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International Journal of Analysis
Volume 2013 (2013), Article ID 784398, 9 pages
http://dx.doi.org/10.1155/2013/784398
Research Article

A Note on Generalized Hardy-Sobolev Inequalities

Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

Received 16 September 2012; Accepted 20 November 2012

Academic Editor: Tohru Ozawa

Copyright © 2013 T. V. Anoop. 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 are concerned with finding a class of weight functions so that the following generalized Hardy-Sobolev inequality holds: , for some , where is a bounded domain in . By making use of Muckenhoupt condition for the one-dimensional weighted Hardy inequalities, we identify a rearrangement invariant Banach function space so that the previous integral inequality holds for all weight functions in it. For weights in a subspace of this space, we show that the best constant in the previous inequality is attained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its refinement due to Hansson.

1. Introduction

In this paper, we are interested to find the best suitable function space for the weights so that the following generalized Hardy-Sobolev inequality holds: for some , where is a bounded domain in . We say that is admissible, if the previous inequality holds. We are also interested to find a class of admissible functions that ensures the best constant in (1) which is attained for some .

Let us first recall the classical Hardy inequality: By taking with in (2), we get The higher dimensional analogue of the previous inequality is referred to as the Hardy-Sobolev inequality in the literature: where is a domain containing the origin with . Clearly (4) does not hold when or 2, since is not locally integrable for that contains the origin.

For , the Hardy-Sobolev inequality is generalized mainly in two directions, namely, the generalized Hardy-Sobolev inequalities and the improved Hardy-Sobolev inequalities. The first one refers to the inequalities of the form (1) for more general weights instead of the homogeneous weight . The second one relies on the fact that the best constant in (4) is not attained in and hence one anticipates to improve (4) by adding nonnegative terms in the left-hand side. The first major improvement in the Hardy-Sobolev inequality is obtained by Brézis and Vázquez in [1] who have proved the following inequality: Motivated by the previous inequality, several improved Hardy-Sobolev inequalities have been proved, for example see [25].

For , as we pointed out before, the Hardy potential is not admissible for any domain in that contains the origin. In [6], Leray showed that is the right admissible function (analogous to Hardy potential) for . In this paper, we focus on finding a large class of admissible functions including that of Leray’s function or its improvements (by adding nonnegative terms) for the generalized Hardy-Sobolev inequalities in bounded domains of .

The most general sufficient condition (for any dimension) for the generalized Hardy-Sobolev inequalities is given by Maźja [7], in terms of the capacity. We recall that for a compact set , the relative capacity of with respect is defined as First, let us see that Maźja’s capacity condition is very much intrinsic on (1). Let be a positive weight satisfying (1), then for any compact subset , By taking the infimum, we get . Therefore Maźja proved that the previous condition is indeed sufficient for (1) (Theorem 1/2.4.1, page 128 of [7]). Since Maźja’s condition is necessary and sufficient, all the improved Hardy-Sobolev inequalities follow directly from Maźja’s result. However, verifying Maźja’s capacity condition for a general weight function is not an easy task. Thus it is of interest to find certain verifiable conditions for the generalized Hardy-Sobolev inequalities by other means.

One such verifiable condition for is obtained by Visciglia in [8]. He proved that (1) holds for weights in the Lorentz space . The embedding of into the Lorentz space played a key role in his result. The case is more subtle, for example, (1) does not hold when and , see [9]. In this paper we obtain a verifiable condition for admissible functions for bounded domains in , using one-dimensional weighted Hardy inequalities and certain rearrangement inequalities.

A general one dimensional weighted Hardy inequality has the following form: For an excellent review on the weighted Hardy inequalities, we refer to [10] by Maligranda et al. Many necessary and sufficient conditions on , , , for holding (9) are available in the literature, see [1113]. Here we make use of the so called Muckenhoupt condition [13] for obtaining a class of weight functions satisfying (1). For a measurable function , we denote its decreasing rearrangement by and the maximal function of is denoted by , that is, . Now we define Indeed, is a rearrangement invariant Banach function space with the norm see [14]. More details of the space are given in Section 3. Now we state our main theorem.

Theorem 1. Let be a bounded domain in and let . Then is admissible and

Having obtained , a class of admissible functions, one would like to know for which among them the best constant in (1) is attained in . Many authors have addressed this question when , see [8, 15] and the references therein. For , Maźja has a sufficient condition (see 2.4.2 of [7]) in terms of capacity. Here we consider the weights in a subclass of so that the best constant in (1) is attained in . For a bounded domain (analogous to space in [15]) we define We show that the best constant in (1) is attained in , when , where is the positive part of . More precisely, we have the following theorem.

Theorem 2. Let and . Define Then is attained for some .

The Moser-Trudinger embedding of into the Orlicz space , the Orlicz space generated by the Orlicz function , can be used to show that functions are admissible. An embedding of , finer than that of Moser-Trudinger, is established independently by Hansson [16], Brézis, and Wainger [17]. As anticipated, this embedding gives a bigger class of admissible functions than . In [18], the authors used this finer embedding and showed that the Lorentz-Zygmund space is admissible. We are not going to use any embeddings for proving that functions are admissible, instead we use some rearrangement inequalities and one dimensional weighted Hardy inequalities. We would like to stress that the admissibility of functions can be used to give alternate proofs for the Moser-Trudinger embedding and its refinement due to Hansson.

The rest of the paper is organised as follows. In Section 2, we recall definition and properties of decreasing rearrangement. Further, we state some classical inequalities that will be used in the subsequent sections. We discuss the properties of the space and give examples of function spaces contained in in Section 3. In Section 4, we give a proof for Theorem 1. The last section contains a proof of Theorem 2.

2. Preliminaries

In this section, we recall the definition and some of the properties of symmetrization and certain inequalities concerning symmetrization that we use in the subsequent sections. For further details on symmetrization, we refer to the books [1921].

Let be a domain. Let .

Then the distribution function of is defined as where denotes the Lebesgue measure of a set . Now we define the decreasing rearrangement as The Schwarz symmetrization of is given by where is the measure of unit ball in and is the open ball, centered at the origin with same measure as .

Next we give some inequalities concerning the distribution function and rearrangement of a function.

Proposition 3. Let be a domain and let and be measurable functions on . Then,(a), ;(b)if and only if.

The map is not subadditive. However, we obtain a subadditive function from , namely, the maximal function of defined by The subadditivity of with respect to helps us to define norms in certain function spaces.

Finally, in the following proposition we state two important inequalities concerning Schwarz symmetrization (decreasing rearrangement).

Proposition 4. Let be a domain in with . Let and be two measurable functions on and let . Then one has the following inequalities.(a)The Hardy-Littlewood inequality: (b)The Polya-Szegö inequality:

Next we state a necessary and sufficient condition for one dimensional weighted Hardy inequality due to Muckenhoupt (see 4.17, [10]).

Proposition 5. Let and be nonnegative measurable functions such that . Let and let be the conjugate exponent of . Then for any , holds for all measurable function if and only if Moreover, if is the best constant in (21), then

3. A Space for Admissible Functions

In this section, we define the function space and give its relation with certain classical function spaces. For a bounded domain , we define One can verify that is a Banach function space with the norm The nomenclature refers to the fact that is the maximal rearrangement invariant Banach function space (Lorentz M-space) on with the fundamental function . Next we give some examples of function spaces in .

Recall, for a bounded domain is the Orlicz space generated by the Orlicz function , that is, A similar analysis as in Lemma 6.2 of [14] gives Using this equivalent definition, we show that is contained in .

Proposition 6. Let be a bounded domain in . Then .

Proof. Let , then using the definition of and the monotonicity of we obtain Now by taking the supremum over in the previous inequality, we obtain the desired fact.

This inclusion is strict as seen in the following example.

Example 7. Let . For small, let Then for , does not belong to but belongs to .

For a bounded domain , the Zygmund space is defined as From the following proposition we see that is contained in .

Proposition 8. Let be a bounded domain in . Then for a measurable function , the following inequality holds:

Proof. Let . Then and the last inequality follows as .

Remark 9. In [18], using Hansson’s embedding, the authors showed that functions are admissible. Since , their result also follows from Theorem 1 without using Hansson’s embedding. Since functions are admissible, Theorem 2.5 of [22] shows that the spaces and are equivalent. Thus, Theorem 1 indeed follows from Hansson’s embedding as in [18]. However, our proof for Theorem 1 relies only on certain classical rearrangement inequalities and the Muckenhoupt condition for 1-dimensional weighted Hardy inequalities.

In the next proposition, we show that the weights considered in [2] for the improved Hardy-Sobolev inequalities are in and hence belong to as well.

Proposition 10. Let be a bounded domain in and let . Then for ,

Proof. Let , and . Note that , , and hence A straightforward calculation gives where the last inequality follows since . Therefore Hence the proof.

Next we verify that weights in satisfy Maźja’s capacity condition.

Proposition 11. Let be a bounded domain in , , and . Then satisfies Maźja’s capacity condition, that is,

Proof. Using Polya-Szegö inequality, we can easily verify that . Further, (see, e.g., page 106 of [7]). Now for a compact set , Therefore,

In the next section, we see that the space almost characterizes the radial weights satisfying Maźja’s condition (Theorem 13).

4. The Generalized Hardy-Sobolev Inequalities

In this section, we give a proof for Theorem 1. Further, when for some , we show that all radial and radially decreasing admissible weights necessarily lie in . First we have the following theorem on one dimensional weighted Hardy inequalities.

Theorem 12. Let be a bounded domain in and let . Then

The previous inequality is known for more general weights (even for measures), see [1113]. Note that when , satisfies Muckenhoupt condition (22) and hence the inequality follows from Proposition 5. We prove Theorem 12 by adapting the proof of Theorem 4, chapter 4 of [10].

Proof. For the simplicity, we let and . Since , we have Now using the Hölder inequality we obtain Therefore, The equality in the last step follows from Fubini’s theorem on the interchange of the order of integration. Next we estimate the innermost integral on the right-hand side of the previous inequality using (41): (using (41)).
Now by substituting back into (43), we get the result.

Proof of Theorem 1. Let . From the inequalities given in Proposition 4, it is clear that the inequality holds, if Since , we can rewrite the previous inequality as Now by Theorem 12, we see that the previous inequality holds with .

In the following theorem, we show that our condition is almost necessary for the generalized Hardy-Sobolev inequality.

Theorem 13. Let and let be such that is positive, radial, and radially decreasing. If is admissible, then .

Proof. Let be admissible and let be such that We use certain test functions in to estimate . For , let Clearly and Therefore Further, since is radial and radially decreasing, we get Now (48), (51), and (52) yield Thus by substituting in the previous inequality and noting that , we get Hence and .

Next we see how one can obtain Moser-Trudinger embedding and Hansson’s embedding using Theorem 1.

Remark 14 (an alternate proof for some classical embeddings). From Theorem 1, for each , we have the generalized Hardy-Sobolev inequality (i)Moser-Trudinger embedding: since , there exists such that Thus from (55) we have The previous inequality shows that for each is a continuous linear functional on with . In other words, (the dual space of ) and In particular, for each and also (ii)Hansson’s embedding: since , there exists such that As before, for each with , that is, Thus for and

5. The Best Constant in the Hardy-Sobolev Inequalities

In this section, we give a proof of Theorem 2 using a direct variational method. For , we define the following: Recall that It is easy to see that , where . Note that, for an admissible and is the best constant in (1). Thus the best constant in (1) is attained for some if has a minimizer on . We show that, under the assumptions of Theorem 2, admits a minimizer on .

First we prove the following compactness theorem.

Lemma 15. Let be a bounded domain in and let . Then the map is compact.

Proof. For we show that . First we estimate the following: Since is bounded, from (55) we get that is bounded. Thus that if . Let . For a given , since , we choose such that Therefore Now For sufficiently large , the first integral can be made smaller than since is embedded compactly in and . Thus we conclude that .

Proof of Theorem 2. Since and , there exists such that (see for e.g., Proposition 4.2 of [23]) and and hence the set is nonempty. Let be a minimizing sequence of on , that is, Using the coercivity of , the reflexivity of , and the compactness of the embedding of into , we can further assume that converges weakly and almost everywhere to some . Since , from the previous lemma we get For , we write Now use the Fatou’s lemma to obtain Thus we get . Set . Now the homogeneity and the weak lower semicontinuity of yields the following: Thus equality must hold at each step and hence . This shows that and .

Next we give an equivalent definition for the space . Recall that is the closure of in .

Theorem 16. Let be a bounded domain in . Then the following statements are equivalent: (i), (ii) and .

Proof. We show that . Let and in . Note that for each is bounded and hence Let be given. We show that there exists such that for all . Since in , there exists such that Using the subadditivity of the maximal operator , we get Note that from (76), there exists such that for all . Now (78) yields the result as . Let be given. For proving , we show that there exists such that . Since , we get such that Let Clearly . Since is continuously embedded in and is dense in , we can choose so that Next we estimate . Let us compute the distribution function : For computing the symmetric rearrangement , we set . If , for all , we get the desired result as . Next we calculate , when . For , note that implies that (if , then , a contradiction). Thus for , using (83) we have Now for , again using (83), we see that ,  for all and hence , for all . Thus Therefore, Since , from (80), we have . Now (82) yields and hence the proof is done.

Next we give example of functions in .

Example 17. We have already seen that (Proposition 6). In fact, since is dense in and the inclusion of in is continuous. Also, using Theorem 16, one can verify that , for .

Remark 18 (an open problem). Whether all the admissible functions are in or not, by setting , the problem can be rephrased as whether implies or not.

Remark 19. For and (not necessarily bounded), using the Muckenhoupt condition (22) and the inequalities given in Proposition 4, one can show that (as in Theorem 1) is admissible if satisfies The previous condition shows that one can obtain Visciglia’s result [8] without using the Lorentz-Sobolev embedding. In fact, one can give an alternate proof for the Lorentz Sobolev embedding of into the Lorentz space using similar arguments as in Remark 14.

Acknowledgments

The author thanks Professor Mythily Ramaswamy for her encouragement and useful discussions. He also thanks Prof. A. K. Nandakumaran and Prof. Prashanth K. Srinivasan for their continuous support and interest. The author also thanks the unknown referees for their suggestions that greatly improved this paper This work has been supported by UGC under Dr. D. S. Kothari postdoctoral fellowship scheme, Grant no. 4-2/2006 (BSR)/13-573/2011(BSR).

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