic cubes {Q k,j } for which

For each integer k we let

a k < ‖w‖ L(log L) m ,Q k,j

≤ 2 n a k .

C k = {Q ∈ ∆ : a k < ‖w‖ L(log L) m ,Q ≤ a k+1 }.

Every dyadic cube Q for which ‖w‖ L(log L) m ,Q ≠ 0 belongs to exactly one C k .

Furthermore, if Q ∈ C k , it follows that Q ⊂ Q k,j for some j. Then

∫

|T b,m

Φ

f(y)|w(y)dy

R n ∑ ∑

≤ C‖b‖ m BMO ‖w‖ L(log L) m ,Q|Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q

k∈Z Q∈C k

∑

≤ C‖b‖ m BMO a ∑ ∑

k+1 |Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q

k∈Z j∈Z Q∈C k :Q⊂Q k,j

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** 1239

∑

≤ C‖b‖ m BMO ‖w‖ L(log L) m ,Q k,j

k,j

∑

|Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q

Q:Q⊂Q k,j

∑

≤ C‖b‖ m BMO ‖w‖ L(log L) m ,Q k,j

˜Φ(δ(1 + ε)l(Qk,j ))|3Q k,j |‖f‖ L(log L) m ,3Q k,j

,

k,j

where the last inequality will follow if we show that there is a constant C such

that for any dyadic cube Q 0 ,

(7)

However, if l(Q 0 ) = 2 −ν 0

, then

(8)

∑

|Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q

Q:Q⊂Q 0

≤ C ˜Φ(δ(1 + ε)l(Q 0 ))|3Q 0 |‖f‖ L(log L) m ,3Q 0

.

∑

|Q| · Φ( l(Q)

2 )|3Q|‖f‖ L(log L) m ,3Q

Q:Q⊂Q 0

= ∑ ∑

2 −νn Φ(2 −ν−1 )|3Q|‖f‖ L(log L) m ,3Q

ν≥ν 0 l(Q)=2 −ν

Q⊂Q 0

= ∑

∑

2 −νn Φ(2 −ν−1 ) |3Q|‖f‖ L(log L) m ,3Q.

ν≥ν 0

By the inequality (2), we have

∑

≤

l(Q)=2 −ν

Q⊂Q 0

|3Q|‖f‖ L(log L) m ,3Q

l(Q)=2 −ν

Q⊂Q 0

∑

l(Q)=2 −ν

Q⊂Q 0

{

|3Q| inf µ + µ

µ>0 |3Q|

∫

3Q

(|f(z)|) }

Ψ m dz

µ

{

∫

(|f(z)|) }

≤ inf |3Q 0 |µ + Cµ Ψ m dz

µ>0

3Q 0

µ

{

≤ C|3Q 0 | inf µ + µ ∫

(|f(z)|) }

Ψ m dz

µ>0 |3Q 0 | 3Q 0

µ

≤ C|3Q 0 |‖f‖ L(log L) m ,3Q 0

,

because of bounded overlap. Now, by condition (1), we have Φ(2 −ν ) ≤ CΦ(2 −ν )

for ν ∈ Z, and then

∑

2 −νn Φ(2 −ν−1 ) ≤ C ∑

2 −νn Φ(2 −ν−1 )

ν≥ν 0 ν≥ν 0

1240 WENMING LI

= C ∑ ∫

ν≥ν 0

∫

≤ C

δ(1−ε)2 −ν−1 1. Our argument will be based on duality and

the result in (i). In fact we will prove something sharper than (6): if δ > 0,

there is a constant C such that for any weight w and all f,

(9)

∫

|T b,m

Φ

f(y)|p w(y)dy ≤ C

R n

∫

R n |f(y)| p M(M L(log L) (m+1)p−1+δ ,˜Φ p )w(y)dy,

where M L(log L) (m+1)p−1+δ ,˜Φ

denotes the maximal operator associated to ˜Φ p

p

and

B(t) = t log(e + t) (m+1)p−1+δ , t > 0.

Selecting δ > 0 such that (m + 1)p − 1 + δ = [(m + 1)p] we get (6).

By (5), there is a constant C so that for all g ≥ 0,

∫

∫

|T b,m

Φ

f(y)|g(y)w(y)1/p dy ≤ C M L(log L) mf(y)M Ψm ,˜Φ (gw1/p )(y)dy.

R n R n

We choose A(t) ≈ t p′ (log t) −1−(p′ −1)δ and C(t) ≈ t p (log t) (m+1)p−1+δ for large

t. It is easy to check that A ∈ B p ′ and

Ψ −1

m (t) ≈ A −1 (t) × C −1 (t).

**WEIGHTED** **INEQUALITIES** **FOR** **COMMUTATORS** 1241

Noticing that Ψ m ∈ B p , by the generalized Hölder inequality (4) and Lemma

2.2 (iii), we can continue with

∫

≤ C M L(log L) mf(y)M C,˜Φ(w 1/p )(y)M A g(y)dy

R n

≤ C( ∫ R n (M L(log L) mf(y)) p (M C,˜Φ(w 1/p )(y)) p dy) 1/p ( ∫ R n M A g(y) p′ dy

≤ C( ∫ ) 1/p

(M L(log L) mf(y)) p M L(log L) (m+1)p−1+δ ,˜Φ

(w)(y)dy

p

R n

( ∫ ) 1/p

′

× g(y) p′ dy

R n

) 1/p

′

≤ C( ∫ 1/p (

|f(y)| p M(M L(log L) (m+1)p−1+δ ,˜Φ

)w(y)dy) ∫ ) 1/p

′

g(y) p′ dy .

p

R n R n

Thus (9) is true. This concludes the proof of Theorem 2.3.

Acknowledgments. The author would like to express his gratitude to the

referee for his very valuable comments and suggestions.

References

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[3] R. O’Neil, Fractional integration in Orlicz spaces. I., Trans. Amer. Math. Soc. 115 (1965),

300–328.

[4] C. Pérez, Sharp L p -weighted Sobolev inequalities, Ann. Inst. Fourier (Grenoble) 45

(1995), no. 3, 809–824.

[5] , Two weighted inequalities for potential and fractional type maximal operators,

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[6] , On sufficient conditions for the boundedness of the Hardy-Littlewood maximal

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College of Mathematics and Information Science

Hebei Normal University

Shijiazhuang, 050016, Hebei, P. R. China

E-mail address: lwmingg@sina.com

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