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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 984259, 12 pages
Sobolev Embeddings for Generalized Riesz Potentials of Functions in Morrey Spaces over Nondoubling Measure Spaces
1Department of Mathematics and Information Sciences, Tokyo Metropolitan University, Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan
2Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan
Received 16 December 2012; Accepted 12 February 2013
Academic Editor: Alfonso Montes-Rodriguez
Copyright © 2013 Yoshihiro Sawano and Tetsu Shimomura. 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.
Our aim in this paper is to deal with the Sobolev embeddings for generalized Riesz potentials of functions in Morrey spaces over nondoubling measure spaces.
In this paper, we show that many endpoint results about the Adams theorem still hold in the nondoubling setting and that the integral kernel can be generalized to a large extent. In , in the setting of the Lebesgue measure, for , recall that Adams considered and proved the boundedness of the fractional integral operator given by The operator is also called the fractional integral operator or the Riesz potential. We denote by the ball with center and of radius , and by its Lebesgue measure, that is, , where is the volume of the unit ball in . Let be a bounded open subset of . We denote its diameter by ;
For , we define the integral mean over by Let . If is a positive function on the interval satisfying the doubling condition (see (23)), then we define the Morrey space to be the family of all for which there is a positive constant such that The norm of is defined by the infimum of the constants satisfying the inequality above. When , is denoted by .
A direct consequence of this notation is that for and .
Much about the case is known. Recall that the Adams theorem about the boundedness of fractional integral operators [1, Theorem 3.1] asserts that provided the parameters , , satisfy See also research papers [2–4, 6–16] and a survey .
Meanwhile, only a few results are known for the case . Trudinger [17, Theorem 1] proved that if then for some constant ; this implies that the operator is bounded from to . See also Serrin  for an alternative proof. Recently, the boundedness of Riesz potentials from to Orlicz-Morrey spaces was shown in . This result extends [20, 21]. One of the reasons why the case when is difficult is the failure of the boundedness of the Hardy-Littlewood maximal operator . In connection with this failure, we do not have Littlewood-Paley characterization. Due to these two difficulties, the case when is hard to analyze. However, from the point of PDEs, we are faced with analyzing the quantity in connection of the Kato condition, where is the potential operator of the operator . See [22, Section 2], for example. Consequently, despite the difficulty arising from harmonic analysis, the case when occurs naturally. As another evidence that the case when is of importance, we recall that the space appears naturally in the following sharp maximal inequalities [23, Theorem 4.7], [24, Theorem 1.3], and [25, Theorem 1.2]: let and . Then, there exists a constant such that for any measurable function , where is the sharp maximal operator due to Fefferman and Stein . A disadvantage of using the Littlewood-Paley theory is that we lose the integrability of functions a little when we consider the inequality where is a Littlewood-Paley patch. By choosing a smooth function such that , recall that we can define the th Littlewood-Paley patch by for . Note that (13) is a direct consequence of the translation invariance of the space . But this loss caused by (13) is quite big. Note that fails. See the appendix for a proof. When , an approach using the Littlewood-Paley patch is taken effectively . Indeed, for all . However, for the case when , due to the fact that the estimate (13) is essential when we consider the Littlewood-Paley patch, we prefer to avoid the Littlewood-Paley patch. See [28–43] for a huge amount of culmination of this approach.
Instead of using the Littlewood-Paley patch, we still have a good approach for the case when . Just make a closer look at the integral kernel. Our method being simple enough, there is no need to stick to the geometric structure of . Our result relies completely only upon the positivity of the integral kernel. So, here and below, we work on a separable metric space equipped with a nonnegative Radon measure , where we do not postulate any other condition on . By , we denote the open ball centered at of radius . While, given a point and in , we write for the distance of the points and , and we write for the distance of the points and in . We assume that and that for and for simplicity. In the present paper, we do not postulate on the “so-called” doubling condition. Recall that a Radon measure is said to be doubling, if there exists a constant such that for all and . Otherwise is said to be nondoubling. In connection with the -covering lemma, the doubling condition had been a key condition in harmonic analysis.
Our aim in this paper is to show that, for the case , the operator and its generalization are bounded from Morrey spaces to Orlicz-Morrey spaces, or, to generalized Hölder spaces, whose definitions will be given in the next section, in the nondoubling setting. Our result extends the results in [17–21]. The definition of is the following: let be a function from to itself and satisfy for all sufficiently small . We do not have to postulate the doubling condition on . See Remark 3 for an example which fails the doubling condition. We define where . Instead of using we discuss defined above. This modification will be necessary in Lemma 9 for example. An example in [44, Section 2] shows that is less likely to be bounded in general, although there does not exist a proof. We refer to  for an attempt of definining fractional integral operators by using the underlying measure .
Note that (18) is necessary in order that the image by of , the indicator functions of the balls, belongs to at least when is the Lebesgue measure. Indeed, if for any sufficiently small . Then, for such that , we have by using the spherical coordinate.
We organize the remaining part of the present paper as follows. In Section 2, we set up some notations. Section 3 is devoted to stating our main results fully based on the notations in Section 2. Some auxiliary lemmas are collected in Section 4. Finally, theorems in the present paper are proven in Section 5.
2. Notation and Terminologies
Let be the set of all continuous functions from to itself with the doubling condition, that is, there exists a constant such that We call the smallest number satisfying (23) the doubling constant of . Note that in view of [46, page 445] and [47, (1.2)], the doubling condition on is a natural one. For , we define the Morrey space as follows: with the norm Then, a routine argument shows that is a Banach space. Due to the fact that is a geometrically doubling space, we can prove that for all . See [48, Proposition 1.1] for a technique used to prove this inequality. Note here that if , and is bounded above on , then in particular, if there exists a constant such that for all , then with equivalent norms. A ball testing shows the following.
Proposition 1. The function is bounded above on if when .
Here and below, we write to indicate that there exists a constant independent of Morrey functions such that . The symbol stands for .
Proof. According to [49, Proposition A], for any ball contained in , we have If , in the sense of sets, then by the closed graph theorem and the doubling condition on and , we conclude If we combine (29) and (30), then we obtain that is bounded above on .
Let us consider the family of all continuous, increasing, convex, and bijective functions from to itself. For , the Orlicz space is defined by where If , are equivalent in the sense that there exists a constant with for all , then we see easily that with equivalent norms. If for large , then will be denoted by respectively.
For and , the Orlicz-Morrey space is defined by where (see [50, 51]). Then, again it is routine to prove that is a norm and that is a Banach space. Note that the space is a special case of Orlicz-Morrey spaces when .
For such that is bounded, the generalized Hölder space is defined by where Then, is a norm modulo constants and thereby is a Banach space. Since is bounded, every is bounded. If as , then every is continuous. For details, we refer to .
3. Main Results
In this section, we state our main theorems, whose proofs are given in Section 5.
Throughout this paper, let be a bounded open set in and denote by , the doubling constant of .
Let us begin with the following result, which is the one of Gunawan type .
Theorem 2. Let be a measurable function such that there exist , , such that Let , and define for . Then, there exists a constant such that for , and , where is a constant depending only on , , , and .
Remark 3. (1) Here it is not significant for us to choose 16; it counts that any number will do as long as it is small enough.
(2) The number 4 in the right-hand side seems to be essential. According to [44, Section 2], it can happen that the norms are not equivalent for .
(3) In view of [53, Lemma 2.5], we see that falls under the scope of Theorem 2. Indeed, Nagayasu and Wadade showed that the kernel which corresponds to satisfies This means that we have (41) with and . Note that implies (41). See also [54, Remark 2.2].
We now state a result for Orlicz-Morrey spaces.
Theorem 5. Let be measurable functions such that there exist , , such that and that
Let . Assume
and that is continuous and decreasing.
Define for . If satisfies then there exists a constant such that for , and , where is a constant depending only on , , , , , and .
Remark 6. Note that is bijective from to by the assumptions in the theorem. Indeed, by the definition of above, is a decreasing function. In addition, , showing that is bijective.
Finally, we shall show a result of Gunawan type about continuity.
Theorem 7. Let be a measurable function such that there exist , , such that Let . Assume the following condition on . There are and such that whenever . Assume in addition the Dini condition If then is bounded from to . More precisely, where is a constant depending only on , , , , , and .
Note that if and , then is bounded.
4. Preliminary Lemmas
Lemma 8. Let be a measurable function such that there exist , , such that
Let . Then
where is a constant depending only on , , , and .
Moreover, if , then where is a constant depending only on , , , , and .
Proof. If and , then a geometric observation shows
Set . Then, by virtue of the doubling condition on , we have
where is a constant depending only on , , and .
Consequently, since , which proves (58).
We choose , so that . Then, we have Thus, since , , being constants, (59) follows.
Lemma 9. Let be a measurable function such that there exist , , such that and that Then, for all ,
Proof. By Fubini's theorem and the dyadic decomposition of the ball, we have Since satisfies (66), we have as required.
5. Proofs of the Theorems
We are now ready to prove our theorems.
Proof of Theorem 2. Let and . By the positivity of the kernel, we may assume that . We write for and . By Lemma 9, we have Meanwhile, by Lemma 8 we have Hence, it follows from (71) and (72) that where depends only on , , , and .
Proof of Theorem 5. By Theorem 2, we have
for and .
Let . For and , since is decreasing, we have by Lemma 8 Hence, in view of the definition of , we have Now let Observe that by definition.
We claim that Indeed, when , we have . Hence, When , we have . Hence, Consequently our claim (79) is justified.
It follows from (76) and (79) that By (49), we obtain Hence, taking , we establish Since is decreasing and we see that for all . Hence, with the aid of (74), we have which proves (50).
A. Disproof of (15)
Inequality (15) can be disproved in terms of Besov spaces and Triebel-Lizorkin spaces. Let satisfy Define for . For parameters and and for , the Besov norm and the Triebel-Lizorkin norm are defined by respectively, and for and for , the Besov norm and the Triebel-Lizorkin norm are defined by Meanwhile, by denoting the set of all polynomials, for parameters and and for a distribution , the homogeneous Besov norm and the homogeneous Triebel-Lizorkin norm are defined by respectively. Also, for and , one defines respectively.
Let , , and . The inhomogeneous Besov space (resp. the homogeneous Besov space ) is defined to be the set of all (resp. ) for which the norm (resp. ) is finite, when , . Likewise, for ,