Weighted Hardy's inequalities and Kolmogorov-type operators

TL;DR

This paper establishes general conditions for weighted Hardy inequalities with respect to probability measures, determines the optimal constants, and explores their connection to Kolmogorov equations with singular potentials, including existence of solutions.

Contribution

It provides new criteria for weighted Hardy inequalities involving probability measures and links these to Kolmogorov operators with singular potentials, including optimal constant determination.

Findings

Derived conditions for weighted Hardy inequalities with probability measures.

Identified the optimal constant for the inequality.

Connected inequalities to the existence of solutions for perturbed Kolmogorov equations.

Abstract

We give general conditions to state the weighted Hardy inequality \[ c\int_{\mathbb{R}^N}\frac{\varphi^2} {|x|^2}d\mu\leq\int_{\mathbb{R}^N}|\nabla \varphi |^2 d\mu+C\int_{\mathbb{R}^N} \varphi^2d\mu,\quad \varphi\in C_c^{\infty}(\mathbb{R}^N),\,c\leq c_{0,\mu}, \] with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential \[ Lu+Vu=\left(\Delta u+\frac{\nabla \mu}{\mu}\cdot \nabla u\right)+\frac{c}{|x|^2}u \] for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\nabla \mu}{\mu}= -|x|^{m-2}x$ with $m> 0$.

Equations

$$c{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}d\mu \le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^{2}d\mu +C{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu ,\phi \in C_c^{\infty }({\mathbb{R}}^N),c\le c_{0,\mu },$$

$$Lu+Vu=\left(\Delta u+\frac{\nabla \mu }{\mu }\cdot \nabla u\right)+\frac{c}{|x{|}^2}u$$

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

$$a_{\mu }(u,v)={\int }_{{\mathbb{R}}^N}\nabla u\cdot \nabla \overline{v}d\mu$$

$$a_{\mu }(u,v)=-{\int }_{{\mathbb{R}}^N}Lu\overline{v}d\mu ,u,v\in C_c^{\infty }({\mathbb{R}}^N).$$

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

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

$$\Vert u(t){\Vert }_{L_{\mu }^2}\le M{e}^{\omega t}\Vert u_0{\Vert }_{L_{\mu }^2},t\ge 0$$

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

$$U_{\mu }:=\frac{1}{4}{\left|\frac{\nabla \mu }{\mu }\right|}^2-\frac{1}{2}\frac{\Delta \mu }{\mu }.$$

$$c_{0,\mu }:=\underset{x\to 0}{\mathrm{liminf}}\left(c_0(N)-|x{|}^2{U}_{\mu }\right)$$

$$U:=U_{\mu }-\frac{1}{|x{|}^2}\underset{x\to 0}{\mathrm{limsup}}|x{|}^2{U}_{\mu }$$

$$|x{|}^{2}U(x)\le \frac{1}{4}\frac{1}{|\log |x|{|}^2},\forall x\in B_{R_0}.$$

$$\underset{\lambda \to 0^{+}}{\mathrm{limsup}}\lambda {\int }_{B_1}|x{|}^{\lambda -N_0}d\mu =+\infty ,$$

$$\Vert u(t){\Vert }_{L_{\mu }^2}\le M{e}^{\omega t}\Vert u_0{\Vert }_{L_{\mu }^2},t\ge 0$$

$$\mu (x)=k{e}^{-b|x{|}^m}$$

$$Au=\Delta u-bm|x{|}^{m-2}x\cdot \nabla u+\frac{c}{|x{|}^2}u,$$

$${\left(\frac{N-\beta -2}{2}\right)}^2{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{|x{|}^2}|x{|}^{-\beta }dx\le {\int }_{{\mathbb{R}}^N}|\nabla \phi {|}^2|x{|}^{-\beta }dx,\phi \in H^1({\mathbb{R}}^N),\beta <N-2.$$

$$\mu \sim {\left(\log \frac{1}{|x|}\right)}^{\alpha },$$

$${\int }_K{\left|\nabla {\mu }^{\frac{1}{2}}\right|}^{2}dx={\int }_K\frac{1}{4}\frac{|\nabla \mu {|}^2}{\mu }dx\le \frac{1}{4}{\left({\int }_K{\left|\frac{\nabla \mu }{\mu }\right|}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_K{\left|\nabla \mu \right|}^{2}dx\right)}^{\frac{1}{2}}<\infty ,$$

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

$$u\in C([0,T],L_{\mu }^2),Vu\in L^1(B_R\times (0,T),d\mu dt)and$$

$${\int }_0^T{\int }_{{\mathbb{R}}^N}u(-{\partial }_t\varphi -L\varphi )d\mu dt-{\int }_{{\mathbb{R}}^N}{u}_0\varphi (\cdot ,0)d\mu ={\int }_0^T{\int }_{{\mathbb{R}}^N}Vu\varphi d\mu dt$$

$$\int L\phi d\mu =0\text{ for every }\phi \in C_c^{\infty }({\mathbb{R}}^N).$$

$${\int }_{{\mathbb{R}}^N}Lu\overline{v}d\mu =-\int \nabla u\cdot \nabla \overline{v}d\mu ,\text{ for every }u,v\in C_c^{\infty }({\mathbb{R}}^N).$$

$$\int Lf\overline{g}d\mu =-\int \nabla f\cdot \nabla \overline{g}d\mu .$$

$$\Vert \nabla u_n-\nabla u_m{\Vert }_{\mu }^2$$

$$\le \Vert u_n-u_m{\Vert }_{\mu }\Vert L{u}_n-L{u}_m{\Vert }_{\mu }.$$

$$-\int L{u}_n\overline{\phi }d\mu =\int \nabla u_n\cdot \nabla \overline{\phi }d\mu$$

$$-\int Lu\overline{\phi }d\mu =\int G\cdot \nabla \overline{\phi }d\mu .$$

$$\int {\partial }_i{u}_n\overline{\psi }dx=-\int u_n{\partial }_i\overline{\psi }dx.$$

Full text

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with respect to a probability measure $d\mu$. Moreover, the optimality of the constant $c_{0,\mu}$ is given. The inequality is related to the following Kolmogorov equation perturbed by a singular potential

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for which the existence of positive solutions to the corresponding parabolic problem can be investigated. The hypotheses on $d\mu$ allow the drift term to be of type $\frac{\nabla\mu}{\mu}=-|x|^{m-2}x$ with $m>0$.

The 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)

1. Introduction

We denote by $L$ be the Kolmogorov operator

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defined on smooth functions. In the standard setting one considers $\mu\in C_{loc}^{1,\alpha}\left(\mathbb{R}^{N}\right)$ for some $\alpha\in(0,1)$ and $\mu(x)>0$ for all $x\in\mathbb{R}^{N}$. In this case the elliptic operator $L$ has coefficients belonging to $C_{loc}^{\alpha}\left(\mathbb{R}^{N}\right)$. Therefore, one can associate to $L$ a semigroup $\{T(t)\}_{t\geq 0}$ (not necessary strongly continuous) in the space of bounded continuous functions, cf. [5]. Considering the measure $d\mu=\mu(x)dx$ and the weighted space $L^{2}_{\mu}:=L^{2}(\mathbb{R}^{N},d\mu)$, the operator $L$ can also be defined via the bilinear form

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on $H^{1}_{\mu}=H^{1}(\mathbb{R}^{N},d\mu)$, the Sobolev space of functions whose weak derivatives belong to $L^{2}_{\mu}.$ Indeed, integrating by parts we get

[TABLE]

In particular, $\int_{\mathbb{R}^{N}}Lu\,d\mu=0$ for every $u\in C_{c}^{\infty}(\mathbb{R}^{N})$. Then $d\mu$ is an invariant measure of $\{T(t)\}$ and hence $\{T(t)\}$ can be extended to a positivity preserving and analytic $C_{0}$-semigroup on $L^{2}_{\mu}$ (see for example [5]).

Recently in [12] and [11], a special class of operators of type $L$ perturbed by the inverse square potential $V(x)=\frac{c}{|x|^{2}}$ was considered and the associated evolution equation

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was studied.

It is well known that the potential $\frac{c}{|x|^{2}},\,c>0,$ is highly singular in the sense that it belongs to a borderline case where the strong maximum principle and Gaussian bounds fail, cf. [3]. Moreover, it is not in the Kato class potentials. If $V\leq\frac{C}{|x|^{2-\varepsilon}}$, then the initial value problem is well-posed. But for $\varepsilon=0$ the problem $\frac{\partial u}{\partial t}=\Delta u+V$ may not have positive solutions. In [4] Baras and Goldstein show that the evolution equation associated to $\Delta+V$ admits a unique positive solution if $c\leq c_{0}(N):=\left(\frac{N-2}{2}\right)^{2}$ and no positive solutions exist if $c>c_{0}(N)$ (see also [7]). When it exists, the solution is exponentially bounded, on the contrary, if $c>c_{0}(N)$, there is the so called instantaneous blowup phenomena.

Replacing the Laplacian by the Kolmogorov operator $L$ a similar behaviour was obtained in [12]. The result was given using a relation between the weak solution of $(P_{V})$ and the bottom of the spectrum of the operator $-(L+V)$

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Cabré and Martel in [7] show that the boundedness of $\lambda_{1}(\Delta+V)$ is a necessary and sufficient condition for the existence of positive and exponentially bounded in time solutions to the associated initial value problem. This result was extended in [12] to the operator $L+V$.

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 a weighted multipolar Hardy inequality and related existence and nonexistence results were stated in [8]. In such a case the invariant measure for these operators is $d\mu=\mu_{A}(x)dx=Ke^{-\frac{1}{2}\sum_{i=1}^{n}\langle A(x-a_{i}),x-a_{i}\rangle}dx$.

Assuming that $0<\mu\in C^{1+\alpha}_{loc}({\mathbb{R}}^{N})$ is a probability density on ${\mathbb{R}}^{N}$ we recall the following result, see [12, Theorem 2.1].

Theorem 1.1**.**

If $0\leq V\in L_{loc}^{1}({\mathbb{R}}^{N})$. Then the following hold:

1. (i)

*If $\lambda_{1}(L+V)>-\infty$, then there exists a nonnegative weak solution *

$u\in C([0,\infty),L^{2}_{\mu})$* of $(P_{V})$ satisfying*

[TABLE]

for some constants $M\geq 1$ and $\omega\in{\mathbb{R}}$. 2. (ii)

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

The existence of positive solutions to $(P_{V})$ is related to Hardy’s inequality on the weighted space $L^{2}_{\mu}$. The nonexistence of solutions is due to the optimality of the constant in the Hardy inequality. Therefore, studying the bottom of the spectrum is equivalent to studying the weighted Hardy inequality

[TABLE]

and the sharpness of the best constant possible.

A special case is given when $\mu_{A}(x)=Ke^{-\frac{1}{2}\langle Ax,x\rangle}$, where $A$ is a positive real Hermitian $N\times N$ matrix and $K$ is a normalizing constant. The operator $L$ becomes the well-known symmetric Ornstein-Uhlenbeck operator $Lu=\Delta u-Ax\cdot\nabla u$. Using this approach it was proved in [12] that if $\mu_{A}(x)=Ke^{-\frac{1}{2}\langle Ax,x\rangle}$, then there exists an exponentially bounded weak solution $u\in C([0,\infty),L^{2}_{\mu_{A}})$ provided that $0\leq c\leq c_{0}(N)$, and no positive exponentially bounded weak solution exists if $c>c_{0}(N)$.

The result was generalized in [11] for the density measure $\mu(x)=Ke^{-\sigma(x)}$ with $c_{1}|x|^{2}\leq\sigma(x)\leq c_{2}|x|^{2}$. Furthermore, under more general hypotheses on $d\mu$ the argument was extended to a larger class of Kolmogorov operators, including the case $\sigma(x)\sim|x|^{m}$ with $m>2$ and $c\leq c_{\sigma}$, for a suitable constant $c_{\sigma}$ which is not the optimal one.

In this paper we generalize these results for a larger class of measures $d\mu$, including the case $\mu(x)=Ke^{-\frac{|x|^{m}}{m}}$ with $m>0$ and obtaining also the optimality of the constant. We observe that such $\mu$ requires to satisfy more general hypotheses, which allow degeneracy at one point. Indeed, for $0<m<1$ such a measure does not belong to $C^{1,\alpha}_{loc}(\mathbb{R}^{N})$ since the gradient is not bounded in [math]. Then, we will consider measures which are not necessarily $(1,\alpha)$-Hölderian in the whole space but such that $\mu\in C^{1,\alpha}_{loc}(\Omega)\ \textrm{where}\ \Omega:=\mathbb{R}^{N}\setminus\{0\}$. Firstly, we need that the unperturbed operator $L$ generates a semigroup. Hence, further conditions on $\mu$ are required in order to guarantee generation results on $L^{2}_{\mu}$. We consider the following hypotheses.

Hypothesis $(H1)$:

1. i)

$\mu\in C^{1,\alpha}_{loc}(\Omega)$; 2. ii)

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

By [2, Corollary 3.7] we have that the closure of $(L,C_{c}^{\infty}(\mathbb{R}^{N}))$ on $L^{2}_{\mu}$ generates a strongly continuous Markov semigroup $T(t)$ on $L^{2}_{\mu}$, which is also analytic. Thanks to this result we can claim that, under Hypothesis $(H1)$, Theorem 1.1 holds.

The second step is, then, to obtain a weighted Hardy inequality. To this purpose we observe that the operator $L+V$ in $L^{2}_{\mu}$ is equivalent to the Schrödinger operator $H=\Delta+(U_{\mu}+V)$ in $L^{2}({\mathbb{R}^{N}})$, where

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Indeed, taking the transformation $T\varphi=\frac{1}{\sqrt{\mu}}\varphi$ we have $L+V=THT^{-1}$. Now, roughly speaking, if $V=\frac{c}{|x|^{2}}$ we expect Hardy’s inequality to hold if $U_{\mu}+\frac{c}{|x|^{2}}\leq\frac{c_{0}(N)}{|x|^{2}}$ in a neighbourhood of the origin, that is $c\leq c_{0}(N)-|x|^{2}U_{\mu}$.

Thus, we consider the following hypothesis on $\mu(x)$.

Hypothesis $(H2)$:

1. i)

$\mu\geq 0$, $\mu^{1/2}\in H^{1}_{loc}(\mathbb{R}^{N})$, $\Delta\mu\in L^{1}_{loc}(\mathbb{R}^{N})$; 2. ii)

the constant

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is finite; 3. iii)

for every $R>0$ the function

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is bounded from above in $\mathbb{R}^{N}\setminus B_{R}$; 4. iv)

there exists a $R_{0}>0$ such that

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Under the assumption $(H2)$ we obtain the weighted Hardy inequality (2) for any $c\leq c_{0,\mu}$. If condition ${\rm iv)}$ of $(H2)$ is not satisfied we still obtain the weighted Hardy inequality if we only assume

**Hypothesis $(H2^{\prime})$: **

• $\bullet$

Conditions $\rm i),ii),iii)$ of $(H2)$ hold.

In this case the constant $c_{0,\mu}$ is not achieved and we obtain (2) for any $c<c_{0,\mu}.$

As regards the optimality, we consider the following hypothesis.

**Hypothesis $(H3)$: **

• i)

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

• ii)

There exists \sup_{\delta\in\mathbb{R}}\big{\{}\frac{1}{|x|^{\delta}}\in L^{1}_{loc}(\mathbb{R}^{N},d\mu)\big{\}}=:N_{0}.

Under condition $(H3)$ Hardy’s inequality does not hold if $c>c_{0}(N_{0})=\left(\frac{N_{0}-2}{2}\right)^{2}$. If, instead, we have

**Hypothesis $(H3^{\prime})$: ** Conditions $i)$ and $ii)$ of $(H3)$ hold and

• iii)

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then the inequality does not hold if $c\geq c_{0}(N_{0})$.

It is obvious that we have the best result when $(H2)$ and $(H3)$ (respectively $(H2^{\prime})$ and $(H3^{\prime})$) hold and the constant $c_{0,\mu}$ coincides with the constant $c_{0}(N_{0})$.

Therefore, we can state our main results.

Theorem 1.2**.**

Assume assumptions $(H2)$ and $(H3)$ and $c_{0,\mu}=c_{0}(N_{0})$. Then the weighted Hardy inequality (2) holds if and only if $c\leq c_{0}(N_{0})$.

Theorem 1.3**.**

Assume assumptions $(H2^{\prime})$ and $(H3^{\prime})$ with $N_{0}>2$ and $c_{0,\mu}=c_{0}(N_{0})$. Then (2) holds if and only if $c<c_{0}(N_{0})$.

Finally, putting together the weighted Hardy inequality (2) and Theorem 1.1, we can state the following.

Theorem 1.4**.**

Assume that hypotheses $(H1)$, $(H2)$ and $(H3)$ hold with $c_{0,\mu}=c_{0}(N_{0})$ and $0\leq V(x)\leq\frac{c}{|x|^{2}}$, then the following assertions are satisfied:

1. (i)

*If $0\leq c\leq c_{0}(N_{0})=\left(\frac{N_{0}-2}{2}\right)^{2}$, then there exists a weak solution $u\in C([0,\infty),L^{2}_{\mu})$ of $(P_{V})$ satisfying *

[TABLE]

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

If $c>c_{0}(N_{0})$, then for any $0\leq u_{0}\in L^{2}_{\mu},\,u_{0}\neq 0,$ there is no positive weak solution of $(P_{V})$ with $V(x)=\frac{c}{|x|^{2}}$ satisfying (3).

If instead, assumptions $(H2^{\prime})$ and $(H3^{\prime})$ with $N_{0}>2$ are fulfilled, the same statement holds but the constant $c_{0}(N_{0})$ is not achieved.

These hypotheses on the measure $\mu$ allow us to treat the case

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for $b,m>0$, associated to the operator

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since one can see that $\mu$ satisfies $(H1)$, $(H2)$ and $(H3)$ with constant $c_{0,\mu}=c_{0}(N_{0})=c_{0}(N)$. Therefore, for such $\mu$, Theorem 1.4 holds.

Moreover, under the same assumptions, one can also consider the measures $\mu_{\beta}(x)=k\frac{1}{|x|^{\beta}}e^{-b|x|^{m}}$. For such measures Hypothesis $(H1)$ is not fulfilled, however we obtain the weighted Hardy inequality with best constant. Indeed, also in this case, the constant of $(H2)$ coincides with the best constant of $(H3)$ and it depends upon the parameter $\beta$ with $\beta<N-2$. We have explicitly $N_{0}=N-\beta$ and $c_{0,\mu}=c_{0}(N_{0})=\left(\frac{N-\beta-2}{2}\right)^{2}$.

For $\mu(x)=\frac{1}{|x|^{\beta}}$ we recover the well known Caffarelli-Nirenberg inequality

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For measures behaving like a logarithm near the origin

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we obtain (2) with constant $c_{0}(N_{0})=c_{0}(N)$. If $\alpha\leq 0$ the constant is achieved and it is the best one. Indeed, $\mu$ satisfies $(H1)$, $(H2)$ and $(H3)$ provided that $\alpha\leq 0$. If instead $\alpha>0$ then $\mu$ satisfies $(H1)$, $(H2^{\prime})$ and $(H3^{\prime})$. So, the constant is not achieved, but it still is the best one.

Finally, we also provide an example in which the constant in (2) of $(H2^{\prime})$ does not coincide with the optimal one of $(H3)$.

2. Weak solutions and bottom of the spectrum

In this section we prove that, under condition $(H1)$ on $\mu$, Theorem 1.1 holds. Firstly we observe that $C_{c}^{\infty}(\mathbb{R}^{N})$ is dense in $H^{1}_{\mu}$. This is given by the condition $\mu^{1/2}\in H^{1}_{loc}(\mathbb{R}^{N})$ (see [15, Theorem 1.1] ) which is ensured by Hypothesis $ii)$ of $(H1)$. Indeed, $\mu\in H^{1}_{loc}(\mathbb{R}^{N})$ implies $\mu\in L^{1}_{loc}(\mathbb{R}^{N})$ and $\nabla\mu\in L^{2}_{loc}(\mathbb{R}^{N})$. Moreover, $\frac{\nabla\mu}{\mu}\in L_{loc}^{r}(\mathbb{R}^{N})$ implies $\frac{\nabla\mu}{\mu}\in L_{loc}^{2}(\mathbb{R}^{N})$. Then $\mu^{\frac{1}{2}}\in L^{2}_{loc}(\mathbb{R}^{N})$ and

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for every compact set $K\subset\mathbb{R}^{N}$. Moreover, one also obtains that $C_{c}^{\infty}\left(\Omega\right)$ is dense in $H^{1}_{\mu}$ by Hypothesis $ii)$ of $(H1)$ (see the Appendix).

Now, we precise the definition of weak solutions. Let us recall the problem

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We say that $u$ is a weak solution to $(P_{V})$ if, for each $T,\,R>0$, we have

[TABLE]

[TABLE]

for all $\phi\in W_{loc}^{2,1}(Q_{T})$ having compact support with $\phi(\cdot,T)=0$, where $B_{R}$ denotes the open ball of ${\mathbb{R}}^{N}$ of radius $R$ and center [math] and for $T>0$, we write $Q_{T}$ for ${\mathbb{R}}^{N}\times[0,T]$.

Theorem 1.1 is based on Cabré-Martel’s idea and it was proved in [12, Theorem 2.1] for measures $\mu$ belonging to $C^{1,\alpha}_{loc}(\mathbb{R}^{N})$. The proof relies on certain properties of the operator $L$ and its corresponding semigroup $T(t)$ in $L^{2}_{\mu}$. Furthermore, the strict positivity on compact sets of $T(t)u_{0}$, if $0\leq u_{0}\in L^{2}_{\mu}\setminus\{0\}$ is required.

Hence, in order to claim that Theorem 1.1 holds in our situation, we only have to ensure that these properties hold. This is stated in Proposition 2.1 and Lemma 2.2 below.

We recall that the measure $\mu(dx)$ is the infinitesimally invariant measure for the operator $L$, i.e.

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Moreover, the operator $L$ is symmetric on $C_{c}^{\infty}(\mathbb{R}^{N})$, i.e.

[TABLE]

Hence, by [2, Corollary 3.7], we have that the closure of $(L,C_{c}^{\infty}(\mathbb{R}^{N}))$ on $L^{2}_{\mu}$ generates a strongly continuous Markov semigroup $T(\cdot)$ on $L^{2}_{\mu}$, which is also analytic.

Let $(L,D(L))$ be the self-adjoint operator defined by the closure of $(L,C_{c}^{\infty}(\mathbb{R}^{N}))$ on $L^{2}_{\mu}$.

Proposition 2.1**.**

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

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• iii)

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

Proof.

i) and ii). Let $u\in D(L)$. Then there exists $u_{n}\in C_{c}^{\infty}(\mathbb{R}^{N})$ such that $u_{n}\to u$ in $L^{2}_{\mu}$ and $Lu_{n}\to Lu$ in $L^{2}_{\mu}.$ By (5) we have

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Then, $\nabla u_{n}$ converges to a function $G\in\left(L_{\mu}^{2}\right)^{N}$. On the other hand, one has

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for every $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$. Taking the limit for $n\to\infty$, since $Lu_{n}\to Lu$ and $\nabla u_{n}\to G$, we have

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By a density argument, this holds true for any $\varphi\in H^{1}_{\mu}.$

It remains to show that the components $G_{i}$ of $G$ are the weak derivatives of $u.$ Fixing $\psi\in C_{c}^{\infty}(\mathbb{R}^{N})$, one obtains

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Then, taking the limit as $n\to\infty$, one has $\int G_{i}\overline{\psi}\,dx=-\int u\partial_{i}\overline{\psi}\,dx.$

Indeed,

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where $\frac{1}{C_{\psi}}=\inf_{x\in supp\psi}\mu(x).$ Similarly we have $\lim_{n\to\infty}\int u_{n}\partial_{i}\overline{\psi}\,dx=\int u\partial_{i}\overline{\psi}\,dx$.

Assertion iii) follows from the analyticity of the semigroup $T(\cdot)$ (cf. [9, Theorem II.4.6], [10, Lemma 1.3.3]).

∎

Since $C_{c}^{\infty}(\Omega)$ is dense in $H^{1}_{\mu}$, where $\Omega=\mathbb{R}^{N}\setminus\{0\}$, we can prove the following Lemma.

Lemma 2.2**.**

Let $V$ be a nonnegative function belonging to $L^{1}_{loc}(\mathbb{R}^{N})$. Let $u$ be a nonnegative weak solution of $(P_{V})$. Then, for every compact set $K\subset\Omega$ and $t>0$ there exists $c(t)>0$ (not depending on $V)$ such that

[TABLE]

Proof.

Let $u_{0}\in C_{c}^{\infty}(\mathbb{R}^{N})$ nonnegative and let $u$ be a nonnegative weak solution of $(P_{V})$. Let $C_{R}=B_{R}\setminus\overline{B}_{1/R}$ such that $K\subset C_{R}$ and let $\varphi\in C_{c}^{\infty}(C_{R})$ such that $0\leq\varphi\leq 1$.

Consider the problem

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Then Problem $(Pb)$ admits a solution $v\in C^{2+\alpha,1+\frac{\alpha}{2}}(\overline{C}_{R}\times[0,T])$. Moreover,

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where $G$ is a strictly positive and continuous function on $(0,+\infty)\times C_{R}\times C_{R}$. Let $c(t)=\min_{(x,y)\in K\times K}G(t,x,y)$. We have for every $x\in K$

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Furthermore, $v$ is a weak solution to $v_{t}=Lv$ in $C_{R}$. In particular, for all $\phi\in W^{2,1}_{2}(C_{R}\times[0,T])$ with $\phi(\cdot,0)\geq 0$ having compact support with $\phi(\cdot,T)=0$, we have

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Comparing with (4), one obtains

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Fix $T,\,R>0$, $0\leq\psi\in C_{c}^{\infty}(C_{R}\times[0,T])$ and consider the parabolic problem

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By [13, Theorem IV.9.1], one obtains a solution $0\leq\phi\in W^{2,1}_{2}(C_{R}\times(0,T))$. By a standard argument, one can insert the solution $\phi$ in (6). Therefore,

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for all $0\leq\psi\in C_{c}^{\infty}(C_{R}\times[0,T])$. Thus,

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Since the last inequality holds for any $\varphi\in C_{c}^{\infty}(C_{R})$ one obtains

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∎

Therefore, we can state the following theorem, for which we omit the proof because it is similar to that of Theorem 1.1, see [12, Theorem 2.1].

Theorem 2.3**.**

Assume that $\mu$ satisfies Hypothesis $(H1)$. Let $0\leq V\in L_{loc}^{1}({\mathbb{R}}^{N})$. Then the following hold:

1. (i)

*If $\lambda_{1}(L+V)>-\infty$, then there exists a nonnegative weak solution *

$u\in C([0,\infty),L^{2}_{\mu})$* of $(P_{V})$ satisfying*

[TABLE]

for some constants $M\geq 1$ and $\omega\in{\mathbb{R}}$. 2. (ii)

If $\lambda_{1}(L+V)=-\infty$, then for any $0\leq u_{0}\in L^{2}_{\mu}\setminus\{0\},$ there exists no nonnegative weak solution of $(P_{V})$ satisfying (7).

3. The Hardy inequality

Let $d\mu$ be a positive measure (not necessary a probability measure) with density $\mu(x)$. Let us recall the definition of $c_{0,\mu}$ and the potential $U$. We set

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and

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Consider

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So, we have

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We start by proving of the following improved Hardy inequality.

Proposition 3.1**.**

Assume $\rm i),ii)$ of $(H2)$. Then, for any $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$, the following inequality holds

[TABLE]

Proof.

One has

[TABLE]

By Minkowski’s inequality for integrals and by a change of variables we have

[TABLE]

Hence,

[TABLE]

Then, (9) follows from the relation

[TABLE]

∎

We recall, under assumption $(i)$ of $(H2)$, that $C_{c}^{\infty}({\mathbb{R}^{N}})$ is dense in $H_{\mu}^{1}$. If moreover $U$ is bounded from above, the result below is a direct consequence of (9).

Corollary 3.2**.**

Assume $\rm i),ii)$ of $(H2)$ and assume that there exists $C_{\mu}\in\mathbb{R}$ such that $U\leq C_{\mu}$. Then, for any $\varphi\in H^{1}_{\mu}$,

[TABLE]

If instead of the boundedness from above of $U$ in $\mathbb{R}^{N}$, we assume $U$ bounded only for $|x|$ large enough (that is $(H2^{\prime})$), then the following result holds.

Theorem 3.3**.**

Let us assume that Hypothesis $(H2^{\prime})$ holds, then for every $c<c_{0,\mu}$ there exists $C_{\mu}$ such that for any $\varphi\in H^{1}_{\mu}$ the weighted Hardy inequality holds

[TABLE]

Proof.

Since $\limsup_{x\to 0}|x|^{2}U=0$, it follows that for every $\varepsilon>0$ there exists $R_{\varepsilon}>0$ such that $U\leq\frac{\varepsilon}{|x|^{2}}$ for all $x\in B_{R_{\varepsilon}}$. Moreover, there exists $C_{\mu}$ depending on $R_{\varepsilon}$ such that $U\leq C_{\mu}$ for every $x\in B^{c}_{R_{\varepsilon}}.$ Then, by Proposition 3.1, we have

[TABLE]

The result follows by taking $c=c_{0,\mu}-\varepsilon$. ∎

We look now for weaker conditions with respect to the boundedness from above for $U$ in $\mathbb{R}^{N}$ in order to get (10). To this purpose, we have to consider improved Hardy’s inequalities.

The first step is to state a relation between the weighted Hardy inequality and a special improved Hardy’s inequality.

Lemma 3.4**.**

Assume $\rm i),ii)$ of $(H2)$, and the improved Hardy inequality

[TABLE]

Then, the weighted Hardy inequality holds

[TABLE]

Proof.

Let $\varphi\in C_{c}^{\infty}(\mathbb{R}^{N})$ and set $u:=\varphi\sqrt{\mu}\in H^{1}(\mathbb{R}^{N})$ with compact support. By (11), which holds by density for such a function $u$, integrating by parts and recalling the expression (8) for $U$, one obtains

[TABLE]

Then, inequality (12) follows. ∎

Now, our aim is to prove (11). Brezis and Vázquez in [6] proved the following inequality

[TABLE]

with $q<\frac{2N}{N-2}$, for a bounded domain $D$ and for every $u\in H^{1}_{0}(D)$. From this, by Hölder’s inequality, it follows an inequality of type

[TABLE]

with potential $U$ belonging to $L_{loc}^{p}(\mathbb{R}^{N})$ for $p>\frac{N}{2}$. This gives us the desired result (12), but forces us to suppose $U_{\mu}\in L_{loc}^{N/2+\varepsilon}(\mathbb{R}^{N})$.

Therefore, in order to prove (11), we will refer to the following improved Hardy inequality, see [14], [1].

Theorem 3.5**.**

For any $u\in C_{c}^{\infty}(B_{1})$ the following inequality holds

[TABLE]

Now, we suppose that $\mu$ and $U$ satisfy condition $(H2)$. We finally obtain the weighted Hardy inequality (12).

Theorem 3.6**.**

Assume that Hypothesis $(H2)$ holds. Then for any $\varphi\in H^{1}_{\mu}$, the following inequality holds

[TABLE]

Proof.

By Proposition 3.4, we need to prove that

[TABLE]

By Hypothesis $(H2)$ on $U$, there exists a $R_{0}\leq 1$ (otherwise one takes $R_{0}=1$) such that $U\leq\frac{1}{4}\frac{1}{|x|^{2}|\log|x||^{2}}$ in $B_{R_{0}}$. Then, for $u\in C_{c}^{\infty}(\mathbb{R}^{N})$, by Theorem 3.5 and a change of variables, one has

[TABLE]

Let $u\in C_{c}^{\infty}(\mathbb{R}^{N})$ and $\theta\in C^{\infty}({\mathbb{R}^{N}})$, $0\leq\theta\leq 1$, such that $\theta=1$ in $B_{\frac{R_{0}}{2}}$ e $\theta=0$ in $B^{c}_{R_{0}}$, and $\frac{1}{2}\frac{|\nabla\theta|^{2}}{\theta}-\Delta\theta\leq M$. Note that such a function exists. For instance, one can consider a translation and a dilatation of the function $\theta_{0}(s)=ce^{-\frac{1}{1-s^{2}}}$ for $|s|\leq 1$ and equals to 0 for $|s|\geq 1$.

Therefore, by (3) and Hypotesis $(H2)$, one obtains

[TABLE]

∎

Remark 3.7**.**

For $N_{e}\in\mathbb{R}$ define $c_{0}(N_{e}):=\left(\frac{N_{e}-2}{2}\right)^{2}.$ Take $c_{0,\mu}=c_{0}(N_{e})$ if $c_{0,\mu}\geq 0$ and $N_{e}=2$ if $c_{0,\mu}<0$. Then, by Theorem 3.6 (resp. Theorem 3.3), Inequality (2) with constant $c\leq c_{0}(N_{e})$ (resp. $c<c_{0}(N_{e})$) holds provided that $(H2)$ (respectively $(H2^{\prime})$) is satisfied.

4. Optimality of the constant in the weighted Hardy inequality

In this section we give conditions in order to prove the sharpness of the constant $c$ in (2).

Theorem 4.1**.**

Let us assume Hypothesis $(H3)$. Then, there exists a function in $H^{1}_{\mu}$ for which the weighted Hardy inequality (2) does not hold if $c>c_{0}(N_{0})$.

Proof.

Let $\gamma$ be such that $\max\{-\sqrt{c},-\frac{N_{0}}{2}\}<\gamma<\min\{\frac{-N_{0}+2}{2},0\}$ so that $\gamma^{2}<c$ and from the definition of $N_{0}$ it follows that $|x|^{2\gamma}\in L^{1}_{loc}(\mathbb{R}^{N},d\mu)$ and $|x|^{2\gamma-2}\notin L^{1}_{loc}(\mathbb{R}^{N},d\mu)$.

Let $n\in\mathbb{N}$ and $\vartheta\in C_{c}^{\infty}(\mathbb{R}^{N})$, $0\leq\vartheta\leq 1$, $\vartheta=1$ in $B_{1}$ and $\vartheta=0$ in $B_{2}^{c}$. Set $\varphi_{n}(x)=\min\{|x|^{\gamma}\vartheta(x),n^{-\gamma}\}$. We observe that

[TABLE]

The functions $\varphi_{n}$ are in $H^{1}_{\mu}$.

Let us assume that $c>c_{0}(N_{0})$, then we have to prove that the bottom of the spectrum of the operator $-\Delta+\frac{\nabla\mu}{\mu}\cdot\nabla-\frac{c}{|x|^{2}}$

[TABLE]

is $-\infty$. To this purpose we have

[TABLE]

On the other hand,

[TABLE]

Taking into account (4) and (15) we have

[TABLE]

We observe that $\gamma^{2}-c<0$. Taking the limit $n\to\infty$ we get

[TABLE]

Hence $\lambda_{1}=-\infty$. ∎

Therefore we obtain the following result.

Theorem 4.2**.**

Assume hypotheses $(H2)$ and $(H3)$ with $c_{0,\mu}=c_{0}(N_{0})$. Then for any $\varphi\in H^{1}_{\mu}$ the following inequality holds

[TABLE]

and $c_{0}(N_{0})$ is the best constant.

If Hypotesis $(H3^{\prime})$ with $N_{0}>2$ holds, one obtains the following theorem.

Theorem 4.3**.**

Let us assume Hypothesis $(H3^{\prime})$ with $N_{0}>2$. Then, there exists a function in $H^{1}_{\mu}$ for which the weighted Hardy inequality (2) does not hold if $c=c_{0}(N_{0})=\left(\frac{N_{0}-2}{2}\right)^{2}$.

Proof.

Let $\gamma$ be such that $0>\gamma>\frac{-N_{0}+2}{2}$ so that $|x|^{2\gamma}\in L^{1}_{loc}(\mathbb{R}^{N},d\mu)$ and $|x|^{2\gamma-2}\in L^{1}_{loc}(\mathbb{R}^{N},d\mu)$ and $\gamma^{2}<c_{0}(N_{0})$. Let $\vartheta\in C_{c}^{\infty}(\mathbb{R}^{N})$, $0\leq\vartheta\leq 1$, $\vartheta=1$ in $B_{1}$ and $\vartheta=0$ in $B_{2}^{c}$. Set $\varphi_{\gamma}(x)=|x|^{\gamma}\vartheta(x)$. We observe that $\varphi_{\gamma}\in H^{1}_{\mu}$.

Let us set $c=c_{0}(N)$. We have to prove that

[TABLE]

is $-\infty$. One has

[TABLE]

On the other hand,

[TABLE]

Taking into account (4) and (17), we have

[TABLE]

Now, taking the limit $\gamma\to\left(\frac{-N_{0}+2}{2}\right)^{+}$, by Hypothesis $(H3^{\prime})$, one obtains

[TABLE]

Then $\lambda_{1}=-\infty$. ∎

Therefore, we obtain the following result.

Theorem 4.4**.**

Assume hypotheses $(H2^{\prime})$ and $(H3^{\prime})$ with $N_{0}>2$ and $c_{0,\mu}=c_{0}(N_{0})$. Then for every $\varphi\in H^{1}_{\mu}$ the following inequality holds

[TABLE]

for all $c<c_{0}(N_{0})$ and $c_{0}(N_{0})$ is the best constant.

In conclusion, we have proved Theorem 4.2 and 4.4 which, together with Theorem 2.3, give Theorem 1.4.

5. Examples

We finally give some examples of measures for which the weighted Hardy inequality holds.

Proposition 5.1**.**

Let $d\mu=\rho(x)dx$ with $\rho(x)=e^{-b|x|^{m}}$, $m>0,\,b\geq 0,$ and $N\geq 3$. Then there exists a positive constant $C$ such that for all $u\in H^{1}_{\mu}$ the following inequality

[TABLE]

holds with best constant.

Proof.

This measure satisfies assumptions $i)$ and $ii)$ of $(H2)$. Then, by a simple computation one obtains

[TABLE]

and $\lim_{x\to 0}|x|^{2}U_{\mu}(x)=0$. Therefore, $c_{0,\mu}=c_{0}(N)$, $U_{\mu}=U$ is a bounded from above far from [math] and assumptions $(H2)$ and $(H3)$ are satisfied with $N_{0}=N$. Then the assertion follows from Theorem 4.2. Moreover, $\mu$ satisfies also Hypothesis $(H1)$. ∎

Proposition 5.2**.**

Let us consider $d\mu=\frac{1}{|x|^{\beta}}\rho(x)dx$ with $\rho(x)$ as in Proposition 5.1, $N\geq 3$ and $\beta<N-2$. Then there exists a constant $C\geq 0$ such that for all $u\in H^{1}_{\mu}$

[TABLE]

*holds with best constant. *

Proof.

This measure satisfies assumptions $i)$ and $ii)$ of $(H2)$ if $\beta<N-2$. Moreover, by a simple computation, one has $c_{0,\mu}=c_{0}(N-\beta)$ and

[TABLE]

Then $(H2)$ and $(H3)$ hold with $c_{0,\mu}=c_{0}(N-\beta)$, $N_{0}=N-\beta$. The result follows from Theorem 4.2. ∎

By taking $b=0$ in Proposition 5.2 we can also state the Caffarelli-Nirenberg inequality.

Corollary 5.3**.**

If $\beta<N-2$ and $N\geq 3$ then the following inequality holds

[TABLE]

The following is an example for a weight which behaves like logarithm near [math].

Proposition 5.4**.**

Let $\theta\in C_{c}^{\infty}({\mathbb{R}^{N}})$ with $0\leq\theta\leq 1$ such that $\theta=1$ on $B_{1/2}$ and $\theta=0$ on $B_{1}^{c}$. Assume that

[TABLE]

• $\bullet$

If $\alpha\leq 0$, then there exists a constant $C\geq 0$ such that for all $u\in H^{1}_{\mu}$

[TABLE]

holds with best constant.

• $\bullet$

If $\alpha>0$, then there exist $c,C\geq 0$ such that for all $u\in H^{1}_{\mu}$

[TABLE]

holds for every $c<c_{0}(N)$ and $c_{0}(N)$ is the best constant.

Proof.

One has that $\mu$ satisfies assumptions $(H2^{\prime})$ and $(H3)$ with $c_{0,\mu}=c_{0}(N)$ and $N_{0}=N$ for all $\alpha\in\mathbb{R}$. Moreover, $\mu$ satisfies $(H2)$ if and only if $\alpha\leq 0$ and $\mu$ satisfies $(H3^{\prime})$ if and only if $\alpha>0$. Indeed, by a simple computation, one obtains

[TABLE]

We have $\limsup_{x\to 0}|x|^{2}U_{\mu}=0$, then the constant in Hardy’s inequality is $c_{0}(N)$. By (18) it is easy to check that assumption $(H2)$ is satisfied if and only if $\alpha\leq 0.$ Then, for $\alpha\leq 0$ Theorem 4.2 applies.

Now, to show that $(H3^{\prime})$ is satisfied for $\alpha>0$ it suffices to prove that

[TABLE]

for some positive $r<\frac{1}{2}$. To this purpose we have

[TABLE]

The second term is uniformly bounded for every $\lambda>0$. As regards the first term, it grows to infinity for $\lambda\to 0^{+}$ if and only if $\alpha-1>-1.$ Therefore, for $\alpha>0$, the assertion follows from Theorem 4.4. ∎

Finally, in the following example the constants $c_{0,\mu}$ and $c_{0}(N)$ do not coincide.

Proposition 5.5**.**

Let $\mu$ be the density $\mu(x)=\mu_{1}\chi_{B_{1/2}}(x)+\mu_{2}\chi_{B^{c}_{1/2}}(x)$ where

[TABLE]

and $\mu_{2}\in C^{2}_{b}(\mathbb{R}^{N})$ a nonnegative smooth function such that $\mu\in H^{2}_{loc}(\mathbb{R}^{N})$. Then $\mu$ satisfies $(H1)$, $(H2^{\prime})$ and $(H3)$ with $c_{0,\mu}<c_{0}(N)$.

Proof.

We have

[TABLE]

Moreover, it can be seen that $\limsup_{x\to 0}|x|^{2}U_{\mu}(x)>0$. Hence, $c_{0,\mu}<c_{0}(N)$. Since $\mu$ is bounded and positive, we have

[TABLE]

for some $r>0$ and for every $\gamma>0$ and $\int_{B_{r}}\frac{1}{|x|^{N}}\,d\mu=\infty$. Then $N_{0}=N.$ ∎

6. Appendix

Let $d\mu$ be a positive measure and $\Omega:=\mathbb{R}^{N}\setminus\{0\}$. Set $L^{p}_{\mu}=L^{p}(\mathbb{R}^{N},d\mu)$ and $\|u\|_{p,\mu}=\left(\int_{\mathbb{R}^{N}}|u|^{p}\,d\mu\right)^{\frac{1}{p}}$ for $p\in[1,\infty)$.

Proposition 6.1**.**

Let $W_{\mu}^{1,p}=\overline{C_{c}^{\infty}(\mathbb{R}^{N})}^{\|\cdot\|_{W_{\mu}^{1,p}}}$, where $\|u\|_{W_{\mu}^{1,p}}=\|u\|_{p,\mu}+\|\nabla u\|_{p,\mu}$. If

[TABLE]

then $C_{c}^{\infty}(\Omega)$ is dense in $W^{1,p}_{\mu}$.

Proof.

Let $u\in C_{c}^{\infty}(\mathbb{R}^{N})$. We have to approximate $u$ with functions in $C_{c}^{\infty}(\Omega)$ with respect to the norm $\|\cdot\|_{W_{\mu}^{1,p}}$.

Let $\vartheta\in C_{b}^{\infty}(\mathbb{R}^{N})$ such that $\chi_{B^{c}_{1}}\leq\vartheta\leq\chi_{B^{c}_{\frac{1}{2}}}$, and set $\vartheta_{n}(x)=\vartheta(nx)$. We observe that $\vartheta_{n}\to 1$ pointwisely in $\Omega$ and $\|\nabla\vartheta_{n}\|_{\infty}\leq Cn$. We have

[TABLE]

The first term of the right hand side converges to [math] by dominated convergence. As regards the second one we have

[TABLE]

So, the first integral converges to [math] by dominated convergence, the last one by Condition (19). Thus,

[TABLE]

∎

Corollary 6.2**.**

Let $p<N$. If $\mu\in W^{1,p}_{\emph{loc\,}}\left(\mathbb{R}^{N}\right)$ then $C_{c}^{\infty}\left(\Omega\right)$ is dense in $W_{\mu}^{1,p}$.

Proof.

By Proposition 6.1, it suffices to verify Condition (19). Let us observe first that since $\mu\in W^{1,p}_{\emph{loc\,}}\left(\mathbb{R}^{N}\right)$, by Sobolev’s embedding theorem, it follows that $\mu\in L_{\emph{loc\,}}^{p^{*}}(\mathbb{R}^{N})$ with $p^{*}=\frac{Np}{N-p}$.

Then,

[TABLE]

One can easily verify that $\frac{N(p-1)}{p}+1-p>0$ if $p<N.$ ∎