Journal of Function Spaces

Volume 2014 (2014), Article ID 274521, 14 pages

http://dx.doi.org/10.1155/2014/274521

Research Article

The Boundedness of Intrinsic Square Functions on the Weighted Herz Spaces

Hua Wang

College of Mathematics and Econometrics, Hunan University, Changsha 410082, China

Received 23 September 2013; Accepted 4 November 2013; Published 12 January 2014

Academic Editor: Dachun Yang

Copyright © 2014 Hua Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley -function, and -function on the weighted Herz spaces with general weights.

1. Introduction and Main Results

Let and . The classical square function (Lusin area integral) is a familiar object. If is the Poisson integral of , where denotes the Poisson kernel in , then we define the classical square function (Lusin area integral) by (see [1, 2])

where denotes the usual cone of aperture one:

Similarly, we can define a cone of aperture for any :

and corresponding square function

The Littlewood-Paley -function (could be viewed as a “zero-aperture” version of ) and the -function (could be viewed as an “infinite aperture” version of ) are defined, respectively, by (see, e.g., [3, 4])

The modern (real-variable) variant of can be defined in the following way (here we drop the subscript if ). Let be real, radial and have support contained in , and let . The continuous square function is defined by (see, e.g., [5, 6])

In 2007, Wilson [7] introduced a new square function called intrinsic square function which is universal in a sense (see also [8]). This function is independent of any particular kernel , and it dominates pointwise all the above-defined square functions. On the other hand, it is not essentially larger than any particular . For , let be the family of functions defined on such that has support containing in , , and for all ,

For and , we set

Then we define the intrinsic square function of (of order ) by the following formula:

We can also define varying-aperture versions of by the formula

The intrinsic Littlewood-Paley -function and the intrinsic -function will be given, respectively, by

In [8], Wilson showed the following weighted boundedness of the intrinsic square functions.

Theorem A. Let , , and . Then there exists a constant independent of such that

Moreover, in [9], Lerner obtained sharp norm inequalities for the intrinsic square functions in terms of the characteristic constant of for all . For further discussions about the boundedness of intrinsic square functions on various function spaces, we refer the readers to [10–17].

Before stating our main results, let us first recall some definitions about the weighted Herz and weak Herz spaces. For more information about these spaces, one can see [18–22] and the references therein. Let and let for any . Denote for , if , and , where is the characteristic function of the set . For any given weight function on and , we denote by the space of all functions satisfying

Definition 1 (see [19]). Let , , and , and let , be two weight functions on .

(a)The homogeneous weighted Herz space is defined by

where

(b)The nonhomogeneous weighted Herz space is defined by

where

For any , , and any measurable function on , we set . Let for and .

Definition 2 (see [21]). Let , , and , and let , be two weight functions on .

(c)A measurable function on is said to belong to the homogeneous weighted weak Herz space if

(d)A measurable function on is said to belong to the nonhomogeneous weighted weak Herz space if

Obviously, if , then for any . We also have when and , where

Thus, weighted (weak) Herz spaces are generalizations of the weighted (weak) Lebesgue spaces. The main purpose of this paper is to consider the boundedness of intrinsic square functions on weighted Herz spaces with weights. At the extreme case, we will also prove that these operators are bounded from the weighted Herz spaces to the weighted weak Herz spaces. Our main results in the paper are formulated as follows.

Theorem 3. Let , , , , and . Then is bounded on provided that and satisfy either of the following:

 (i) , , and ; (ii) , , , and .

Theorem 4. Let , , , , and . If , , and , then is bounded from into .

Theorem 5. Let , , , , and . If , then is bounded on provided that and satisfy either of the following:

 (i) , , and ; (ii) , , , and .

Theorem 6. Let , , , , and . If , , , and , then is bounded from into .

In [7], Wilson also showed that for any , the functions and are pointwise comparable, with comparability constants depending only on and . Thus, as direct consequences of Theorems 3 and 4, we obtain the following.

Corollary 7. Let , , , , and . Then is bounded on provided that and satisfy either of the following:

 (i) , , and ; (ii) , , , and .

Corollary 8. Let , , , , and . If , , and , then is bounded from into .

2. Weights

The classical weight theory was first introduced by Muckenhoupt in the study of weighted boundedness of Hardy-Littlewood maximal functions in [23]. A weight is a nonnegative, locally integrable function on , that denotes the ball with the center and radius . For any ball and , denotes the ball concentric with whose radius is times as long. For a given weight function and a measurable set , we also denote the Lebesgue measure of by and set weighted measure . We say that is in the Muckenhoupt class with , if there exists a constant such that for every ball ,

For the endpoint case , , if

where is a positive constant which is independent of the choice of . The smallest value of such that the above inequalities hold is called the characteristic constant of and denoted by . If there exist two constants and such that the following reverse Hölder inequality holds:

then we say that satisfies the reverse Hölder condition of order and write . It is well known that if with , then for all . Moreover, if with , then there exists such that .

The following properties for weights will be repeatedly used in this paper.

Lemma 9 (see [24]). Let with . Then, for any ball , there exists an absolute constant such that

In general, for any , one has

where does not depend on nor on .

Lemma 10 (see [1, 24]). Let , , and . Then there exist constants such that

for any measurable subset of a ball .

Throughout this paper, always denotes a positive constant which is independent of the main parameters involved but may vary from line to line.

3. Proofs of Theorems 3 and 4

Proof of Theorem 3. We only need to show the theorem for the homogeneous case because the proof of the nonhomogeneous result is similar and so is omitted here. Let . Following [25], for any , we decompose as

Since () is a sublinear operator, then we can write

Since and , then . By Theorem A and Lemma 9, we have

For the term , we first use Minkowski's inequality to derive

For any , , and , we have

For any , , and with , by a direct computation, we can easily see that

Thus, by using inequality (31) and Minkowski's inequality, we deduce

Denote the conjugate exponent of by . Applying Hölder's inequality and the condition, we can deduce that

Substituting inequality (34) into (33), we thus obtain

Here, we will consider two cases. For the case of , using the well-known inequality and changing the order of summation, we find that

Moreover, it follows immediately from Lemma 9 that

Since when and for , by Lemma 10, we can get

Therefore

where the last inequality holds since . On the other hand, for the case of , we will use Hölder's inequality to obtain

Using the same arguments as above, we can also prove the following estimates under the assumption that :

Hence

Summarizing the above estimates for the term , we obtain that for every ,

Let us now turn to estimate the last term . In this case, for any , , and with , it is easy to check that

Then it follows from inequality (31) and Minkowski's inequality that

This estimate together with (34) implies that

Hence

Now we will consider the following two cases again. For the case of , by using the inequality and changing the order of summation, we obtain

Since , then there exists such that for . Thus, by Lemma 10 again, we can get

where . Therefore, we have

where in the last inequality we have used the fact that under our assumption (i) or (ii). On the other hand, for the case of , an application of Hölder's inequality gives us that

By using the same arguments as for , we are able to prove that the following two series are convergent by an absolute constant under the assumption (i) or (ii):

Consequently

From the above discussions for the term , we know that, for any ,

Summing up the above estimates for , , and , we complete the proof of Theorem 3.

Proof of Theorem 4. Let . For any , as in the proof of Theorem 3, we will split into three parts:

Then, for any given , we have

Applying Chebyshev’s inequality, Theorem A, and Lemma 9, we obtain

For any , it follows from inequalities (33) and (34) that

By using Lemma 9, inequality (38), and the fact that , we deduce that

Moreover, since , then we have that, for any ,

Set . If , then the inequality

holds trivially. Now we suppose that . First it is easy to verify that . Then, for any fixed , we are able to find a maximal positive integer such that

Hence

Because , then by Lemma 10, with the same notations as in (49), we can get

Therefore

On the other hand, it follows from inequalities (34) and (45) that

In the present situation, since with , then it follows from inequality (49) that

Furthermore, recalling that , then, for any , we have

Repeating the arguments used for the term , we can also obtain

Combining the above estimates for , , and and then taking the supremum over all , we finish the proof of Theorem 4.

4. Proofs of Theorems 5 and 6

In order to prove the main theorems of this section, let us first establish the following results.

Proposition 11. Let , , and with . Then, for any , one has

Proof. Since , then, by Lemma 9, we know that for any ,

Therefore

Taking square-roots on both sides of the above inequality, we are done.

Proposition 12. Let , , and with . Then, for any , one has

Proof. For any and , it is easy to see that

Since , then, by duality, we have

For , we denote the weighted maximal operator by ; that is,

where the supremum is taken over all balls which contain . Then, by Lemma 9, we can get

Substituting inequality (77) into (75) and then using Hölder's inequality together with the boundedness of , we thus obtain

This estimate together with (74) implies the desired result.

Proposition 13. Let , , and with . Then, for any , one has

Proof. We will adopt the same method given in [26]. For any , set and . We also set

Observe that . Thus, for any ,

The weighted weak type estimate of yields

To estimate II, we now claim that the following inequality holds:

Assuming the claim for the moment, then, it follows from Chebyshev's inequality and inequality (83) that

Hence

Changing the order of integration yields

Combining the above estimate (86) with (82) and taking th root on both sides, we are done. So it remains to prove inequality (83). Set and . For each given , by Lemma 9, we thus have

It is not difficult to check that and . In fact, for any , there exists a point such that . Then we can deduce that

Note that . So we have

Hence

which is equivalent to

The above inequality implies in particular that there is a point . In this case, we have with , which implies that . Thus we obtain

Therefore

which is exactly what we want. This completes the proof of Proposition 13.

We are now in a position to give the proofs of the main theorems.

Proof of Theorem 5. From the definition of , we readily see that

Let . We decompose as in the proof of Theorem 3; then we have

Note that when . Since and , then . Applying Propositions 11–13, Theorem A, and inequality (94), we obtain

From the above estimate (96) and Lemma 9, it follows that

For any , , , and with , then, by a simple calculation, we can easily deduce that

Thus, by inequality (31) and Minkowski's inequality, we get

Moreover, by using Minkowski's inequality, (34), and (99), we obtain

Consequently

where the last inequality holds under our assumption . On the other hand, for any , , , and with , it is easy to verify that

Then it follows from inequality (31) and Minkowski's inequality that

Furthermore, by Minkowski's inequality, (34), and (103), we have

Therefore

where the last inequality also holds since . Following along the same lines as in Theorem 3, we can also show that

Summing up the above estimates for , , and , we complete the proof of Theorem 5.

Proof of Theorem 6. Let . We set as in the proof of Theorem 4; then for any given , we can write

Since when , applying Chebyshev's inequality, Lemma 9, and (96), we obtain

For the term , when , then it follows from (94), (99), (34), and the fact that

For the last term , when , by using (94), (103), (34), and the fact that , we get

The rest of the proof is exactly the same as that of Theorem 4, and we finally obtain

Combining the above estimates for , , and and then taking the supremum over all , we conclude the proof of Theorem 6.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

R. F. Gundy and R. L. Wheeden, “Weighted integral inequalities for nontangential maximal function, Lusin area integral, and Walsh-Paley series,” Studia Mathematica, vol. 49, no. 2, pp. 107–124, 1974.

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970.

B. Muckenhoupt and R. L. Wheeden, “Norm inequalities for the Littlewood-Paley function ,” Transactions of the American Mathematical Society, vol. 191, pp. 95–111, 1974.

E. M. Stein, “On some functions of Littlewood-Paley and Zygmund,” Bulletin of the American Mathematical Society, vol. 67, pp. 99–101, 1961.

S. Y. A. Chang, J. M. Wilson, and T. H. Wolff, “Some weighted norm inequalities concerning the schrödinger operators,” Commentarii Mathematici Helvetici, vol. 60, no. 1, pp. 217–246, 1985.

S. Chanillo and R. L. Wheeden, “Some weighted norm inequalities for the area integral,” Indiana University Mathematics Journal, vol. 36, pp. 277–294, 1987.

M. Wilson, “The intrinsic square function,” Revista Matematica Iberoamericana, vol. 23, no. 3, pp. 771–791, 2007.

M. Wilson, Weighted Littlewood-Paley Theory and Exponential-Square Integrability, vol. 1924 of Lecture Notes in Math, Springer, New York, NY, USA, 2007.

A. K. Lerner, “Sharp weighted norm inequalities for Littlewood-Paley operators and singular integrals,” Advances in Mathematics, vol. 226, no. 5, pp. 3912–3926, 2011.

J. Z. Huang and Y. Liu, “Some characterizations of weighted Hardy spaces,” Journal of Mathematical Analysis and Applications, vol. 363, no. 1, pp. 121–127, 2010.

Y. Y. Liang, E. Nakai, D. C. Yang, and J. Q. Zhang, “Boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces,” Banach Journal of Mathematical Analysis, vol. 8, pp. 221–268, 2014.

Y. Y. Liang and D. C. Yang, “Intrinsic square function characterizations of Musielak-Orlicz Hardy spaces,” Transactions of the American Mathematical Society. In press

H. Wang, “Intrinsic square functions on the weighted Morrey spaces,” Journal of Mathematical Analysis and Applications, vol. 396, no. 1, pp. 302–314, 2012.

H. Wang, “Weak type estimates for intrinsic square functions on weighted Morrey spaces,” Analysis in Theory and Applications, vol. 29, pp. 104–119, 2013.

H. Wang, “Boundedness of intrinsic square functions on the weighted weak Hardy spaces,” Integral Equations and Operator Theory, vol. 75, no. 1, pp. 135–149, 2013.

H. Wang and H. P. Liu, “The intrinsic square function characterizations of weighted Hardy spaces,” Illinois Journal of Mathematics, vol. 56, no. 2, pp. 367–381, 2012.

H. Wang and H. P. Liu, “Weak type estimates of intrinsic square functions on the weighted Hardy spaces,” Archiv der Mathematik, vol. 97, no. 1, pp. 49–59, 2011.

Y. Komori and K. Matsuoka, “Boundedness of several operators on weighted Herz spaces,” Journal of Function Spaces and Applications, vol. 7, no. 1, pp. 1–12, 2009.

S. Z. Lu and D. C. Yang, “The decomposition of weighted Herz space on and its applications,” Science in China A, vol. 38, no. 2, pp. 147–158, 1995.

S. Z. Lu and D. C. Yang, “Hardy-Littlewood-Sobolev theorems of fractional integration on Herz-type spaces and its applications,” Canadian Journal of Mathematics, vol. 48, no. 2, pp. 363–380, 1996.

S. Z. Lu, K. Yabuta, and D. C. Yang, “Boundedness of some sublinear operators in weighted Herz-type spaces,” Kodai Mathematical Journal, vol. 23, no. 3, pp. 391–410, 2000.

L. Tang and D. C. Yang, “Boundedness of vector-valued operators on weighted Herz spaces,” Approximation Theory and its Applications, vol. 16, no. 2, pp. 58–70, 2000.

B. Muckenhoupt, “Weighted norm inequalities for the Hardy maximal function,” Transactions of the American Mathematical Society, vol. 165, pp. 207–226, 1972.

J. Garcia-Cuerva and J. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, The Netherlands, 1985.

S. Z. Lu, D. C. Yang, and G. E. Hu, Herz Type Spaces and Their Applications, Science Press, Beijing, China, 2008.

A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, NY, USA, 1986.