Endpoint Estimates for a Class of Littlewood-Paley Operators with Nondoubling Measures

Abstract

Let be a positive Radon measure on which may be nondoubling. The only condition that satisfies is for all , , and some fixed constant . In this paper, we introduce the operator related to such a measure and assume it is bounded on . We then establish its boundedness, respectively, from the Lebesgue space to the weak Lebesgue space , from the Hardy space to and from the Lesesgue space to the space . As a corollary, we obtain the boundedness of in the Lebesgue space with .

1. Introduction

A positive Radon measure on is said to be doubling if there exists some constant such that for all , . It is well known that the doubling condition is an essential assumption in many results of classical Calderón-Zygmund theory. However in the recent years, it has been shown that a big part of the classical theory remains valid if the doubling assumption on is substituted by the growth condition as follows:

(1.1)

for all , where is some fixed number with . For example, In 2001, Tolsa in [1, 2] investigated the weak (1,1) inequality for singular integrals, the Littlewood-Paley theory and the theorem with nondoubling measures. In 2002, García-Cuerva and Gatto [3] investigated the boundedness properties of fractional integral operators associated to nondoubling measures. In 2005, Hu et al. [4] studied the multilinear commutators of singular integrals with nondoubling measures. Since 2007, Hu et al. [5] have proved some boundedness results of Marcinkiewicz integrals with nondoubling measures on some function spaces.

On the other hand, let be a function on such that there exist positive constants , , , and satisfying

(a) and ,

(b),

(c) for .

For this , we define the Littlewood-Paley's -function with nondoubling measures as follows:

(1.2)

where and .

Note that if we replace by in the above definition and when is the Poisson kernel, we obtain classical -function defined and studied by Stein [6] and later by Fefferman [7], where the weak (1,1) with and weak with boundedness of function were obtained. In the same paper, Fefferman [7] also established the bounds of for and . For the more generalized -function defined by (1.1), the boundedness is also well known (see, e.g., [8, pages 309–318]). On the other hand, inspired by the works of Sakamoto and Yabuta in 1999, the first author in this paper studied parametric -function systematically in his PhD thesis [9]. Later, in 2008, Lin and Meng [10] gave some results on parametric -function with nondoubling measures. But their result only valid for , one cannot obtain the results for classical operators even for or in the classical case studied by Stein in 1961 [6].

In this paper, we will study the properties of operator with nondoubling measures on some function spaces under the conditions (a)–(c).

First, before stating our main results, we give some notation and definitions, let be a closed cube with sides parallel to the axes. Denote its side length by and its center by . Given and , we say is -doubling if , where is the cube concentric with with side length . If are not specified, by a doubling cube we mean a -doubling cube. For any cube , we denote by the smallest doubling cube which contains and has the same center as .

Given two cubes in , set

(1.3)

where is the first integer such that and

(1.4)

with , where is the first integer such that .

In the article of [1, page 95], we know that with constants that may depend on and . The following atomic Hardy space was introduced by Tolsa in [11].

Definition 1.1.

For a fixed , a function is called an atomic block if

(1)there exists some cube such that ;

(2);

(3)there are fucntions with supports in cubes and numbers such that

(1.5)

Define

(1.6)

We say that if there are atomic blocks such that with . The norm of is defined by

(1.7)

where the infimum is taken over all the possible decompositions of in atomic blocks.

It was shown by Tolsa that the space was proved to be the Hardy space in [11] with equivalent norms. We will denote the space and the norm , respectively, by and for convenience. He also proved that the dual space of is the following space .

Definition 1.2.

Let be a fixed constant. A function is said to be in the space if there exists some constant such that for any cube centered at some point of

(1.8)

and for any two doubling cubes

(1.9)

where denotes the mean value of over cube . The minimal constant above is defined to be the norm of in the space and denoted by .

Tolsa in [11] proved that the definition of the space and are independent of the choice of . The following space was introduced in [12]. It is easy to see that .

Definition 1.3.

A function is said to be in the space if there exists some positive constant such that for any doubling cube ,

(1.10)

and for any two -doubling cubes ,

(1.11)

The minimal constant as above is defined to be the norm of in the space , we denote it by .

In this paper, we always assume that and are considered as they are defined at the beginning of this paper. Our main results are as follows.

Theorem 1.4.

Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded from to .

Theorem 1.5.

Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded from to .

Theorem 1.6.

Let be a function on , satisfying (a)–(c), , . If is bounded on , then for , is either infinite everywhere or finite almost everywhere. More precisely, if is finite at some point , then is -finite almost everywhere and

(1.12)

Corollary 1.7.

Let be a function on , satisfying (a)–(c), , . If is bounded on , then it is also bounded on for any .

Remark 1.8.

It is natural to consider the similar problems with more general rough kernels. However, even in the doubling measure case, if we take in (1.1) (in this case, is defined and studied by [13]), from the results in [8], we know that it is impossible to give similar results as above for Littlewood-Paley function even for . In fact, by the counter example in [13], even the boundedness does not hold. In this sense, the condition we assumed on is necessary and reasonable. On the other hand, in 2008, Lin and Meng [10] gave some results on parametric -function with nondoubling measures. In fact the results in [10] are only valid for . By the same reason as above, one cannot obtain the result when which in this case, the operator coincides with the classical operator studied by Torchinsky and Wang in [13] and it is a generalization of the classical operators studied by Stein and Fefferman.

Remark 1.9.

Even in the classical case, the index is sharp for weak boundedness; see [6] for detail.

We arrange our paper as follows, in Section 2, we give and prove some key lemmas. The proof of our main theorems will be given in Section 3. Throughout this paper, the letter will denote a positive constant that may vary at each occurrence but is independent of the essential variables. will always denote that there exists a constant , such that .

2. Main Lemmas

We need two lemmas given by Tolsa.

Lemma 2.1 (see [11]).

If are concentric cubes such that there are no -doubling cubes with of the form , , with , then

(2.1)

where depends only on , , , .

Lemma 2.2 (see [11]).

For any and any then we have one has the following:

(a)there exists a family of almost disjoint cubes (that means , depends only on ) such that

(a.1);

(a.2) for any ;

(a.3) a.e. on ,

(b)for each , let be the smallest ()-doubling cube of the form , , with that , and let , then there exists a family of functions with and with constant sign satisfying

(b.1);

(b.2), where is some constant;

(b.3).

To prove our theorems, we prepare another two key lemmas.

For any subset , we denote

(2.2)

It is easy to see that

(2.3)

Lemma 2.3.

Let and be the same as in Lemma 2.2, and let be center. Then for any and ,

(2.4)

Lemma 2.4.

Let and be the same as in Lemma 2.2, and let be center. For any and , then

(2.5)

Proof of Lemma 2.3.

Denote

(2.6)

Set

(2.7)

It is easy to see

(2.8)

We first estimate . Set

(2.9)

We have

(2.10)

Note that and , then , . These two inequalities will always be used in the following proof, so we will not mention them every time.

For any , we have that , , , , . Then

(2.11)

For any , we get . Then

(2.12)

For , we obtain that , . Therefore

(2.13)

For , we have that , . Therefore

(2.14)

Thus, we get

(2.15)

Next we estimate . Set

(2.16)

For any , there exist two constants , such that . Since , , we then have that

(2.17)

For any , the following inequalities hold: , , and . It follows that

(2.18)

Next we estimate . Set

(2.19)

Then

(2.20)

For any , we can get that and there exist two constants and such that . Then

(2.21)

For any , we can get and . Then

(2.22)

Next we estimate . Set

(2.23)

If , we obtain , and

(2.24)

Therefore

(2.25)

If , we have and

(2.26)

Therefore

(2.27)

If , we have and . Then

(2.28)

The proof of Lemma 2.3 is finished.

Proof of Lemma 2.4.

Denote

(2.29)

Let and divide into four parts

(2.30)

Then .

Since and , we have that , .

We first estimate . Set

(2.31)

Then

(2.32)

For any , we have that and . Then we get

(2.33)

and therefore

(2.34)

For any , we have , . It follows that

(2.35)

If , we will get . It follows that

(2.36)

If , we obtain that . Thus we can get

(2.37)

Next we estimate . Set

(2.38)

Then

(2.39)

For any , the inequalities and hold, and we have

(2.40)

For any , the inequalities and hold, and we can get

(2.41)

Next we estimate . Let

(2.42)

Then

(2.43)

Since , we can choose small enough such that and .

Now denote

(2.44)

and for a subset , we denote

(2.45)

It is easy to see that .

For any , we have , . It follows that

(2.46)

The last inequality holds because and we choose small enough such that , and then we can get , which leads to the above inequality. Using these inequalities, we obtain

(2.47)

If , we get . It follows that

(2.48)

Next we estimate . Set

(2.49)

Then

(2.50)

For any , we have . Then we get

(2.51)

If , we get and . Then we can obtain

(2.52)

If , we get . Then

(2.53)

For any , the inequalities , and hold, from which we obtain

(2.54)

If , we have . Then

(2.55)

If , we have and it follows that

(2.56)

If , we get and . Then we have

(2.57)

3. Proof of Theorems

Proof of Theorem 1.4.

To prove Theorem 1.4, we will choose and .

Let and . Applying Lemma 2.2 to and , we obtain a family of almost disjoint cubes . With the notation , , the same as in Lemma 2.2, we can decompose , with that and . And can be decomposed as . By (a.1) of Lemma 2.2, we have .

Thus, to prove that is of weak type , we only need to prove

(3.1)

Since and , we only need to show that both and satisfy the inequality (2.4).

For , it follows from (b.1) of Lemma 2.2 that , and we have . Using -boundedness of and (a.3) and (b.2) from Lemma 2.2, we obtain

(3.2)

To prove that satisfies inequality (3.1), it suffices to show that

(3.3)

Since , we have that

(3.4)

If we can prove

(3.5)

then we finish the proof of Theorem 1.4.

We first estimate and divide it into two parts

(3.6)

By the assumption of boundedness of , and the fact that is the doubling cube, it is easy to see that

(3.7)

Next we estimate . By the Minkowski's inequality, Fubini's theorem and condition (b) of that , we have the following estimate:

(3.8)

Choose small enough such that , and .

For any subset , we denote

(3.9)

Let be center. Using the above notations, it is easy to see that

(3.10)

By Lemma 2.3, we will have the following inequalities:

(3.11)

(3.12)

where we used Lemma 2.1 to estimate

(3.13)

We now estimate . Let . Recall that , and

(3.14)

We first estimate . By the Minkowski's inequality, Fubini's theorem and property (b) of , we can obtain that

(3.15)

If we prove

(3.16)

for any and , then

(3.17)

and by Lemma 2.2, we conclude that .

Now choose small enough such that , and . Then for any and , note that

(3.18)

Next we estimate . By property (c) of and , we have

(3.19)

since . Using the above inequality, Minkowski's inequality and Fubini's theorem, we get

(3.20)

So by Lemma 2.4, for and any

(3.21)

Then we get

(3.22)

and we will have . With the estimates of

(3.23)

we obtain that

(3.24)

Thus,

(3.25)

Therefore we finish the proof of Theorem 1.4.

Proof of Theorem 1.5.

Note that the definition of is independent of the choice of the constant , we can assume that , still with . By Theorem 1.4, the operator is bounded from to . By a standard argument, we only need to prove that for any atomic block with supp Write

(3.26)

By definition of , we have . By (3.25) in the proof of Theorem 1.4, we can get

(3.27)

To estimate , let be as in Definition 1.1 and write

(3.28)

By the assumption of boundedness of and the Holder inequality,

(3.29)

By (3.11) in the proof of Theorem 1.4,

(3.30)

(3.31)

where is a cube and has the same center as and , is the center of . From (3.29) to (3.30), we use the conclusion

(3.32)

As a matter of fact, let and then by the definition of , we get that

(3.33)

Using (3.30) and the fact that

(3.34)

we have

(3.35)

From (3.29) and (3.35), we can obtain that

(3.36)

We have

(3.37)

Combining (3.27) and (3.37), we finish the proof of Theorem 1.5.

Proof of Theorem 1.6.

Now we begin to prove Theorem 1.6. First we claim that there is a positive constant such that for any and -doubling cube , the following inequality holds:

(3.38)

To prove (3.38), for each fixed cube , let be the smallest ball which contains and has the same center as . Then . We decompose as

(3.39)

By the Hölder's inequality and boundedness of , we have

(3.40)

We denote by the radius of . Note that for any and for any , then by Minkowski's inequality, we have

(3.41)

Since we have the following estimate:

(3.42)

We claim

(3.43)

Then

(3.44)

By (3.42), (3.44), and (3.45), we can obtain

(3.45)

The method to prove (3.43) is quite similar as to prove (2.4) in the proof of Theorem 1.4 and we omit it.

Thus, to prove (3.38), we only need to prove for any , the following inequality holds:

(3.46)

We note that can be looked as a vector valued Calderón-Zygmund singular integral operator in the following Hilbert space :

(3.47)

In fact, can be written by

(3.48)

where . Also note that for ,

(3.49)

where . We will divide into four parts, namely,

(3.50)

Then

(3.51)

This can be obtained from the same idea used before, see also the main step in [14], here we omit the proof of it.

From (3.38), we get that for , if for some point , then is -finite almost everywhere and in this case we have

(3.52)

provided that is a -doubling cube.

To prove , we still need to prove that satisfies

(3.53)

for any two -doubling cubes .

Let and set , then

(3.54)

Again, we can get the conclusion under the nondoubling condition which is similar to the proof of (2.2) in [14] that

(3.55)

For any and each fixed , similar to the proof of (2.4) from Lemma 2.3, we have that

(3.56)

Therefore we have

(3.57)

Next we estimate . By an estimate similar to (3.45), we have

(3.58)

By (3.54), (3.55), (3.57), and (3.58), we can get that

(3.59)

Take mean value over for and over for , we get

(3.60)

Therefore

(3.61)

Since by Hölder's inequality, the boundedness assumption of and the fact that is -doubling, we have that

(3.62)

which completes the proof of Theorem 1.6.

Using Theorems  1.4 and 1.5 and [4, Theorem  3.1], Corollary 1.7 is obvious.