A class of weighted Hardy inequalities and applications to evolution problems

TL;DR

This paper establishes a class of weighted Hardy inequalities involving Kolmogorov operators with inverse square potentials, demonstrating the optimality of constants and exploring existence and nonexistence results for related evolution problems in a probabilistic setting.

Contribution

It introduces a new class of weighted Hardy inequalities for Kolmogorov operators with inverse square potentials, extending classical results to a probabilistic framework with optimal constants.

Findings

Proved the optimality of the Hardy inequality constant.

Derived existence and nonexistence results for evolution problems.

Extended Cabré-Martel's approach to Kolmogorov operators.

Abstract

\begin{abstract} We state the following weighted Hardy inequality \begin{equation*} c_{o, \mu}\int_{{\R}^N}\frac{\varphi^2 }{|x|^2}\, d\mu\le \int_{{\R}^N} |\nabla\varphi|^2 \, d\mu + K \int_{\R^N}\varphi^2 \, d\mu \quad \forall\, \varphi \in H_\mu^1 %\qquad c\le c_\mu, \end{equation*} in the context of the study of the Kolmogorov operators \begin{equation*} Lu=\Delta u+\frac{\nabla \mu}{\mu}\cdot\nabla u \end{equation*} perturbed by inverse square potentials and of the related evolution problems. The function $\mu$ in the drift term is a probability density on $\R^N$. We prove the optimality of the constant $c_{o, \mu}$ and state existence and nonexistence results following the Cabr\'e-Martel's approach \cite{CabreMartel} extended to Kolmogorov operators. \end{abstract}

Equations

$$c_{o,\mu }{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}d\mu \le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +K{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu \forall \phi \in H_{\mu }^1$$

$$Lu=\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u$$

$$Lu=\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u,$$

$$(P)\quad\left\{\begin{array}[]{ll}\partial_{t}u(x,t)=Lu(x,t)+V(x)u(x,t),\quad\,x\in{\mathbb{R}}^{N},t>0,\\ u(\cdot,0)=u_{0}\geq 0\in L_{\mu}^{2}.\end{array}\right.$$

$${\lambda }_1(L+V):=\underset{\phi \in H_{\mu }^1\setminus \{0\}}{\inf }\left(\frac{{\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu -{\int }_{{\mathbb{R}}^N}V{\phi }^{2}d\mu }{{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu }\right)$$

$${\int }_{{\mathbb{R}}^N}V{\phi }^{2}d\mu \le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +K{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu \forall \phi \in H_{\mu }^1,K>0$$

$$f_{\epsilon }=(\epsilon +|x{|}^2{)}^{\frac{\alpha }{2}},\alpha <0,\epsilon >0,$$

$$\frac{\nabla f_{\epsilon }}{f_{\epsilon }}\cdot \nabla \mu =\frac{\alpha x}{\epsilon +|x{|}^2}\cdot \nabla \mu \le \left(k_1+\frac{k_2\alpha }{\epsilon +|x{|}^2}\right)\mu$$

$$c{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}d\mu \le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +k_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu ,$$

$$\begin{array}{rl}& {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu ={\int }_{{\mathbb{R}}^N}|\nabla (\psi f_{\epsilon }){|}^{2}d\mu \\ & ={\int }_{{\mathbb{R}}^N}|\nabla \psi f_{\epsilon }+\nabla f_{\epsilon }\psi {|}^{2}d\mu \\ & ={\int }_{{\mathbb{R}}^N}|\nabla \psi {|}^2{f}_{\epsilon }^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\psi }^2|\nabla f_{\epsilon }{|}^{2}d\mu +2{\int }_{{\mathbb{R}}^N}{f}_{\epsilon }\psi \nabla \psi \cdot \nabla f_{\epsilon }d\mu \\ & ={\int }_{{\mathbb{R}}^N}|\nabla \psi {|}^2{f}_{\epsilon }^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\psi }^2|\nabla f_{\epsilon }{|}^{2}d\mu \\ & -{\int }_{{\mathbb{R}}^N}{\psi }^2|\nabla f_{\epsilon }{|}^{2}d\mu -{\int }_{{\mathbb{R}}^N}{f}_{\epsilon }^2{\psi }^2\frac{\Delta f_{\epsilon }}{f_{\epsilon }}d\mu -{\int }_{{\mathbb{R}}^N}{f}_{\epsilon }^2{\psi }^2\frac{\nabla f_{\epsilon }}{f_{\epsilon }}\cdot \nabla \mu dx.\end{array}$$

$$\Delta f_{\epsilon }=\frac{\alpha (N-2+\alpha )|x{|}^2+\alpha \epsilon N}{(\epsilon +|x{|}^2{)}^{2-\frac{\alpha }{2}}}$$

$$\begin{array}{rl}{\int }_{{\mathbb{R}}^N}|& \nabla \phi {|}^{2}d\mu \ge -{\int }_{{\mathbb{R}}^N}\frac{\Delta f_{\epsilon }}{f_{\epsilon }}{\phi }^{2}d\mu -{\int }_{{\mathbb{R}}^N}\frac{\nabla f_{\epsilon }}{f_{\epsilon }}\cdot \nabla \mu {\phi }^{2}dx\\ & \ge -\left[\alpha (N-2)+{\alpha }^2\right]{\int }_{{\mathbb{R}}^N}\frac{|x{|}^2}{(\epsilon +|x{|}^2{)}^2}{\phi }^{2}d\mu -\epsilon \alpha N{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{(\epsilon +|x{|}^2{)}^2}d\mu \\ & -k_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu -k_2\alpha {\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{\epsilon +|x{|}^2}d\mu \\ & =[-\alpha (N-2+k_2)-{\alpha }^2]{\int }_{{\mathbb{R}}^N}\frac{|x{|}^2}{(\epsilon +|x{|}^2{)}^2}{\phi }^{2}d\mu \\ & -\epsilon \alpha (N+k_2){\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{(\epsilon +|x{|}^2{)}^2}d\mu -k_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu .\end{array}$$

$${\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +k_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu \ge c{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}d\mu ,$$

$$\underset{\alpha }{\max }[-\alpha (N+k_2-2)-{\alpha }^2]={\left(\frac{N+k_2-2}{2}\right)}^2=:c_o(N+k_2),$$

$${\int }_{{\mathbb{R}}^N}V{\phi }^{2}dx\le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}dx$$

$$-\frac{\Delta f}{f}\ge V\forall x\in {\mathbb{R}}^N.$$

$$\Vert \cdot {\Vert }_{H_{\mu }^1}^2:=\Vert \cdot {\Vert }_{L_{\mu }^2}^2+\Vert \nabla \cdot {\Vert }_{L_{\mu }^2}^2$$

$$c{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}d\mu \le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +k_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu ,$$

$$\mu (x)\ge C{e}^{-\frac{k_1}{2|\alpha |}|x{|}^2}|x{|}^{k_2-\frac{k_1}{|\alpha |}\epsilon },\text{ for}|x|\ge r_0,$$

$$\frac{\alpha \rho }{\epsilon +{\rho }^2}{\mu }^{\prime }(\rho )\le \left(k_1+\frac{k_2\alpha }{\epsilon +{\rho }^2}\right)\mu (\rho ),$$

$${\mu }^{\prime }(\rho )\ge a(\rho )\mu (\rho )$$

$$a(\rho )=\left[\frac{k_1}{\alpha }\left(\frac{\epsilon +{\rho }^2}{\rho }\right)+\frac{k_2}{\rho }\right].$$

$$\mu (r)\ge \mu (r_0)e^{{\int }_{r_0}^{r}a(s)ds}=\mu (r_0){\left(\frac{r}{r_0}\right)}^{k_2-\frac{k_1}{|\alpha |}\epsilon }{e}^{-\frac{k_1}{2|\alpha |}(r^2-r_0^2)}\text{ for}r\ge r_0,$$

$$\mu (r)\ge \frac{\mu (r_0)}{r_0^{k_2-\frac{k_1}{|\alpha |}\epsilon }}{e}^{\frac{k_1}{2|\alpha |}{r}_0^2}{r}^{k_2-\frac{k_1}{|\alpha |}\epsilon }{e}^{-\frac{k_1}{2|\alpha |}{r}^2}\text{ for}r\ge r_0.$$

$$\mu (x)=\frac{1}{|x{|}^{\gamma }}{e}^{-\delta |x{|}^m},\delta \ge 0,\gamma <N-2$$

$$\frac{\alpha (-\gamma -\delta m|x{|}^m)}{\epsilon +|x{|}^2}\le k_1+\frac{\alpha k_2}{\epsilon +|x{|}^2}$$

$$-(\alpha \gamma +\alpha k_2+k_1\epsilon )-\alpha \delta m|x{|}^m-k_1|x{|}^2\le 0.$$

$${\left(\frac{N-2-\gamma }{2}\right)}^2{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}|x{|}^{-\gamma }dx\le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^2|x{|}^{-\gamma }dx\forall \phi \in H_{\mu }^1.$$

$$\underset{|x|\to 0}{\lim }\frac{\mu (|x|)}{|x{|}^{k_2}}=l,l>0,$$

$$\varphi_{\varepsilon}(x)=\left\{\begin{array}[]{ll}(\varepsilon+|x|)^{\eta}&\text{ if }|x|\in[0,1[,\\ (\varepsilon+|x|)^{\eta}\theta(x)&\text{ if }|x|\in[1,2[,\\ 0&\text{ if }|x|\in[2,+\infty[,\end{array}\right.$$

Full text

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in the context of the study of the Kolmogorov operators

[TABLE]

perturbed by inverse square potentials and of the related evolution problems. The function $\mu$ in the drift term is a probability density on $\mathbb{R}^{N}$. We prove the optimality of the constant $c_{o,\mu}$ and state existence and nonexistence results following the Cabré-Martel’s approach [8] extended to Kolmogorov operators.

The first two authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilitá e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Keywords: Weighted Hardy inequality, optimal constant, Kolmogorov operators, singular potentials.

1. Introduction

This paper on weighted Hardy inequalities fits in the framework of the study of Kolmogorov operators on smooth functions

[TABLE]

with $\mu$ probability density on $\mathbb{R}^{N}$, and of the related evolution problems

[TABLE]

The operator $L$ in $(P)$ is perturbed by the singular potential $V(x)=\frac{c}{|x|^{2}}$, $c>0$, and $L_{\mu}^{2}:=L(\mathbb{R}^{N},d\mu)$, with $d\mu(x)=\mu(x)dx$.

The interest in inverse square potentials of type $V\sim\frac{c}{|x|^{2}}$ relies in their criticality: the strong maximum principle and Gaussian bounds fail (see [2]). Furthermore interest in singular potentials is due to the applications to many fields, for example in many physical contexts as molecular physics [22], quantum cosmology (see e.g. [5]), quantum mechanics [4] and combustion models [18].

The operator $\Delta+V$, $V(x)=\frac{c}{|x|^{2}}$, has the same homogeneity as the Laplacian and do not belong to the Kato’s class, then $V$ cannot be regarded as a lower order perturbation term.

A remarkable result stated in 1984 by P. Baras and J. A. Goldstein in [3] shows that the evolution problem $(P)$ with $L=\Delta$ admits a unique positive solution if $c\leq c_{o}=\left(\frac{N-2}{2}\right)^{2}$ and no positive solutions exist if $c>c_{o}$. When it exists, the solution is exponentially bounded, on the contrary, if $c>c_{o}$, there is the so-called instantaneous blowup phenomenon.

In order to extend these results to Kolmogorov operators the technique must be different.

A result analogous to that stated in [3] has been obtained in 1999 by X. Cabré and Y. Martel [8] for more general potentials $0\leq V\in L_{loc}^{1}(\mathbb{R}^{N})$ with a different approach.

To state the existence and nonexistence results we follow the Cabré-Martel’s approach. We use the relation between the weak solution of $(P)$ and the bottom of the spectrum of the operator $-(L+V)$

[TABLE]

with $H^{1}_{\mu}$ suitable weighted Sobolev space.

When $\mu=1$ Cabré and Martel showed that the boundedness of $\lambda_{1}(\Delta+V)$, $0\leq V\in L_{loc}^{1}({\mathbb{R}}^{N})$, is a necessary and sufficient condition for the existence of positive exponentially bounded in time solutions to the associated initial value problem. Later in [19, 9] similar results have been extended to Kolmogorov operators. The proof uses some properties of the operator $L$ and of its corresponding semigroup in $L_{\mu}^{2}(\mathbb{R}^{N})$.

For Ornstein-Uhlenbeck type operators $Lu=\Delta u-\sum_{i=1}^{n}A(x-a_{i})\cdot\nabla u$, $a_{i}\in\mathbb{R}^{N}$, $i=1,\dots,n$, perturbed by multipolar inverse square potentials, weighted multipolar Hardy inequalities and related existence and nonexistence results were stated in [10]. In such a case, the invariant measure for these operators is $d\mu=\mu_{A}(x)dx=Ke^{-\frac{1}{2}\sum_{i=1}^{n}\left\langle A(x-a_{i}),x-a_{i}\right\rangle}dx$.

There is a close relation between the estimate of the bottom of the spectrum $\lambda_{1}(L+V)$ and the weighted Hardy inequality with $V(x)=\frac{c}{|x|^{2}}$, $c\leq c_{o,\mu}$,

[TABLE]

with the best possible constant $c_{o,\mu}$.

In particular the existence of positive solutions to $(P)$ is related to the Hardy inequality (1.1) and the nonexistence is due to the optimality of the constant $c_{o,\mu}$.

The main results in the paper are, in Section 2, the weighted Hardy inequality (1.1) with measures which satisfy fairly general conditions and the optimality of the constant $c_{o,\mu}$ in Section 3.

The proof of the weighted Hardy inequality is different from the others in literature. It is based on the introduction of a suitable $C^{\infty}$-function and it can be used to prove inequality (1.1) with $0\leq V\in L^{1}_{loc}(\mathbb{R}^{N})$ of a more general type, in other words Hardy type inequalities.

In [9] the authors state a weighted Hardy inequality using a different approach and improved Hardy inequalities. This requires suitable conditions on $\mu$. Our technique, with different assumptions on $\mu$, allows us to achieve the best constant (cf. [9, Theorem 3.3]) for a wide class of functions $\mu$. To state the optimality of the constant in the estimate we need further assumptions on $\mu$ as usually it is done. We find a suitable function $\varphi$ for which the inequality (1.1) doesn’t hold if $c>c_{o,\mu}$, this is a crucial point in the proof. The way of estimate the bottom of the spectrum is close to the one used in [9]. We remark that the inequality obtained under our hypotheses applies in the context of weighted multipolar Hardy inequalities stated in the forthcoming paper [11].

Finally we state an existence and nonexistence result in Section 4 following the Cabré-Martel’s approach and using some results stated in [19, 9] when the function $\mu$ belongs to $C^{1,\lambda}_{loc}(\mathbb{R}^{N})$ or belongs to $C^{1,\lambda}_{loc}(\mathbb{R}^{N}\setminus\{0\})$, for some $\lambda\in(0,1)$.

Some classes of functions $\mu$ satisfying the hypotheses of the main Theorems are given in Section 2.

2. Weighted Hardy inequalities

Let $\mu$ a weight function in $\mathbb{R}^{N}$. We define the weighted Sobolev space $H^{1}_{\mu}=H^{1}(\mathbb{R}^{N},\mu(x)dx))$ as the space of functions in $L^{2}_{\mu}:=L^{2}(\mathbb{R}^{N},\mu(x)dx)$ whose weak derivatives belong to $(L_{\mu}^{2})^{N}$.

As first step we consider the following conditions on $\mu$ which we need to state a preliminary weighted Hardy inequality.

• $H_{1})$

$\quad\mu\geq 0$, $\mu\in L^{1}_{loc}(\mathbb{R}^{N})$;

• $H_{2})$

$\quad\nabla\mu\in L_{loc}^{1}(\mathbb{R}^{N})$;

• $H_{3})$

there exist constants $k_{1},k_{2}\in\mathbb{R}$, $k_{2}>2-N$, such that if

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it holds

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for any $\varepsilon>0$.

The condition $H_{3})$ contains the requirement that the scalar product $\alpha x\cdot\frac{\nabla\mu}{\mu}$ is bounded in $B_{R}$, $R>0$, while $\frac{\alpha x}{\varepsilon+|x|^{2}}\cdot\frac{\nabla\mu}{\mu}$ is bounded in $\mathbb{R}^{N}\setminus B_{R}$, where $B_{R}$ is a ball of radius $R$ centered in zero.

The reason we use the function $f_{\varepsilon}$, introduced in [16], will be clear in the proof of the weighted Hardy inequality which we will state below. Finally we observe that we need the condition $k_{2}>2-N$ to apply Fatou’s lemma in the proof of Theorem 2.1.

Theorem 2.1**.**

Under conditions $H_{1})$–$H_{3})$ there exists a positive constant $c$ such that

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for any function $\varphi\in C^{\infty}_{c}(\mathbb{R}^{N})$, where $c\in(0,c_{o}(N+k_{2})]$ with $c_{o}(N+k_{2})=\left(\frac{N+k_{2}-2}{2}\right)^{2}$.

Proof..

As first step we start from the integral of the square of the gradient of the function $\varphi$. Then we introduce $\psi=\frac{\varphi}{f_{\varepsilon}}$, with $f_{\varepsilon}$ defined in $H_{3})$, and integrate by parts taking in mind $H_{1})$ and $H_{2})$.

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Observing that

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and using hypothesis $H_{3})$ we deduce that

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The constant $-\alpha(N-2+k_{2})-\alpha^{2}$ is greater than zero since $\alpha<0$ and $k_{2}>2-N$, so by Fatou’s lemma we state the following estimate letting $\varepsilon\to 0$

[TABLE]

with $c=-\alpha(N-2+k_{2})-\alpha^{2}$. Finally we observe that

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attained for $\alpha_{o}=-\frac{N+k_{2}-2}{2}$.∎

Remark** 2.2****.**

In an alternative way we can define $f_{\varepsilon}$ in $H_{3})$ setting $\alpha=\alpha_{o}$ and get the estimate (2.1) with $c=c_{o}(N+k_{2})$. Although the result goes in the same direction, in the proof we point out that $c_{o}(N+k_{2})$ is the maximum value of the constant $c$.

Remark** 2.3****.**

In the case $\mu=1$ we obtain the classical Hardy inequality. We remark that if in the proof we introduce a function $f\in C^{\infty}(\mathbb{R}^{N})$ in place of $f_{\varepsilon}$, the inequality (2.3) can be used to get Hardy type inequalities

[TABLE]

where the potential $V=V(x)\in L^{1}_{loc}(\mathbb{R}^{N})$, $V(x)\geq 0$, is such that

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Operators perturbed by potentials of a more general type, for which the generation of semigroups was stated, have been investigated, for example, in [12, 13, 14] when $\mu=1$. For functions $\mu\neq 1$ such that $k_{2}\neq 0$ we have to modify the condition $H_{3})$ to get the Hardy type inequality (2.4) with respect to the measure $d\mu$.

Now we suppose that

• $H_{4})$

$\quad\mu\geq 0$, $\sqrt{\mu}\in H^{1}_{loc}(\mathbb{R}^{N})$;

• $H_{5})$

$\quad\mu^{-1}\in L_{loc}^{1}(\mathbb{R}^{N})$.

Let us observe that in the hypotheses $H_{4})$-$H_{5})$ the space $C^{\infty}_{c}(\mathbb{R}^{N})$ is dense in $H^{1}_{\mu}$ and $H^{1}_{\mu}$ is the completion of $C^{\infty}_{c}(\mathbb{R}^{N})$ with respect to the Sobolev norm

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(see [24]). For some interesting papers on density of smooth functions in weighted Sobolev spaces and related questions we refer, for example, to [20, 17, 6, 21, 15, 25, 7].

So we can deduce the following result from Theorem 2.1 by density argument.

Theorem 2.4**.**

Under conditions $H_{2})$–$H_{5})$ there exists a positive constant $c$ such that

[TABLE]

for any function $\varphi\in H^{1}_{\mu}$, where $c\in(0,c_{o}(N+k_{2})]$ with $c_{o}(N+k_{2})=\left(\frac{N+k_{2}-2}{2}\right)^{2}$.

We give some examples of functions $\mu$ which satisfy the hypotheses of Theorem 2.4.

We remark that, in the hypotheses $\mu=\mu(|x|)\in C^{1}$ for $|x|\in[r_{0},+\infty[$, $r_{0}>0$, a class of weight functions $\mu$ which satisfies $H_{3})$ is the following

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where $C$ is a constant depending on $\mu(r_{0})$ and $r_{0}$.

Indeed, in the case of radial functions, $\mu(x)=\mu(|x|)$, if we set $|x|=\rho$ the condition $H_{3})$ states that $\mu$ satisfies the following inequality

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which implies

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with

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Integrating in $[r_{0},r]$ we get

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from which

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Example** 2.5****.**

Another class of weight functions satisfying $H_{3})$, when $k_{1}=k_{2}=0$, consists of the bounded increasing functions, as, for example, $\cos e^{-|x|^{2}}$. Such a function verifies the requirements in the Theorem 2.4.

In the following example we consider a wide class of functions which contains the Gaussian measure and polynomial type measures. A class of functions which behaves as $\frac{1}{|x|^{\gamma}}$ when $|x|$ goes to zero.

Example** 2.6****.**

We consider the following weight functions

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and state for which values of $\gamma$ and $m$ the functions in (2.7) are “good” functions to get the weighted Hardy inequality (2.5).

The weight $\mu$ satisfies $H_{2})$, $H_{4})$ and $H_{5})$ if $\gamma>-N$. The condition $H_{3})$

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is fulfilled if

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In the case $\delta=0$ we only need to require that $\gamma\leq-k_{2}-\frac{k_{1}}{\alpha}\varepsilon$ and we are able to get the Caffarelli-Niremberg inequality

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While if $\gamma=0$ the inequality (2.5) holds, for $k_{1}$ large enough, with $k_{2}=0$ if $m=2$ and with $k_{2}<0$ if $m<2$.

In general to get (2.8) we need the following conditions on parameters and on the constant $k_{1}$:

• $i)$

$\qquad\gamma\in(-N,-k_{2}]$, $\delta=0$, $k_{1}=0$,

• $ii)$

$\qquad\gamma\in(-N,-k_{2}]$, $k_{1}\geq-2\alpha\delta$, $m=2$,

• $iii)$

$\qquad\gamma\in(-N,-k_{2})$, $k_{1}\geq\tilde{k}_{1}$, $m<2$,

where $\tilde{k}_{1}=\frac{\frac{m}{2}\left(1-\frac{m}{2}\right)^{\frac{2}{m}-1}(-\alpha\delta m)^{\frac{2}{m}}}{[\alpha(\gamma+k_{2})]^{\frac{2}{m}-1}}$, to get the inequality (2.5).

Example** 2.7****.**

The function $\mu(x)=[\log(1+|x|)]^{-\gamma}$, for $\gamma<N-2$, behaves as $\frac{1}{|x|^{\gamma}}$ when $|x|$ goes to 0. So can state the weighted Hardy inequality (2.5) with $k_{1}=0$ and $\gamma\in(-N,-k_{2}]$ as in the previous example.

3. Optimality of the constant

To state the optimality of the constant $c_{o}(N+K_{2})$ in the estimate (2.5) we need further assumptions on $\mu$ as usually it is done. We remark that in the proof of optimality the choice of the function $\varphi$ plays a fundamental role.

We suppose

• $H_{6})$

$\frac{\mu(x)}{|x|^{\delta}}\in L^{1}_{loc}(\mathbb{R}^{N})$ iff $\delta\leq N+k_{2}$.

We observe that the condition $H_{6})$ is necessary for the technique used to estimate the bottom of the spectrum of the operator $-L-V$ in the proof of the optimality. For example the functions $\mu$ such that

[TABLE]

verify $H_{6})$.

The result below states the optimality of the constant $c_{o}(N+k_{2})$ in the Hardy inequality.

Theorem 3.1**.**

In the hypotheses of Theorem 2.4 and if $H_{6})$ holds, for $c>c_{o}(N+k_{2})=\left(\frac{N+k_{2}-2}{2}\right)^{2}$ the inequality (2.5) doesn’t hold for any $\varphi\in H_{\mu}^{1}$.

Proof..

Let $\theta\in C_{c}^{\infty}(\mathbb{R}^{N})$ a cut-off function, $0\leq\theta\leq 1$, $\theta=1$ in $B_{1}$ and $\theta=0$ in $B_{2}^{c}$. We introduce the function

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where $\varepsilon>0$ and the exponent $\eta$ is such that

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The function $\varphi_{\varepsilon}$ belongs to $H_{\mu}^{1}$ for any $\varepsilon>0$.

For this choice of $\eta$ we obtain $\eta^{2}<c$, $|x|^{2\eta}\in L_{loc}^{1}(\mathbb{R}^{N},d\mu)$ and $|x|^{2\eta-2}\notin L_{loc}^{1}(\mathbb{R}^{N},d\mu)$.

Let us assume that $c>c_{o}(N+k_{2})$. Our aim is to prove that the bottom of the spectrum of the operator $-(L+V)$

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is $-\infty$. For this purpose we estimate at first the numerator in (3.1).

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where $C_{1}=\left(2\eta^{2}+2\|\nabla\theta\|_{\infty}\right)\int_{B_{1}^{c}}d\mu$.

Furthermore

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Putting together (3.2) and (3.3) we get from (3.1)

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Letting $\varepsilon\to 0$ in the numerator above, taking in mind that $|x|^{2\eta}\in L^{1}_{loc}(\mathbb{R}^{N},d\mu)$ and Fatou’s lemma, we obtain

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and, then, $\lambda_{1}=-\infty$. ∎

4. Kolmogorov operators and existence and nonexistence results

In the standard setting one considers $\mu\in C^{1,\lambda}_{loc}(\mathbb{R}^{N})$ for some $\lambda\in(0,1)$ and $\mu>0$ for any $x\in\mathbb{R}^{N}$.

We consider Kolmogorov operators

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on smooth functions, where the probability density $\mu$ in the drift term is not necessarily $(1,\lambda)$-Hölderian in the whole space but belongs to $C^{1,\lambda}_{loc}(\mathbb{R}^{N}\setminus\{0\})$.

These operators arise from the bilinear form integrating by parts

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The purpose is to get existence and nonexistence results for weak solutions to the the initial value problem on $L^{2}_{\mu}$ corresponding to the operator $L$ perturbed by an inverse square potential

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where $V(x)=\frac{c}{|x|^{2}}$, with $c>0$.

We say that $u$ is a weak solution to ($P$) if, for each $T,R>0$, we have

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and

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for all $\phi\in W_{2}^{2,1}(\mathbb{R}^{N}\times\left[0,T\right])$ having compact support with $\phi(\cdot,T)=0$, where $B_{R}$ denotes the open ball of $\mathbb{R}^{N}$ of radius $R$ centered at [math]. For any $\Omega\subset\mathbb{R}^{N}$, $W_{2}^{2,1}(\Omega\times(0,T))$ is the parabolic Sobolev space of the functions $u\in L^{2}(\Omega\times(0,T))$ having weak space derivatives $D_{x}^{\alpha}u\in L^{2}(\Omega\times(0,T))$ for $|\alpha|\leq 2$ and weak time derivative $\partial_{t}u\in L^{2}(\Omega\times(0,T))$ equipped with the norm

[TABLE]

Let us assume that the function $\mu$ is a probability density on $\mathbb{R}^{N}$, $\mu>0$. In the hypothesis

• $H_{7})$

$\quad\mu\in C_{loc}^{1,\lambda}(\mathbb{R}^{N})$, $\lambda\in(0,1)$

it is known that the operator $L$ with domain

[TABLE]

is the weak generator of a not necessarily $C_{0}$-semigroup in $C_{b}(\mathbb{R}^{N})$. Since $\int_{\mathbb{R}^{N}}Lu\,d\mu=0$ for any $u\in C_{c}^{\infty}(\mathbb{R}^{N})$, then $d\mu=\mu(x)dx$ is the invariant measure for this semigroup in $C_{b}(\mathbb{R}^{N})$. So we can extend it to a positivity preserving and analytic $C_{0}$-semigroup $\{T(t)\}_{t\geq 0}$ on $L^{2}_{\mu}$, whose generator is still denoted by $L$ (see [23]).

When the assumptions on $\mu$ allow degeneracy at one point, we require the following conditions to get $L$ generates a semigroup:

• $H_{8})$

$\quad\mu\in C_{loc}^{1,\lambda}(\mathbb{R}^{N}\setminus\{0\})$, $\lambda\in(0,1)$, $\mu\in H^{1}_{loc}(\mathbb{R}^{N})$, $\frac{\nabla\mu}{\mu}\in L^{r}_{loc}(\mathbb{R}^{N})$ for some $r>N$, and $\inf_{x\in K}\mu(x)>0$ for any compact set $K\subset\mathbb{R}^{N}$.

So by [1, Corollary 3.7]), we have that the closure of $(L,C_{c}^{\infty}(\mathbb{R}^{N}))$ on $L^{2}_{\mu}$ generates a strongly continuous and analytic Markov semigroup $\{T(t)\}_{t\geq 0}$ on $L^{2}_{\mu}$.

We observe that the function $e^{-\delta|x|^{m}}$ fully satisfies the condition $H_{8})$ while $\cos e^{-|x|^{2}}$ is $(1,\lambda)$-Hölderian in $\mathbb{R}^{N}$ (see Examples in Section 2).

For weight functions $\mu$ satisfying assumption $H_{7})$ or $H_{8})$ there are some interesting properties regarding the semigroup $\{T(t)\}_{t\geq 0}$ generated by the operator $L$. These properties listed in the Proposition below are well known under hypothesis $H_{7})$ (see [23]) and have been proved in [9] if $\mu$ satisfies $H_{8})$.

Proposition 4.1**.**

Assume that $\mu$ satisfies $H_{7})$ or $H_{8})$. Then the following assertions hold:

• (i)

$D(L)\subset H^{1}_{\mu}$.

• (ii)

For every $f\in D(L),\,g\in H^{1}_{\mu}$ we have

[TABLE]

• (iii)

$T(t)L^{2}_{\mu}\subset D(L)$* for all $t>0$.*

The following Theorem stated in [19] for functions $\mu$ satisfying condition $H_{7})$, was proved in [9] for functions $\mu$ under condition $H_{8})$.

Theorem 4.2**.**

Let $0\leq V(x)\in L^{1}_{loc}(\mathbb{R}^{N})$. Assume that the weight function $\mu$ satisfies $H_{4})$, $H_{5})$ and $H_{8})$. Then the following assertions hold:

• (i)

If $\lambda_{1}(L+V)>-\infty$, then there exists a positive weak solution $u\in C([0,\infty),L^{2}_{\mu})$ of $(P)$ satisfying

[TABLE]

for some constants $M\geq 1$ and $\omega\in{\mathbb{R}}$.

• (ii)

If $\lambda_{1}(L+V)=-\infty$, then for any $0\leq u_{0}\in L^{2}_{\mu}\setminus\{0\},$ there is no positive weak solution of $(P)$ satisfying (4.2).

To get existence and nonexistence of solutions to $(P)$ we put together the weighted Hardy inequality (2.4), Theorem 3.1 and Theorem 4.2. So we can state the following result.

Theorem 4.3**.**

Assume that the weight function $\mu$ satisfies hypotheses $H_{2})$–$H_{6})$, $H_{8})$ and $0\leq V(x)\leq\frac{c}{|x|^{2}}$. The following assertions hold:

1. (i)

If $0\leq c\leq c_{o}(N+k_{2})=\left(\frac{N+k_{2}-2}{2}\right)^{2}$, then there exists a positive weak solution $u\in C([0,\infty),L^{2}_{\mu})$ of $(P)$ satisfying

[TABLE]

for some constants $M\geq 1$, $\omega\in{\mathbb{R}}$, and any $u_{0}\in L^{2}_{\mu}$. 2. (ii)

If $c>c_{o}(N+k_{2})$, then for any $0\leq u_{0}\in L^{2}_{\mu},\,u_{0}\neq 0,$ there is no positive weak solution of $(P)$ with $V(x)=\frac{c}{|x|^{2}}$ satisfying (4.3).