Abstract
We introduce new classes of weights and functions associated with a nondecreasing function of upper type with and obtain the weighted norm inequalities for Calderón-Zygmund operators of -type and their commutators.
1. Introduction
For a nonnegative and nondecreasing function mapping from to , we shall mean that it is of upper type with , if there exists a constant such that for all and . In this paper, we always assume that .
Let be a continuous associated with the kernel function from to that satisfies for all and not in the support of is said to be associated with Here, the kernel is defined on and satisfies that for any and some the size conditionand for some the regularity conditionswhenever andwhenever We also suppose that is associated with and it can be extended to be a bounded extension on ,In this paper, we will consider weighted estimates for these operators and their commutators.
Remark 1. It is clear that if satisfies (3), (4), (5), and (6) then falls within the scope of the Calderón-Zygmund theory. Then has an extension that maps into , and by interpolation, also maps into itself for
Remark 2. When , from [1–3], we know that the pseudodifferential operators with the symbol defined by satisfy (3), (4), (5), and (6), where denotes the set of all smooth functions on satisfying for all multi-indices and .
Definition 3. A weight will always mean a positive function which is locally integrable. We say that a weight belongs to the class for and , if there is a positive constant such that, for all cubes . We also say that a nonnegative function satisfies the condition if there exists a constant for all cubes , We also write , , and .
Remark 4. We remark that coincides with Muckenhoupt’s class of weights for all However, in general, the class is strictly larger than the class for all On the other hand, when is a constant function, also coincides with for any In particular, if , .
Remark 5. It should be noted that may not be a doubling measure; that is, for each positive constant , there exists at least a cube such that For example, let . If , then and is a nondoubling measure, but ; see [1, 4]. We also need to point out that these weights have many differences from Orobitg-Pérez’s undoubling weights [5] because they are against the reverse Hölder inequality by Proposition 15 (v).
Next we state our main theorem for these operators satisfying (3)–(6) as follows.
Theorem 6. Assume that satisfies (3), (4), (5), and (6). Let with Then is bounded from to for and to
We also consider the commutator of Coifman-Rochberg-Weiss defined by Since our operator is stronger than the standard Carlderón-Zygmund operator, we will generalize the symbol from usual to .
Definition 7. A locally integrable function is said to be in with and , if there exists a positive constant such that for any cube where is the average of on . A norm for , denoted by , is given by the infimum of the constants satisfying (13).
Remark 8. Clearly for and if or We define In [6], Morvidone proved that these spaces are independent of the scale ; that is, in the sense of norm. So we denote simply.
Remark 9. Recently Harboure et al. [7] established an intimate relationship between these spaces and -Carleson measures similar to well-known Fefferman-Stein’s theorem. Precisely, let be a function in with a null integral. if and only if is a -Carleson measure; that is, , where is a Carleson box.
Theorem 10. Assume that satisfies (3), (4), (5), and (6). Let for and with . Then there exists a positive constant such that for any
To consider the endpoint case, we introduce the Orlicz norm. For and a cube on , by the Luxemburg norm of it means that We recall that if and only if .
Theorem 11. Assume that satisfies (3), (4), (5), and (6). Let for and with . Then there exists a positive constant such that for any
Remark 12. One of our original motivations is to procure some detailed operators that satisfy our conditions like pseudodifferential operators. Restricted with our knowledge, we have found nothing to date.
2. Some Preliminaries and Notations
In this section, we begin with defining the weighted maximal function by
Proposition 13. Let . Thenand for ,
Proof. By the Marcinkiewicz interpolation theorem and the fact that is bounded on , we only need to prove (18). For any , let . Fixing , then there exists a cube such that Thus, covers By Vitali’s lemma, there exists a class of disjoint cubes such that and
By the boundedness of and the following covering lemma, we obtain some properties for .
Lemma 14 (see [8]). There exists a sequence of points in so that the family of cubes where , , satisfies the following: (a).(b)There exists a constant such that for any .
Proposition 15. The following statements hold: (i)If , then .(ii) if and only if , where .(iii)If , , then for any .(iv)If for , then Particularly, let for any measurable set ,(v)If with , then there exist positive numbers , , and so that for all cubes (vi)If with then there exists such that .
Proof. (i), (ii), and (iii) are easy from the definition. We only prove (iv). In fact, note that ; by Hölder’s inequality and condition we then have for all For , we note that thus it concludes (iv).
For (v), by Lemma 2.4 [1], there exist positive constants and , such thatfor any with On the other hand, let , with and , where and is the sequence of Lemma 14. It is easy to see that . Now let be the constant in (27). By using Lemma 14, condition, and Hölder’s inequality, we obtain the desired inequality with ; here is sufficiently big. And (vi) is a consequence of (v).
Then we also define maximal functions associated with and by From (iv) in Proposition 15, we have that for . From this and using (18) and (19), we can get the following result.
Proposition 16. Let and suppose that . If , then the equality Furthermore, let and if and only if
Proposition 17. Let and and suppose that . There exists a constant such that
As a consequence of Proposition 17, we have the following.
Corollary 18. Let and Let be a radial, positive function with compact support and total integral . Set . Then (i) for and ;(ii), as , almost every for ;(iii), as , almost every for .
Next we define the dyadic maximal function and sharp maximal function and by for Then we have the following.
Lemma 19. Let be a locally integrable function on , , , and . Then may be written as a disjoint union of maximal dyadic cubes with Moreover, we have that for almost all and .
Proof. The proof is standard; see Lemma 1 on P. 150 of [9].
Lemma 20. Let and . For a locally integrable function , and for and positive , then there exist positive constants and such that we have the following inequality:for all , where .
Proof. The proof is very similar to that of Lemma [1].
Corollary 21. Let , , and . Then there exists a constant such that
Note that a.e. for any . By Corollary 21, we have the following.
Corollary 22. Let , , , and , and then
For and the Young function , we define the maximal function by
Lemma 23. Let and be locally integrable. Then there exist positive constants and independent of and such that
Proof. On the one hand, we take with which implies On the other hand, for any and any fixed cube , write , with . Thus Since for any with , we have Noting that for any two , then
The following proposition is a technical result. Its proof copies almost verbatim the proof of Proposition 3 [10] and is omitted.
Proposition 24. Let . If then for all cube for all .
Proposition 25. Supposing that is in , there exist positive constants and such that
Proof. The proof is classical; we refer to Proposition [11].
As we know that is also a Young function, the corresponding average is denoted by Then there is a generalized Hölder inequalityBy Proposition 25 and (48), we can get
3. Proofs of Theorems
The proof of Theorem 6 needs the following technical lemma.
Lemma 26. Assume that satisfies (3), (4), (5), and (6). Let and Then there exists a constant such that for all and all where is a variant of dyadic sharp maximal operator
Proof. Fix and let a dyadic cube , and we write . When , since , we have the inequality Applying Kolmogorov’s inequality to the term and the weak boundedness of , we have Set , for in the regularity in (4) and (5); we have Thus .
When , by Hölder’s inequality with , we have Similar to , it follows that with Kolmogorov’s inequality to the term and the weak (1,1) boundedness of . For , taking in the size condition (3), we have The proof is complete.
Proof of Theorem 6. From Proposition 17 and Lemmas 20 and 26, the standard statement gets the proof of Theorem 6.
To prove Theorem 10, we establish the following pointwise estimate firstly.
Lemma 27. Assume that satisfies (3), (4), (5), and (6). Let for , , and . Then for almost all where is a variant of dyadic maximal operator
Proof. Fix and a dyadic cube , we decompose the function into with , where . For , we can write When , Since , we have Choosing any , by the Hölder inequality and Proposition 24, it estimates that For , we recall that is a Calderón-Zygmund operator and is of weak type . By Kolmogorov’s inequality, (49), and Proposition 24, we then have and the last inequality holds by . It turns to the term . Choosing and using (4), (5), and (49) and Proposition 24, we get that for any and Thus .
When , since , we have Choosing any and denoting , by the Hölder inequality and Proposition 24, the same estimate with is obtained that Also, since , by Kolmogorov’s inequality, (49), and Proposition 24, we have The last term remains. Choosing and using (3) and (49) and Proposition 24, we get that for any and We complete the proof.
Proof of Theorem 10. By Corollary 22, Lemma 27, Lemma 23, and Theorem 6, we have
Lemma 28. Let and . There exists a positive such that, for any function and ,
Proof. Let be any compact subset in . For , there exists a cube with such thatby Vitali covering lemma, we can choose a finite family of disjoint cubes such that and satisfies (70). Thus From this, by (iv) in Proposition 15 with and , we obtain that Thus, (69) holds.
Lemma 29. Let , , and . Then there exists a positive constant such that for any smooth function with compact support
Proof. Similar to the proof of Lemma in [12] or Lemma in [1], we can get the desired result by Proposition 16, Lemma 27, and Lemma 23. We omit the details here.
Proof of Theorem 11. By homogeneity, we need only to show that Now, since is submultiplicative, we have by Lemma 28 and Lemma 29
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work is supported by Young Foundation of Education Department of Hubei Province (no. Q20162504).