BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION ON HERZ-TYPE SPACES

2000,20B(4):461-470

..4atherhi(c;a.9'cientia

1~4mJJ1~m BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION ON HERZ-TYPE SPACES 1 Liu Zongguang ( ;t1J ~ 7L ) Department of Mathematics, Zhejiang University, Hangzhou 310028, China Abstract

This paper establishes some strong type and weak type estimates for commu-

tator [b, I,] on Herz-type spaces, where b E BMO( R n with 0

)

and I, is a fractional integration

< 1 < n.

Key words Herz space, weak Herz space, Herz-type Hardy space, fractional integration, BMO(R n ) function, commutator 1991 MR Subject Classification 42B20, 35J05

1

Introduction

Let b E BMO(Rn ) and I, be a fractional integration with 0 < 1 < n. The commutator [b, I,] generated by b and I, is defined by

[b,II]f(x) == b(x)I,f(x) - I,(bf)(x). S.Chanillo[2] stated that the operator [b, II] is a bounded operator from

LPl (R n )

on [JJ2 (R n )

for 1/p2 == 1/p1 - lin and 1 < P1 < nil. Recently, Lu Shanzhen and Yang Dachun[9] studied the commutator [b, I,] on Herz spaces and a new class of Herz-type Hardy spaces introduced by them and established the corresponding results in [2] on these spaces. In this paper, we take further studies on commutator [b, II] on Herz-type spaces. In Section 2 of this paper, we obtain some strong type estimates for [b, I,] on Herz spaces and Herz-type Hardy spaces. In Section 3 of this paper, we establish some weak type L log" L inequalities and some weak type boundedness for [b, II] on Herz spaces. First, let us introduce some notations. For k E Z, let Bk == {x E R" : Ixl :::; 2k } , Ck == B k \ Bk-1 and Xk == characteristic function of the set C k • The Herz space is defined as follows.

Definition

1.1[1]

Let a E R,O

< P < 00 and 0 < q < 00.

(a) The homogeneous Herz space is defined by

1 Received

Apr.27,1999

XCk

denote the

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(b) The nonhomogeneous Herz space is defined by

For k E Z and measurable function f(x) on Rn, let mk(A,f) = I{x E Ck : If(x)1

for kEN, let mk(A, f) = mk(A, f) and mO(A, f) space is defined as follows.

= I{x

E B« : If(x)1

> A}I.

> A}I;

The weak Herz

Definition 1.2[5] let a E R, 0 < p < 00 and 0 < q < 00. (1) A measurable function f( x) on R" is said to belong to the homogeneous weak Herz space Wk:,P(R n ) , if

(2) A measurable function f(x) on R" is said to belong to the nonhomogeneous weak Herz space WK:,P(R n ) , if

Definition 1.3[9] Let a E R, sEN U {O}, 1 < q < 00, l/q + l/q' = 1, b E Li;c(Rn). A function a(x) on R" is said to be a central (a, q, s; b)-atom, if supp a(x) C B(O, r) for some r > 0; (i) (ii) IlaIILq(Rn) ~ IB(O, r)l-a/n = c-:», (iii) JRn x f3a(x)dx = JRn x f3a(x)b(x)dx = 0, '11/31 ~ s, where /3 = (/31,/32'··· ,/3n) E (N U {o})n, x f3 ~ 1xg2 •• • x~n. Definition 1.4[9] Let a E R, sEN U {O}, 0 < p < 00, 1 < q < 00, l/q + l/q' = 1 and b E Lr;c(Rn). A temperate distribution f is said to belong to H k;,t,· (R n ) (or H (R n)),

xf

if it can be written as

f = .

f:

Ajaj (or

J=-OO

is a central (a, q, s; b)-atom with supp aj C define the quasinorm on H

k;,t,· (R n )

f = .

f: Ajaj) in the 8'(R

J=O

B~

(or H

and

j=~oo IAj IV < 00

K;,t'· (R n ))

«s:

n

)

sense, where each aj

( or j~O IAj IV < 00).

We

as

where infimum is taken over all central atomic decompositions of

f.

Throughout this paper, C, Co and C 1 denote constants that are independent of the main parameters involved but whose values may differ from line to line. The expression A

rv

B means

that there are C, Co > o such that Co ~ AIB ~ C, for any power exponent p with 1 < p < 00, we define the conjugate exponent p' = pl(p - 1). For b E BMO(Rn ) , bk denotes its mean value over B k and Ilbll* denotes its norm in BMO(R n ) .

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The Strong Type Estimates In this section, we obtain some strong type estimates for commutator [b, I,] on Herz spaces

and Herz-type Hardy spaces. The main results are following theorems. Theorem 2.1 Let b E BMO(R n ) and I, be a fractional integration with 0

< 1.< n. If

1 < q1 < nfl, 1/q2 = 1/q1(1 - Ip1/n), -n/q1 + I < (}:1 < n(l - 1/q1), 0:2 = (}:1 + l(p1/q1 - 1) and 0 < P1 ~ min(P2,q1), then commutator [b,I,] is a bounded operator from k~/'Pl(Rn) to k~22'P2(Rn).

Proof Because K~./'Pl (R n) C K~22'P2 (R n) with P1 ~ P2, we only need to prove that [b, I,] is a bounded operator from k~l'Pl(Rn) to k~2'Pl(Rn). •

Let f(x) E K~l,Pl(Rn), then f(x)

= L: 00

j=-oo

Ajaj(x), where Aj

aj(x) = 2J""Hx)~r
)

j='2;oo IAj IPI

1/Pl

=

IIfllk:I' ,P' (R n r

= 2JolllfXjIILql(Rn) and .

lI aj IIL

q,

(Rn)

= 2- j OI and

Thus

Let l/qo = 1/q1 -lin, then 1/q2 = l/qo + l/n(l- P1/Q1). Using the well known fact that [b, I,] is a bounded oprator from Lql (R n ) to Lqo (R n ) in [2] and the Holder inequality, we obtain

When 0

< P1

and when P1

~

1, it is easy to see that

> 1, by the Holder inequality, we have

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ACTA MATHEMATICA SCIENTIA

j+2

00

< C { j~ IAjlPI (k~OO 2(k- )OIPJ/2)

~

C( L 00

IAjIPI)

1/Pl

j

=

C1lfllk;t

P1

Vo1.20 Ser.B

} 1/Pl

(Rn),

j=-oo

Now we estimate E 1 • It is easy to see that

Because Iy - zl2: lyl-lzl2: 2 k- 1 - 2j 2: 2 k- 3 for y E Ck,Z E Bj,j

S

k -

3, we can get that

This deduces that

[xk (b -

b

k

)1I aJ·11 Lq2 (Rn) < -

C2-k(n(1-1/ql)+I(Pl/ql-1»+j(n(1-1/ql)-Ol)llbll

*.

By the Holder inequality,

I

ll(b- bk)ajll£l(Rn)

(n -

ll

~

C2- k(n -

l)

S

C(k - j)2-k(n-l)+j(n(1-1/ql)-ol)l/bll*·

III((b - bk)aj)(Y) ~ C2-

k

(II (b -

bj )aj IILI(Rn) + Ibk - bj

Illaj 1I£1(Rn))

This follows that

Noticing that a1 < n(1 - l/q1) and a2 = a1 estimate E 2 in this theorem, we obtain

+ 1(P1/ql -

1) and using the similar method to

Thus we complete the proof of Theorem 2.1. When «i 2: n(1 - l/q1), the result of Theorem 2.1 is not true. k~l'Pl (Rn) by H k~~'tl'S (R n), we get the following

It needs to replace

Theorem 2.2 Let b E BMO(R n ) and I, be a fractional integration with 0 < 1 < n. If 1 < q1 < nil, a1 2: n(I-I/q1), l/q2 = I/Ql(l-lp1/ n), 0 < PI S min(Q1,P2), a2 = a1 +1(P1/Ql- 1) n 2 and s + 1 > a1 + n( 1/Q1 - 1), then [b, I,] is a bounded operator H K~~'tl,8 (R n) to 2 ,P2(R ).

k:

Proof Similar to the proof of Theorem 2.1, we only need prove 02,Pl(R n). from HK°I,Pl,8(Rn) to K q2 ql,b

[b~

I,] is a bounded operator

Liu: BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION

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465

00

Let f E HK;11,'t1,S(R n), then f(x) == . I: Ajaj(x) in the S'(R n) sense, where each aj is a )=-00

central (O:l,Pb s; b)-atom with supp aj C

rx JRn

i3 aj (x )b(x )dx

= 0, VI.8I:S

e; i.e.

IlajIILQ'(Rn):S

C2- j o , and

Ln x i3aj(x)dx

=

s. Thus

Similar to the estimate of E 2 in Theorem 2.1, we have

For F 1 , we need pointwise estimate for \[b,I,]aj(x)\ with z E Ck,j ~ k - 3,

In the second inequality, we have used the s-order vanishing moments of aj and the s-order

Taylor expansion of [z - yl'-n at x with 8 1,82 E (0,1), and x E Ck, Y E Cj, this deduces that

Noticing s

+ 1 > (};1 + n(1/q1 -

Ix -

8i yl ~

Ixl- Iyl > Ixl/2

for

1) and similar to the estimate E 1 in Theorem 2.1, we get

By the definition of quasinorm of elements in the space H k01'bP1,s (R n ), we get ql,

This completes the proof of the Theorem 2.2. Remark 1 In this section, we only discuss the problems associate with the homogeneous

Herz-type spaces. In fact, the similar results related to nonhomogeneous Herz-type spaces also hold, To limit the length of this paper, we omit the details.

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Tile Weak Type Estimates In this section, we obtain, at first, some weak type L log" L inequalities for [b, I,] on the

nonhomogeneous Herz spaces, then we introduce the so-called condition E and find the fact that, if b E BMO( R n ) satisfies the condition [, and the parameters of these Herz spaces satisfy certain conditions, the commutator [b, I,] is a bounded operator from the homogeneous and nonhomogeneous Herz spaces to corresponding weak Herz spaces. Theorem 3.1 Let b E BMO(R n ) and I, be a fractional integration with 0 < I < n.

=

If 0 < P1 :::; 1, P1 :::; P2, 1 < q1 < nfl, 1/q2 1/q1(1 - lp1/ n), a1 = n(l - 1/q1) and a2 = a1 + l(P1/q1 - 1), for any f E K;/,Pl(Rn) and any A > 0, then there exists a constant C > 0 independent of f and A, such that

{E OO

)P2/q2}1/

2 02P2 mk A, [b,I] f I k

(

P2

k=O

Proof It is easy to see that

{~2k02P21nk(.X'[b,I,]fY

2/q2

CllfIIKa1,Pl(Rn) ( 1 + log + CllfIIKa1'Pl(Rn)) < ql qlA · A

r

/P2 ::;

{~2k02Plmk(.X'

[b,I,]fy

tl q2

r

lPl

for P1 :::; P2. Thus, we only need to prove this theorem in the case that P1 = P2. Let f E K;/,Pl(R n ) , we can get the same decomposition for f(x). Denote ko = [4/(nI)] + 1, using the fact that [b,I,] is a bounded operator from Lql(Rn) to Lqo(R n) and 1/q2 =

l/qo

+ l/n(l -

P1/q1) with l/qo = 1/q1 -lin, we have

{k~l 2k02Plmk (A, [b,I,]fytlQ2 } llpl C

::; -:xllfIILql(Rn) (

::; -:x E IAj 12-

C o o .1 0 j=O

k o-1

E 2 OIPl) k

k=O 1

C (

< -:x

<

00

j=O

00

)

1/PI

·

j=O

k~O 2k02Pl mk ( A, [b, I'] f

c{ k~O +c{ k~O

C

< -:xII EAjajIILql(Rn)

E IAj IPI

and

{

l/PI

v:}

2k02Plmk (A/2, [b,I,]

llpl

(~Ajaj) ytlQ2

2k02Plmk (A/2, [b,Il]

r

lPl

(j~Z Ajaj) y.tQ2 } llpl

= G1 + G2 • Noticing the fact that [b,Iz] is a bounded operator from Lql (R n) to

Gz

L90 (Rn),

< C {k~O 2kolPlmk ( A/2, [b,I,] (j~Z Ajaj) ytlQO } llpl

we get

Liu: BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION

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467

For G 1 , we need the pointwise estimate for I[b, I,]aj (x) I for x E Ck, 0 ~ j ~ k - 3. It is easy to see the fact that, when x E C k , Y E B j with 0 ~ j ~ k - 3, the following conditions are satisfied:

Ix - yl ~

k

2

-

2

,

L.laj(y)ld y::; G and L.lb(Y) - bjllaj(y)dyl ::; J

Gllbll.

J

Thus

i[b, I,]aj(x) I = I L.

(b(X?x-_b;r1~~j(Y)dyl

J

< Ib(x) - b·1 [ -

J

laj(y)1 dy +

lB' Ix J

[ Ib(y) - bjllaj(y)ldy Ix - yln-l

lB. J

1+ C2- (n- I)llbll* i, 1+ Ck2- (n- I)l\bll* ·

~ C2- k (n- l) Ib(x) - bj

~ C2- k (n- l) Ib(x) -

yln-l

k

k

Thus we get

By the John-Nirenberg inequality, we obtain

For the estimate of H 2 , we use a well known inquality that log2 x ~ x/2 when x ~ 4 and get the following simply claim: Claim If there is u > 1, such that 2X / x ~ u for some x ~ 4, then 2X ~ Cu log2 u.

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Noticing k( n - 1) 2:: 4 when k

I{

Vo1.20 Ser.B

2:: ko, if

c. : Ck2- k (n - l ) lI bll* :E IAj I > A/4} Ii= 0, 00

x E

j=O

we get

2k (n -

l)

1 < k(n -I)

This deduces that 2

k(n-l)

<

(~

C

00

~ I"b"* ~ IAjl·

t,IAjl)

log+ (

~

t,IAjl).

Let K A be the maximal integer of k's satisfing the last inequality, we have

By the definition of quasinorm of element in

HK~lPl(Rn),

we obtain

This finishes the proof of Theorem 3.1. Theorem 3.2 Let b E BMO(Rn) and I, be a fractional integration with 0 < I < n.

If 0 < Pl ~ 1, Pl ~ P2, 1 < ql < nil, 11q2 = 11ql ., lin and al = n(1 - 1Iql), for any f E K~/,Pl(Rn) and any A > 0, then there exists a constant C > 0 independent of f and A, such that

The proof of this theorem is similar to the proof of Theorem 3.1, we omit the details. Now, we introduce the so-called condition E, Definition 3.1 It is said that b E BMO(Rn) satisfies the condition [" if for any k,j E Z with j ~ k - 3 and any z E Ck, there exists a constant C > 0 only dependent on n, such that

where bj is the mean value of b(it) over B i: Remark 2 The condition [, is nontrival. Example Any odd function b E BMO(R l ) satisfies the condition [, because it is easy to see that bj = 0 for any j E Z. By [3], we can find out many unbounded odd functions belong to BMO(Rl ) .

No.4

Liu: BOUNDEDNESS OF COMMUTATORS OF FRACTIONAL INTEGRATION

469

Theorem 3.3 Let b E BMO(RU ) and satisfy the condition E and I, be a fractional integration with 0

< I < n. If 0 < P1 ~ 1, P1 ~ P2, 1 < q1 < n/l~ 1/q2 == 1/q1(1-1p1/q1), a1 ==

n(l - 1/q1) and a2 == a1 + l(p1/q1 - 1), then [b,Iz] is a bounded operator from i<~/'Pl (R U) to Wk~ll'Pl(RU). For nonhomogeneous Herz-type spaces, the similar result holds.

Proof Because Wk~22'PI(RU) C Wk~22'P2(RU) with P1 ~ P2, we only need to prove that [b,Iz] is a bounded operator from k~/ ,PI (Rn) to W i<~22'PI (R n). Let f E i<~/'PI(Rn), we can get the same decemposition for f(x). Then, we get

Similar to the estimate for E 2 in Theorem 2.1, we have

For 11 , similar to the estimate for G 1 in Theorem 3.1, when x E Ck,j

Hb, Iz]aj(x) I ~ C2- k (n ~ C2- k(n -

~

k - 3, we have

- bj I + C2- k (n-I)llbll* k l ) Ib(x) - bk I + C2- (n - l ) Ilbll*· l ) Ib(x)

In the last inequality, we have used the condition E for b. Thus, we get k-3

I[b, I,] (

L

j=-oo

Ajaj) (x)

I ~ C2-

k

(n - l )

II/IIK:,' ,P, (Rn) (Ib(x)

This deduces that

For J 1, similar to the estimate H 1 in Theorem 3.1, we have

- bk

I+ II bll. ).

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I

: C2-k(n-llllbll.lltllk;,',Pl(Rnl} :I 0, then 2k( n - ll :S fll bll.lltllk;t P 1 (R n) " Let K>.. be the maximal integer satisfing the last inequality, we get

X

E

C~,

This finishes the proof of Theorem 3.3. Theorem 3.4 Let b E BMO(Rn ) and satisfy the condition

.c

and I, be a fractional

integration with 0 < 1 < n. If 0 < Pl ~ 1,Pl ~ p2,1 < ql < nll,1/q2 == l/ql - lin and == n(l- l/ql), then [b,I,] is a bounded operator from k~/'Pl(Rn) to WK~~/'P2(Rn). For nonhomogeneous Herz-type space, the similar result holds. The proof of this theorem is similar to that of theorem 3.3.

Ql

Acknowledgement The author wishes to express his deep thanks to his advisor Professor Wang Silei for constant encouragement and guidance. References 1 Baernstein II A, Sawyer E T. Embedding and multiplier theorems for HP(Rn). Memoirs of Amer Math Soc, 1985,53(318) 2 Chanillo S. A note on commutators. Indiana Univ Math J, 1982, 31: 7-16 3 Coifman R R, Rochberg R. Another charact.erization of BMO. Proc Amer Math Soc, 1980,79: 249-254 4 Garcia-Cuerva J, Herrero M -J L. A theory of Hardy spaces associated to Herz spaces. Proc London Math Soc, 1994,69: 605-628 5 Hu Guoen, Lu Shanzhen, Yang Dachun. The weak Herz spaces. Joural of Beijing Normal University (Natural Science), 1997, 26: 417-428 6 Hu Guoen, Lu Shanzhen, Yang Dachun. The applications of weak Herz spaces. Adv in Math (China), 1997, 26: 417-428 7 Lu Shanzhen, Yang Dachun. The decomposition of the weighted Herz spaces and its application. Science in China (Ser.A), 1995, 38: 147-158 8 Lu Shanzhen, Yang Dachun. The weighted Herz-type Hardy spaces and its applications. Science in China(Ser.A), 1995, 38: 662-673 9 Lu Shanzhen, Yang Dachun. The Continuity of commutators on Herz-type spaces. Michigen Math J, 1997, 44: 255-281 10 Lu Shanzhen, Yang Dachun. Some Hardy spaces associated with the Herz spaces and their wavelet characterizations. Beijing Shifan Daxue Xuebao, 1993,29: 10-19 (Chinese)