Báo cáo hóa học: " Research Article Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures" pot

Hindawi Publishing CorporationJournal of Inequalities and Applications Volume 2008, Article ID 141379, 25 pages doi:10.1155/2008/141379 Research Article Boundedness of Parametrized Littl

Trang 1

Hindawi Publishing Corporation

Journal of Inequalities and Applications

Volume 2008, Article ID 141379, 25 pages

doi:10.1155/2008/141379

Research Article

Boundedness of Parametrized Littlewood-Paley

Operators with Nondoubling Measures

Haibo Lin 1 and Yan Meng 2

1 School of Mathematical Sciences, Beijing Normal University,

Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

2 School of Information, Renmin University of China, Beijing 100872, China

Received 2 April 2008; Accepted 30 July 2008

Recommended by Siegfried Carl

parametrized g λ∗functionM∗,ρ λ is bounded on L p μ for p ∈ 2, ∞ with the assumption that the

kernel of the operatorM∗,ρ λ satisfies some H ¨ormander-type condition, and is bounded from L1μ into weak L1μ with the assumption that the kernel satisfies certain slightly stronger H¨ormander-

type condition As a corollary,M∗,ρ λ with the kernel satisfying the above stronger H ¨ormander-type

condition is bounded on L p μ for p ∈ 1, 2 Moreover, the authors prove that for suitable indexes

ρ and λ,M∗,ρ λ is bounded from L∞μ into RBLOμ the space of regular bounded lower oscillation

into L1μ if the kernel satisfies the above stronger H¨ormander-type condition The corresponding

properties for the parametrized area integralMρ

Sare also established in this paper

Commons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

1 Introduction

Let μ be a nonnegative Radon measure on Rd which only satisfies the following growth

condition that for all x∈ Rd and all r > 0:

μ

B x, r≤ C0r n , 1.1

where C0 and n are positive constants and n ∈ 0, d, and Bx, r is the open ball centered

at x and having radius r Such a measure μ may be nondoubling We recall that a measure

μ is said to be doubling, if there is a positive constant C such that for any x ∈ suppμ and

r > 0, μ Bx, 2r ≤ CμBx, r It is well known that the doubling condition on underlying

Trang 2

2 Journal of Inequalities and Applications

measures is a key assumption in the classical theory of harmonic analysis However, inrecent years, many classical results concerning the theory of Calder ´on-Zygmund operatorsand function spaces have been proved still valid if the underlying measure is a nonnegativeRadon measure onRd which only satisfies1.1 see 1 8 The motivation for developingthe analysis with nondoubling measures and some examples of nondoubling measures can

be found in9 We only point out that the analysis with nondoubling measures played astriking role in solving the long-standing open Painlev´e’s problem by Tolsa in10

Let K be a μ-locally integrable function onRd×Rd \{x, y : x y} Assume that there exists a positive constant C such that for any x, y∈ Rd with x / y,

where ρ ∈ 0, ∞ The parametrized area integral M ρ

S and g λ∗ function M∗,ρ λ are defined,respectively, by

 {y, t : y ∈ R d , t > 0 }, ρ ∈ 0, ∞, and λ ∈ 1, ∞ It is easy to verify that if μ is the d-dimensional Lebesgue measure inRd, and

K x, y Ωx − y

|x − y| d−1 1.7

withΩ homogeneous of degree zero and Ω ∈ Lipα S d−1 for some α ∈ 0, 1, then K satisfies

1.2 and 1.3 Under these conditions, Mρin1.4 is just the parametrized Marcinkiewiczintegral introduced by H ¨ormander in11, and Mρ

SandM∗,ρ λ as in1.5 and 1.6, respectively,

are the parametrized area integral and the parametrized g∗λfunction considered by Sakamotoand Yabuta in12 We point out that the study of the Littlewood-Paley operators is motivated

by their important roles in harmonic analysis and PDE13,14 Since the Littlewood-Paley

Trang 3

Journal of Inequalities and Applications 3

operators of high dimension were first introduced by Stein in15, a lot of papers focus onthese operators, among them we refer to16–21 and their references

When ρ 1, the operator Mρ as in 1.4 is just the Marcinkiewicz integral withnondoubling measures in22, where the boundedness of such operator in Lebesgue spacesand Hardy spaces was established under the assumption that Mρ is bounded on L2μ.

Throughout this paper, we always assume that the parametrized Marcinkiewicz integralwith nondoubling measuresMρas in1.4 is bounded on L2μ By a similar argument in

22, it is easy to obtain the boundedness of the parametrized Marcinkiewicz integral Mρ

with ρ ∈ 0, ∞ from L1μ into weak L1μ, from the Hardy space H1μ into L1μ, and from L∞μ into RBLOμ the space of regular bounded lower oscillation functions; see

Definition 2.5 below As a corollary, it is easy to see that Mρ is bounded on L p μ with

we will prove that M∗,ρ λ as in 1.6 is bounded from H1μ into L1μ And in the last

section, the corresponding results for the parametrized area function Mρ S as in 1.5 areestablished

For a cube Q ⊂ Rd we mean a closed cube whose sides parallel to the coordinate

axes and we denote its side length by lQ and its center by x Q Let α > 1 and β > α n We

say that a cube Q is an α, β-doubling cube if μαQ ≤ βμQ, where αQ denotes the cube with the same center as Q and lαQ αlQ For definiteness, if α and β are not specified,

by a doubling cube we mean a 2, 2 d1 -doubling cube Given two cubes Q ⊂ R in R d,set

K Q,R≡ 1

N Q,R k1

where N Q,R is the smallest positive integer k such that l2 k Q ≥ lR see 23

In what follows, C denotes a positive constant that is independent of main parameters

involved but whose value may diﬀer from line to line Constants with subscripts, such as C1,

do not change in diﬀerent occurrences We denote simply by A B if there exists a positive

constant C such that A ≤ CB; and A∼B means that A B and B A For a μ-measurable set E, χ E denotes its characteristic function For any p ∈ 1, ∞, we denote by pits conjugate

index, namely, 1/p 1/p 1.

2 Boundedness ofM∗,ρ λ in Lebesgue spaces

This section is devoted to the behavior of the parametrized g λ∗ function M∗,ρ λ in Lebesguespaces

Theorem 2.1 Let K be a μ-locally integrable function on R d× Rd \ {x, y : x y} satisfying 1.2

and1.3, and let M ∗,ρ λ be as in1.6 with ρ ∈ 0, ∞ and λ ∈ 1, ∞ Then for any p ∈ 2, ∞, M ∗,ρ λ

is bounded on L p μ.

Trang 4

To obtain the boundedness ofM∗,ρ λ in L p μ with p ∈ 1, 2, we introduce the following condition on the kernel K, that is, for some fixed σ > 2,

and2.1, and let M ∗,ρ λ be as in1.6 with ρ ∈ n/2, ∞ and λ ∈ 2, ∞ Then M ∗,ρ λ is bounded from

L1μ into weak L1μ, namely, there exists a positive constant C such that for any β > 0 and any

f ∈ L1μ,

μ x∈ Rd :M∗,ρ λ fx > β≤ C

β f L1μ 2.2

By the Marcinkiewicz interpolation theorem, and Theorems 2.1 and 2.2, we can

immediately obtain the L p μ-boundedness of the operator M ∗,ρ λ for p ∈ 1, 2.

Corollary 2.3 Under the same assumption of Theorem 2.2 ,M∗,ρ λ is bounded on L p μ for any p ∈

1, 2.

Remark 2.4 We point out that it is still unclear if condition 2.1 in Theorem 2.2 and

Corollary 2.3can be weakened

Now we turn to discuss the property of the operator M∗,ρ λ in L∞μ To this end,

we need to recall the definition of the space RBLOμ the space of regular bounded loweroscillation functions

Definition 2.5 Let η ∈ 1, ∞ A μ-locally integrable function f on R dis said to be in the spaceRBLOμ if there exists a positive constant C such that for any η, ηd1 -doubling cube Q,

m Q f − ess inf

x∈Q f x ≤ C, 2.3and for any twoη, η d1 -doubling cubes Q ⊂ R,

cubes with η ∈ 1, ∞ in the definition of RBLOμ Moreover, it was proved in 25 that the

definition is independent of the choices of the constant η ∈ 1, ∞ The space RBLOμ is a

subspace of RBMOμ which was introduced by Tolsa in 23

Trang 5

Theorem 2.7 Let K be a μ-locally integrable function on R d× Rd \ {x, y : x y} satisfying

1.2 and 1.3, and let M ∗,ρ λ be as in1.6 with ρ ∈ 0, ∞ and λ ∈ 1, ∞ Then for any f ∈ L∞μ,

M∗,ρ λ f is either infinite everywhere or finite almost everywhere More precisely, if M ∗,ρ λ f is finite

at some point x0∈ Rd , thenM∗,ρ λ f is finite almost everywhere and

M∗,ρ

where the positive constant C is independent of f.

We point out thatTheorem 2.7is also new even when μ is the d-dimensional Lebesgue

measure onRd

In the rest part ofSection 2, we will prove Theorems2.1,2.2, and2.7, respectively ToproveTheorem 2.1, we first recall some basic facts and establish a technical lemma For η > 1, let M ηbe the noncentered maximal operator defined by

Lemma 2.8 Let K be a μ-locally integrable function on R d × Rd \ {x, y : x y} satisfying

1.2 and 1.3, and η ∈ 1, ∞ Let M ρ be as in1.4 and M ∗,ρ λ be as in1.6 with ρ ∈ 0, ∞ and

λ ∈ 1, ∞ Then for any nonnegative function φ, there exists a positive constant C such that for all

Trang 6

For any fixed y∈ Rd and t > 0, write

Combining the estimates for E1and E2yields2.9, which completes the proof ofLemma 2.8

Proof of Theorem 2.1 For the case of p 2, choosing φx 1 inLemma 2.8, then we easilyobtain that

holds in this case

For the case of p ∈ 2, ∞, let q be the index conjugate to p/2 Then fromLemma 2.8

and the H ¨older inequality, it follows that

Trang 7

To proveTheorem 2.2, we need the following Calder ´on-Zygmund decomposition withnondoubling measuressee 23 or 26

Lemma 2.9 Let p ∈ 1, ∞ For any f ∈ L p μ and λ > 0 (λ> 2 d1 f L1μ / μ if μ < ∞), one

has the following.

a There exists a family of almost disjoint cubes {Q j}j (i.e.,

Remark 2.10 From the proof of the Calder ´on-Zygmund decomposition with nondoubling

measures see 23 or 26, it is easy to see that if we replace R j with Rj, the smallest

6√d,6√dn1-doubling cube of the form 6√dk Q j k ∈ N, the above conclusions a and

b still hold Here and hereafter, when we mention R jinLemma 2.9we always mean Rj

Proof of Theorem 2.2 Let f ∈ L1μ and β> 2 d1 f L1μ / μ note that if 0 < β ≤

2d1 f L1μ / μ, the estimate 2.2 obviously holds ApplyingLemma 2.9to f at the level β,

where ω j , ϕ j , Q j , and R jare the same as inLemma 2.9 It is easy to see thatg L∞μ β and

g L1μ f L1μ By the boundedness ofM∗,ρ λ in L2μ, we easily obtain that

μ x∈ Rd :M∗,ρ λ gx > β≤ β−2M∗,ρ

λ g2

L2μ β−1f L1μ 2.20

Trang 8

Froma ofLemma 2.9, it follows that

Trang 9

It is easy to see that for any x∈ Rd \R∗

j , y ∈ 4R jwith|y−x| < t and z ∈ R j,|x−x Q j|−2√dl R j ≤

|x − y| < t and |y − z| < 4√dl R j This fact along the Minkowski inequality and 1.2 leads to

Notice that for any z ∈ R j , x ∈ Rd \ R∗

j and y ∈ Rd \ 4R j,|y − z|∼|y − x Q j |, and |x − x Q j | <

Trang 10

On the other hand, it is easy to verify that for any y∈ Rd \ 4R j and t > |y − x Q j| 2√dl R j,

R j ⊂ {z : |y − z| ≤ t} and |x − x Q j | < 2t Choose 0 < < min{1/2, λ − 2n/2, ρ − n/2, σ/2 − 1}

we always take to satisfy this restriction in our proof The vanishing moment of b j on R j

and the Minkowski inequality give us that

t/l

R j22

logy − x Q

Trang 11

Trang 12

where C 8√de 22/ Note that for any y ∈ Rd \ B j and z ∈ R jwith|y − z| ≤ t ≤ |y − x|,

then|y − z|∼|y − x Q j | and |y − x Q j | ≤ t √dl R j Consequently,

Trang 13

A trivial computation involving the fact that|x − y| > |x − x Q j |/2 for any x ∈ R d \ R∗

R j ⊂ {z : |y − z| ≤ t} and t |x − y| ≥ |x − x Q j | C l R j Thus, from the vanishing moment of

b j on R j , it follows that

Trang 14

|y−x|

|y−x Qj |C lR j

log

L3b j

L1μ 2.41Combining the estimates for L1, L2, and L3yields that

which along with the estimates for F1and F2leads to2.24

Now we turn to prove the estimate2.25 Observe that if supp h ⊂ I for some cube

I, then by1.2, we have that for any s > 1 and any x ∈ R d \ sI,

Trang 15

As for M2z, notice that for any x, y, z ∈ R d satisfying |y − x| < t and 2|y − z| ≤ |x − z|,

|x − z|/2 < t From this fact and ρ ∈ n/2, ∞, it follows that for any x ∈ R d \ sI and z ∈ I,

To estimate M3z, we first have that for any x, y, z ∈ R dsatisfying 2|y − z| ≤ |x − z|, 2|y − x| ≥

|x − z|, and |y − x| ≤ 3|x − z|/2 Consequently, for any x ∈ R d \ sI and z ∈ I,

Trang 16

This fact together with2.48 tells us that

m Q

M∗,ρ λ f− m R

M∗,ρ λ f K Q,R f L∞μ 2.53

We first verify2.52 For each fixed cube Q, let B be the smallest ball which contains

Q and has the same center as Q Denote by r the radius of B Decompose f as

f x fxχ 8B x fxχRd \8B x ≡ f1x f2x, 2.54and write

Trang 17

x∈QM∗,ρ λ,∞

Obviously, for any x ∈ Q, z ∈ 8B, and y ∈ R d with |x− y| > 16r, |x− y|∼|y − z| Some

computation involving this fact and1.2 yields that

Trang 18

Thus, the proof of the estimate2.52 can be reduced to proving that for any x, x∈ Q,

M∗,ρ λ,∞

Trang 19

Now we prove thatM∗,ρ λ f satisfies 2.53 Let Q ⊂ R be any two 16√d,16√dd1

-doubling cubes Set N ≡ N Q,R 1 For any x ∈ Q and any y ∈ R, write

The Minkowski inequality involving the fact that for any y ∈ 2k−1 Q and z ∈ 2k1 Q\ 2k Q,

|y − z|∼|z − x Q | and t ≥ |y − z| ≥ 2 k−2 l Q gives us that

V1 f L∞μ

N Q,R k2

Trang 20

To estimate V3, we first have that for any x ∈ Q, z ∈ 2 k1 Q\ 2k Q, and y ∈ Rd \ 2k2 Q,

|y − x Q |∼|y − x|∼|y − z| and |z − x Q| 2k l Q This fact and the Minkowski inequality state

that

V3 f L∞μ

N Q,R k2

then Theorems 2.2 and 2.7 still hold Therefore, applying the interpolation theorem see

23, Theorem 7.1 between the endpoint estimates that M∗,ρ λ is bounded from L∞μ into

RBLOμ, which is a subspace of RBMOμ, and the boundedness of M∗,ρ λ in L2μ, we can

Trang 21

obtain thatM∗,ρ λ as in1.6 is bounded on L p μ for p ∈ 2, ∞ with the kernel satisfies 1.2and1.3 On the other hand, it follows from the Marcinkiewicz interpolation theorem that

M∗,ρ λ as in1.6 is also bounded on L p μ for p ∈ 1, 2 with the kernel satisfying 1.2 and

2.1

3 Boundedness ofM∗,ρ λ in Hardy spaces

In this section, we will prove that the operatorM∗,ρ λ as in1.6 is bounded from H1μ into

L1μ To state our result, we first recall the definition of the space H1μ via the “grand”

maximal function characterization of Tolsasee 29

ii 0 ≤ ϕy ≤ 1/|y − x| n for all y∈ Rd,

iii |∇ϕy| ≤ 1/|y − x| n1 for all y∈ Rd

Definition 3.2 The Hardy space H1μ is defined to be the set of all functions f ∈ L1μ

satisfying that+

Rd f dμ 0 and MΦf ∈ L1μ Moreover, we define the norm of f ∈ H1μ by

f H1μ ≡ f L1μMΦf

L1μ 3.2

and2.1, and M ∗,ρ λ be as in1.6 with ρ ∈ n/2, ∞ and λ ∈ 2, ∞ Then, M ∗,ρ λ is bounded from

H1μ into L1μ.

We begin with the proof of Theorem 3.3with the atomic characterization of H1μ

established by Tolsa in23

Definition 3.4 Let η ∈ 1, ∞ and p ∈ 1, ∞ A function b ∈ L1

locμ is called to be an atomic

A function f ∈ L1μ is said to belong to the space H 1,p

atbμ if there exist atomic blocks b i such that f ≡ ∞

i1 b i with

i |b i|H 1,p

atbμ < ∞ The H 1,p

atbμ norm of f is defined by

f H 1,p μ≡ infi |b i|H 1,p μ, where the infimum is taken over all the possible decompositions

of f in atomic blocks.

Trang 11

Trang 12

b j on R j , it follows that

Trang 14

14 Journal of. ..

Trang 16

This fact together with 2.48 tells us that

Tiêu đề	Boundedness of Parametrized Littlewood-Paley Operators with Nondoubling Measures
Tác giả	Haibo Lin, Yan Meng
Trường học	Beijing Normal University
Chuyên ngành	Mathematics
Thể loại	bài báo nghiên cứu
Năm xuất bản	2008
Thành phố	Beijing

Định dạng
Số trang	25
Dung lượng	613,88 KB