# A Note on Generalized Hardy-Sobolev Inequalities

Authors: T. V. Anoop
Journal: International Journal of Analysis
 (2013)

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## Abstract

We are concerned with finding a class of weight functionsg so that the following generalized Hardy-Sobolev inequality holds: ∫Ωgu2≤C∫Ω|∇u|2,u∈H01(Ω), for some C>0, where Ω is a bounded domain in ℝ2. 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.


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## Body

## 1. Introduction

In this paper, we are interested to find the best suitable function space for the weightsg so that the following generalized Hardy-Sobolev inequality holds:
(1)∫Ωgu2≤C∫Ω|∇u|2,u∈H01(Ω),
for some C>0, where Ω is a bounded domain in ℝN. We say that g 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 u∈H01(Ω).Let us first recall the classical Hardy inequality:(2)∫0∞‍[1x∫0x‍f(s)ds]2dx≤C2∫0∞‍f(x)2dx,f≥0.
By taking f=u′ with u(0)=0 in (2), we get
(3)∫0∞‍1x2u(x)2dx≤C2∫0∞‍|u′(x)|2dx.
The higher dimensional analogue of the previous inequality is referred to as the Hardy-Sobolev inequality in the literature:
(4)∫Ω‍1|x|2u(x)2dx≤C2∫Ω‍|∇u(x)|2dx,u∈H01(Ω),
where Ω⊂ℝN is a domain containing the origin with N≥3. Clearly (4) does not hold when N=1 or 2, since 1/|x|2 is not locally integrable for Ω that contains the origin.ForN≥3, 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 1/|x|2. The second one relies on the fact that the best constant in (4) is not attained in H01(Ω) 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:
(5)(N-22)2∫Ω‍1|x|2u(x)2dx+λΩ∫Ω‍u(x)2dx≤∫Ω‍|∇u(x)|2dx,u∈H01(Ω).
Motivated by the previous inequality, several improved Hardy-Sobolev inequalities have been proved, for example see [2–5].ForN=2, as we pointed out before, the Hardy potential 1/|x|2 is not admissible for any domain in ℝ2 that contains the origin. In [6], Leray showed that 1/|x|2|log(x)|2 is the right admissible function (analogous to Hardy potential) for Ω=B(0;1)⊂ℝ2. 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 ℝ2.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 E⊂Ω, the relative capacity of E with respect Ω is defined as
(6)cap(E;Ω):=inf{∫Ω‍|∇u|2:u∈𝒞c∞(Ω),u∣E≥1}.
First, let us see that Maźja’s capacity condition is very much intrinsic on (1). Let g be a positive weight satisfying (1), then for any compact subset Ω,
(7)∫E‍g(x)dx≤∫Ω‍gu2≤C∫Ω‍|∇u|2,u∈𝒞c∞(Ω),u∣E≥1.
By taking the infimum, we get ∫E‍g(x)dx≤Ccap(E;Ω). Therefore
(8)supE⊂⊂Ω∫E‍g(x)dxcap(E;Ω)≤C.
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 forN≥3 is obtained by Visciglia in [8]. He proved that (1) holds for weights in the Lorentz space L(N/2,∞). The embedding of H01(Ω) into the Lorentz space L(2*,2) played a key role in his result. The case N=2 is more subtle, for example, (1) does not hold when Ω=ℝ2 and ∫ℝ2‍g>0, see [9]. In this paper we obtain a verifiable condition for admissible functions for bounded domains in ℝ2, using one-dimensional weighted Hardy inequalities and certain rearrangement inequalities.A general one dimensional weighted Hardy inequality has the following form:(9)(∫ab‍[∫0x‍f(s)ds]qu(x)dx)1/q≤Cp,q(∫0∞‍f(x)pv(x)dx)1/p,f≥0.
For an excellent review on the weighted Hardy inequalities, we refer to [10] by Maligranda et al. Many necessary and sufficient conditions on u, v, p, q for holding (9) are available in the literature, see [11–13]. Here we make use of the so called Muckenhoupt condition [13] for obtaining a class of weight functions satisfying (1). For a measurable function g, we denote its decreasing rearrangement by g* and the maximal function of g* is denoted by g**, that is, g**(t)=(1/t)∫0t‍g*(s)ds. Now we define
(10)ℳlogL:={gmeasurable:sup0<t<|Ω|tlog(|Ω|t)g**(t)<∞}.
Indeed, ℳlogL is a rearrangement invariant Banach function space with the norm
(11)∥g∥ℳlogL=sup0<t<|Ω|tlog(|Ω|t)g**(t),
see [14]. More details of the space ℳlogL are given in Section 3. Now we state our main theorem.Theorem 1.
LetΩ be a bounded domain in ℝ2 and let g∈ℳlogL. Then g is admissible and
(12)∫Ω‍gu2≤∥g∥ℳlogLπ∫Ω‍|∇u|2,∀u∈H01(Ω).Having obtainedℳlogL, a class of admissible functions, one would like to know for which among them the best constant in (1) is attained in H01(Ω). Many authors have addressed this question when N≥3, see [8, 15] and the references therein. For N=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 ℳlogL so that the best constant in (1) is attained in H01(Ω). For a bounded domain Ω⊂ℝ2 (analogous to space ℱN/2 in [15]) we define
(13)ℱ1:=𝒞c∞(Ω)¯inℳlogL.
We show that the best constant in (1) is attained in H01(Ω), when g+∈ℱ1, where g+ is the positive part of g. More precisely, we have the following theorem.Theorem 2.
Letg∈ℳlogL and g+∈ℱ1∖{0}. Define
(14)λ1(g)=inf{∫Ω‍|∇u|2∫Ω‍gu2:u∈H01(Ω),∫Ω‍gu2>0}.
Then λ1(g) is attained for some u∈H01(Ω).The Moser-Trudinger embedding ofH01(Ω) into the Orlicz space LA(Ω), the Orlicz space generated by the Orlicz function A(t)=et2-1, can be used to show that LlogL functions are admissible. An embedding of H01(Ω), 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 LlogL. In [18], the authors used this finer embedding and showed that the Lorentz-Zygmund space L1,∞(logL)2 is admissible. We are not going to use any embeddings for proving that ℳlogL functions are admissible, instead we use some rearrangement inequalities and one dimensional weighted Hardy inequalities. We would like to stress that the admissibility of ℳlogL 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 Section2, 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 ℳlogL and give examples of function spaces contained in ℳlogL 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 [19–21].LetΩ⊂ℝN be a domain. Let Ef(s)={x:|f(x)|>s},s>0.Then thedistribution functionαf of f is defined as
(15)αf(s):=|Ef(s)|,fors>0,
where |A| denotes the Lebesgue measure of a set A⊂ℝN. Now we define the decreasing rearrangementf* as
(16)f*(t):=inf{s>0:αf(s)≤t},t>0.
The  Schwarz symmetrization of f is given by
(17)f⋆(x)=f*(ωN|x|N),∀x∈Ω⋆,
where ωN is the measure of unit ball in ℝN 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.Proposition 3.
LetΩ⊂ℝN be a domain and let f and g be measurable functions on Ω. Then,(a)
f*(αf(s))≤s, αf(f*(t))≤t;(b)
t<αf(s)if and only ifs<f*(t).The mapf→f* is not subadditive. However, we obtain a subadditive function from f*, namely, the maximal function f** of f* defined by
(18)f**(t)=1t∫0t‍f*(τ)dτ,t>0.
The subadditivity of f** with respect to f 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.
LetΩ be a domain in ℝN with N≥2. Let fand g be two measurable functions on Ω and let u∈H01(Ω). Then one has the following inequalities.(a)
The Hardy-Littlewood inequality:(19)∫Ω‍|f(x)g(x)|dx≤∫Ω⋆‍f⋆(x)g⋆(x)dx=∫0|Ω|‍f*(t)g*(t).(b)
The Polya-Szegö inequality:(20)N2ωN2/N∫0|Ω|‍t2-2/N[u*′(t)]2dt=∫Ω⋆‍|∇u⋆(x)|2dx≤∫Ω‍|∇u(x)|2dx.Next we state a necessary and sufficient condition for one dimensional weighted Hardy inequality due to Muckenhoupt (see 4.17, [10]).Proposition 5.
Letu and v be nonnegative measurable functions such that v>0. Let 1<p<∞ and let p′ be the conjugate exponent of p. Then for any a>0,
(21)∫0a‍|∫ta‍f(s)ds|pu(t)dt≤C∫0a‍|f(s)|pv(s)ds,
holds for all measurable function f if and only if
(22)A=sup0<t<a(∫0t‍u(s)ds)1/p(∫ta‍v(s)1-p′ds)1/p′<∞.
Moreover, if Cb is the best constant in (21), then
(23)A≤Cb≤p1/p[p′]1/p′A.


## 3. A Space for Admissible Functions

In this section, we define the function spaceℳlogL and give its relation with certain classical function spaces. For a bounded domain Ω⊂ℝ2, we define
(24)ℳlogL:={umeasurable:sup0<t<|Ω|u**(t)tlog(|Ω|t)<∞}.
One can verify that ℳlogL is a Banach function space with the norm
(25)∥u∥ℳlogL:=sup0<t<|Ω|u**(t)tlog(|Ω|t).
The nomenclature refers to the fact that ℳlogL is the maximal rearrangement invariant Banach function space (Lorentz M-space) on Ω with the fundamental function tlog(|Ω|/t). Next we give some examples of function spaces in ℳlogL.Recall, for a bounded domainΩ⊂ℝ2,LlogL is the Orlicz space generated by the Orlicz function tlog+(t), that is,
(26)LlogL:={umeasurable:∫Ω‍u(x)log+(u(x))dx<∞}.
A similar analysis as in Lemma 6.2 of [14] gives
(27)LlogL={umeasurable:∫0|Ω|‍u*(t)log(|Ω|t)dt<∞}.
Using this equivalent definition, we show that LlogL is contained in ℳlogL.Proposition 6.
LetΩ be a bounded domain in ℝ2. Then LlogL⊂ℳlogL.Proof.
Letf∈LlogL, then using the definition of f** and the monotonicity of log(|Ω|/t) we obtain
(28)f**(t)tlog(|Ω|t)=log(|Ω|t)∫0t‍f*(s)ds≤∫0t‍log(|Ω|s)f*(s)ds.
Now by taking the supremum over t∈(0,|Ω|) in the previous inequality, we obtain the desired fact.This inclusion is strict as seen in the following example.Example 7.
LetΩ=B(0;1)⊂ℝ2. For δ small, let
(29)g(x)=χB(0;δ)(x)|x|2(log(x))2(log|log(x)|)β.
Then for 0<β≤1, g does not belong to LlogL but belongs to ℳlogL.For a bounded domainΩ⊂ℝ2, the Zygmund space L1,∞(logL)2 is defined as
(30)L1,∞(logL)2:={fmeasurable:sup0<t≤|Ω|t[log(e|Ω|t)]2f*(t)<∞}.
From the following proposition we see that L1,∞(logL)2 is contained in ℳlogL.Proposition 8.
LetΩ be a bounded domain in ℝ2. Then for a measurable function f, the following inequality holds:
(31)f**(t)tlog(|Ω|t)≤sup0<s≤tf*(s)s[log(|Ω|s)]2,t∈(0,|Ω|).Proof.
Lett∈(0,|Ω|). Then
(32)f**(t)tlog(|Ω|t)=log(|Ω|t)∫0t‍f*(s)ds=log(|Ω|t)∫0t‍f*(s)s×[log(|Ω|s)]21s[log(|Ω|/s)]2ds≤sup0<s≤tf*(s)s[log(|Ω|s)]2,
and the last inequality follows as log(|Ω|/t)∫0t‍(1/(s[log(|Ω|/s)]2)ds=1.Remark 9.
In [18], using Hansson’s embedding, the authors showed that L1,∞(logL)2 functions are admissible. Since L1,∞(logL)2⊂ℳlogL, their result also follows from Theorem 1 without using Hansson’s embedding. Since ℳlogL functions are admissible, Theorem 2.5 of [22] shows that the spaces L1,∞(logL)2 and ℳlogL 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 L1,∞(logL)2 and hence belong to ℳlogL as well.Proposition 10.
LetΩ be a bounded domain in ℝ2 and let R≥supx∈Ω|x|e. Then for γ≥2,
(33)1|x|2[log(R/|x|)]γ∈L1,∞(logL)2.Proof.
Letg(x)={|x|2[log(R/|x|)]γ}-1,x∈B(0;R), and f(x)=g(x)χΩ(x). Note that f(x)≤g(x), x∈B(0;R), and hence
(34)f*(t)≤g*(t),t∈(0,R).
A straightforward calculation gives
(35)g*(t)=π{t[log(R2πt)]γ}-1=2γπ{t[log(R2πt)]γ}-1≤2γπ{t[log(e|Ω|t)]γ}-1,
where the last inequality follows since R≥supx∈Ω|x|e. Therefore
(36)t[log(e|Ω|t)]2g*(t)≤2γπ[log(e|Ω|t)]2-γ.
Hence the proof.Next we verify that weights inℳlogL satisfy Maźja’s capacity condition.Proposition 11.
LetΩ be a bounded domain in ℝ2, g∈ℳlogL, and g≥0. Then g satisfies Maźja’s capacity condition, that is,
(37)supK⊂⊂Ω∫K‍gdxcap(K,Ω)<∞.Proof.
Using Polya-Szegö inequality, we can easily verify thatcap(K⋆,Ω⋆)≤cap(K,Ω). Further, cap(K⋆,Ω⋆)=(π/2)log(|Ω|/|K|)-1 (see, e.g., page 106 of [7]). Now for a compact set K,
(38)∫K‍g(x)dxcap(K,Ω)≤2πlog(|Ω||K|)∫K⋆‍g⋆(x)dx≤2πlog(|Ω||K|)∫0|K|‍g*(s)ds.
Therefore,
(39)supK⊂⊂Ω∫K‍g(x)dxcap(K,Ω)≤2πsup0<t<|Ω|log(|Ω|t)∫0t‍g*(s)ds=2π∥g∥ℳlogL<∞.In the next section, we see that the spaceℳlogL 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, whenΩ=B(0;r) for some r>0, we show that all radial and radially decreasing admissible weights necessarily lie in ℳlogL. First we have the following theorem on one dimensional weighted Hardy inequalities.Theorem 12.
LetΩ be a bounded domain in ℝ2 and let g∈ℳlogL. Then
(40)∫0|Ω|‍(∫t|Ω|‍f(s)ds)2g*(t)dt≤4∥g∥ℳlogL∫0|Ω|‍s[f(s)]2ds,f≥0.The previous inequality is known for more general weights (even for measures), see [11–13]. Note that when g∈ℳlogL, g* 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 leta=|Ω| and ∥g∥=∥g∥ℳlogL. Since g∈ℳlogL, we have
(41)log(at)(∫0t‍g*(τ)dτ)≤∥g∥,∀t∈(0,a).
Now using the Hölder inequality we obtain
(42)(∫ta‍f(s)ds)2=(∫ta‍f(s)s[log(as)]1/41s[log(a/s)]1/4ds)2≤∫ta‍[f(s)]2s[log(as)]1/2ds×∫ta‍1s[log(a/s)]1/2ds=2[log(at)]1/2∫ta‍[f(s)]2s[log(as)]1/2ds.
Therefore,
(43)∫0a‍(∫ta‍f(s)ds)2g*(t)dt≤2∫0a‍[∫ta‍[f(s)]2s[log(as)]1/2ds]×[log(at)]1/2g*(t)dt=2∫0a‍[∫0s‍[log(at)]1/2g*(t)dt][f(s)]2×s[log(as)]1/2ds,
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):
(44)∫0s‍[log(at)]1/2g*(t)dt≤∥g∥∫0s‍(∫0t‍g*(τ)dτ)(-1)/2g*(t)dt=2∥g∥∫0s‍ddt(∫0t‍g*(τ)dτ)1/2dt=2∥g∥(∫0s‍g*(τ)dτ)1/2≤2∥g∥[log(a/s)]1/2.
(using (41)).
Now by substituting back into (43), we get the result.Proof of Theorem1.
Letg∈ℳlogL. From the inequalities given in Proposition 4, it is clear that the inequality
(45)∫Ω‍gu2≤C∫Ω‍|∇u|2,∀u∈H01(Ω)
holds, if
(46)∫0|Ω|‍g*(t)u*(t)2≤4πC∫0|Ω|‍t[u*′(t)]2dt,∀u∈H01(Ω).
Since u*(|Ω|)=0, we can rewrite the previous inequality as
(47)∫0|Ω|‍(∫t|Ω|‍[-u*′(s)]ds)2g*(t)≤4πC∫0|Ω|‍t[-u*′(t)]2dt.
Now by Theorem 12, we see that the previous inequality holds with C=∥g∥ℳlogL/π.In the following theorem, we show that our condition is almost necessary for the generalized Hardy-Sobolev inequality.Theorem 13.
LetΩ=B(0;R)⊂ℝ2 and let g∈L1(Ω) be such that g is positive, radial, and radially decreasing. If g is admissible, then g∈ℳlogL.Proof.
Letg be admissible and let C>0 be such that
(48)∫Ω‍gu2≤C∫Ω‍|∇u|2,∀u∈H01(Ω).
We use certain test functions in H01(Ω) to estimate ∥g∥ℳlogL. For r>0, let
(49)ur(x)={log(Rr),|x|≤r,log(R|x|),|x|≥r.
Clearly ur∈H01(Ω) and
(50)∇ur(x)={0,|x|≤r,-x|x|2,|x|≥r.
Therefore
(51)∫Ω‍|∇ur|2=2π∫rR‍1t2tdt=2πlog(Rr).
Further, since g is radial and radially decreasing, we get
(52)∫Ω‍gur2≥[log(Rr)]2∫B(0;r)‍g(x)dx=[log(Rr)]2∫0πr2‍g*(t)dt.
Now (48), (51), and (52) yield
(53)∫0πr2‍g*(s)ds≤2πC1log(R/r).
Thus by substituting t=πr2 in the previous inequality and noting that 2log(R/r)=log(|Ω|/πr2), we get
(54)tlog(|Ω|t)g**(t)≤4πC,∀t∈(0,|Ω|).
Hence g∈ℳlogL and ∥g∥ℳlogL≤4πC.Next we see how one can obtain Moser-Trudinger embedding and Hansson’s embedding using Theorem1.Remark 14 (an alternate proof for some classical embeddings).
From Theorem1, for each g∈ℳlogL, we have the generalized Hardy-Sobolev inequality
(55)∫Ω‍gu2≤∥g∥ℳlogLπ∫Ω‍|∇u|2,∀u∈H01(Ω).(i)
Moser-Trudinger embedding: sinceLlogL⊂ℳlogL, there exists C1>0 such that
(56)∥g∥ℳlogL≤C1π∥g∥LlogL.
Thus from (55) we have
(57)∫Ω‍gu2≤C1∥g∥LlogL∫Ω‍|∇u|2,∀u∈H01(Ω).
The previous inequality shows that for each u∈H01(Ω),u2 is a continuous linear functional on LlogL with ∥u2∥[LlogL]*≤C1∥∇u∥22. In other words, u2∈Lexp (the dual space of LlogL) and
(58)∫Ω‍e(u2/C1∥∇u∥22)-1≤1.
In particular, for each u∈H01(Ω),u∈Lexp2 and also
(59)∥u∥Lexp2≤C1∥∇u∥2,∀u∈H01(Ω).(ii)
Hansson’s embedding: sinceL1,∞(logL)2⊂ℳlogL, there exists C2>0 such that
(60)∥g∥ℳlogL≤C2π∥g∥L1,∞(logL)2.
As before, for each u∈H01(Ω),u2∈[L1,∞(logL)2]*(=L∞,1(logL)-2) with ∥u2∥[L1,∞(logL)2]*≤C2∥∇u∥22,  that is,
(61)∫0|Ω|‍[u*(t)]2t[log(e|Ω|/t)]2≤C2∥∇u∥22,∀u∈H01(Ω).
Thus for u∈H01(Ω),u∈L∞,2(logL)-1 and
(62)∥u∥L∞,2(logL)-1≤C2∥∇u∥2,∀u∈H01(Ω).


## 5. The Best Constant in the Hardy-Sobolev Inequalities

In this section, we give a proof of Theorem2 using a direct variational method. For g∈ℳlogL, we define the following:
(63)ℳ:={u∈H01(Ω):∫Ω‍gu2=1},(64)G+(u):=∫Ω‍g+u2,u∈H01(Ω).
Recall that
(65)λ1(g)=inf{∫Ω‍|∇u|2∫Ω‍gu2:u∈H01(Ω),∫Ω‍gu2>0}.
It is easy to see that λ1(g)=inf{J(u):u∈ℳ}, where J(u)=∫Ω‍|∇u|2. Note that, for an admissible g,λ1(g)>0 and 1/λ1(g) is the best constant in (1). Thus the best constant in (1) is attained for some u∈H01(Ω) if J has a minimizer on ℳ. We show that, under the assumptions of Theorem 2, J admits a minimizer on ℳ.First we prove the following compactness theorem.Lemma 15.
LetΩ be a bounded domain in ℝ2 and let h∈ℱ1. Then the map
(66)H(u)=∫Ω‍hu2,u∈H01(Ω)
is compact.Proof.
Forun⇀u we show that H(un)→H(u). First we estimate the following:
(67)|H(un)-H(u)|≤∫Ω‍|h||un2-u2|≤∫Ω‍(|h||un+u|)(|h||un-u|)≤(∫Ω‍|h|(un+u)2)1/2×(∫Ω‍|h|(un-u)2)1/2.
Since un is bounded, from (55) we get that (∫Ω‍|h|(un+u)2)1/2 is bounded. Thus H(un)→H(u) that if ∫Ω‍|h|(un-u)2→0. Let m=supn∥∇(un-u)∥2. For a given ε>0, since |h|∈ℱ1, we choose hε∈𝒞c∞(Ω) such that
(68)∥|h|-hε∥ℳlogL<εm.
Therefore
(69)∫Ω‍(|h|-hε)(un-u)2≤∥|h|-hε∥ℳlogL∫Ω‍|∇(un-u)|2<ε.
Now
(70)∫Ω‍|h|(un-u)2=∫Ω‍hε(un-u)2+∫Ω‍(|h|-hε)(un-u)2≤∫Ω‍hε(un-u)2+ε.
For sufficiently large n, the first integral can be made smaller than ε since H01(Ω) is embedded compactly in L2(Ω) and hε∈𝒞c∞(Ω). Thus we conclude that ∫Ω‍|h|(un-u)2→0.Proof of Theorem2.
Sinceg∈Lloc1(Ω) and g+≠0, there exists φ∈Cc∞(Ω) such that ∫Ω‍gφ2>0 (see for e.g., Proposition 4.2 of [23]) and and hence the set ℳ is nonempty. Let {un} be a minimizing sequence of J on ℳ, that is,
(71)limn→∞J(un)=λ1(g)=infu∈ℳJ(u).
Using the coercivity of J, the reflexivity of H01(Ω), and the compactness of the embedding of H01(Ω) into L2(Ω), we can further assume that {un} converges weakly and almost everywhere to some u∈H01(Ω). Since g+∈ℱ1, from the previous lemma we get
(72)limn→∞∫Ω‍g+un2=∫Ω‍g+u2.
For un∈ℳ, we write
(73)∫Ω‍g-un2=∫Ω‍g+un2-1.
Now use the Fatou’s lemma to obtain
(74)∫Ω‍g-u2≤∫Ω‍g+u2-1.
Thus we get ∫Ω‍gu2≥1. Set u~:=u/(∫Ω‍gu2)1/2. Now the homogeneity and the weak lower semicontinuity of J yields the following:
(75)λ1(g)≤J(u~)=J(u)∫Ω‍gu2≤J(u)≤liminfnJ(un)=λ1(g).
Thus equality must hold at each step and hence ∫Ω‍gu2=1. This shows that u∈ℳ and J(u)=λ1(g).Next we give an equivalent definition for the spaceℱ1. Recall that ℱ1 is the closure of 𝒞c∞(Ω) in ℳlogL.Theorem 16.
LetΩ be a bounded domain in ℝ2. Then the following statements are equivalent: (i)
g∈ℱ1,(ii)
g∈ℳlogL and limt→0g**(t)tlog(|Ω|/t)=0.Proof.
(
i
)
⇒
(
i
i
) We show that limt→0g**(t)tlog(|Ω|/t)=0. Let gn∈𝒞c∞(Ω) and gn→g in ℳlogL. Note that for each  gn,gn**(t) is bounded and hence
(76)limt→0gn**(t)tlog(|Ω|t)=0,∀n∈ℕ.
Let ε>0 be given. We show that there exists δ>0 such that g**(t)tlog(|Ω|/t)<ε, for all t∈(0,δ). Since gn→g in ℳlogL, there exists n0∈ℕ such that
(77)sup0<t<|Ω|(g-gn0)**(t)tlog(|Ω|t)<ε.
Using the subadditivity of the maximal operator f→f**, we get
(78)g**(t)tlog(|Ω|t)≤(g-gn0)**(t)tlog(|Ω|t)+gn0**(t)tlog(|Ω|t)<ε+gn0**(t)tlog(|Ω|t).
Note that from (76), there exists δ>0 such that gn0**(t)tlog(|Ω|/t)<ε, for all t∈(0,δ). Now (78) yields the result as
(79)g**(t)tlog(|Ω|t)<2ε,∀t∈(0,δ).(ii)⇒(i). Let ε>0 be given. For proving g∈ℱ1, we show that there exists g~ε∈𝒞c∞(Ω) such that ∥g-g~ε∥ℳlogL<2ε. Since limt→0g**(t)tlog(|Ω|/t)=0, we get δ>0 such that
(80)g**(t)tlog(|Ω|t)<ε,∀t∈(0,δ].
Let
(81)Aε:={x∈Ω:|g(x)|<g*(δ)},gε=gχAε.
Clearly gε∈L2(Ω). Since L2(Ω) is continuously embedded in ℳlogL and 𝒞c∞(Ω) is dense in L2(Ω), we can choose  g~ε∈𝒞c∞(Ω) so that
(82)∥gε-g~ε∥ℳlogL<ε.
Next we estimate ∥g-gε∥ℳlogL. Let us compute the distribution function αg-gε:
(83)αg-gε(s)=αgχAεc(s)={αg(s)s≥g*(δ),αg(g*(δ))s<g*(δ).
For computing the symmetric rearrangement (g-gε)*(t), we set δ1=αg-gε(g*(δ))=αg(g*(δ)). If δ1=0,αg-gε(s)=0, for all s>0, we get the desired result as ∥g-gε∥ℳlogL=0. Next we calculate (g-gε)*(t), when δ1>0. For 0<t<δ1, note that αg-gε(s)≤t implies that s≥g*(δ) (if s<g*(δ), then δ1=αg-gε(g*(δ))≤αg-gε(s)≤t, a contradiction). Thus for t<δ1, using (83) we have
(84)(g-gε)*(t)=inf{s:αg-gε(s)≤t}=inf{s≥g*(δ):αg-gε(s)≤t}=inf{s:αg(s)≤t}=g*(t).
Now for t≥δ1, again using (83), we see that αg-gε(s)≤αg(g*(δ))=δ1≤t,  for all s<g*(δ) and hence (g-gε)*(t)=0,  for all t≥δ1. Thus
(85)(g-gε)**(t)={g**(t)t<δ1,δ1tg**(δ1)t≥δ1.
Therefore,
(86)(g-gε)**(t)tlog(|Ω|t)≤{g**(t)tlog(|Ω|t)t<δ1,g**(δ1)δ1log(|Ω|δ1)t≥δ1.
Since δ1≤δ, from (80), we have ∥g-gε∥ℳlogL<ε. Now (82) yields ∥g-g~ε∥ℳlogL<2ε and hence the proof is done.Next we give example of functions inℱ1.Example 17.
We have already seen thatLlogL⊂ℳlogL (Proposition 6). In fact, LlogL⊂ℱ1 since 𝒞c∞(Ω) is dense in LlogL and the inclusion of LlogL in ℳlogL is continuous. Also, using Theorem 16, one can verify that 1/|x|2[log(R/|x|)]γ∈ℱ1, for γ>2.Remark 18 (an open problem).
Whether all the admissible functions are inℳlogL or not, by setting λ1(g⋆)=inf{(∫Ω‍|∇u⋆|2/∫Ω‍g⋆u⋆2):u∈H01(Ω),∫Ω‍gu2>0}, the problem can be rephrased as whether λ1(g)>0 implies λ1(g⋆)>0 or not.Remark 19.
ForN≥3 and Ω⊂ℝN (not necessarily bounded), using the Muckenhoupt condition (22) and the inequalities given in Proposition 4, one can show that (as in Theorem 1) g is admissible if g satisfies
(87)sup0<t<|Ω|t2/Ng**(t)<∞.
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 𝒟01,2(Ω) into the Lorentz space L(2*,2) using similar arguments as in Remark 14.



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*Source: 784398-2013-01-08.xml*