ANote on Generalized Hardy-Sobolev Inequalities

We are concerned with �nding a class of weight functions gg so that the following generalized Hardy-Sobolev inequality holds: ∫Ω gggg 2 ≤ CC∫Ω |∇gg| , gg u uu0(Ω), for some CC C 0, whereΩ is a bounded domain in R . 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 re�nement due to Hansson.


Introduction
In this paper, we are interested to �nd the best suitable function space for the weights  so that the following generalized Hardy-Sobolev inequality holds: for some   0, where Ω is a bounded domain in ℝ  .We say that  is admissible, if the previous inequality holds.We are also interested to �nd a class of admissible functions that ensures the best constant in (1) which is attained for some    1 0 (Ω).Let us �rst recall the classical Hardy inequality: (3) e higher dimensional analogue of the previous inequality is referred to as the Hardy-Sobolev inequality in the literature: where Ω ⊂ ℝ  is a domain containing the origin with   .
Clearly (4) does not hold when   1 or 2, since 1/|| 2 is not locally integrable for Ω 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.e �rst one refers to the inequalities of the form (1) for more general weights instead of the homogeneous weight 1/|| 2 .e second one relies on the fact that the best constant in (4) is not attained in  1 0 (Ω) and hence one anticipates to improve (4) by adding nonnegative terms in the le�-hand side.e �rst ma�or 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 [2][3][4][5].
For   2, as we pointed out before, the Hardy potential 1/|| 2 is not admissible for any domain in ℝ 2 that contains the origin.In [6], Leray showed that 1/|| 2 | log()| 2 is the right admissible function (analogous to Hardy potential) for Ω =     2 .In this paper, we focus on �nding 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  2 .
e 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   Ω, the relative capacity of  with respect Ω is de�ned 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 Ω, By taking the in�mum, we get ∫     cap Ω.erefore Maźja proved that the previous condition is indeed sufficient for (1) (eorem 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.us it is of interest to �nd certain veri�able conditions for the generalized Hardy-Sobolev inequalities by other means.One such veri�able condition for  ≥  is obtained by Visciglia in [8].He proved that (1) holds for weights in the Lorentz space 2, ∞.e embedding of    Ω into the Lorentz space 2 * , 2 played a key role in his result.e case  = 2 is more subtle, for example, (1) does not hold when Ω =  2 and ∫  2   , see [9].In this paper we obtain a veri�able condition for admissible functions for bounded domains in  2 , 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 [11][12][13].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, Having obtained ℳ log , a class of admissible functions, one would like to know for which among them the best constant in (1) is attained in    Ω.Many authors have addressed this question when  ≥ , see [8,15] and the references therein.For  = 2, Maźja has a sufficient condition (see 2.4.2 of [7]) in terms of capacity.Here we consider the weights in a subclass of ℳ log  so that the best constant in (1) is attained in    Ω.For a bounded domain Ω   2 (analogous to space ℱ 2 in [15]) we de�ne We show that the best constant in (1) is attained in    Ω, when  +  ℱ  , where  + is the positive part of .More precisely, we have the following theorem.eorem 2. Let   ℳ log  and  +  ℱ  ⧵ {}.�e�ne en    is attained for some      Ω.
e Moser-Trudinger embedding of    Ω into the Orlicz space   Ω, the Orlicz space generated by the Orlicz function  =   2  − , can be used to show that  log  functions are admissible.An embedding of    Ω, �ner 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 �ner embedding and showed that the Lorentz-Zygmund space  1,∞ (  2 is admissible.We are not going to use any embeddings for proving that ℳ   functions are admissible, instead we use some rearrangement inequalities and one dimensional weighted Hardy inequalities.We would like to stress that the admissibility of ℳ   functions can be used to give alternate proofs for the Moser-Trudinger embedding and its re�nement due to Hansson.e rest of the paper is organised as follows.In Section 2, we recall de�nition 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 ℳ   and give examples of function spaces contained in ℳ   in Section 3. In Section 4, we give a proof for eorem 1. e last section contains a proof of eorem 2.

Preliminaries
In this section, we recall the de�nition 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 [19][20][21].
en the distribution function   of  is de�ned as where  denotes the Lebesgue measure of a set  ⊂ ℝ  .Now we de�ne the decreasing rearrangement  * as  * (  inf       ( ≤  ,    (16) e Schwarz symmetrization of  is given by where   is the measure of unit ball in ℝ  and Ω ⋆ is the open ball, centered at the origin with same measure as Ω.
Next we give some inequalities concerning the distribution function and rearrangement of a function.Finally, in the following proposition we state two important inequalities concerning Schwarz symmetrization (decreasing rearrangement).

Proposition 4.
Let Ω be a domain in ℝ  with   2. Let  and  be two measurable functions on Ω and let    1  (Ω.en one has the following inequalities. Next we state a necessary and sufficient condition for one dimensional weighted Hardy inequality due to Muckenhoupt (see 4.17, [10]).Remark 9.In [18], using Hansson's embedding, the authors showed that  , log  2 functions are admissible.Since  , log  2 ⊂ ℳ log , their result also follows from eorem 1 without using Hansson's embedding.Since ℳ log  functions are admissible, eorem 2.5 of [22] shows that the spaces  , log  2 and ℳ log  are equivalent.us, eorem 1 indeed follows from Hansson's embedding as in [18].However, our proof for eorem 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  , log  2 In the next section, we see that the space ℳ log  almost characterizes the radial weights satisfying Maźja's condition (eorem 13).

The Generalized Hardy-Sobolev Inequalities
In this section, we give a proof for eorem 1.Further, when Ω = ( ) for some   , we show that all radial and radially decreasing admissible weights necessarily lie in ℳ log .First we have the following theorem on one dimensional weighted Hardy inequalities.(40) e previous inequality is known for more general weights (even for measures), see [11][12][13].Note that when   ℳ log ,  * satis�es Muckenhoupt condition ( 22) and hence the inequality follows from Proposition 5. We prove eorem 12 by adapting the proof of eorem 4, chapter 4 of [10].(47) Now by eorem 12, we see that the previous inequality holds with  =   log  /.
In the following theorem, we show that our condition is almost necessary for the generalized Hardy-Sobolev inequality.eorem 13.Let Ω = (0 )   2 and let    1 (Ω) be such that  is positive, radial, and radially decreasing.If  is admissible, then    log .
Proof.Let  be admissible and let   0 be such that We use certain test functions in Hence    log  and   log  ≤ 4.
Next we see how one can obtain Moser-Trudinger embedding and Hansson's embedding using eorem 1.
Remark 14 (an alternate proof for some classical embeddings).From eorem 1, for each    log , we have the generalized Hardy-Sobolev inequality (i) Moser-Trudinger embedding: since  log    log , there exists  1  0 such that us from (55) we have e previous inequality shows that for each    1 0 (Ω),  2 is a continuous linear functional on  log  with  2  [ log  * ≤  1 ∇ 2  2 .In other words,  2   exp (the dual space of  log ) and In particular, for each    1 0 (Ω),    exp 2 and also (ii) Hansson's embedding: since

The Best Constant in the Hardy-Sobolev Inequalities
In this section, we give a proof of eorem 2 using a direct variational method.For   ℳ log , we de�ne the following: Recall that It is easy to see that  1 ( = inf( ∶   ℳ, where ( = ∫ Ω |∇| 2 .Note that, for an admissible ,  1 ( > 0 and 1 1 ( is the best constant in (1).us the best constant in (1) is attained for some    1 0 (Ω if  has a minimizer on ℳ.We show that, under the assumptions of eorem 2,  admits a minimizer on ℳ.
First we prove the following compactness theorem.
Next we give example of functions in ℱ 1 .
Example 17.We have already seen that  log   ℳ log  (Proposition 6).In fact,  log   ℱ 1 since  ∞  (Ω) is dense in  log  and the inclusion of  log  in ℳ log  is continuous.Also, using eorem 16, one can verify that 1 2 [log()]   ℱ 1 , for   2.
(87) e 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  1,2 0 (Ω) into the Lorentz space (2 * , 2) using similar arguments as in Remark 14.

eorem 12 .
Let Ω be a bounded domain in ℝ 2 and let   ℳ log .en

Proposition 5 .
Let  and  be nonnegative measurable functions such that   .Let 1    ∞ and let  ′ be the conjugate exponent of .enfor any   , e nomenclature refers to the fact that ℳ log  is the maximal rearrangement invariant Banach function space (Lorentz M-space) on Ω with the fundamental function  logΩ.Next we give some examples of function spaces in ℳ log .Recall, for a bounded domain Ω ⊂ ℝ 2 ,  log  is the Orlicz space generated by the Orlicz function log + , that is, Let Ω be a bounded domain in ℝ 2 .en for a measurable function , the following inequality holds: ( ,(21)holds for all measurable function  if and only if �sing this equivalent de�nition, we show that  log  is contained in ℳ log .Proposition 6.Let Ω be a bounded domain in ℝ 2 .enlog⊂ ℳ log .Proof.Let    log , then using the de�nition of  * * and the monotonicity of logΩ we obtain *  .(28)Now by taking the supremum over   0, Ω in the previous inequality, we obtain the desired fact.isinclusion is strict as seen in the following example.Example 7. Let Ω  0  ⊂ ℝ 2 .For  small, let 2  *    .(30)From the following proposition we see that  , log  2 is contained in ℳ log .Proposition 8.