Research Article A Note on Generalized Hardy-Sobolev Inequalities

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1 Hindawi Publishing Corporation International Journal of Analysis Volume 3, Article ID , 9 pages hp://dx.doi.org/.55/3/ Research Article A Note on Generalized Hardy-Sobolev Inequalities T.V.Anoop Department of Mathematics, Indian Institute of Science, Bangalore 56, India Correspondence should be addreed to T. V. Anoop; anoop@math.iisc.ernet.in Received 6 September ; Accepted November Academic Editor: Tohru Ozawa Copyright 3 T. V. Anoop. is is an open acce article distributed under the Creative Commons Aribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We are concerned with nding a cla of weight functions gg so that the following generalized Hardy-Sobolev inequality holds: gggg CCC uuu, uuuuu (), for some CCCC, where 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 aained. Our method gives an alternate way of proving the Moser-Trudinger embedding and its re nement due to Hanon.. Introduction In this paper, we are interested to nd the best suitable function space for the weights gg so that the following generalized Hardy-Sobolev inequality holds: gggg CCCC uu, uuuuu (), () for some CCCC, where is a bounded domain in R NN. We say that gg is admiible, if the previous inequality holds. We are also interested to nd a cla of admiible functions that ensures the best constant in () which is aained for some uuuuu (). Let us rst recall the claical Hardy inequality: xx xx ff () dddd dddd d dd By taking fffff with uuuuu u u in (), we get xx uu(xx) dddd d dd ff(xx) ddddd dd d dd () uu (xx) ddddd (3) e higher dimensional analogue of the previous inequality is referred to as the Hardy-Sobolev inequality in the literature: xx uu(xx) dddd d dd uu (xx) ddddd dd d dd (), (4) where R NN is a domain containing the origin with NNNN. Clearly (4) does not hold when NNNNor, since / xxx is not locally integrable for that contains the origin. For NN N N, 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 () for more general weights instead of the homogeneous weight / xxx. e second one relies on the fact that the best constant in (4) is not aained in HH () 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 [] who have proved the following inequality: NNNN xx uu(xx) ddddddd uu(xx) dddd uu (xx) ddddd dd d dd (). Motivated by the previous inequality, several improved Hardy-Sobolev inequalities have been proved, for example see [ 5]. For NNNN,aswepointedoutbefore,theHardypotential / xxx is not admiible for any domain in R that contains the origin. In [6], Leray showed that / xxx log(xxxx is the right admiible function (analogous to Hardy potential) for (5)

2 International Journal of Analysis = BBBBB BB B B. In this paper, we focus on nding a large cla of admiible functions including that of Leray s function or its improvements (by adding nonnegative terms) for the generalized Hardy-Sobolev inequalities in bounded domains of R. 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 EE E E, the relative capacity of EE with respect is de ned as cap (EEE E) = inf uu uuuuu cc (), uuu EE. (6) First, let us see that Maźja s capacity condition is very much intrinsic on (). Let gg be a positive weight satisfying (), then for any compact subset, gg (xx) dddd d dd gggg EE CC uu, uuuuu cc (), uuu EE. By taking the in mum, we get EE gggggggggg g gg cap(eee EE. erefore (7) sup EE gg (xx) dddd EEEEE cap (EEE E) CCC (8) Maźja proved that the previous condition is indeed sufficient for () ( eorem /.4., page 8 of [7]). Since Maźja s condition is neceary 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 NN N N is obtained by Visciglia in [8].Heprovedthat() holds for weights in the Lorentz space LLLLLLLL LL. e embedding of HH () into the Lorentz space LLLL, ) played a key role in his result. e case NN N N is more subtle, for example, () does not hold when = R and R gg g g, see [9]. In this paper we obtain a veri able condition for admiible functions for bounded domains in R, using one-dimensional weighted Hardy inequalities and certain rearrangement inequalities. A general one dimensional weighted Hardy inequality has the following form: bb xx qq /qq ff () dddd uu (xx) dddd /pp CC ppppp ff(xx) pp vv (xx) dddd, ff f ff For an excellent review on the weighted Hardy inequalities, we refer to[] by Maligranda et al. Many neceary and sufficient conditions on uu, vv, pp, qq for holding (9) are available in the literature, see [ 3]. Here we make use of the so (9) called Muckenhoupt condition [3] forobtainingaclaof weight functions satisfying (). For a measurable function gg, we denote its decreasing rearrangement by gg and the maximal function of gg is denoted by gg, that is, gg (t t (/t t gg (s. Now we de ne M log LL = gg measurable sup t <t gg () <. () Indeed, M log LL is a rearrangement invariant Banach function space with the norm gg M log LL = sup t <t gg (), () see [4]. More details of the space M log LL are given in Section 3.Nowwestateourmaintheorem. eorem. Let be a bounded domain in R and let ggg M log LL. en gg is admiible and gggg gg M log LL uu, uuuuu (). () ππ Having obtained M log LL, a cla of admiible functions, onewouldliketoknowforwhichamongthemthebest constant in () is aained in HH (). Many authors have addreed this question when NN N N, see [8, 5] and the references therein. For NN N N, Maźja has a sufficient condition (see.4. of [7]) in terms of capacity. Here we consider the weights in a subcla of M log LL so that the best constant in () isaainedinhh (). For a bounded domain R (analogous to space F NNNN in [5]) we de ne F = CC cc () in M log LLL (3) Weshowthahebestconstantin() isaainedinhh (), when gg + F, where gg + is the positive part of gg. More precisely, we have the following theorem. eorem. Let gg g g ggg gg and gg + F {}. e ne λλ gg = inf uu uuuuu gggg () gggg, >. (4) en λλ (ggg is aained for some uuuuu (). e Moser-Trudinger embedding of HH () into the Orlicz space LL AA (), the Orlicz space generated by the Orlicz function AAAAAA A AA,canbeusedtoshowthatLLLLL LL functions are admiible. An embedding of HH (), ner than that of Moser-Trudinger, is established independently by Hanon [6], Brézis, and Wainger [7]. As anticipated, this embedding gives a bigger cla of admiible functions

3 International Journal of Analysis 3 than LL LLL LL. In[8], the authors used this ner embedding and showed that the Lorentz-Zygmund space LL, (log LLL is admiible. We are not going to use any embeddings for proving that M log LL functions are admiible, instead we use some rearrangement inequalities and one dimensional weighted Hardy inequalities. We would like to stre that the admiibility of M log LL functions can be used to give alternate proofs for the Moser-Trudinger embedding and its re nement due to Hanon. e rest of the paper is organised as follows. In Section, we recall de nition and properties of decreasing rearrangement. Further, we state some claical inequalities that will be used in the subsequent sections. We discu the properties of the space M log LL and give examples of function spaces contained in M log LL in Section 3. In Section 4, wegivea proof for eorem. e last section contains a proof of eorem.. 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 [9 ]. Let R NN be a domain. Let EE ff (s s s s s s s. en the distribution function αα ff of ff is de ned as αα ff () = EE ff (), for s (5) where AAA denotes the Lebesgue measure of a set AAAA NN. Now we de ne the decreasing rearrangement ff as ff () = inf s ff (), t (6) e Schwarz symmetrization of ff isgivenby ff (xx) =ff ωω NN xx NN, xxxx, (7) where ωω NN is the measure of unit ball in R NN and istheopen 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 R NN be a domain and let ff and gg be measurable functions on. en, (a) ff (αα ff ( s, αα ff (ff ( t ; (b) t ff (sifandonlyifs (t. e map ff f ff is not subadditive. However, we obtain a subadditive function from ff, namely, the maximal function ff of ff de ned by ff () = ff (ττ) ddddd dd d dd (8) e subadditivity of ff with respect to ff helps us to de ne 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 R NN with NNNN. Let ff and gg betwomeasurablefunctionson and let uuuuu (). enonehasthefollowinginequalities. (a) e Hardy-Lilewood inequality: ff (xx) gg (xx) ddddddd ff (xx) gg (xx) dddd (b) e Polya-Szegö inequality: NN ωω /NN NN = ff () gg (). /NN uu () dddd d uu (xx) dddd uu (xx) ddddd (9) () Next we state a neceary and sufficient condition for one dimensional weighted Hardy inequality due to Muckenhoupt (see 4.7, []). Proposition 5. Let uu and vv be nonnegative measurable functions such that vvvv. Let <ppppand let pp be the conjugate exponent of pp. enforany, pp ff () dddd uu () dddd d dd ff () pp vv () ddddd () holds for all measurable function ff ifandonlyif AA A AAA <t /pp uu () dddd Moreover, if CC bb is the best constant in (), then vv() pp dddd/pp <. () AAAAA bb pp /pp pp /pp AAA (3) 3. A Space for Admiible Functions In this section, we de ne the function space M log LL and give its relation with certain claical function spaces. For a bounded domain R, we de ne M log LL = uu measurable sup uu () t <. <t (4) One can verify that M log LL is a Banach function space with the norm uu = sup uu () t <t. (5)

4 4 International Journal of Analysis e nomenclature refers to the fact that M log LL is the maximal rearrangement invariant Banach function space (Lorentz M-space) on with the fundamental function t. Next we give some examples of function spaces in M log LL. Recall, for a bounded domain R, LL LLL LL is the Orlicz space generated by the Orlicz function t + (t, that is, LL LLL LL LL uu measurable uu (xx) log + (uu (xx)) dddd d d. (6) A similar analysis as in Lemma 6. of [4] gives LL LLL LL = uu measurable uu () log dddd d d. (7) sing this equivalent de nition, we show that LL LLL LL is contained in M log LL. Proposition 6. Let beaboundeddomaininr. en LL LLL LL L L LLL LL. Proof. Let ff f ff fff ff, then using the de nition of ff and the monotonicity of log( /t we obtain ff () t = log ff () dddd log ff () ddddd (8) Now by taking the supremum over t t in the previous inequality, we obtain the desired fact. is inclusion is strict as seen in the following example. Example 7. Let = BBBBB BB B B. For δδ small, let χχ BBBBBBBB (xx) gg (xx) = xx log (xx). ββ (9) log log (xx) en for <ββββ, gg does not belong to LL LLL LL but belongs to M log LL. For a bounded domain R, the Zygmund space LL, (log LLL is de ned as LL, log LL =ff measurable sup log <t ee ff () <. (3) From the following proposition we see that LL, (log LLL is contained in M log LL. Proposition 8. Let be a bounded domain in R. enfora measurable function ff, the following inequality holds: ff () t sup ff () log <s, t(, ). Proof. Let t t. en ff () t = log ff () dddd = log ff () (3) log log ( /) dddd sup ff () log <s, (3) and the last inequality follows as log( /t t (/(s )dddd d d. Remark 9. In [8], using Hanon s embedding, the authors showed that LL, (log LLL functions are admiible. Since LL, (log LLL M log LL, their result also follows from eorem without using Hanon s embedding. Since M log LL functions are admiible, eorem.5 of [] shows that the spaces LL, (log LLL and M log LL are equivalent. us, eorem indeed follows from Hanon s embedding as in [8]. However, our proof for eorem relies only on certain claical rearrangement inequalities and the Muckenhoupt condition for -dimensional weighted Hardy inequalities. In the next proposition, we show that the weights considered in [] for the improved Hardy-Sobolev inequalities are in LL, (log LLL and hence belong to M log LL as well. Proposition. Let be a bounded domain in R and let RR R RRR xxxx xxxxx. enforγγγγ, xx log (RRRRRRR ) γγ LL, log LL. (33) Proof. Let gggggg g ggggg [log(rrrrrrrrr γγ }, xx x xxxxx xxx, and ffffff f ffffffff (xxx. Notethatffffff f ffffff, xx x xxxxx xxx, and hence ff () gg (), t(, RR). (34)

5 International Journal of Analysis 5 A straightforward calculation gives gg () =ππ log RR ππ γγ = γγ ππlog RR γγ ππ γγ ππlog γγ ee, (35) where the last inequality follows since RR R RRR xxxx xxxxx. erefore log Hence the proof. ee gg () γγ γγ ee ππlog. (36) Next we verify that weights in M log LL satisfy Maźja s capacity condition. Proposition. Let beaboundeddomaininr, gg g M log LL, and gg g g. en gg satis es s capacit condition, that is, sup KK gggggg cap (KKK K) <. (37) KKKKK Proof. Using Polya-Szegö inequality, we can easily verify that cap(kk, ) cap(kkk KK. Further, cap(kk, ) = (πππππ πππ ππππππππππ (see, e.g., page 6 of [7]). Now for a compact set KK, erefore, KK gg (xx) dddd cap (KKK K) ππ log KK KK gg (xx) dddd KKK log ππ KK gg () ddddd (38) sup KK gg (xx) dddd KKKKK cap (KKK K) ππ sup log <t gg () dddd (39) = ππ gg M log LL <. In the next section, we see that the space M log LL almost characterizes the radial weights satisfying Maźja s condition ( eorem 3). 4. The Generalized Hardy-Sobolev Inequalities In this section, we give a proof for eorem. Further, when = BBBBB BBB for some rrrr, we show that all radial and radially decreasing admiible weights necearily lie in M log LL. First we have the following theorem on one dimensional weighted Hardy inequalities. eorem. Let be a bounded domain in R and let ggg M log LL. en ff () dddd gg () dddd 4gg M log LL ff () ddddd dd d dd (4) e previous inequality is known for more general weights (even for measures), see [ 3]. Note that when gg g g ggg gg, gg satis es Muckenhoupt condition () and hence the inequality follows from Proposition 5. We prove eorem by adapting the proof of eorem 4, chapter 4 of []. Proof. For the simplicity, we let a a and ggg g gggg M log LL. Since gg g g ggg gg, we have log gg (ττ) dddd gg, t (, ). (4) Now using the Hölder inequality we obtain ff () dddd erefore, = ff () log /4 dddd /4 log (a) ff () log / dddd log (a) / dddd =log / ff () log / ddddd ff () dddd gg () dddd ff () log / dddd log / gg () dddd = log / gg () ddddff () log / ddddd (4) (43) e equality in the last step follows from Fubini s theorem on the interchange of the order of integration. Next we estimate

6 6 International Journal of Analysis the innermost integral on the right-hand side of the previous inequality using (4): log / gg () dddd gg =gg gg ( )/ (ττ) dddd gg () dddd dd dddd =gg gg / (ττ) dddd gg log (a) /. gg / (ττ) dddd dddd (using (4)). Now by substituting back into (43), we get the result. (44) Proof of eorem. Let gg g g ggg gg. From the inequalities given in Proposition 4,it is clear that the inequality holds, if gggg CC uu, uu u uu () (45) gg () uu () 4ππππ uu () ddddd ddd d dd (). (46) Since uu ( ) =,wecanrewritethepreviousinequalityas uu () dddd gg () 4ππππ uu () ddddd (47) Now by eorem, we see that the previous inequality holds with CC C CCCC M log LL /ππ. In the following theorem, we show that our condition is almost neceary for the generalized Hardy-Sobolev inequality. eorem 3. Let = BBBBB BBB B B and let ggggg () be such that gg is positive, radial, and radially decreasing. If gg is admiible, then gg g g ggg gg. Proof. Let gg be admiible and let CCCCbe such that gggg CCCC uu, uuuuu (). (48) We use certain test functions in HH () to estimate ggg M log LL. For rrrr, let log RR rr, uu rr (xx) = log RR xx, xx rrr xx rrr (49) Clearly uu rr HH () and erefore uu rr (xx) =, xx rrr xx xx, xx rrr (5) uu rr RR = ππ rr t t t RR. (5) rr Further, since gg is radial and radially decreasing, we get gggg rr log RR rr gg (xx) dddd BBBBBBBB Now (48), (5),and(5) yield ππππ = log RR ππππ rr gg () ddddd (5) gg () dddd d ddddd log (RRRRR). (53) us by substituting t in the previous inequality and noting that log(rrrrrr R RRRRRRRRRRRR ), we get t gg () 4πππππ πππ π (, ). (54) Hence gg g g ggg gg and ggg M log LL 4ππππ. Next we see how one can obtain Moser-Trudinger embedding and Hanon s embedding using eorem. Remark 4 (an alternate proof for some claical embeddings). From eorem, foreachgg g g ggg gg, we have the generalized Hardy-Sobolev inequality gggg gg M log LL uu, uuuuu (). (55) ππ (i) Moser-Trudinger embedding: since LL LLL LL L M log LL, there exists CC >such that gg M log LL CC ππgg LL LLL LL. (56) us from (55)wehave gggg CC gg LL LLL LL uu, uuuuu (). (57) e previous inequality shows that for each uu u HH (), uu is a continuous linear functional on LL LLL LL with uu [LL LLL LLL CC uuu. In other words, uu LL exp (the dual space of LL LLL LL) and ee (uu /CC uuu ). (58) In particular, for each uuuuu (), uu u uu exp and also uu LLexp CC uu, uuuuu (). (59)

7 International Journal of Analysis 7 (ii) Hanon s embedding: since LL, (log LLL M log LL, there exists CC >such that gg M log LL CC ππgg LL, (log LLL. (6) As before, for each uu u uu (), uu [LL, (log LLL ] (= LL, (log LLL ) with uu [LL, (log LLL ] CC uuu, that is, uu () log (eeeeeeee) CC uu, uuuuu (). (6) us for uuuuu (), uu u uu, (log LLL and uu LL, (log LLL CC uu, uuuuu (). (6) 5. The Best Constant in the Hardy-Sobolev Inequalities Inthiection,wegiveaproofof eorem using a direct variational method. For gg g g ggg gg, we de ne the following: Recall that M = uu uuu () gggg =, (63) GG + (uu) = gg + uu, uuuuu (). (64) λλ gg = inf uu uuuuu gggg () gggg, >. (65) Itiseasytoseethatλλ (ggg g gggggggggg g gg g gg, where JJJJJJ J J uuu. Note that, for an admiible ggg gg (ggg g g and /λλ (ggg is the best constant in (). us the best constant in () is aained for some uuuuu () if JJ has a minimizer on M. We show that, under the aumptions of eorem, JJ admits a minimizer on M. First we prove the following compactne theorem. Lemma 5. Let be a bounded domain in R and let h F. en the map is compact. HH (uu) = huu, uuuuu () (66) Proof. For uu nn uuweshowthathhhhh nn ) HHHHHH. First we estimate the following: HH uu nn HH(uu) h uu nn uu h uu nn +uu h uu nn uu h uu nn +uu / / h uu nn uu. (67) Since uu nn is bounded, from (55)wegehat( h (uu nn + uuu ) / is bounded. us HHHHH nn ) HHHHHH that if h (uu nn uuu. Let mm m mmm nn (uu nn uuuu.foragivenεεεε, since h F, we choose h εε CC cc () such that erefore Now h h εε M log LL < εε mm. (68) h h εε uu nn uu h h εε M log LL uu nn uu < εεε (69) h uu nn uu = h εε uu nn uu + h h εε uu nn uu h εε uu nn uu + εεε (7) For sufficiently large nn, the rst integral can be made smaller than εε since HH () is embedded compactly in LL () and h εε CC cc (). us we conclude that h (uu nn uuu. Proof of eorem. Since ggggg loc() and gg +, there exists φφφφφ cc () such that gggg >(see for e.g., Proposition 4. of [3]) and and hence the set M is nonempty. Let {uu nn } be a minimizing sequence of JJ on M, that is, lim nnnn JJ uu nn =λλ gg = inf uuuu JJ (uu). (7) Using the coercivity of JJ, the re exivity of HH (), and the compactne of the embedding of HH () into LL (), we can further aume that {uu nn } converges weakly and almost everywhere to some uuuuu (). Since gg + F, from the previous lemma we get lim gg + uu nnnn nn = gg + uu. (7) For uu nn M,wewrite gg uu nn = gg + uu nn. (73) NowusetheFatou slemmatoobtain gg uu gg + uu. (74) us we get gggg. Set uu uu uuuuu gggg ) /. Now the homogeneity and the weak lower semicontinuity of JJ yields the following: JJ (uu) λλ gg JJ(uu) = JJ(uu) lim inf JJ uu gggg nn nn =λλ gg. (75) usequalitymustholdateachstepandhence gggg =. is shows that uuuuand JJJJJJ J JJ (ggg.

8 8 International Journal of Analysis Next we give an e uivalent de nition for the space F. Recall that F is the closure of CC cc () in M log LL. eorem 6. Let be a bounded domain in R. en the following statements are equivalent: (i) gggg, (ii) gg g g ggg gg and lim gg (t t t t. Proof. (iii i iiiiii Weshowthatlim gg (t t t t. Let gg nn CC cc () and gg nn ggin M log LL.Notethatforeach gg nn,gg nn (t is bounded and hence lim gg nn () t =, nn n nn (76) Let εεεεbegiven. We show that there exists δδδδsuchthat gg (t t t t for all t t t. Since gg nn gg in M log LL, there exists nn Nsuchthat sup ggggg nn () t <t < εεε (77) Using the subadditivity of the maximal operator ff f ff, we get gg () t ggggg nn () t +gg nn () t <εεεεε nn () t. (78) Note that from (76), there exists δδ δ δ such that gg nn (t t t t for all t t t. Now (78) yields the result as gg () t < εεε εεε ε (, δδ). (79) (iiiii i iiii. Let εεεεbe given. For proving gggg, we show that there existsgg εε CC cc () such that gg ggg εε < εε. Since lim gg (t t t t, we get δδδδsuch that Let gg () t < εεε εεε ε (, δδ]. (8) AA εε = xx xxxgg (xx) < gg (δδ), gg εε = gggg AAεε. (8) Clearly gg εε LL (). Since LL () is continuously embedded in M log LL and CC cc () is dense in LL (), we can choosegg εε CC cc () so that gg εε gg εε M log LL < εεε (8) Next we estimate ggggg εε. Let us compute the distribution function αα gggggεε : αα gggggεε () =αα ggggaa εε cc () = αα gg () s (δδ), αα gg gg (δδ) s (δδ). (83) For computing the symmetric rearrangement (ggggg εε ) (t, we set δδ =αα gggggεε (gg (δδδδ δ δδ gg (gg (δδδδ. If δδ =, αα gggggεε (s s s, for all, we get the desired result as ggggg εε =. Next we calculate (gg g gg εε ) (t, when δδ >. For <t, note that αα gggggεε (s s implies that s (δδδ (if s (δδδ, then δδ =αα gggggεε (gg (δδδδ δ δδ gggggεε (s s, a contradiction). us for t, using (83)wehave ggggg εε () = inf s gggggεε () = inf s (δδ) αα gggggεε () = inf s gg () =gg (). (84) Now for t, again using (83),weseethatαα gggggεε (s s αα gg (gg (δδδδ δ δδ, for all s (δδδ and hence (ggggg εε ) (t t, for all t. us erefore, ggggg εε () = gg () t, δδ gg δδ t. ggggg εε () t gg () t t, gg δδ δδ log t δδ. (85) (86) Since δδ δδ, from (8),wehave gg g gg εε < εε. Now (8) yields gg ggg εε < εε andhencetheproofisdone. Next we give example of functions in F. Example 7. We have already seen that LL LLL LL L L LLL LL (Proposition 6). In fact, LL LLL LL L L since CC cc () is dense in LL LLL LL and the inclusion of LL LLL LL in M log LL is continuous. Also, using eorem 6, one can verify that / xxx [log(rrrrrrrrr γγ F, for γγγγ. Remark 8 (an open problem). Whether all the admiible functions are in M log LL or not, by seing λλ (gg ) = inf{( uu / gg uu ) uu u uu (), gggg > }, the problem can be rephrased as whether λλ (ggg g g implies λλ (gg )>or not. Remark 9. For NN N N and R NN (not necearily bounded), using the Muckenhoupt condition () and the

9 International Journal of Analysis 9 inequalities given in Proposition 4, one can show that(as in eorem ) gg is admiible if gg satis es sup /NN gg () <. (87) <t e previous condition shows that one can obtain Visciglia s result [8] without using the Lorentz-Sobolev embedding. In fact,onecangiveanalternateproofforthelorentzsobolev embedding of DD, () into the Lorentz space LLLL, ) using similar arguments as in Remark 4. Acknowledgments e author thanks Profeor Mythily Ramaswamy for her encouragement and useful discuions. He also thanks Prof. A. K. Nandakumaran and Prof. Prashanth K. Srinivasan for their continuous support and interest. e author also thanks the unknown referees for their suggestions that greatly improved this paper is work has been supported by UGC under Dr. D. S. Kothari postdoctoral fellowship scheme, Grant no. 4-/6 (BSR)/3-573/(BSR). References [] H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Revista Matemática de la Universidad Complutense de Madrid, vol., no., pp , 997. [] Adimurthi, N. Chaudhuri, and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proceedings of the American Mathematical Society, vol. 3, no., pp ,. [3] N. Chaudhuri and M. Ramaswamy, Existence of positive solutions of some semilinear elliptic equations with singular coefficients, Proceedings of the Royal Society of Edinburgh A, vol. 3, no. 6, pp ,. [4] S. Filippas and A. Tertikas, Optimizing improved Hardy inequalities, Journal of Functional Analysis, vol. 9, no., pp ,. [5] N. Ghououb and A. Moradifam, On the best poible remaining term in the Hardy inequality, Proceedings of the National Academy of Sciences of the United States of America, vol. 5, no. 37, pp , 8. [6] J. Leray, Etude de diverses equations integrales nonlineaire et de quelques prob-lemes que pose lhydrodinamique, Journal de Mathématiques Pures et Appliquées, vol., no. 9, pp. 8, 933. [7] V. G. Maz ja, Sobolev Spaces, Springer Series in Soviet Mathematics, Springer, Berlin, Germany, 985, Translated from the Ruian by T. O. Shaposhnikova. [8] N. Visciglia, A note about the generalized Hardy-Sobolev inequality with potential in LL ppppp (R nn ), Calculus of Variations and Partial Differential Equations, vol. 4, no., pp , 5. [9] C. Cosner, K. J. Brown, and J. Fleckinger, Principal eigenvalues for problems with inde nite weight function on R nn, Proceedings of the American Mathematical Society, vol.9,no.,pp , 99. [] L. Maligranda, L.-E. Peron, and A. Kufner, e Hardy Inequality: About its History and Some Related Results, 7. [] G. Talenti, Oervazioni sopra una clae di disuguaglianze, Rendiconti del Seminario Matematico e Fisico di Milano, vol. 39, pp. 7 85, 969. [] G. Tomaselli, A cla of inequalities, Bolleino Della Unione Matematica Italiana, vol., no. 4, pp. 6 63, 969. [3] B. Muckenhoupt, Hardy s inequality with weights, Studia Mathematica, vol. 44, pp. 3 38, 97, Collection of articles honoring the completion by Antoni Zygmund of 5 years of scienti c activity, I. [4] C. Benne and R. Sharpley, Interpolation of Operators, vol. 9 of Pure and Applied Mathematics, Academic Pre, Boston, Ma, USA, 988. [5] T. V. Anoop, M. Lucia, and M. Ramaswamy, Eigenvalue problems with weights in Lorentz spaces, Calculus of Variations and Partial Differential Equations, vol. 36, no. 3, pp , 9. [6] K. Hanon, Imbedding theorems of Sobolev type in potential theory, Mathematica Scandinavica, vol. 45, no., pp. 77, 979. [7] H. Brézis and S. Wainger, A note on limiting cases of Sobolev embeddings and convolution inequalities, Communications in Partial Differential Equations, vol. 5, no. 7, pp , 98. [8] M. Krbec and T. Scho, Superposition of imbeddings and Fefferman s inequality, Bolleino della Unione Matematica Italiana B, vol., no. 3, pp , 999. [9] D. E. Edmunds and W. D. Evans, Hardy Operators, Function Spaces and Embeddings, Springer Monographs in Mathematics, Springer, Berlin, Germany, 4. [] S. Kesavan, Symmetrization & Applications, vol. 3 of Series in Analysis, World Scienti c Publishing, Hackensack, NJ, USA, 6. [] M. Lo and E. H. Lieb, Analysis, vol. 4 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, nd edition,. [] D. E. Edmunds and H. Triebel, Sharp Sobolev embeddings and related Hardy inequalities: the critical case, Mathematische Nachrichten, vol. 7, pp. 79 9, 999. [3] B. Kawohl, M. Lucia, and S. Prashanth, Simplicity of the principal eigenvalue for inde nite quasilinear problems, Advances in Differential Equations, vol., no. 4, pp , 7.