A Note on Generalized Hardy-Sobolev Inequalities

Abstract

We are concerned with finding a class of weight functions 𝑔 so that the following generalized Hardy-Sobolev inequality holds: ∫_Ω𝑔𝑢²≤𝐶∫_Ω|∇𝑢|²,𝑢∈𝐻¹₀(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.

1. Introduction

In this paper, we are interested to find the best suitable function space for the weights 𝑔 so that the following generalized Hardy-Sobolev inequality holds: _Ω𝑔𝑢²≤𝐶_Ω|∇𝑢|²,𝑢∈𝐻¹₀(Ω), (1) for some 𝐶>0, where Ω is a bounded domain in ℝ^𝑁. 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∈𝐻¹₀(Ω).

Let us first recall the classical Hardy inequality: ^∞₀1𝑥^𝑥₀𝑓(𝑠)𝑑𝑠²𝑑𝑥≤𝐶₂^∞₀𝑓(𝑥)²𝑑𝑥,𝑓≥0. (2) By taking 𝑓=𝑢^′ with 𝑢(0)=0 in (2), we get ^∞₀1𝑥²𝑢(𝑥)²𝑑𝑥≤𝐶₂^∞₀𝑢^′(𝑥)²𝑑𝑥. (3) The higher dimensional analogue of the previous inequality is referred to as the Hardy-Sobolev inequality in the literature: _Ω1|𝑥|²𝑢(𝑥)²𝑑𝑥≤𝐶_2Ω|∇𝑢(𝑥)|²𝑑𝑥,𝑢∈𝐻¹₀(Ω), (4) where Ω⊂ℝ^𝑁 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.

For 𝑁≥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 𝐻¹₀(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: 𝑁−22²_Ω1|𝑥|²𝑢(𝑥)²𝑑𝑥+𝜆_ΩΩ𝑢(𝑥)²≤𝑑𝑥_Ω|∇𝑢(𝑥)|²𝑑𝑥,𝑢∈𝐻¹₀(Ω). (5) Motivated by the previous inequality, several improved Hardy-Sobolev inequalities have been proved, for example see [2–5].

For 𝑁=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 cap(𝐸;Ω)∶=inf_Ω|∇𝑢|²∶𝑢∈𝒞^∞_𝑐(Ω),𝑢∣_𝐸≥1. (6) 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, _𝐸𝑔(𝑥)𝑑𝑥≤_Ω𝑔𝑢²≤𝐶_Ω|∇𝑢|²,𝑢∈𝒞^∞_𝑐(Ω),𝑢∣_𝐸≥1. (7) By taking the infimum, we get ∫_𝐸𝑔(𝑥)𝑑𝑥≤𝐶cap(𝐸;Ω). Therefore sup_𝐸⊂⊂Ω∫_𝐸𝑔(𝑥)𝑑𝑥cap(𝐸;Ω)≤𝐶. (8) 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 𝑁≥3 is obtained by Visciglia in [8]. He proved that (1) holds for weights in the Lorentz space 𝐿(𝑁/2,∞). The embedding of 𝐻¹₀(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: ^𝑏_𝑎^𝑥₀𝑓(𝑠)𝑑𝑠^𝑞𝑢(𝑥)𝑑𝑥^1/𝑞≤𝐶_𝑝,𝑞^∞₀𝑓(𝑥)^𝑝𝑣(𝑥)𝑑𝑥^1/𝑝,𝑓≥0. (9) 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)∫^𝑡₀𝑔^∗(𝑠)𝑑𝑠. Now we define ℳlog𝐿∶=𝑔measurable∶sup_|Ω|0<𝑡<|Ω|𝑡log𝑡𝑔^∗∗(𝑡)<∞. (10) Indeed, ℳlog𝐿 is a rearrangement invariant Banach function space with the norm 𝑔_ℳlog𝐿=sup_|Ω|0<𝑡<|Ω|𝑡log𝑡𝑔^∗∗(𝑡), (11) see [14]. More details of the space ℳlog𝐿 are given in Section 3. Now we state our main theorem.

Theorem 1. Let Ω be a bounded domain in ℝ² and let 𝑔∈ℳlog𝐿. Then 𝑔 is admissible and _Ω𝑔𝑢²≤𝑔_ℳlog𝐿𝜋_Ω|∇𝑢|²,∀𝑢∈𝐻¹₀(Ω). (12)

Having obtained ℳlog𝐿, a class of admissible functions, one would like to know for which among them the best constant in (1) is attained in 𝐻¹₀(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 𝐻¹₀(Ω). For a bounded domain Ω⊂ℝ² (analogous to space ℱ_𝑁/2 in [15]) we define ℱ₁∶=𝒞^∞_𝑐(Ω)inℳlog𝐿. (13) We show that the best constant in (1) is attained in 𝐻¹₀(Ω), when 𝑔⁺∈ℱ₁, where 𝑔⁺ is the positive part of 𝑔. More precisely, we have the following theorem.

Theorem 2. Let 𝑔∈ℳlog𝐿 and 𝑔⁺∈ℱ₁⧵{0}. Define 𝜆₁𝑔∫=inf^Ω|∇𝑢|²∫_Ω𝑔𝑢²∶𝑢∈𝐻¹₀(Ω),_Ω𝑔𝑢²>0. (14) Then 𝜆₁(𝑔) is attained for some 𝑢∈𝐻¹₀(Ω).

The Moser-Trudinger embedding of 𝐻¹₀(Ω) into the Orlicz space 𝐿_𝐴(Ω), the Orlicz space generated by the Orlicz function 𝐴(𝑡)=𝑒^𝑡2−1, can be used to show that 𝐿log𝐿 functions are admissible. An embedding of 𝐻¹₀(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 𝐿^1,∞(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 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 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].

Let Ω⊂ℝ^𝑁 be a domain. Let 𝐸_𝑓(𝑠)={𝑥∶|𝑓(𝑥)|>𝑠},𝑠>0.

Then the distribution function 𝛼_𝑓 of 𝑓 is defined as 𝛼_𝑓(𝑠)∶=𝐸_𝑓(𝑠),for𝑠>0, (15) where |𝐴| denotes the Lebesgue measure of a set 𝐴⊂ℝ^𝑁. Now we define the decreasing rearrangement 𝑓^∗ as 𝑓^∗(𝑡)∶=inf𝑠>0∶𝛼_𝑓(𝑠)≤𝑡,𝑡>0. (16) The Schwarz symmetrization of 𝑓 is given by 𝑓^⋆(𝑥)=𝑓^∗𝜔_𝑁|𝑥|^𝑁,∀𝑥∈Ω^⋆, (17) where 𝜔_𝑁 is the measure of unit ball in ℝ^𝑁 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. Let Ω⊂ℝ^𝑁 be a domain and let 𝑓 and 𝑔 be measurable functions on Ω. Then,(a)𝑓^∗(𝛼_𝑓(𝑠))≤𝑠, 𝛼_𝑓(𝑓^∗(𝑡))≤𝑡;(b)𝑡<𝛼_𝑓(𝑠)if and only if𝑠<𝑓^∗(𝑡).

The map 𝑓→𝑓^∗ is not subadditive. However, we obtain a subadditive function from 𝑓^∗, namely, the maximal function 𝑓^∗∗ of 𝑓^∗ defined by 𝑓^∗∗(𝑡)=1𝑡^𝑡₀𝑓^∗(𝜏)𝑑𝜏,𝑡>0. (18) 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. Let Ω be a domain in ℝ^𝑁 with 𝑁≥2. Let 𝑓and 𝑔 be two measurable functions on Ω and let 𝑢∈𝐻¹₀(Ω). Then one has the following inequalities.(a)The Hardy-Littlewood inequality: _Ω𝑓(𝑥)𝑔(𝑥)𝑑𝑥≤_Ω⋆𝑓^⋆(𝑥)𝑔^⋆(𝑥)=𝑑𝑥_0|Ω|𝑓^∗(𝑡)𝑔^∗(𝑡). (19)(b)The Polya-Szegö inequality: 𝑁²𝜔_𝑁^2/𝑁₀^|Ω|𝑡^2−2/𝑁𝑢^∗′(𝑡)²𝑑𝑡=_Ω⋆∇𝑢^⋆(𝑥)²≤𝑑𝑥_Ω|∇𝑢(𝑥)|²𝑑𝑥. (20)

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

Proposition 5. Let 𝑢 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, ^𝑎₀^𝑎_𝑡𝑓(𝑠)𝑑𝑠^𝑝𝑢(𝑡)𝑑𝑡≤𝐶^𝑎₀𝑓(𝑠)^𝑝𝑣(𝑠)𝑑𝑠, (21) holds for all measurable function 𝑓 if and only if 𝐴=sup_0<𝑡<𝑎^𝑡₀𝑢(𝑠)𝑑𝑠^1/𝑝𝑎_𝑡𝑣(𝑠)^1−𝑝′𝑑𝑠^1/𝑝′<∞. (22) Moreover, if 𝐶_𝑏 is the best constant in (21), then 𝐴≤𝐶_𝑏≤𝑝^1/𝑝𝑝^′1/𝑝′𝐴. (23)

3. A Space for Admissible Functions

In this section, we define the function space ℳlog𝐿 and give its relation with certain classical function spaces. For a bounded domain Ω⊂ℝ², we define ℳlog𝐿∶=𝑢measurable∶sup_|Ω|0<𝑡<𝑢^∗∗(𝑡)|Ω|𝑡log𝑡<∞. (24) One can verify that ℳlog𝐿 is a Banach function space with the norm ‖𝑢‖_ℳlog𝐿∶=sup_|Ω|0<𝑡<𝑢^∗∗(𝑡)|Ω|𝑡log𝑡. (25) 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 domain Ω⊂ℝ²,𝐿log𝐿 is the Orlicz space generated by the Orlicz function 𝑡log⁺(𝑡), that is, 𝐿log𝐿∶=𝑢measurable∶_Ω𝑢(𝑥)log⁺(𝑢(𝑥))𝑑𝑥<∞. (26) A similar analysis as in Lemma 6.2 of [14] gives =𝐿log𝐿𝑢measurable∶₀^|Ω|𝑢^∗(𝑡)|Ω|log𝑡𝑑𝑡<∞. (27) Using this equivalent definition, we show that 𝐿log𝐿 is contained in ℳlog𝐿.

Proposition 6. Let Ω be a bounded domain in ℝ². Then 𝐿log𝐿⊂ℳlog𝐿.

Proof. Let 𝑓∈𝐿log𝐿, then using the definition of 𝑓^∗∗ and the monotonicity of log(|Ω|/𝑡) we obtain 𝑓^∗∗(𝑡)|Ω|𝑡log𝑡|Ω|=log𝑡^𝑡₀𝑓^∗(𝑠)≤𝑑𝑠_𝑡0|Ω|log𝑠𝑓^∗(𝑠)𝑑𝑠. (28) 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. Let Ω=𝐵(0;1)⊂ℝ². For 𝛿 small, let 𝑔(𝑥)=𝜒_𝐵(0;𝛿)(𝑥)|𝑥|²log(𝑥)²loglog(𝑥)^𝛽. (29) Then for 0<𝛽≤1, 𝑔 does not belong to 𝐿log𝐿 but belongs to ℳlog𝐿.

For a bounded domain Ω⊂ℝ², the Zygmund space 𝐿^1,∞(log𝐿)² is defined as 𝐿^1,∞log𝐿²∶=𝑓measurable∶sup_{|Ω|0<𝑡≤}𝑡𝑒|Ω|log𝑡²𝑓^∗(𝑡)<∞. (30) From the following proposition we see that 𝐿^1,∞(log𝐿)² is contained in ℳlog𝐿.

Proposition 8. Let Ω be a bounded domain in ℝ². Then for a measurable function 𝑓, the following inequality holds: 𝑓^∗∗(𝑡)|Ω|𝑡log𝑡≤sup_{0<𝑠≤𝑡}𝑓^∗(𝑠)𝑠|Ω|log𝑠²,𝑡∈(0,|Ω|). (31)

Proof. Let 𝑡∈(0,|Ω|). Then 𝑓^∗∗(𝑡)|Ω|𝑡log𝑡|Ω|=log𝑡^𝑡₀𝑓^∗(𝑠)|Ω|𝑑𝑠=log𝑡^𝑡₀𝑓^∗(𝑠)𝑠×|Ω|log𝑠²1𝑠log(|Ω|/𝑠)²𝑑𝑠≤sup_{0<𝑠≤𝑡}𝑓^∗(𝑠)𝑠|Ω|log𝑠², (32) and the last inequality follows as log(|Ω|/𝑡)∫^𝑡₀(1/(𝑠[log(|Ω|/𝑠)]²)𝑑𝑠=1.

Remark 9. In [18], using Hansson’s embedding, the authors showed that 𝐿^1,∞(log𝐿)² functions are admissible. Since 𝐿^1,∞(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 𝐿^1,∞(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 𝐿^1,∞(log𝐿)² and hence belong to ℳlog𝐿 as well.

Proposition 10. Let Ω be a bounded domain in ℝ² and let 𝑅≥sup_𝑥∈Ω|𝑥|𝑒. Then for 𝛾≥2, 1|𝑥|²log(𝑅/|𝑥|)^𝛾∈𝐿^1,∞log𝐿². (33)

Proof. Let 𝑔(𝑥)={|𝑥|²[log(𝑅/|𝑥|)]^𝛾}⁻¹,𝑥∈𝐵(0;𝑅), and 𝑓(𝑥)=𝑔(𝑥)𝜒_Ω(𝑥). Note that 𝑓(𝑥)≤𝑔(𝑥), 𝑥∈𝐵(0;𝑅), and hence 𝑓^∗(𝑡)≤𝑔^∗(𝑡),𝑡∈(0,𝑅). (34) A straightforward calculation gives 𝑔^∗(𝑡)=𝜋𝑡log𝑅²𝜋𝑡^𝛾−1=2^𝛾𝜋𝑡𝑅log²𝜋𝑡^𝛾−1≤2^𝛾𝜋𝑡𝑒|Ω|log𝑡^𝛾−1, (35) where the last inequality follows since 𝑅≥sup_𝑥∈Ω|𝑥|𝑒. Therefore 𝑡𝑒|Ω|log𝑡²𝑔^∗(𝑡)≤2^𝛾𝜋𝑒|Ω|log𝑡^2−𝛾. (36) Hence the proof.

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

Proposition 11. Let Ω be a bounded domain in ℝ², 𝑔∈ℳlog𝐿, and 𝑔≥0. Then 𝑔 satisfies Maźja’s capacity condition, that is, sup_𝐾⊂⊂Ω∫_𝐾𝑔𝑑𝑥cap(𝐾,Ω)<∞. (37)

Proof. Using Polya-Szegö inequality, we can easily verify that cap(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, ∫_𝐾𝑔(𝑥)𝑑𝑥()≤2cap𝐾,Ω𝜋|Ω|log|𝐾|_𝐾^⋆𝑔^⋆(𝑥)≤2𝑑𝑥𝜋|Ω|log|𝐾|₀^|𝐾|𝑔^∗(𝑠)𝑑𝑠. (38) Therefore, sup_𝐾⊂⊂Ω∫_𝐾𝑔(𝑥)𝑑𝑥()≤2cap𝐾,Ω𝜋sup_|Ω|0<𝑡<|Ω|log𝑡^𝑡₀𝑔^∗(𝑠)=2𝑑𝑠𝜋𝑔_ℳlog𝐿<∞. (39)

In the next section, we see that the space ℳlog𝐿 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 Ω=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. Let Ω be a bounded domain in ℝ² and let 𝑔∈ℳlog𝐿. Then ₀^|Ω|_𝑡^|Ω|𝑓(𝑠)𝑑𝑠²𝑔^∗(𝑡)𝑔𝑑𝑡≤4_ℳlog𝐿0^|Ω|𝑠𝑓(𝑠)²𝑑𝑠,𝑓≥0. (40)

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 let 𝑎=|Ω| and ‖𝑔‖=‖𝑔‖_ℳlog𝐿. Since 𝑔∈ℳlog𝐿, we have 𝑎log𝑡^𝑡₀𝑔^∗(𝜏)𝑑𝜏≤𝑔,∀𝑡∈(0,𝑎). (41) Now using the Hölder inequality we obtain ^𝑎_𝑡𝑓(𝑠)𝑑𝑠²=^𝑎_𝑡𝑓(𝑠)√𝑠𝑎log𝑠^1/41√𝑠log(𝑎/𝑠)^1/4𝑑𝑠²≤^𝑎_𝑡𝑓(𝑠)²𝑠𝑎log𝑠^1/2×𝑑𝑠_𝑎𝑡1𝑠log(𝑎/𝑠)^1/2𝑎𝑑𝑠=2log𝑡^1/2𝑎_𝑡𝑓(𝑠)²𝑠𝑎log𝑠^1/2𝑑𝑠. (42) Therefore, ^𝑎₀^𝑎_𝑡𝑓(𝑠)𝑑𝑠²𝑔^∗(𝑡)𝑑𝑡≤2^𝑎₀^𝑎_𝑡𝑓(𝑠)²𝑠𝑎log𝑠^1/2×𝑎𝑑𝑠log𝑡^1/2𝑔^∗(𝑡)𝑑𝑡=2^𝑎₀^𝑠₀𝑎log𝑡^1/2𝑔^∗(𝑡)𝑑𝑡𝑓(𝑠)²𝑎×𝑠log𝑠^1/2𝑑𝑠, (43) 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): ^𝑠₀𝑎log𝑡^1/2𝑔^∗(𝑡)≤𝑑𝑡𝑔^𝑠₀^𝑡₀𝑔^∗(𝜏)𝑑𝜏^(−1)/2𝑔^∗(𝑡)𝑑𝑡=2𝑔^𝑠₀𝑑𝑑𝑡^𝑡₀𝑔^∗(𝜏)𝑑𝜏^1/2𝑑𝑡=2𝑔^𝑠₀𝑔^∗(𝜏)𝑑𝜏^1/2≤2𝑔log(𝑎/𝑠)^1/2. (44) (using (41)).
Now by substituting back into (43), we get the result.

Proof of Theorem 1. Let 𝑔∈ℳlog𝐿. From the inequalities given in Proposition 4, it is clear that the inequality _Ω𝑔𝑢²≤𝐶_Ω|∇𝑢|²,∀𝑢∈𝐻¹₀(Ω) (45) holds, if ₀^|Ω|𝑔^∗(𝑡)𝑢^∗(𝑡)²≤4𝜋𝐶₀^|Ω|𝑡𝑢^∗′(𝑡)²𝑑𝑡,∀𝑢∈𝐻¹₀(Ω). (46) Since 𝑢^∗(|Ω|)=0, we can rewrite the previous inequality as ₀^|Ω|_𝑡^|Ω|−𝑢^∗′(𝑠)𝑑𝑠²𝑔^∗(𝑡)≤4𝜋𝐶₀^|Ω|𝑡−𝑢^∗′(𝑡)²𝑑𝑡. (47) 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. Let Ω=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. Let 𝑔 be admissible and let 𝐶>0 be such that _Ω𝑔𝑢²≤𝐶_Ω|∇𝑢|²,∀𝑢∈𝐻¹₀(Ω). (48) We use certain test functions in 𝐻¹₀(Ω) to estimate ‖𝑔‖_ℳlog𝐿. For 𝑟>0, let 𝑢_𝑟(𝑥)=𝑅log𝑟,|𝑥|𝑅≤𝑟,log|𝑥|,|𝑥|≥𝑟. (49) Clearly 𝑢_𝑟∈𝐻¹₀(Ω) and ∇𝑢_𝑟(𝑥)=|𝑥|−𝑥0,≤𝑟,|𝑥|²,|𝑥|≥𝑟. (50) Therefore _Ω∇𝑢_𝑟²=2𝜋^𝑅_𝑟1𝑡²𝑅𝑡𝑑𝑡=2𝜋log𝑟. (51) Further, since 𝑔 is radial and radially decreasing, we get _Ω𝑔𝑢²_𝑟≥𝑅log𝑟²_𝐵(0;𝑟)𝑔(𝑥)=𝑅𝑑𝑥log𝑟^2𝜋𝑟2₀𝑔^∗(𝑡)𝑑𝑡. (52) Now (48), (51), and (52) yield ^𝜋𝑟2₀𝑔^∗(𝑠)1𝑑𝑠≤2𝜋𝐶().log𝑅/𝑟 (53) Thus by substituting 𝑡=𝜋𝑟² in the previous inequality and noting that 2log(𝑅/𝑟)=log(|Ω|/𝜋𝑟²), we get |Ω|𝑡log𝑡𝑔^∗∗(𝑡)≤4𝜋𝐶,∀𝑡∈(0,|Ω|). (54) Hence 𝑔∈ℳlog𝐿 and ‖𝑔‖_ℳlog𝐿≤4𝜋𝐶.

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 𝑔∈ℳlog𝐿, we have the generalized Hardy-Sobolev inequality _Ω𝑔𝑢²≤𝑔_ℳlog𝐿𝜋_Ω|∇𝑢|²,∀𝑢∈𝐻¹₀(Ω). (55)(i)Moser-Trudinger embedding: since 𝐿log𝐿⊂ℳlog𝐿, there exists 𝐶₁>0 such that 𝑔_ℳlog𝐿≤𝐶₁𝜋𝑔_𝐿log𝐿. (56) Thus from (55) we have _Ω𝑔𝑢²≤𝐶₁𝑔_{𝐿log𝐿Ω}|∇𝑢|²,∀𝑢∈𝐻¹₀(Ω). (57) The previous inequality shows that for each 𝑢∈𝐻¹₀(Ω),𝑢² is a continuous linear functional on 𝐿log𝐿 with ‖𝑢²‖_{[𝐿log𝐿]}^∗≤𝐶₁‖∇𝑢‖²₂. In other words, 𝑢²∈𝐿_exp (the dual space of 𝐿log𝐿) and _Ω𝑒^{(𝑢2/𝐶1‖∇𝑢‖22)}−1≤1. (58) In particular, for each 𝑢∈𝐻¹₀(Ω),𝑢∈𝐿_exp2 and also ‖𝑢‖_𝐿2exp≤𝐶₁‖∇𝑢‖₂,∀𝑢∈𝐻¹₀(Ω). (59)(ii)Hansson’s embedding: since 𝐿^1,∞(log𝐿)²⊂ℳlog𝐿, there exists 𝐶₂>0 such that 𝑔_ℳlog𝐿≤𝐶₂𝜋𝑔_{𝐿1,∞(log𝐿)}². (60) As before, for each 𝑢∈𝐻¹₀(Ω),𝑢²∈[𝐿^1,∞(log𝐿)²]^∗(=𝐿^∞,1(log𝐿)⁻²) with ‖𝑢²‖_{[𝐿1,∞(log𝐿)}²_]^∗≤𝐶₂‖∇𝑢‖²₂, that is, ₀^|Ω|𝑢^∗(𝑡)²𝑡log(𝑒|Ω|/𝑡)²≤𝐶₂‖∇𝑢‖²₂,∀𝑢∈𝐻¹₀(Ω). (61) Thus for 𝑢∈𝐻¹₀(Ω),𝑢∈𝐿^∞,2(log𝐿)⁻¹ and ‖𝑢‖_𝐿^∞,2_(log𝐿)⁻¹≤𝐶₂‖∇𝑢‖₂,∀𝑢∈𝐻¹₀(Ω). (62)

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 𝑔∈ℳlog𝐿, we define the following: ℳ∶=𝑢∈𝐻¹₀(Ω)∶_Ω𝑔𝑢²,𝐺=1₊(𝑢)∶=_Ω𝑔⁺𝑢²,𝑢∈𝐻¹₀(Ω). (63)(64) Recall that 𝜆₁𝑔∫=inf^Ω|∇𝑢|²∫_Ω𝑔𝑢²∶𝑢∈𝐻¹₀(Ω),_Ω𝑔𝑢²>0. (65) 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∈𝐻¹₀(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. Let Ω be a bounded domain in ℝ² and let ℎ∈ℱ₁. Then the map 𝐻(𝑢)=_Ωℎ𝑢²,𝑢∈𝐻¹₀(Ω) (66) is compact.

Proof. For 𝑢_𝑛⇀𝑢 we show that 𝐻(𝑢_𝑛)→𝐻(𝑢). First we estimate the following: 𝐻𝑢_𝑛−𝐻(𝑢)≤_Ω|ℎ|𝑢²_𝑛−𝑢²≤_Ω|ℎ|𝑢_𝑛+𝑢|ℎ|𝑢_𝑛≤−𝑢_Ω|ℎ|𝑢_𝑛+𝑢^21/2×_Ω|ℎ|𝑢_𝑛−𝑢^21/2. (67) Since 𝑢_𝑛 is bounded, from (55) we get that (∫_Ω|ℎ|(𝑢_𝑛+𝑢)²)^1/2 is bounded. Thus 𝐻(𝑢_𝑛)→𝐻(𝑢) that if ∫_Ω|ℎ|(𝑢_𝑛−𝑢)²→0. Let 𝑚=sup_𝑛‖∇(𝑢_𝑛−𝑢)‖². For a given 𝜀>0, since |ℎ|∈ℱ₁, we choose ℎ_𝜀∈𝒞^∞_𝑐(Ω) such that |ℎ|−ℎ_{𝜀ℳlog𝐿}<𝜀𝑚. (68) Therefore _Ω|ℎ|−ℎ_𝜀𝑢_𝑛−𝑢²≤|ℎ|−ℎ_{𝜀ℳlog𝐿Ω}∇𝑢_𝑛−𝑢²<𝜀. (69) Now _Ω|ℎ|𝑢_𝑛−𝑢²=_Ωℎ_𝜀𝑢_𝑛−𝑢²+_Ω|ℎ|−ℎ_𝜀𝑢_𝑛−𝑢²≤_Ωℎ_𝜀𝑢_𝑛−𝑢²+𝜀. (70) For sufficiently large 𝑛, the first integral can be made smaller than 𝜀 since 𝐻¹₀(Ω) is embedded compactly in 𝐿²(Ω) and ℎ_𝜀∈𝒞^∞_𝑐(Ω). Thus we conclude that ∫_Ω|ℎ|(𝑢_𝑛−𝑢)²→0.

Proof of Theorem 2. Since 𝑔∈𝐿¹_loc(Ω) and 𝑔⁺≠0, there exists 𝜑∈𝐶^∞_𝑐(Ω) such that ∫_Ω𝑔𝜑²>0 (see for e.g., Proposition 4.2 of [23]) and and hence the set ℳ is nonempty. Let {𝑢_𝑛} be a minimizing sequence of 𝐽 on ℳ, that is, lim_𝑛→∞𝐽𝑢_𝑛=𝜆₁𝑔=inf_𝑢∈ℳ𝐽(𝑢). (71) Using the coercivity of 𝐽, the reflexivity of 𝐻¹₀(Ω), and the compactness of the embedding of 𝐻¹₀(Ω) into 𝐿²(Ω), we can further assume that {𝑢_𝑛} converges weakly and almost everywhere to some 𝑢∈𝐻¹₀(Ω). Since 𝑔⁺∈ℱ₁, from the previous lemma we get lim_𝑛→∞Ω𝑔⁺𝑢²_𝑛=_Ω𝑔⁺𝑢². (72) For 𝑢_𝑛∈ℳ, we write _Ω𝑔⁻𝑢²_𝑛=_Ω𝑔⁺𝑢²_𝑛−1. (73) Now use the Fatou’s lemma to obtain _Ω𝑔⁻𝑢²≤_Ω𝑔⁺𝑢²−1. (74) Thus we get ∫_Ω𝑔𝑢²≥1. Set 𝑢∶=𝑢/(∫_Ω𝑔𝑢²)^1/2. Now the homogeneity and the weak lower semicontinuity of 𝐽 yields the following: 𝜆₁𝑔(𝑢)=𝐽(𝑢)≤𝐽∫_Ω𝑔𝑢²≤𝐽(𝑢)≤liminf_𝑛𝐽𝑢_𝑛=𝜆₁𝑔. (75) 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 space ℱ₁. Recall that ℱ₁ is the closure of 𝒞^∞_𝑐(Ω) in ℳlog𝐿.

Theorem 16. Let Ω be a bounded domain in ℝ². Then the following statements are equivalent: (i)𝑔∈ℱ₁, (ii)𝑔∈ℳlog𝐿 and lim_𝑡→0𝑔^∗∗(𝑡)𝑡log(|Ω|/𝑡)=0.

Proof. (𝑖)⇒(𝑖𝑖) We show that lim_𝑡→0𝑔^∗∗(𝑡)𝑡log(|Ω|/𝑡)=0. Let 𝑔_𝑛∈𝒞^∞_𝑐(Ω) and 𝑔_𝑛→𝑔 in ℳlog𝐿. Note that for each 𝑔_𝑛,𝑔_𝑛^∗∗(𝑡) is bounded and hence lim_𝑡→0𝑔_𝑛^∗∗(𝑡)|Ω|𝑡log𝑡=0,∀𝑛∈ℕ. (76) 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 𝑔_𝑛→𝑔 in ℳlog𝐿, there exists 𝑛₀∈ℕ such that sup_|Ω|0<𝑡<𝑔−𝑔_𝑛0^∗∗(𝑡)|Ω|𝑡log𝑡<𝜀. (77) Using the subadditivity of the maximal operator 𝑓→𝑓^∗∗, we get 𝑔^∗∗(𝑡)|Ω|𝑡log𝑡≤𝑔−𝑔_𝑛0^∗∗(𝑡)|Ω|𝑡log𝑡+𝑔_𝑛^∗∗₀(𝑡)|Ω|𝑡log𝑡<𝜀+𝑔_𝑛^∗∗₀(𝑡)|Ω|𝑡log𝑡. (78) Note that from (76), there exists 𝛿>0 such that 𝑔_𝑛^∗∗₀(𝑡)𝑡log(|Ω|/𝑡)<𝜀, for all 𝑡∈(0,𝛿). Now (78) yields the result as 𝑔^∗∗(𝑡)|Ω|𝑡log𝑡<2𝜀,∀𝑡∈(0,𝛿). (79)(𝑖𝑖)⇒(𝑖). Let 𝜀>0 be given. For proving 𝑔∈ℱ₁, we show that there exists 𝑔_𝜀∈𝒞^∞_𝑐(Ω) such that ‖𝑔−𝑔_𝜀‖_ℳlog𝐿<2𝜀. Since lim_𝑡→0𝑔^∗∗(𝑡)𝑡log(|Ω|/𝑡)=0, we get 𝛿>0 such that 𝑔^∗∗(𝑡)|Ω|𝑡log𝑡<𝜀,∀𝑡∈(0,𝛿]. (80) Let 𝐴_𝜀∶=𝑥∈Ω∶𝑔(𝑥)<𝑔^∗(𝛿),𝑔_𝜀=𝑔𝜒_𝐴𝜀. (81) Clearly 𝑔_𝜀∈𝐿²(Ω). Since 𝐿²(Ω) is continuously embedded in ℳlog𝐿 and 𝒞^∞_𝑐(Ω) is dense in 𝐿²(Ω), we can choose 𝑔_𝜀∈𝒞^∞_𝑐(Ω) so that 𝑔_𝜀−𝑔_{𝜀ℳlog𝐿}<𝜀. (82) Next we estimate ‖𝑔−𝑔_𝜀‖_ℳlog𝐿. Let us compute the distribution function 𝛼_{𝑔−𝑔𝜀}: 𝛼_{𝑔−𝑔𝜀}(𝑠)=𝛼_{𝑔𝜒𝐴𝜀𝑐}(𝑠)=𝛼_𝑔(𝑠)𝑠≥𝑔^∗(𝛿),𝛼_𝑔𝑔^∗(𝛿)𝑠<𝑔^∗(𝛿). (83) For computing the symmetric rearrangement (𝑔−𝑔_𝜀)^∗(𝑡), we set 𝛿₁=𝛼_{𝑔−𝑔𝜀}(𝑔^∗(𝛿))=𝛼_𝑔(𝑔^∗(𝛿)). If 𝛿₁=0,𝛼_{𝑔−𝑔𝜀}(𝑠)=0, for all 𝑠>0, we get the desired result as ‖𝑔−𝑔_𝜀‖_ℳlog𝐿=0. Next we calculate (𝑔−𝑔_𝜀)^∗(𝑡), when 𝛿₁>0. For 0<𝑡<𝛿₁, note that 𝛼_{𝑔−𝑔𝜀}(𝑠)≤𝑡 implies that 𝑠≥𝑔^∗(𝛿) (if 𝑠<𝑔^∗(𝛿), then 𝛿₁=𝛼_{𝑔−𝑔𝜀}(𝑔^∗(𝛿))≤𝛼_{𝑔−𝑔𝜀}(𝑠)≤𝑡, a contradiction). Thus for 𝑡<𝛿₁, using (83) we have 𝑔−𝑔_𝜀^∗(𝑡)=inf𝑠∶𝛼_{𝑔−𝑔𝜀}(𝑠)≤𝑡=inf𝑠≥𝑔^∗(𝛿)∶𝛼_{𝑔−𝑔𝜀}(𝑠)≤𝑡=inf𝑠∶𝛼_𝑔(𝑠)≤𝑡=𝑔^∗(𝑡). (84) Now for 𝑡≥𝛿₁, again using (83), we see that 𝛼_{𝑔−𝑔𝜀}(𝑠)≤𝛼_𝑔(𝑔^∗(𝛿))=𝛿₁≤𝑡,  for all 𝑠<𝑔^∗(𝛿) and hence (𝑔−𝑔_𝜀)^∗(𝑡)=0, for all 𝑡≥𝛿₁. Thus 𝑔−𝑔_𝜀^∗∗(𝑡)=𝑔^∗∗(𝑡)𝑡<𝛿₁,𝛿₁𝑡𝑔^∗∗𝛿₁𝑡≥𝛿₁. (85) Therefore, 𝑔−𝑔_𝜀^∗∗(𝑡)|Ω|𝑡log𝑡≤𝑔^∗∗(𝑡)|Ω|𝑡log𝑡𝑡<𝛿₁,𝑔^∗∗𝛿₁𝛿₁|Ω|log𝛿₁𝑡≥𝛿₁. (86) Since 𝛿₁≤𝛿, from (80), we have ‖𝑔−𝑔_𝜀‖_ℳlog𝐿<𝜀. Now (82) yields ‖𝑔−𝑔_𝜀‖_ℳlog𝐿<2𝜀 and hence the proof is done.

Next we give example of functions in ℱ₁.

Example 17. We have already seen that 𝐿log𝐿⊂ℳlog𝐿 (Proposition 6). In fact, 𝐿log𝐿⊂ℱ₁ since 𝒞^∞_𝑐(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 in ℳlog𝐿 or not, by setting 𝜆₁(𝑔^⋆)=inf{(∫_Ω|∇𝑢^⋆|²/∫_Ω𝑔^⋆𝑢^⋆2)∶𝑢∈𝐻¹₀(Ω),∫_Ω𝑔𝑢²>0}, the problem can be rephrased as whether 𝜆₁(𝑔)>0 implies 𝜆₁(𝑔^⋆)>0 or not.

Remark 19. For 𝑁≥3 and Ω⊂ℝ^𝑁 (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 sup_|Ω|0<𝑡<𝑡^2/𝑁𝑔^∗∗(𝑡)<∞. (87) 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 𝒟₀^1,2(Ω) into the Lorentz space 𝐿(2^∗,2) 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).