A class of weighted Hardy inequalities and applications to evolution problems

Abstract

We state the following weighted Hardy inequality:

$$\begin{aligned} c_{o, \mu }\int _{{\mathbb {R}}^N}\frac{\varphi ^2 }{|x|^2}\, \mathrm{d}\mu \le \int _{{\mathbb {R}}^N} |\nabla \varphi |^2 \, \mathrm{d}\mu + K \int _{\mathbb {R}^N}\varphi ^2 \, \mathrm{d}\mu \quad \forall \, \varphi \in H_\mu ^1, \end{aligned}$$

in the context of the study of the Kolmogorov operators:

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

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 (Cabré and Martel in C R Acad Sci Paris 329 (11): 973–978, 1999) extended to Kolmogorov operators.

1 Introduction

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

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

where $\mu$ is a probability density on $\mathbb {R}^N$, and of the related evolution problems:

$$\begin{aligned} (P)\quad \left\{ \begin{array}{ll} \partial _tu(x,t)=Lu(x,t)+V(x)u(x,t),\quad \,x\in {\mathbb {R}}^N,\quad t>0,\\ u(\cdot ,0)=u_0\ge 0\in L_\mu ^2. \end{array} \right. \end{aligned}$$

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

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 [23], quantum cosmology (see, e.g., [5]), quantum mechanics [4] and combustion models [19].

The operator $\Delta +V$, $V(x)=\frac{c}{|x|^{2}}$, has the same homogeneity as the Laplacian and does not belong to the Kato 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 $c\le c_o=\left( \frac{N-2}{2} \right) ^{2}$ 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 blow-up 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 $0\le V\in L_{\mathrm{loc}}^1(\mathbb {R}^N)$ 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 $-(L+V)$:

$$\begin{aligned} \lambda _1(L+V):=\inf _{\varphi \in H^1_\mu {\setminus } \{0\}} \left( \frac{\int _{{\mathbb {R}}^N}|\nabla \varphi |^2\,\mathrm{d}\mu -\int _{{\mathbb {R}}^N}V\varphi ^2\,\mathrm{d}\mu }{\int _{{\mathbb {R}}^N}\varphi ^2\,\mathrm{d}\mu } \right) \end{aligned}$$

where $H^1_\mu$ is the suitable weighted Sobolev space.

When $\mu =1$, Cabré and Martel showed that the boundedness of $\lambda _1(\Delta +V)$, $0\le V\in L_{\mathrm{loc}}^1({\mathbb {R}}^N)$, is a necessary and sufficient condition for the existence of positive exponentially bounded in time solutions to the associated initial value problem. Later in [9, 20], 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, $Lu=\Delta u - \sum _{i=1}^{n}A(x-a_i)\cdot \nabla u$, $a_i\in \mathbb {R}^N$, $i=1,\ldots , n$, perturbed by multipolar inverse square potentials, weighted multipolar Hardy inequalities and related existence and nonexistence results were stated in [11]. In such a case, the invariant measure for these operators is $\mathrm{d}\mu =\mu _A (x) \mathrm{d}x =Ke^{-\frac{1}{2}\sum _{i=1}^{n}\left\langle A(x-a_i), x-a_i\right\rangle }\mathrm{d}x$.

There is a close relation between the estimate of the bottom of the spectrum $\lambda _1(L+V)$ and the weighted Hardy inequality with $V(x)=\frac{c}{|x|^2}$, $c\le c_{o,\mu }$,

$$\begin{aligned} \int _{\mathbb {R}^N}V\,\varphi ^2\,\mathrm{d}\mu \le \int _{\mathbb {R}^N}|\nabla \varphi |^2\mathrm{d}\mu + K\int _{\mathbb {R}^N} \varphi ^2\mathrm{d}\mu \quad \forall \,\varphi \in H^1_\mu ,\qquad K>0 \end{aligned}$$

(1)

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

In particular, the existence of positive solutions to (P) is related to the Hardy inequality (1) and the nonexistence is due to the optimality of the constant $c_{o, \mu }$.

The main results in the paper are, in Sect. 2, the weighted Hardy inequality (1) with measures which satisfy fairly general conditions and the optimality of the constant $c_{o, \mu }$ in Sect. 3.

The proof of the weighted Hardy inequality is different from the others in the literature. It is based on the introduction of a suitable $C^\infty$ function, and it can be used to prove inequality (1) with $0\le V\in L^1_{\mathrm{loc}}(\mathbb {R}^N)$ 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) does not hold if $c>c_{o,\mu }$, and this is a crucial point in the proof. The way to 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 [12].

Finally, we state an existence and nonexistence result in Sect. 4 following the Cabré–Martel’s approach and using some results stated in [9, 20] when the function $\mu$ belongs to $C^{1,\lambda }_{\mathrm{loc}}(\mathbb {R}^N)$ or belongs to $C^{1,\lambda }_{\mathrm{loc}}(\mathbb {R}^N{\setminus }\{0\})$, for some $\lambda \in (0,1)$.

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

2 Weighted Hardy inequalities

Let $\mu$ be a weight function in $\mathbb {R}^N$. We define the weighted Sobolev space $H^1_\mu =H^1(\mathbb {R}^N, \mu (x)\mathrm{d}x)$ as the space of functions in $L^2_\mu :=L^2(\mathbb {R}^N, \mu (x)\mathrm{d}x)$ 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:

($H_1$):

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

($H_2$):

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

($H_3$):

   there exist constants $k_1, k_2\in \mathbb {R}$, $k_2>2-N$, such that if

$$\begin{aligned} f_\varepsilon =(\varepsilon +|x|^{2})^{\frac{\alpha }{2}}, \quad \alpha < 0 , \quad \varepsilon >0, \end{aligned}$$

it holds

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

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 [17], 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 1.

Theorem 1

Under conditions ($H_1$–$H_3$), there exists a positive constant c such that

$$\begin{aligned} c \int _{\mathbb {R}^N} \frac{\varphi ^2}{|x|^2}\,\mathrm{d}\mu \le \int _{\mathbb {R}^N}|\nabla \varphi |^2\,\mathrm{d}\mu +k_1\int _{\mathbb {R}^N}\varphi ^2\,\mathrm{d}\mu , \end{aligned}$$

(2)

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 ($H_2$).

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N}|\nabla \varphi |^2\, \mathrm{d}\mu&= \int _{\mathbb {R}^N}|\nabla (\psi f_\varepsilon )|^2\, \mathrm{d}\mu \\&= \int _{\mathbb {R}^N}|\nabla \psi f_\varepsilon +\nabla f_\varepsilon \psi |^2\, \mathrm{d}\mu \\&= \int _{\mathbb {R}^N} |\nabla \psi |^2f_\varepsilon ^2\, \mathrm{d}\mu +\int _{\mathbb {R}^N} \psi ^2|\nabla f_\varepsilon |^2\, \mathrm{d}\mu +2\int _{\mathbb {R}^N}f_\varepsilon \psi \nabla \psi \cdot \nabla f_\varepsilon \, \mathrm{d}\mu \\&= \int _{\mathbb {R}^N} |\nabla \psi |^2f_\varepsilon ^2\, \mathrm{d}\mu +\int _{\mathbb {R}^N}\psi ^2 |\nabla f_\varepsilon |^2 \, \mathrm{d}\mu \\&\quad - \int _{\mathbb {R}^N} \psi ^2|\nabla f_\varepsilon |^2 \, \mathrm{d}\mu - \int _{\mathbb {R}^N}f_\varepsilon ^2\psi ^2 \frac{\Delta f_\varepsilon }{f_\varepsilon }\, \mathrm{d}\mu - \int _{\mathbb {R}^N}f_\varepsilon ^2\psi ^2 \frac{\nabla f_\varepsilon }{f_\varepsilon } \cdot \nabla \mu \, \mathrm{d}x. \end{aligned} \end{aligned}$$

(3)

Observing that

$$\begin{aligned} \Delta f_\varepsilon =\frac{\alpha (N-2+\alpha )|x|^2+\alpha \varepsilon N}{(\varepsilon +|x|^{2})^{2-\frac{\alpha }{2}}} \end{aligned}$$

and using hypothesis ($H_3$), we deduce that

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^N}|\nabla \varphi |^2\, \mathrm{d}\mu&\ge -\int _{\mathbb {R}^N}\frac{\Delta f_\varepsilon }{f_\varepsilon } \varphi ^2\, \mathrm{d}\mu -\int _{\mathbb {R}^N} \frac{\nabla f_\varepsilon }{f_\varepsilon } \cdot \nabla \mu \,\varphi ^2\, \mathrm{d}x \\&\ge -\left[ \alpha (N-2)+\alpha ^2\right] \int _{\mathbb {R}^N}\frac{|x|^2 }{(\varepsilon +|x|^2)^2}\varphi ^2\, \mathrm{d}\mu -\varepsilon \alpha N\int _{\mathbb {R}^N}\frac{\varphi ^2}{(\varepsilon +|x|^2)^2}\, \mathrm{d}\mu \\&\quad -k_1 \int _{\mathbb {R}^N}\varphi ^2\,\mathrm{d}\mu - k_2 \alpha \int _{\mathbb {R}^N}\frac{\varphi ^2}{\varepsilon +|x|^2}\, \mathrm{d}\mu \\&= [-\alpha (N-2+k_2)-\alpha ^2]\int _{\mathbb {R}^N}\frac{|x|^2 }{(\varepsilon +|x|^2)^2}\varphi ^2\, \mathrm{d}\mu \\&\quad - \varepsilon \alpha (N+k_2)\int _{\mathbb {R}^N}\frac{\varphi ^2}{(\varepsilon +|x|^2)^2}\, \mathrm{d}\mu - k_1 \int _{\mathbb {R}^N}\varphi ^2\,\mathrm{d}\mu . \end{aligned} \end{aligned}$$

(4)

The constant $-\alpha (N-2+k_2)-\alpha ^2$ is greater than zero for $-(N-2+k_2)<\alpha <0$ and $k_2>2-N$, so by Fatou’s lemma, we state the following estimate letting $\varepsilon \rightarrow 0$:

$$\begin{aligned} \int _{\mathbb {R}^N}|\nabla \varphi |^2\, \mathrm{d}\mu + k_1 \int _{\mathbb {R}^N}\varphi ^2\, \mathrm{d}\mu \ge c\int _{\mathbb {R}^N}\frac{\varphi ^2}{|x|^2}\,\mathrm{d}\mu , \end{aligned}$$

where $c=-\alpha (N-2+k_2)-\alpha ^2$. Finally, we observe that

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

attained for $\alpha _o= -\frac{N+k_2-2}{2}$. $\square$

Remark 1

In an alternative way, we can define $f_\varepsilon$ in ($H_3$) setting $\alpha =\alpha _o$ and get the estimate (2) 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

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 (4) can be used to get Hardy type inequalities:

$$\begin{aligned} \int _{\mathbb {R}^N} V\varphi ^2\,\mathrm{d}x \le \int _{\mathbb {R}^N}|\nabla \varphi |^2\,\mathrm{d}x \end{aligned}$$

(5)

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

$$\begin{aligned} -\frac{\Delta f}{f} \ge V \qquad \forall \, x\in \mathbb {R}^N. \end{aligned}$$

Operators perturbed by potentials of a more general type, for which the generation of semigroups was stated, have been investigated, for example, in [13,14,15] when $\mu =1$ and in [10] in weighted spaces. For functions $\mu \ne 1$ such that $k_2\ne 0$, we have to modify the condition ($H_3$) to get the Hardy type inequality (5) with respect to the measure $\mathrm{d}\mu$.

Now, we suppose that

($H_4$):

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

($H_5$):

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

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:

$$\begin{aligned} \Vert \cdot \Vert _{H^1_\mu }^2 := \Vert \cdot \Vert _{L^2_\mu }^2 + \Vert \nabla \cdot \Vert _{L^2_\mu }^2 \end{aligned}$$

(see [25]). For some interesting papers on density of smooth functions in weighted Sobolev spaces and related questions, we refer, for example, to [6, 7, 16, 18, 21, 22, 26].

So, we can deduce the following result from Theorem 1 by density argument.

Theorem 2

Under conditions ($H_2$–$H_5$), there exists a positive constant c such that

$$\begin{aligned} c \int _{\mathbb {R}^N} \frac{\varphi ^2}{|x|^2}\,\mathrm{d}\mu \le \int _{\mathbb {R}^N}|\nabla \varphi |^2\,\mathrm{d}\mu +k_1\int _{\mathbb {R}^N}\varphi ^2\,\mathrm{d}\mu , \end{aligned}$$

(6)

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.

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:

$$\begin{aligned} \mu (x) \ge C e^{-\frac{k_1}{2|\alpha |}|x|^2}|x|^{k_2-\frac{k_1}{|\alpha |}\varepsilon }, \quad \text { for}\quad |x|\ge r_0, \end{aligned}$$

(7)

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:

$$\begin{aligned} \frac{\alpha \rho }{\varepsilon +\rho ^2}\mu '(\rho ) \le \left( k_1+ \frac{ k_2 \alpha }{\varepsilon +\rho ^2}\right) \mu (\rho ), \end{aligned}$$

which implies

$$\begin{aligned} \mu '(\rho ) \ge a(\rho )\mu (\rho ) \end{aligned}$$

where

$$\begin{aligned} a(\rho )=\frac{k_1}{\alpha }\left( \frac{\varepsilon +\rho ^2}{\rho }\right) +\frac{ k_2}{\rho }. \end{aligned}$$

Integrating in $[r_0, r]$, we get

$$\begin{aligned} \mu (r)\ge \mu (r_0) e^{\int _{r_0}^r a(s)\mathrm{d}s}= \mu (r_0)\left( \frac{r}{r_0}\right) ^{k_2-\frac{k_1}{|\alpha |}\varepsilon } e^{-\frac{k_1}{2|\alpha |}(r^2-r_0^2)} \qquad \text { for}\quad r\ge r_0, \end{aligned}$$

from which we deduce that

$$\begin{aligned} \mu (r)\ge \frac{\mu (r_0)}{ r_0^{k_2-\frac{k_1}{|\alpha |}\varepsilon }} e^{\frac{k_1}{2|\alpha |} r_0^2} r^{k_2 -\frac{k_1}{|\alpha |}\varepsilon }e^{-\frac{k_1}{2|\alpha |} r^2} \qquad \text { for}\quad r\ge r_0. \end{aligned}$$

Example 1

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 of Theorem 2.

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

We consider the following weight functions:

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

(8)

We state the values of $\gamma$ and m for which the functions in (8) are “good”  functions to get the weighted Hardy inequality (6).

The weight $\mu$ satisfies ($H_2$), ($H_4$) and ($H_5$) if $\gamma >-N$. The condition ($H_3$):

$$\begin{aligned} \frac{\alpha (-\gamma -\delta m |x|^m)}{\varepsilon +|x|^2}\le k_1+\frac{\alpha k_2}{\varepsilon +|x|^2}, \end{aligned}$$

is fulfilled if

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

(9)

In the case $\delta =0$, we only need to require that $\gamma \le -k_2-\frac{k_1}{\alpha }\varepsilon$, and we are able to get the Caffarelli–Nirenberg inequality:

$$\begin{aligned} \left( \frac{N-2-\gamma }{2}\right) ^2 \int _{\mathbb {R}^N} \frac{\varphi ^2}{|x|^2}|x|^{-\gamma }\,\mathrm{d}x \le \int _{\mathbb {R}^N}|\nabla \varphi |^2|x|^{-\gamma }\,\mathrm{d}x \qquad \forall \varphi \in H_\mu ^1. \end{aligned}$$

While if $\gamma =0$, the inequality (6) 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 (9), 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):

$\qquad \gamma \in (-N,-k_2]$, $k_1\ge -2\alpha \delta$, $m= 2$,

(iii):

$\qquad \gamma \in (-N,-k_2)$, $k_1\ge {\tilde{k}}_1$, $m<2$,

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 (6).

Example 3

The function $\mu (x)=[\log (1+|x|)]^{-\gamma }$, for $\gamma <N-2$, behaves as $\frac{1}{|x|^\gamma }$ when |x| goes to 0. So, we can state the weighted Hardy inequality (6) 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 (6), 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

($H_6$):

   $\frac{\mu (x)}{|x|^\delta } \in L^1_{\mathrm{loc}}(\mathbb {R}^N)$ iff $\delta \le N+k_2$.

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

$$\begin{aligned} \lim _{|x|\rightarrow 0}\frac{\mu (|x|)}{|x|^{k_2}}=l, \qquad l>0, \end{aligned}$$

verify ($H_6$).

The result below states the optimality of the constant $c_o(N+k_2)$ in the Hardy inequality.

Theorem 3

In the hypotheses ($H_2$–$H_6$), the Hardy inequality (6) does not hold for any $\varphi \in H_\mu ^1$ if $c>c_o(N+k_2)=\left( \frac{N+k_2-2}{2} \right) ^2$.

Proof

Let $\theta \in C_c^\infty (\mathbb {R}^N)$ be a cut-off function, $0 \le \theta \le 1$, $\theta =1$ in $B_1$ and $\theta =0$ in $B_2^c$. We introduce the function:

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

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

$$\begin{aligned} \max \left\{ -\sqrt{c},-\frac{N+k_2}{2} \right\}< \eta < \min \left\{ -\frac{N+k_2-2}{2}, 0 \right\} . \end{aligned}$$

The function $\varphi _\varepsilon$ belongs to $H_\mu ^1$ for any $\varepsilon >0$.

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

Let us assume that $c>c_o(N+k_2)$. In order to state the result, we prove that bottom of the spectrum of the operator $-(L+V)$:

$$\begin{aligned} \lambda _1=\inf _{\varphi \in H^1_\mu {\setminus } \{0\}} \left( \frac{\int _{{\mathbb {R}}^N}|\nabla \varphi |^2\,\mathrm{d}\mu -\int _{{\mathbb {R}}^N}\frac{c}{|x|^2}\varphi ^2\,\mathrm{d}\mu }{\int _{{\mathbb {R}}^N}\varphi ^2\,\mathrm{d}\mu } \right) , \end{aligned}$$

(10)

is $-\infty$. For this purpose, we estimate at first the numerator in (10) with $\varphi =\varphi _\varepsilon$.

$$\begin{aligned}&\int _{\mathbb {R}^N} \left( |\nabla \varphi _\varepsilon |^2 -\frac{c}{|x|^2}\varphi _\varepsilon ^2 \right) \, \mathrm{d}\mu \nonumber \\&\quad = \int _{B_1}\left[ |\nabla (\varepsilon +|x|)^\eta |^2 -\frac{c}{|x|^2}(\varepsilon +|x|)^{2\eta }\right] \, \mathrm{d}\mu \nonumber \\&\qquad +\int _{B_1^c}\left[ |\nabla (\varepsilon +|x|)^\eta \theta |^2 -\frac{c}{|x|^2}(\varepsilon +|x|)^{2\eta }\theta ^2 \right] \, \mathrm{d}\mu \nonumber \\&\quad \le \int _{B_1}\left[ \eta ^2(\varepsilon +|x|)^{2\eta -2} -\frac{c}{|x|^2}(\varepsilon +|x|)^{2\eta } \right] \, \mathrm{d}\mu \nonumber \\&\qquad +\eta ^2\int _{B_1^c} (\varepsilon +|x|)^{2\eta -2}\theta ^2\, \mathrm{d}\mu + \int _{B_1^c}(\varepsilon +|x|)^{2\eta }|\nabla \theta |^2\, \mathrm{d}\mu \nonumber \\&\qquad + 2\eta \int _{B_1^c} \theta (\varepsilon +|x|)^{2\eta -1}\frac{x}{|x|}\cdot \nabla \theta \, \mathrm{d}\mu \nonumber \\&\quad \le \int _{B_1} (\varepsilon +|x|)^{2\eta } \left[ \frac{\eta ^2}{(\varepsilon +|x|)^2} -\frac{c}{|x|^2}\right] \,\mathrm{d}\mu \nonumber \\&\qquad + 2\eta ^2\int _{B_1^c}(\varepsilon +|x|)^{2\eta -2}\theta ^2\,\mathrm{d}\mu + 2\int _{B_1^c}(\varepsilon +|x|)^{2\eta }|\nabla \theta |^2\,\mathrm{d}\mu \nonumber \\&\quad \le \int _{B_1} (\varepsilon +|x|)^{2\eta } \left[ \frac{\eta ^2}{(\varepsilon +|x|)^2} -\frac{c}{|x|^2}\right] \,\mathrm{d}\mu +C_1, \end{aligned}$$

(11)

where $C_1=\left( 2\eta ^2+ 2\Vert \nabla \theta \Vert _\infty \right) \int _{B_1^c}\mathrm{d}\mu$.

Furthermore,

$$\begin{aligned} \int _{\mathbb {R}^N}\varphi _\varepsilon ^2\,\mathrm{d}\mu \ge \int _{B_2{\setminus } B_1 }(\varepsilon + |x|)^{2\eta }\theta ^2\,\mathrm{d}\mu =C_{2,\varepsilon }. \end{aligned}$$

(12)

Putting together (11) and (12), we get from (10):

$$\begin{aligned} \lambda _1 \le \frac{\int _{B_1}(\varepsilon +|x|)^{2\eta }\left[ \frac{\eta ^2}{(\varepsilon +|x|)^2}-\frac{c}{|x|^2} \right] \,\mathrm{d}\mu +C_1}{C_{2,\varepsilon }}. \end{aligned}$$

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

$$\begin{aligned} \lim \limits _{\varepsilon \rightarrow 0}\int _{B_1}(\varepsilon +|x|)^{2\eta }\left[ \frac{\eta ^2}{(\varepsilon +|x|)^2}-\frac{c}{|x|^2} \right] \,\mathrm{d}\mu \le -(c-\eta ^2)\int _{B_1}|x|^{2\eta -2}\,\mathrm{d}\mu =-\infty , \end{aligned}$$

and then, $\lambda _1=-\infty$. $\square$

4 Kolmogorov operators and existence and nonexistence results

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

We consider Kolmogorov operators:

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

(13)

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 $C^{1,\lambda }_{\mathrm{loc}}(\mathbb {R}^N{\setminus }\{0\})$.

These operators arise from the bilinear form integrating by parts:

$$\begin{aligned} a_\mu (u,v)=\int _{{\mathbb {R}}^N}\nabla u\cdot \nabla v\,\mathrm{d}\mu =-\int _{{\mathbb {R}}^N}(Lu)v\,\mathrm{d}\mu . \end{aligned}$$

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

$$\begin{aligned} (P)\quad \left\{ \begin{array}{ll} \partial _tu(x,t)=Lu(x,t)+V(x)u(x,t),\quad \,x\in {\mathbb {R}}^N, t>0,\\ u(\cdot ,0)=u_0\ge 0\in L_\mu ^2, \end{array} \right. \end{aligned}$$

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:

$$\begin{aligned} u\in C(\left[ 0, T \right] , L^2_\mu ), \quad Vu\in L^1(B_R \times \left( 0,T\right) , \mathrm{d}\mu \mathrm{d}t ) \end{aligned}$$

and

$$\begin{aligned} \int _0^T \int _{\mathbb {R}^N}u(-\partial _t\phi - L\phi )\,\mathrm{d}\mu \mathrm{d}t -\int _{\mathbb {R}^N}u_0\phi (\cdot ,0)\,\mathrm{d}\mu = \int _0^T \int _{\mathbb {R}^N} Vu\phi \, \mathrm{d}\mu \mathrm{d}t \end{aligned}$$

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 0. 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 $|\alpha |\le 2$ and weak time derivative $\partial _t u \in L^2(\Omega \times (0,T))$ equipped with the norm:

$$\begin{aligned} \begin{aligned} \Vert u\Vert _{W_2^{2,1}(\Omega \times (0,T))}&:= \left( \phantom {\sum _{1\le |\alpha |\le 2}} \Vert u\Vert _{L^2(\Omega \times (0,T))}^2 + \Vert \partial _t u\Vert _{L^2(\Omega \times (0,T))}^2 \right. \\&\quad \left. \, + \sum _{1\le |\alpha |\le 2} \Vert D^{\alpha }u\Vert _{L^2(\Omega \times (0,T))}^2 \right) ^{\frac{1}{2}}. \end{aligned} \end{aligned}$$

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

($H_7$):

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

it is known that the operator L with domain

$$\begin{aligned} D_{\mathrm{max}}(L)=\lbrace u\in C_b(\mathbb {R}^N)\cap W_{\mathrm{loc}}^{2,p}(\mathbb {R}^N) \text { for all } 1<p<\infty , Lu\in C_b(\mathbb {R}^N)\rbrace \end{aligned}$$

is the weak generator of a not necessarily $C_0$-semigroup in $C_b(\mathbb {R}^N)$. Since $\int _{\mathbb {R}^N}Lu\,\mathrm{d}\mu =0$ for any $u\in C_c^{\infty }(\mathbb {R}^N)$, $\mathrm{d}\mu =\mu (x)\mathrm{d}x$ 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 $\lbrace T(t)\rbrace _{t\ge 0}$ on $L^2_\mu$, whose generator is still denoted by L (see [24]).

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

($H_8$):

$\quad \mu \in C_{\mathrm{loc}}^{1,\lambda }(\mathbb {R}^N{\setminus } \{0\})$, $\lambda \in (0,1)$, $\mu \in H^{1}_{\mathrm{loc}}(\mathbb {R}^N)$, $\frac{\nabla \mu }{\mu }\in L^r_{\mathrm{loc}}(\mathbb {R}^N)$ 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 $\lbrace T(t)\rbrace _{t\ge 0}$ 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 Sect. 2).

For weight functions $\mu$ satisfying assumption ($H_7$) or ($H_8$), there are some interesting properties regarding the semigroup $\{T(t)\}_{t\ge 0}$ generated by the operator L. These properties listed in the Proposition below are well known under hypothesis ($H_7$) (see [24]) and have been proved in [9] if $\mu$ satisfies ($H_8$).

Proposition 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:

  $$\begin{aligned} \int L f g \,\mathrm{d}\mu =-\int \nabla f\cdot \nabla g\,\mathrm{d}\mu . \end{aligned}$$

• (iii) $T(t)L^2_\mu \subset D(L)$ for all $t>0$.

The following Theorem stated in [20] for functions $\mu$ satisfying condition ($H_7$) was proved in [9] for functions $\mu$ under condition ($H_8$).

Theorem 4

Let $0\le V(x)\in L^1_{\mathrm{loc}}(\mathbb {R}^N)$. 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

  $$\begin{aligned} \Vert u(t)\Vert _{L^2_\mu }\le Me^{\omega t}\Vert u_0\Vert _{L^2_\mu },\quad t\ge 0 \end{aligned}$$

  (14)

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

• (ii) If $\lambda _1(L+V)=-\infty$, then for any $0\le u_0\in L^2_\mu {\setminus } \{0\},$ there is no positive weak solution of (P) satisfying (14).

To get existence and nonexistence of solutions to (P), we put together the weighted Hardy inequality (2), Theorems 3 and 4. So, we can state the following result.

Theorem 5

Assume that the weight function $\mu$ satisfies hypotheses ($H_2$–$H_6\mathrm{)}$, ($H_8$) and $0\le V(x)\le \frac{c}{|x|^2}$. The following assertions hold:

• (i) If $0\le c\le c_o(N+k_2)=\left( \frac{N+k_2-2}{2} \right) ^2$, then there exists a positive weak solution $u\in C([0,\infty ),L^2_\mu )$ of (P) satisfying

  $$\begin{aligned} \Vert u(t)\Vert _{L^2_\mu }\le Me^{\omega t}\Vert u_0\Vert _{L^2_\mu },\quad t\ge 0 \end{aligned}$$

  (15)

  for some constants $M\ge 1$, $\omega \in {\mathbb {R}}$, and any $u_0\in L^2_\mu$.

• (ii) If $c> c_o(N+k_2)$, then for any $0\le u_0\in L^2_\mu ,\,u_0\ne 0,$ there is no positive weak solution of (P) with $V(x)=\frac{c}{|x|^2}$ satisfying (15).