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Lipschitz Spaces and Fractional Integral Operators Associated with Nonhomogeneous Metric Measure Spaces
The fractional operator on nonhomogeneous metric measure spaces is introduced, which is a bounded operator from into the space . Moreover, the Lipschitz spaces on nonhomogeneous metric measure spaces are also introduced, which contain the classical Lipschitz spaces. The authors establish some equivalent characterizations for the Lipschitz spaces, and some results of the boundedness of fractional operator in Lipschitz spaces are also presented.
As we know, the theory on spaces of homogeneous type is needed to assume that measure of metric spaces satisfies the doubling measure condition, which means that there exists a constant , such that, for every ball of center and radius , . In recent years, many classical theories have been proved still valid without the assumption of doubling measure condition; see [1–12]. Recall that a Radon measure on is said to only satisfy the polynomial growth condition, if there exists a positive constant such that, for all and , , where is some fixed number in and . The analysis associated with such nondoubling measures is proved to play a striking role in solving the long-standing open Painlevé’s problem by Tolsa . Obviously, the nondoubling measure with the polynomial growth condition may not satisfy the well-known doubling condition, which is a key assumption in harmonic analysis on spaces of homogeneous type. In 2010, Hytönen  introduced a new class of metric measure spaces satisfying both the so-called geometrically doubling and the upper doubling conditions (see the definition below), which are called nonhomogeneous spaces. Recently, many classical results have been proved still valid if the underlying spaces are replaced by the nonhomogeneous spaces of Hytönen (see [4–6, 9–12]).
Let be a nonhomogeneous metric measure space in the sense of Hytönen . In this paper, we establish the definition of fractional operator on nonhomogeneous metric measure spaces, which contains the classical fractional integral operator introduced by García-Cuerva and Gatto , and similar to the definition introduced by Fu et al. , then we get the -boundedness for fractional integral operator on nonhomogeneous metric measure spaces. In Section 3, we also establish the definition of Lipschitz spaces on nonhomogeneous metric measure spaces, which contains the classical Lipschitz spaces. We establish some equivalent characterizations for the Lipschitz spaces. In Section 4, we present some results of the boundedness of fractional operator in Lipschitz spaces.
To state the main results of this paper, we first recall some necessary notions and remarks.
Definition 1 (see ). A metric space is said to be geometrically doubling if there exists some such that, for any ball , there exist a finite ball covering of such that the cardinality of this covering is at most .
Definition 2 (see ). A metric measure space is said to be upper doubling if is a Borel measure on and there exist a dominating function and a positive constant such that, for each , is nondecreasing and
A metric measure space is called a nonhomogeneous metric measure space if is geometrically doubling and is upper doubling.
Remark 3. (i) Obviously, a space of homogeneous type is a special case of upper doubling spaces, where we take the dominating function . On the other hand, the Euclidean space with any Radon measure as in (1) is also an upper doubling space by taking the dominating function .
(ii) Let be upper doubling with being the dominating function on as in Definition 2. It was proved in  that there exists another dominating function such that and, for all with , Thus, in this paper, we always suppose that satisfies (2).
Definition 4 (see ). Let . A ball is called -doubling if .
As stated in lemma of , there exist plenty of doubling balls with small radii and with large radii. In the rest of the paper, unless and are specified otherwise, by an -doubling ball we mean a -doubling with a fixed number , where is viewed as a geometric dimension of the spaces.
Definition 5 (see ). Let . A dominating function is satisfying the -weak reverse doubling condition if, for all and , there exists a number , depending only on and , such that, for all , and, moreover,
Remark 6. (i) It is easy to see that if and satisfies the -weak reverse doubling condition, then also satisfies the -weak reverse doubling condition.
(ii) Assume that . For any fixed , we know that
(ii) It is easy to see that the -weak reverse doubling condition is much weaker than the assumption introduced by Bui and Duong in [4, Subsection 7.3]: there exists such that, for all and , .
Definition 7 (see ). For any two balls , define where is the center of the ball .
Remark 8. The following discrete version, , of defined in Definition 7, was first introduced by Bui and Duong  in nonhomogeneous metric measure spaces, which is more close to the quantity introduced by Tolsa  in the setting of nondoubling measures. For any two balls , let be defined by where and , respectively, denote the radii of the balls and , and denotes the smallest integer satisfying . Obviously, . As was pointed by Bui and Duong , in general, it is not true that .
Definition 9 (see ). Let . A function is said to be in the space RBMO if there exist a positive constant , and for any ball , a number such that
for any two balls and such that ,
The infimum of the positive constants satisfying above two inequalities is defined to be the RBMO norm of and denoted by .
From [14, Lemma 4.6], it follows that the space is independent of .
In this paper, we consider a variant of the fractional integrals from [7, Definition 4.1].
Definition 10. Let and . A function is said to be a fractional kernel of order and regularity if it satisfies the following two conditions:(i)for all with ,
(ii)for all with ,
A linear operator is called fractional integral operator with satisfying (10) and (11),
Remark 11. By taking , it is easy to see that Definition 10 in this paper contains Definition 4.1 introduced by García-Cuerva and Gatto in , and Definition 10 is similar to Definition 1.9 introduced by Fu et al. in .
Definition 12. Let be a fractional kernel of order and regularity , and . We define
where is some fixed point of .
We observe that the integral in (13) converges both locally and at as a consequence of (10), (11), and Hölder’s inequality. Of course the function just defined depends on the election of , but the difference between any two functions obtained in (13) for different elections of is just a constant.
From now on, we will assume that . The results below are also true when .
Now we state the first main theorem of this paper.
Theorem 13. Let and . If satisfy the -weak reverse doubling condition with , then that is, is a bounded operator from into the space .
Next, let us introduce Lipschitz spaces on nonhomogeneous metric measure spaces.
Definition 14. Given that , we say that the function satisfies a Lipschitz condition of order provided that and the smallest constant in inequality (15) will be denoted by . It is easy to see that the linear space with the norm is a Banach space, and we will call it .
The second main result of this paper is the following some equivalent characterizations for the Lipschitz spaces.
Theorem 16. For a function , the conditions , , and are equivalent as follows.(A)There exist some constant and a collection of numbers of , one for each , such that these two properties hold: for any all with radius
and for any ball such that and radius ,
(B)There is a constant such that
for -almost every and in the support of .(C)For any given , , there is a constant , such that for every ball of radius , one has
where and also for any ball such that and radius
In addition, the quantities , , and with a fixed are equivalent.
Now we state the third main result of this paper.
Theorem 17. Let be a fractional kernel. and . If satisfy the -weak reverse doubling condition with , then maps boundedly into .
Theorem 18. Let be a fractional kernel and ; if satisfy the -weak reverse doubling condition with , then maps boundedly into if and only if .
Theorem 19. Let be a fractional kernel and . If satisfy the -weak reverse doubling condition with , then maps RBMO boundedly into if and only if .
Finally, we make some conventions on notation. Throughout the whole paper, stands for a positive constant, which is independent of the main parameters, but it may vary from line to line.
2. Proof of Theorem 13
Proof of Theorem 13. We are going to adapt to our context of the proof given by García-Cuerva and Gatto . Consider
By Hölder’s inequality, if , then ;
which holds even for . We can and assume that . By (5), we can choose such that . Then
By the relation between and , . We use Hölder’s inequality once more to obtain
Then, by applying Tchebichev’s inequality, we have This completes the proof of Theorem 13.
Corollary 20. Let and . If satisfy the -weak reverse doubling condition with , then
Proof. It suffices to apply Marcinkiewicz’s interpolation theorem with indices slightly bigger and slightly smaller than .
3. Proof of Theorem 16
Lemma 21. Let . If , then, for almost every with respect to , there exists a sequence of -doubling balls with , such that
Proof of Theorem 16. . Consider as in the lemma and let , , a sequence of -doubling balls with . Consider
and by (5), Lemma 21, we obtain that
Let and be two points as in the lemma; take any ball with and let . Now define , for , where is the first integer such that . Then
where is independent of and .
A similar argument can be made for the point with any ball such that and . Therefore Take two sequences of -doubling balls and with and . We have
. For , by the properties of function and Hölder’s inequality, we obtain By the similar argument, for any ball such that and radius ,
. Define first . Then (17) is exactly (20). To prove (16), we write This concludes the proof of the theorem.
Remark 22. Theorem 16 is also true if the number 2 in condition is replaced by any fixed . In that case, the proof uses -doubling balls, that is, balls satisfying .
Proof of Theorem 17. Without loss of generality, we assume that . Consider and let be the ball with center and radius . Then, we have
For the first term, by , then ,
The second term is estimated in a similar way after noting that .
Next, by using (2), Hölder’s inequality, and , we get Putting together the three estimates, then maps boundedly into .
Proof of Theorem 18. If , then by the continuity of the operator implies that must be constant; that is, .
On the other hand, we can observe that this implies that
Thus we can write where . For , Similarly, we know that ; using Hölder’s inequality , and letting , then we have In order to estimate , we use (2) to obtain Combining the estimates for , , and , we obtain this finishes the proof.
Proof of Theorem 19. implies that and, equivalently, that
for all .
Take two points and let with . Then For the first term, by Hölder’s inequality with some , Using , the second term can be dealt with in the same way as the ; then . It is easy to see that ; then Thus the proof of Theorem 19 is completed.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Jiang Zhou is supported by the National Science Foundation of China (Grants nos. 11261055 and 11161044) and by the National Natural Science Foundation of Xinjiang (Grants nos. 2011211A005 and BS120104).
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Copyright © 2014 Jiang Zhou and Dinghuai 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.