On the best possible remaining term in the Hardy inequality

Abstract

We give a necessary and sufficient condition on a radially symmetric potential V on a bounded domain Ω of ℝⁿ that makes it an admissible candidate for an improved Hardy inequality of the following type. For every ∈ H¹₀(Ω)

A characterization of the best possible constant c(V) is also given. This result yields easily the improved Hardy's inequalities of Brezis-Vázquez [Brezis H, Vázquez JL (1997) Blow up solutions of some nonlinear elliptic problems. Revista Mat Univ Complutense Madrid 10:443–469], Adimurthi et al. [Adimurthi, Chaudhuri N, Ramaswamy N (2002) An improved Hardy Sobolev inequality and its applications. Proc Am Math Soc 130:489–505], and Filippas-Tertikas [Filippas S, Tertikas A (2002) Optimizing improved Hardy inequalities. J Funct Anal 192:186–233] as well as the corresponding best constants. Our approach clarifies the issue behind the lack of an optimal improvement while yielding the following sharpening of known integrability criteria: If a positive radial function V satisfies lim inf_r→oln(r)∫^r_o,sV(s)ds>−∞,then there exists ρ:=ρ(Ω) > 0 such that the above inequality holds for the scaled potential v_ρ(x)=v(^|x|_ρ).On the other hand, if lim _r→0 ln(r)∫^r_o,sV(s)ds=−∞, then there is no ρ > 0 for which the inequality holds for V_ρ.

Let Ω be a bounded domain in Rⁿ, n ≥ 3, with 0 ∈ Ω. The classical Hardy inequality asserts that for all u ∈ H¹₀(Ω)

This inequality and its various improvements are used in many contexts, such as in the study of the stability of solutions of semilinear elliptic and parabolic equations (1,2), the analysis of the asymptotic behavior of the heat equation with singular potentials (3), as well as in the study of the stability of eigenvalues in elliptic problems such as Schrödinger operators (4).

Now, it is well known that ${\left(\frac{n-2}{2}\right)}^2$ is the best constant for the inequality shown as Eq. 1, and that this constant is, however, not attained in H¹₀(Ω). So, one could anticipate improving this inequality by adding a nonnegative correction term to the right-hand side of the inequality shown as Eq. 1, and indeed, several sharpened Hardy inequalities have been established in recent years (3,5), mostly triggered by the following improvement of Brezis and Vázquez (1). For all u in H¹₀(Ω),

The constant λ_Ω in Eq. 2 is given by

where ω_n and |Ω| denote the volume of the unit ball and Ω, respectively, and z₀ = 2.4048… is the first zero of the Bessel function J₀(z). Moreover, λ_Ω is optimal when Ω is a ball but is—again—not achieved in H¹₀(Ω). This led to one of the open problems mentioned in ref. 1 (Problem 2), which is whether the two terms on the right-hand side of the inequality shown as Eq. 2 (i.e., the coefficients of |u|²) are just the first two terms of an infinite series of correcting terms.

This question was addressed by several authors. In particular, Adimurthi et al. (6) proved that for every integer k, there exists a constant c depending on n, k, and Ω such that for all u ∈ H¹₀(Ω),

where ρ=(sup_{x ∈ Ω}|x|)(e^{ee..e(k–times)}). Here, we have used the notation log⁽¹⁾(.)=log(.)and log^(k)(.)=log(log^(k−1)(.)) for k ≥ 2.

Also motivated by the question of Brezis and Vázquez, Filippas and Tertikas proved in ref. 5 that the inequality can be repeatedly improved by adding to the right-hand side specific potentials that lead to an infinite series expansion of Hardy's inequality. More precisely, by defining iteratively the following functions,

they prove that for any D ≥ sup_{x ∈ Ω}|x|, the following inequality holds for any u ∈ H¹₀(Ω):

Moreover, they proved that $\frac{1}{4}$ is the best constant, which again is not attained in H¹₀(Ω).

In this article, we show that all the above results—and more—follow from a specific characterization of those potentials V that yield an improved Hardy inequality. Here are our main results.

Theorem 1. Let V be a radially symmetric decreasing function on a ball B_ρ in ℝⁿ with radius ρ, n ≥ 2, in such a way that V(x) = v(|x|) for some nonnegative function v on (0,ρ]. The following properties are then equivalent:

1. The ordinary differential equation

   

   has a positive solution on (0,ρ] (possibly y(ρ) = 0).
2. The following improved Hardy inequality holds for u in H¹₀(B_ρ):

   
   Moreover, the best constant b(V): = sup{b;(H_bV)holds} can be characterized by the formula

   

Here is an immediate application of the above characterization.

Corollary 2. Assume n ≥ 2, then there is no strictly positive v in C¹(0,∞) such that the inequality

holds for all u ∈ W^1,2(Rⁿ).

We shall also see that the results of Brezis–Vázquez, Adimurthi et al., and Filippas–Tertikas mentioned above can be easily deduced by simply checking that the potentials V they consider correspond to equations (D_V), where an explicit positive solution can be found.

Our approach turned out to be useful also for determining the best constants in the above-mentioned improvements. Indeed, the case when V ≡ 1 will follow immediately from Theorem 1. A slightly more involved reasoning—but also based of the above characterization—will allow us to recover the best one established by Filippas–Tertikas, and to show that the best constant in the improvement of Adimurthi et al. is actually equal to $\frac{1}{4}$, which was conjectured by Chaudhuri in a recent article (7) where he shows the estimate $\frac{1}{4}$ ≤ c ≤$\frac{1}{2}$.

Because the existence of positive solutions for ordinary differential equations of the form (D_V) is closely related to the oscillatory properties of second-order equations of the form z″(s) + a(s)z(s) = 0, Theorem 1 also allows for the use of the extensive literature on the oscillatory properties of such equations to deduce various interesting results such as the following corollary.

Corollary 3. Let V be a positive radial function on a ball Ω in ℝⁿ, n ≥ 2.

1. If lim_r→0 ln(r)∫^r_o,sV(s)ds> −∞, then there exists α : = α(Ω)>0 such that an improved Hardy inequality (H_Vα) holds for the scaled potential V_α(x): = α² V(αx).

2. If lim_r→0 ln(r)∫^r_o,sV(s)ds=−∞, then there are no β, c > 0, for which(HV_β,c) holds with V_β,c = cV(βx).

The following is a consequence of the above corollary combined with Theorem 1.

Corollary 4. Let Ω be a smooth bounded domain in Rⁿ (n ≥ 2).

1. For any α < 2, there is c > 0 such that the following inequality holds for all u ∈ H¹₀(Ω),

   

   Moreover, the best constant c(α,Ω) ≥ c(α,B_ρ), the latter being the best constant corresponding to the ball of radius ρ = sup_{x ∈ Ω}|x|, which is exactly equal to the largest b such that $y″+\frac{1}{r}y\prime +b\frac{1}{r^{\alpha }}y=0$ has a positive solution on (0,ρ].
2. In particular, the following inequality holds for all u ∈ H¹₀(Ω):

   

   with $c(\Omega ):=c(0,\Omega )\ge \frac{z_0^2}{{({sup}_{x\in \Omega }|x|)}^2}$, the latter being the best constant associated to the same inequality on the ball of radius $\rho ={sup}_{x\in \Omega }\left|x\right|$.

3. If α ≥ 2, then there is no c > 0 for which inequality (H_V) holds for $V_{\alpha ,c}(x)=c\frac{1}{|x{|}^{\alpha }}$.

The above corollary gives another proof of the fact that ${\left(\frac{n-2}{2}\right)}^2$ is the best constant for the classical Hardy inequality. Moreover, an offshoot of our approach are the following Hardy inequalities in the critical dimension two as well as their best constants. Some of these results were also obtained in refs. 4 and 6.

Corollary 5. Let Ω be a smooth domain in R² with 0 ∈ Ω, then we have the following inequalities.

• If D ≥ sup_{x ∈ Ω}|x|, then for u ∈ H₀¹(Ω)

  

  and $\frac{1}{4}$ is the best constant.
• If ρ ≥(sup_{x ∈ Ω}|x|)(e^{ee..e(k – times)}), then for u ∈ H₀¹(Ω)

  

  and $\frac{1}{4}$ is the best constant for every k ≥ 1.
• If α < 2, then for u ∈ H₀¹(Ω)

  

  C_α is the largest c > 0 such that $y″+\frac{1}{r}y\prime +b\frac{1}{r^{\alpha }}y=0$ has a positive solution on $(0,{(\frac{\left|\Omega \right|}{{\omega }_n})}^{\frac{1}{2}})$. Moreover, C_α is the best constant when Ω is a ball.

Two-dimensional Inequalities

In this section, we establish the following improvements of Poincaré and Poincaré–Wirtinger inequalities.

Theorem 6. Let a < b, k is a differentiable function on (a,b), and φ be a strictly positive real valued differentiable function on (a,b). Then, every h ∈ C¹([a,b]) such that

satisfies the following inequality:

Moreover, assuming Eq.12, the equality holds if and only if h(r) = φ(r) for all r ∈ (a,b).

Proof: Define $\psi (r)=h(r)/\phi (r),r\in [a,b]$. Then we can estimate $I={\int }_b^a|h\text{'}(r){|}^{2}k(r)dr$ as follows,

Hence, we have

and Eq. 13 holds. Note that the last inequality is an idendity if and only if h(r) = φ(r) for all r ∈ (a,b).

By applying Theorem 6 to the weight k(r) = r, we obtain the following generalization of the two-dimensional Poincaré inequality.

Corollary 7. (Generalized two-dimensional Poincaré inequality) Let 0 ≤ a < b and φ be a strictly positive real valued differentiable function on (a,b). Then every h ∈ C¹([a,b]) with

satisfies the following inequality:

Moreover, under the assumption of Eq.14, the equality holds if and only if h(r) = φ(r) for all r ∈ (a,b).

By applying Theorem 6 to the weight k(r) = 1, we obtain the following generalization of the two-dimensional Poincaré–Wirtinger inequality.

Corollary 8. (Generalized Poincaré–Wirtinger inequality) Let a < b and φ be a strictly positive real valued differentiable function on (a,b). Then, every h ∈ C¹([a,b]) with

satisfies the following inequality:

Moreover, under assumption of Eq.16, the equality holds if and only if h(r) = φ(r) for all r ∈ (a,b).

Proof of the Main Results

We start with the sufficient condition of Theorem 1 by establishing the following.

Proposition 9. (Improved Hardy inequality) Let Ω be a bounded smooth domain in Rⁿ (n ≥ 2) with 0 ∈ Ω, and set $R={(|\Omega |/{\omega }_n)}^{1\backslash n}$ and $\rho ={sup}_{x\in \Omega }\left|x\right|$. Suppose V is a function on Ω of the form V(x) = v(|x|) where ${\left(\frac{n-2}{2}\right)}^2\frac{1}{r^2}+v(r)$ is a decreasing function on (0,ρ), and such that for some C²-function φ on (0,R), we have

Then for any u ∈ H₀¹(Ω), we have

Proof: We first prove the inequality for smooth radial functions on the ball Ω = B_R. For such $u\in C_0^2(B_R)$, we define $w(r)=u(r)r^{(n-2)/2}$, r = |x|. In view of Corollary 7, we can write

Hence, the inequality shown as Eq. 20 holds for radial smooth positive functions. For a nonradial function u on general domain Ω, we use symmetrization arguments. Let B_R be a ball having the same volume as Ω with $R={(|\Omega |/{\omega }_n)}^{1\backslash n}$ and let u* be the symmetric decreasing rearrangement of the function |u|. Now note that for any $u\in H_0^1(\Omega )$,$u^{*}\in H_0^1(B_R)$. It is well known that the symmetrization does not change the L^p norm, and that it decreases the Dirichlet energy, while increasing the integrals ${\int }_{\Omega }\left\{{\left(\frac{n-2}{2}\right)}^2\frac{1}{|x{|}^2}+V(x)u^2\right\}dx$, because the weight ${\left(\frac{n-2}{2}\right)}^2\frac{1}{|x{|}^2}+V(x)$ is a decreasing function of|x|. Hence, the inequality shown as Eq. 20 holds for every $u\in H_0^1(\Omega )$, where Ω is a smooth bounded domain in Rⁿ, n ≥ 2.

Lemma 1. Let x(r) be a function in C¹(0,R] that is a solution of

where F is a nonnegative continuous function, then ${lim}_{r\downarrow 0}x(r)=0$.

Proof: Divide Eq. 21 by r and integrate once. Then, we have

It follows that ${lim}_{r\downarrow 0}x(r)$ exists. In order to prove that this limit is zero, we claim that ${\int }_r^R\frac{x^2(s)}{s}ds\&lt;\infty$. Indeed, otherwise we have $G(r):={\int }_r^R\frac{x^2(s)}{s}ds\to \infty$. From the inequality shown as Eq. 21, we have

Note that F ≥ 0, and G goes to infinity as r goes to zero. Thus, for r sufficiently small, we have $-rG\prime (r)\ge \frac{1}{2}{G}^2(r)$; hence, $\left(\frac{1}{G(r)}\right)\prime \ge \frac{1}{2}(ln(r))\prime$, which contradicts the fact that G(r) goes to infinity as r tends to zero. Thus, our claim is true, and the limit of x(r) is indeed zero.

Lemma 2. Assume v(r) > 0 for all 0 < r ≤ R. If the equation $\phi ″+\frac{\phi \prime }{r}+v(r)\phi =0$ has a positive solution on some interval (0,R], then we have, necessarily,

Proof: First observe that φ cannot have a local minimum; hence, it is either increasing or decreasing on (0,δ), for δ sufficiently small. Assume φ is increasing; then, $\frac{{\phi }^{″}}{{\phi }^{\prime }}\le -\frac{a}{r}$, thus $\phi \prime \ge \frac{c}{r^a}$ for some c > 0. Therefore, φ(r) → -∞ as r → 0, which is a contradiction. Because, φ cannot have a local minimum, it should be decreasing on (0,R). Hence, φ satisfies the second assertion. To obtain the first, set $x(r)=r\frac{\phi \prime (r)}{\phi (r)}$. One can easily verify that x(r) satisfies the ODE:

where $F(r)=r^{2}v(r)\ge 0$. By Lemma 1, we conclude that ${lim}_{r\downarrow 0}r\frac{\phi \prime (r)}{\phi (r)}={lim}_{r\downarrow 0}x(r)=0$.

Lemma 3. Let V be positive radial potential on the ball Ω of radius R in Rⁿ (n ≥ 2). Assume that for all $u\in H_0^1(\Omega )$,

Then, there exists a C² supersolution to the equation

Proof: By assumption, we have that

is nonnegative. We then consider $({\phi }_n,{\lambda }_1^n)$ to be the first eigenpair for the problem

where $L=-\Delta -{\left(\frac{n-2}{2}\right)}^2\frac{1}{|x{|}^2}-V$, and $B_{\frac{R}{n}}$ is a ball of radius $\frac{R}{n}$, n ≥ 2. The eigenfunctions can be chosen in such a way that φ_n > 0 on $\Omega \backslash B_{\frac{R}{n}}$ and φ_n(b) = 1, for some b ∈ Ω with $\frac{R}{2}\&lt;\left|b\right|\&lt;R$.

Note that ${\lambda }_1^n\downarrow 0$. Harnak's inequality yields that for any compact subset $K,\frac{{\max }_K{\phi }_n}{{\min }_K{\phi }_n}\le C(K)$, with the later constant being independent of φ_n. Also, standard elliptic estimates yields that the family (φ_n) have uniformly bounded derivatives on the compact sets $\Omega -B_{\frac{R}{n}}$. There exists therefore a subsequence ${({\phi }_{n_{l_2}})}_{n_{l_2}}$ of ${({\phi }_n)}_n$ that converges to some ${\phi }_2\in C^2(\Omega \backslash B_{\frac{R}{2}})$. Now consider ${({\phi }_{n_{l_2}})}_{n_{l_2}}$ on $\Omega \backslash B_{\frac{R}{3}}$ and a subsequence ${({\phi }_{n_{l3}})}_{n_{l_3}}$ that converges to ${\phi }_3\in C^2(\Omega \backslash B_{\frac{R}{3}})$, with ${\phi }_3(x)={\phi }_2(x)$ for all $x\in (\Omega \backslash B_{\frac{R}{2}})$. By repeating this argument we get a supersolution $\phi \in C^2(\Omega \backslash \{0\})$ i.e. Lφ ≥ 0, such that φ > 0 on Ω\{0}.

Proof of Theorem 1: The implication 1) implies 2) follows immediately from Proposition 9 and Lemma 2. It is valid for any smooth bounded domain provided v is assumed to be nondecreasing on (0,R). This condition is not needed if the domain is a ball of radius R.

To show that 2) implies 1), we assume that inequality (H_V) holds on a ball Ω of radius R, and then apply Lemma 3 to obtain a C² supersolution for Eq. 24. Now take the surface average of u, that is

is strictly positive. We may assume that the unit ball is contained in Ω (otherwise we just use a smaller ball). By a standard calculation we get

Because u(x) is a supersolution of the inequality shown as Eq. 24, w satisfies the inequality:

for 0 < r < R. Now set $\phi (r)=r^{\frac{n-2}{2}}w(r)$ for 0 < r < R. Using the inequality shown as Eq. 25, a straightforward calculation shows that φ satisfies the following inequality

By standard results we conclude that the equation $y″(r)+\frac{1}{r}y\prime +v(r)y=0$ has actually a positive solution ϕ on (0,R).

To establish the formula shown as Eq. 6, it is clear that by the sufficient condition c(V) ≥ c whenever $y″(r)+\frac{1}{r}y\prime +cv(r)y=0$ has a positive solution on (0,R). On the other hand, the necessary condition yields that $y″+\frac{1}{r}y\prime +cv(r)y=0$ has a positive solution on (0,R). The proof is now complete.

Proof of Corollary 2: Assume the inequality (H_V) holds on all of ℝⁿ. An argument similar to that of Lemma 3 yields a positive solution for $y″(r)+\frac{1}{r}y\prime +v(r)y=0$ on (0,∞). From the proof of lemma 2, we know that y is decreasing on (0,∞). Hence, $\frac{\phi ″(r)}{\phi \prime (r)}\ge -\frac{1}{r}$, which yields ${\phi }^{\prime }(r)\le \frac{c}{r}$, for some c < 0. Thus φ(r) → -∞. This is a contradiction, and the proof is complete.

Applications

We now apply Theorem 1 to recover all previously known improvements of Hardy's inequality in a relatively simple and unified way. For that, we need to investigate whether the ordinary differential equation $y″+\frac{y\prime }{r}+v(r)y(r)=0$, corresponding to a potential v has a positive solution φ on (0,δ) for some δ > 0. In this case, $\psi (r)=\phi \left(\frac{\delta r}{R}\right)$ is a solution for $y″(r)+\frac{1}{r}y\prime +\frac{{\delta }^2}{R^2}v\left(\frac{\delta }{R}r\right)y=0$ on (0,R), which means that the scaled potential $V_{\delta }(x)=\frac{{\delta }^2}{R^2}V\left(\frac{\delta }{R}x\right)$ yields an improved Hardy formula ($(H_{V_{\delta }})$) on a ball of radius R. Here is an immediate consequence.

1) The Brezis–Vázquez improvement (1): Here, we need to show that we can have an improved inequality with a constant potential. In this case, the best constant for which the equation $y″+\frac{y\prime }{r}+cy(r)=0$, has a positive solution on (0,R), with $R={(|\Omega |/{\omega }_n)}^{\frac{1}{n}}$ is $z_0^2{\omega }_n^{2/n}|\Omega {|}^{-2/n}$, where z₀ = 2.4048… is the first zero of the Bessel function J₀(z). For that, we need to show that if α is the first root of an arbitrary solution of the Bessel equation $y″+\frac{y\prime }{r}+y(r)=0$, then we necessarily have α ≤ z₀. To see this, we let x(t) = aJ₀(t) + bY₀(t), where J₀ and Y₀ are the two standard linearly independent solutions of the Bessel equation, and a and b are constants. Assume the first zero of x(t) is larger than z₀. Since the first zero of Y₀ is smaller than z₀, we have a ≥ 0. Also b ≤ 0, because Y₀(t) → -∞ as t → 0. Finally, note that Y₀(z₀) > 0, so that if b > 0, then x(z₀ + ɛ) < 0 for ɛ sufficiently small, which is impossible. This means that b = 0, which is also a contradiction, and we are done proving the result of Brezis–Vázquez mentioned in the introduction.

2) The best constant in the improvement of Adimurthi et al. (6): In this case, one easily sees that the function ${\phi }_k(r)={\left({\pi }_{i=1}^k{log}^{(i)}\frac{\rho }{r}\right)}^{\frac{1}{2}}$ is a solution of the equation

on (0,R), which means that the inequality (H_V) holds for the potential $V_k(x)=\frac{1}{4{\left|x\right|}^2}{\sum }_{j=1}^k{\left({\prod }_{i=1}^j{log}^{(i)}\frac{\rho }{\left|x\right|}\right)}^{-2}$, which yields the result of Adimurthi et al. In the following, we use our characterization to show that the constant appearing in the above improvement is indeed the best constant in the following sense: For 1 ≤ m ≤ k, the infimum over $u\in H_0^1(\Omega )$\{0} of the quantity

is equal to $\frac{1}{4}$.

We proceed by contradiction, and assume that such an infimum is actually equal to $\frac{1}{4}+\lambda$, with λ > 0. From Theorem 1, we deduce that there exists a positive function φ(r) such that

Now set $f(r)=\frac{\phi (r)}{\phi m(r)}>0$ and write

to obtain that

We first claim that f is monotone for r small enough. Indeed, if not, then f′(α_n) = 0 for some sequence ${\left\{{\alpha }_n\right\}}_{n=1}^{\infty }$ that converges to zero, and consequently there exists a sequence ${\left\{{\beta }_n\right\}}_{n=1}^{\infty }$ that also converges to zero, such that f″(β_n) = 0, and f′(β_n) > 0. But this contradicts with the latter identity, which proves our claim. We now consider whether f is increasing or decreasing:

Case 1. Assume f′(r) > 0 for r > 0 sufficiently small. Then, we will have

Integrating once, we get $f\prime (r)\ge \frac{c}{r{\prod }_{j=1}^m{log}^j\left(\frac{\rho }{r}\right)}$, for some c > 0. Hence, lim_{r → 0}f(r) = -∞, which is a contradiction.

Case 2. Assume f′(r) < 0 for r > 0 sufficiently small. Then $\frac{(rf\prime (r))\prime }{rf\prime (r)}\ge \sum _{i=1}^m\frac{1}{r{\prod }_{j=1}^i{log}^j\left(\frac{\rho }{r}\right)}$; thus,

for some c > 0 and r > 0 sufficiently small. On the other hand,

Because f′(r) < 0, there exists l such that f(r) > l > 0 for r > 0 sufficiently small. From the above inequality, we then have

From the inequality shown as Eq. 27, we have lim_a→0 af′ (a) = 0. Hence, $bf\prime (b)\&lt;-\frac{\lambda l}{{\prod }_{j=1}^m{log}^j\left(\frac{\rho }{b}\right)}$, for every b > 0, and for r > 0 sufficiently small, $f\prime (r)\&lt;-\frac{\lambda l}{r{\prod }_{j=1}^m{log}^j\left(\frac{\rho }{r}\right)}$. Therefore lim_r→0f(r) = +∞, and by choosing l large enough (e.g., $l>\frac{c}{\lambda }$, we get to contradict the inequality shown as Eq. 27 and the proof is now complete.

3) The Filippas–Tertikas improvement (5): Let D ≤ sup_{x ∈ Ω} |x|, and define

Using the fact that $X{\prime }_k(r)=\frac{1}{r}{X}_1(r)X_2(r)\dots X_{k-1}(r)X_k^2(r)$ for k = 1,2,…, we get

This means that the inequality (H_V) holds for the potential

which yields the result of Filippas and Tertikas (5). One can again identify the best constant by showing in the same way as above that for all 1 ≤ m ≤ k, the infimum over $u\in H_0^1(\Omega )$\{0} of the quantity

is equal to $\frac{1}{4}$

Remark 1. Theorem 1 characterizes the best constant when Ω is a ball. If now Ω is a general bounded domain Ω, we can obviously get lower and upper bounds for the best constant, by simply noting that

where B_R is the smallest ball containing Ω and B_ρ is the largest ball contained in Ω. Note that the best constant corresponding to the potentials involving iterated logs above is $\frac{1}{4}$ independently of the radius of the ball, and therefore it is the case for any bounded domain. It is however not the case for V = 1, for which the best constant is still not known for general domains.

To prove Corollary 3, we need the following result concerning the existence of nonoscillatory solutions (i.e., those z(s) such that z(s) > 0 for s > 0 sufficiently large) for the second-order ODE

Interesting results in this direction were established by many authors (see refs. 8–11). Here is a typical criterium about the oscillatory properties of Eq. 28:

1. If $lim{sup}_{t\to \infty }t{\int }_t^{\infty }a(s)ds\&lt;\frac{1}{4}$, then Eq. 28 is nonoscillatory.

2. If $lim{inf}_{t\to \infty }t{\int }_t^{\infty }a(s)ds>\frac{1}{4}$, then Eq. 28 is oscillatory.

Now, in order to make the connection between improved Hardy inequalities and the existence of nonoscillatory solutions for Eq. 28, it suffices to notice that the change of variable s = -ln(r) shows that $y"+\frac{1}{r}y\prime +v(r)y=0$ has a positive solution on (0,δ) for some δ > 0 if and only if z″(s) + e^-2sv(e^-s)z(s) = 0 has a strictly positive solution on (b,+∞) for some b > 0. The above criteria, combined with Theorem 1, clearly yield Corollary 3.