Improved Hardy Inequalities with a Class of Weights

Abstract

In the framework of Hardy type inequalities and their applications to evolution problems, the paper deals with local and nonlocal weighted improved Hardy inequalities related to the study of Kolmogorov operators perturbed by singular potentials. The class of weights is wide enough. We focus our attention on weighted Hardy inequalities with potentials obtained by inverse square potentials adding a nonnegative correction term. The method used to get the results is based on the introduction of a suitable vector-valued function and on a generalized vector field method. The local estimates show some examples of this type of potentials and extend some known results to the weighted case.

1. Introduction

The paper deals with local and nonlocal improved Hardy inequalities with a class of weights $\mu$ of a quite general type and with inverse square potentials perturbed by a function V. The paper fits into the context of Hardy type inequalities with weights stated in [1].

Hardy’s inequality was introduced in 1920 [2] in the one-dimensional case (see also [3,4]). The classical Hardy inequality in ${\mathbb{R}}^N$ is (see, e.g., [5,6,7] for historical reviews, and [8])

$$c_o\left(N\right){\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}dx\le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}dx$$

(1)

for any functions $\phi \in H^1\left({\mathbb{R}}^N\right)$, where $c_o\left(N\right)={\left(\frac{N-2}{2}\right)}^2$ is the optimal constant.

Weighted Hardy inequalities with an optimal constant depending on a weight function $\mu$ have been stated in [9,10] with Gaussian measures and inverse square potentials with a single pole and in the multipolar case, respectively. In a setting of more general measures, we refer to [11,12,13,14], in the last papers with multipolar potentials.

In particular, in [12], the authors proved the inequality

$$c_{N,\mu }{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}\mu \left(x\right)dx\le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^2\mu \left(x\right)dx+C_{\mu }{\int }_{{\mathbb{R}}^N}{\phi }^2\mu \left(x\right)dx,$$

(2)

for any functions $\phi$ in a weighted Sobolev space with $c_{N,\mu }={\left(\frac{N+K_{\mu }-2}{2}\right)}^2$ the optimal constant and $K_{\mu }$, $C_{\mu }$ constants depending on $\mu$. For example, when $\mu =\frac{1}{{\left|x\right|}^{\gamma }}$, $\gamma <N-2$, it yields $K_{\mu }=-\gamma$ and $C_{\mu }=0$, whereas if $\mu =e^{-{\delta \left|x\right|}^2}$, $\delta >0$, we get $K_{\mu }=0$ (see [12]).

In this paper, we improve this results by adding a nonnegative correction term in the left-hand side in (2).

In particular, we state sufficient conditions on V to get in ${\mathbb{R}}^N$ the estimate

$$c_{N,\mu }{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +{\int }_{{\mathbb{R}}^N}V{\phi }^{2}d\mu \le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +C_{\mu }{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu ,$$

(3)

for any functions $\phi$ in a suitable Sobolev space with a weight satisfying suitable local integrability assumptions, where $d\mu =\mu \left(x\right)dx$ and $c_{N,\mu }$ is the constant in (2).

The class of weight functions includes Gaussian functions and weights involving a distance to the origin.

To prove the result we use a method based on the introduction of a suitable vector-valued function and on a generalized vector field method (see [15] for some results when the weights satisfy the Hölder condition).

Examples of weight functions are shown in the paper.

In the case of the Lebesgue measure there is a very huge literature on the extension of Hardy’s inequality. In particular, the improved version of the classical Hardy inequality in a bounded domain $\Omega$ in ${\mathbb{R}}^N$, $N\ge 3$,

$${\left(\frac{N-2}{2}\right)}^2{\int }_{\Omega }\frac{{\phi }^2}{{\left|x\right|}^2}dx+c_{\Omega }{\int }_{\Omega }{\phi }^{2}dx\le {\int }_{\Omega }{|\nabla \phi |}^{2}dx$$

(4)

was stated in [16] for all $\phi \in H_0^1(\Omega )$.

Later improvements of the estimate (4) of the type

$${\left(\frac{N-2}{2}\right)}^2{\int }_{\Omega }\frac{{\phi }^2}{{\left|x\right|}^2}dx+{\int }_{\Omega }V{\phi }^{2}dx\le {\int }_{\Omega }{|\nabla \phi |}^{2}dx$$

(5)

can be found, for example, in [17,18,19,20].

We obtain, using the method to get the inequality in the whole space, weighted versions of the local estimate (5) for some functions V, and well-known inequalities when $\mu =1$. In particular, we focus our attention on the inequalities

$$c_{N,\mu }{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +\frac{1}{4}{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^2{|log|x\left|\right|}^2}d\mu \le {\int }_{B_1}{|\nabla \phi |}^{2}d\mu +C_{\mu }{\int }_{B_1}{\phi }^{2}d\mu$$

(6)

and, for $\beta \in (0,2]$,

$$c_{N,\mu }{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +{\beta }^2{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^{2-\beta }}d\mu \le {\int }_{B_1}{|\nabla \phi |}^{2}d\mu +C_{\mu }{\int }_{B_1}{\phi }^{2}d\mu$$

(7)

for any functions $\phi \in C_c^{\infty }\left(B_1\right)$, where $B_1$ is the unit ball in ${\mathbb{R}}^N$. For $\beta =2$, we get the weighted version of (4) with 4 in place of $c_{\Omega }$.

Hardy inequalities are applied in many fields. From a mathematical point of view, a motivation for us to study Hardy inequalities with a weight and related improvements is due to their applications to evolution problems

$$\left(P\right)\left\{\begin{array}{c}{\partial }_{t}u(x,t)=Lu(x,t)+\tilde{V}\left(x\right)u(x,t),x\in {\mathbb{R}}^N,t>0,\hfill \\ u(·,0)=u_0\ge 0\in L_{\mu }^2\hfill \end{array}\right.$$

where $L_{\mu }^2:=L^2({\mathbb{R}}^N,d\mu )$ and L is the Kolmogorov operator

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

(8)

defined on smooth functions, perturbed by singular potentials $\tilde{V}$.

An existence result can be obtained, reasoning as in [11], following Cabré and Martel’s approach based on the relation between the weak solution of $\left(P\right)$ and the estimate of the bottom of the spectrum of the operator $-(L+\tilde{V})$

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

which results from the Hardy inequality. In the case $\mu =1$, Cabré and Martel in [21] showed that the boundedness of ${\lambda }_1(\Delta +\tilde{V})$ was a necessary and sufficient condition for the existence of positive exponentially bounded in time solutions to the associated initial value problem. Later, in [9,11,13], similar results were extended to Kolmogorov operators perturbed by inverse square potentials, and in the latter paper, in the multipolar case. The proof used some properties of the operator L and of its corresponding semigroup in $L_{\mu }^2\left({\mathbb{R}}^N\right)$.

In this paper, we include the existence result for the sake of completeness.

The paper is organized as follows.

In Section 2, we introduce the class of weights and the conditions on the potentials V with some examples. In Section 3, we state the improved weighted Hardy inequality and some consequences. Section 4 is devoted to the weighted local estimates. Finally, in Section 5, we show an application of the estimates to evolution problems.

2. A Class of Weight Functions and Potentials

Let $\mu \ge 0$ be a weight function on ${\mathbb{R}}^N$. We define the weighted Sobolev space $H_{\mu }^1=H^1({\mathbb{R}}^N,\mu \left(x\right)dx)$ as the space of functions in $L_{\mu }^2:=L^2({\mathbb{R}}^N,\mu \left(x\right)dx)$ whose weak derivatives belong to $L_{\mu }^2$.

The class of function $\mu$ is considered to fulfill the conditions

$\left(H_1\right)$

$\left(i\right)\sqrt{\mu }\in H_{loc}^1\left({\mathbb{R}}^N\right)$;

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

Let us observe that under the assumption $\left(i\right)$$\left(H_1\right)$, we get $\mu ,\nabla \mu \in L_{loc}^1\left({\mathbb{R}}^N\right)$. The reason we assume $\left(H_1\right)$ is that we need the density of the space $C_c^{\infty }\left({\mathbb{R}}^N\right)$ in $H_{\mu }^1$ (see e.g., [22]). Thus, we can regard $H_{\mu }^1$ as the completion of $C_c^{\infty }\left({\mathbb{R}}^N\right)$ with respect to the Sobolev norm

$${\parallel ·\parallel }_{H_{\mu }^1}^2:={\parallel ·\parallel }_{L_{\mu }^2}^2+{\parallel \nabla ·\parallel }_{L_{\mu }^2}^2.$$

We introduce in the proof of the Hardy inequality in the next section the function $f=\frac{g}{{\left|x\right|}^{\alpha }}$, with g a radial function, for suitable values of $\alpha$. We need the following condition on g.

$\left(H_2\right)$

$(i){\displaystyle g>0,-\frac{1}{g}\frac{\partial g}{\partial x_j}\sqrt{\mu }\in L_{loc}^2\left({\mathbb{R}}^N\right),\frac{1}{g}\frac{{\partial }^{2}g}{\partial x_j^2}\mu \in L_{loc}^1\left({\mathbb{R}}^N\right)}$;

$(ii){\displaystyle -\frac{\Delta g}{g}+(N-2)\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}\ge 0}$.

The assumption on the potential V in the estimates is the following

$\left(H_3\right)$

$V=V\left(x\right)\in L_{loc}^1\left({\mathbb{R}}^N\right)$ and

$$0\le V\le W:=-\frac{\Delta g}{g}+(N-2)\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}=-\frac{g^{″}}{g}-\frac{g^{\prime }}{\rho g},$$

where $g^{\prime }$, $g^{″}$ are the first and second derivatives with respect to $\rho =\left|x\right|$, respectively.

Under condition $\left(i\right)$ in $H_2$, we can integrate by parts in the proof of the inequality in the next section. The class of radial functions g satisfying $\left(ii\right)$ in $\left(H_2\right)$ is such that

$${\left(g^{\prime }\rho \right)}^{\prime }\le 0$$

(9)

which implies that $g^{\prime }\rho$ is decreasing, so we have

$$g^{\prime }\left(r\right)r\le g^{\prime }\left(r_0\right)r_0,r_0\le r.$$

(10)

If $g\in C^2({\mathbb{R}}^N\backslash \left\{0\right\})$, this condition involves that the function $g\left(r\right)-c_{1}logr$ is decreasing

$$g\left(r\right)-c_{1}logr<g\left(r_0\right)-c_{1}log{r}_0,c_1=g^{\prime }\left(r_0\right)r_0,$$

as we can see integrating (10) in $[r_0,r]$, $r_0>0$.

Functions satisfying condition (9) in $(0,R)$, $R<1$, for example, are the functions $g\left(r\right)={|logr|}^{\beta }$, $\beta \in (0,1)$, $g\left(r\right)=1-r^{\beta }$, $\beta \in (0,2]$.

The functions W such that there exists a positive radial solution of the Bessel equation associated to the potential W

$$g^{″}+\frac{g^{\prime }}{r}+Wg=0$$

(11)

are good functions. We observe that, if $W=1$, the Bessel function $J_0$ is a positive solution of Equation (11).

The author in [19] proved that, under suitable hypotheses, Equation (11) was a necessary and sufficient condition to get an improved Hardy inequality in bounded domains in ${\mathbb{R}}^N$.

A further assumption we need is the following.

$\left(H_4\right)$

There exist constants $K_1,K_2,K_3\in \mathbb{R}$, $K_2>2-N$ and $K_3=K_2$ if $K_2\ne 0$, $K_3\le 0$ if $K_2=0$, such that

$$\left(\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right)·\frac{\nabla \mu }{\mu }\le K_1-\frac{\alpha K_2}{{\left|x\right|}^2}+K_3\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}$$

or, equivalently, such that the function $g=g\left(r\right)$ satisfies the inequality

$$\frac{g^{\prime }}{g}\left(\frac{{\mu }^{\prime }}{\mu }-\frac{K_3}{r}\right)\le K_1+\frac{\alpha }{r}\left(\frac{{\mu }^{\prime }}{\mu }-\frac{K_2}{r}\right).$$

(12)

For g fixed, it is a condition for $\mu$. For example, if $g=1$, weight functions satisfying $\left(H_4\right)$ are the functions

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

(13)

for suitable values of m (see [12]). Conversely, for $\mu$ fixed, $\left(H_4\right)$ represents a condition on g.

Finally, we remark that the weights in (13) fulfill condition $\left(H_1\right)$.

3. Weighted Improved Hardy Inequalities

In this section, we state a weighted improved Hardy inequality in the setting of a more general measure with respect to [15]. This allows us to improve the results in [12] on weighted Hardy inequalities by adding a nonnegative correction term in the estimates.

The method to get the result was introduced in [15] for a class of weights satisfying the Hölder condition. We enlarge the class of weights for which we can state the result. For this class, a weighted Hardy inequality with a different method was stated in [12].

The next result states sufficient conditions to get an improved Hardy inequality with weight.

Theorem 1.

If $\left(H_1\right)$–$\left(H_4\right)$ hold, then we get the estimate

$$\frac{{(N+K_2-2)}^2}{4}{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +{\int }_{{\mathbb{R}}^N}V{\phi }^{2}d\mu \le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +K_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu$$

(14)

for any functions $\phi \in H_{\mu }^1$.

Proof. 

By the density result, we prove (14) for any $\phi \in C_c^{\infty }\left({\mathbb{R}}^N\right)$.

We introduce the vector-valued function

$$F=-\frac{\nabla f}{f}\mu =-\frac{\nabla g}{g}\mu +\alpha \frac{x}{{\left|x\right|}^2}\mu ,$$

where $f=\frac{g}{{\left|x\right|}^{\alpha }}$, $\alpha \in (0,N+K_2-2)$.

We get

$$\mathrm{div}F=-\frac{\Delta f}{f}\mu +{\left|\frac{\nabla f}{f}\right|}^2\mu -\frac{\nabla f}{f}·\nabla \mu ,$$

where

$$\begin{array}{cc}\hfill -\frac{\Delta f}{f}& =-{\left|x\right|}^{\alpha }\Delta \frac{1}{{\left|x\right|}^{\alpha }}-2{\left|x\right|}^{\alpha }\nabla \frac{1}{{\left|x\right|}^{\alpha }}·\frac{\nabla g}{g}-\frac{\Delta g}{g}\hfill \\ & =\frac{\alpha (N-2-\alpha )}{{\left|x\right|}^2}+2\alpha \frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}-\frac{\Delta g}{g}.\hfill \end{array}$$

Now, we observe that $F_j$, ${\displaystyle \frac{\partial F_j}{\partial x_j}\in L_{loc}^1\left({\mathbb{R}}^N\right)}$, where $F_j$ is the jth component of F. Indeed, for any K compact set in ${\mathbb{R}}^N$, by the Hölder and the classical Hardy inequalities, using hypotheses $\left(i\right)$ in $\left(H_1\right)$ on $\mu$ and $\left(i\right)$ in $\left(H_2\right)$ on g, we obtain the following estimate

$$\begin{array}{cc}\hfill {\int }_K\left|F_j\right|dx& \le {\int }_K\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|\mu \left(x\right)dx+\alpha {\int }_K\frac{\mu \left(x\right)}{\left|x\right|}dx\hfill \\ & \le {\int }_K\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|\mu \left(x\right)dx+\alpha {\left({\int }_K\frac{\mu \left(x\right)}{{\left|x\right|}^2}dx\right)}^{\frac{1}{2}}{\left({\int }_K\mu \left(x\right)dx\right)}^{\frac{1}{2}}\hfill \\ & \le {\left({\int }_K{\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|}^2\mu \left(x\right)dx\right)}^{\frac{1}{2}}{\left({\int }_K\mu \left(x\right)dx\right)}^{\frac{1}{2}}\hfill \\ & +\frac{2\alpha }{(N-2)}{\left({\int }_K{\left|\nabla \sqrt{\mu }\right|}^{2}dx\right)}^{\frac{1}{2}}{\left({\int }_K\mu \left(x\right)dx\right)}^{\frac{1}{2}}.\hfill \end{array}$$

To obtain the local integrability of the partial derivative of $F_j$

$$\begin{array}{cc}\hfill \frac{\partial F_j}{\partial x_j}& =\frac{\partial }{\partial x_j}\left(-\frac{1}{g}\frac{\partial g}{\partial x_j}\mu +\alpha \frac{x_j}{{\left|x\right|}^2}\mu \right)=\frac{1}{g^2}{\left(\frac{\partial g}{\partial x_j}\right)}^2\mu -\frac{1}{g}\frac{{\partial }^{2}g}{\partial x_j^2}\mu \hfill \\ & -\frac{1}{g}\frac{\partial g}{\partial x_j}\frac{\partial \mu }{\partial x_j}+\alpha \frac{\mu }{{\left|x\right|}^2}-2\alpha \frac{x_j^2}{{\left|x\right|}^4}\mu +\alpha \frac{x_j}{{\left|x\right|}^2}\frac{\partial \mu }{\partial x_j}\hfill \\ & =d_1+d_2+d_3+d_4+d_5+d_6,\hfill \end{array}$$

(15)

we estimate the terms on the right-hand side in (15). The terms $d_1$ and $d_2$ belong to $L_{loc}^1\left({\mathbb{R}}^N\right)$ by hypotheses,

$$\begin{array}{cc}\hfill {\int }_K\left|d_3\right|dx& \le {\left({\int }_K{\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|}^2\mu \left(x\right)dx\right)}^{\frac{1}{2}}{\left({\int }_K{\left|\frac{1}{\sqrt{\mu }}\frac{\partial \mu }{\partial x_j}\right|}^{2}dx\right)}^{\frac{1}{2}}\hfill \\ & \le 4{\left({\int }_K{\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|}^{2}d\mu \right)}^{\frac{1}{2}}{\left({\int }_K{\left|\nabla \sqrt{\mu }\right|}^{2}dx\right)}^{\frac{1}{2}},\hfill \end{array}$$

and $d_4$ and $d_5$ can be estimated using the Hardy inequality and the hypothesis $\left(i\right)$ in $\left(H_1\right)$ as above. As regards the remaining term, we have

$$\begin{array}{cc}\hfill {\int }_K\left|d_6\right|dx& \le \alpha {\int }_K\frac{\sqrt{\mu }}{\left|x\right|}\frac{1}{\sqrt{\mu }}\left|\frac{\partial \mu }{\partial x_j}\right|dx\hfill \\ & \le 2\alpha {\left({\int }_K\frac{\mu }{{\left|x\right|}^2}dx\right)}^{\frac{1}{2}}{\left({\int }_K{\left|\nabla \sqrt{\mu }\right|}^{2}dx\right)}^{\frac{1}{2}}.\hfill \\ & \le \frac{4\alpha }{(N-2)}\left({\int }_K{\left|\nabla \sqrt{\mu }\right|}^{2}dx\right).\hfill \end{array}$$

Now, we start from the following integral

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\mathrm{div}F{\phi }^{2}dx& ={\int }_{{\mathbb{R}}^N}[\frac{\alpha (N-2-\alpha )}{{\left|x\right|}^2}+2\alpha \frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}\hfill \\ & -\frac{\Delta g}{g}]{\phi }^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right|}^2{\phi }^{2}d\mu \hfill \\ & -{\int }_{{\mathbb{R}}^N}\left(\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right)·\frac{\nabla \mu }{\mu }]{\phi }^{2}d\mu .\hfill \end{array}$$

(16)

The first step is to estimate the integral on the left-hand side in (16) from above. To this aim, we integrate by parts and use Hölder’s and Young’s inequalities to get

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}& \mathrm{div}F{\phi }^{2}d\mu =-2{\int }_{{\mathbb{R}}^N}\phi F·\nabla \phi d\mu \hfill \\ & \le 2{\left({\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu \right)}^{\frac{1}{2}}{\left({\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla f}{f}\right|}^2{\phi }^{2}d\mu \right)}^{\frac{1}{2}}\hfill \\ & \le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla f}{f}\right|}^2{\phi }^{2}d\mu \hfill \\ & ={\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right|}^2{\phi }^{2}d\mu .\hfill \end{array}$$

(17)

On the other hand, starting from (16), by condition $\left(H_4\right)$, we obtain

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\mathrm{div}F{\phi }^{2}d\mu & \ge \alpha (N-2-\alpha ){\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}\left(2\alpha \frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}-\frac{\Delta g}{g}\right){\phi }^{2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right|}^2{\phi }^{2}d\mu -K_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu \hfill \\ & +\alpha K_2{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu -K_3{\int }_{{\mathbb{R}}^N}\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}{\phi }^{2}d\mu \hfill \\ & =\alpha (N+K_2-2-\alpha ){\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}\left[(2\alpha -K_3)\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}-\frac{\Delta g}{g}\right]{\phi }^{2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}{\left|\frac{\nabla g}{g}-\alpha \frac{x}{{\left|x\right|}^2}\right|}^2{\phi }^{2}d\mu -K_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu .\hfill \end{array}$$

(18)

The inequalities (17) and (18) lead us to the estimate

$$\begin{array}{cc}\hfill \alpha (N+& K_2-2-\alpha ){\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +{\int }_{{\mathbb{R}}^N}[(2\alpha -K_3)\frac{x}{{\left|x\right|}^2}·\frac{\nabla g}{g}\hfill \\ & -\frac{\Delta g}{g}]{\phi }^{2}d\mu \le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +K_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu .\hfill \end{array}$$

(19)

The maximum value of the first constant on the left-hand side in (19) is

$$\underset{\alpha }{max}\alpha (N+k_2-2-\alpha )=\frac{{(N+k_2-2)}^2}{4},$$

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

Observing that $2{\alpha }_o-K_3\ge N-2$ and keeping in mind condition $\left(H_3\right)$, we obtain the inequality (14). □

Remark 1.

For $g=1$ and $V=W=0$, we obtain a weighted Hardy inequality. For $g=1$, $\mu =1$ and so $K_2=0$, and the method to get the result in Theorem 1 reduces to the vector field method used in [8] to prove the classical Hardy inequality.

An example of a weight satisfying condition $\left(H_1\right)$ is given by the function $\mu =\frac{1}{{\left|x\right|}^{\gamma }}$, for $\gamma <N-2$. In this case, condition (12) is verified for $K_2,K_3\le -\gamma$ for any $K_1\ge 0$. Then, as a consequence of Theorem 1, we get the inequality

$$\frac{{(N-\gamma -2)}^2}{4}{\int }_{{\mathbb{R}}^N}\frac{{\phi }^2}{{\left|x\right|}^2}{\left|x\right|}^{-\gamma }dx+{\int }_{{\mathbb{R}}^N}V{\phi }^2{\left|x\right|}^{-\gamma }dx\le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^2{\left|x\right|}^{-\gamma }dx$$

(20)

for any functions $\phi \in H_{\mu }^1$. For $V=W=0$, the inequality above is the Caffarelli–Niremberg inequality.

Finally, as a direct consequence of Theorem 1, we deduce the following result concerning a class of general weighted Hardy inequalities for V satisfying $\left(H_3\right)$

$${\int }_{{\mathbb{R}}^N}V{\phi }^{2}d\mu \le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +K_1{\int }_{{\mathbb{R}}^N}{\phi }^{2}d\mu$$

for any functions $\phi \in H_{\mu }^1$.

4. Local Estimates

In this section, we state some local weighted estimates by means of the method used to prove Theorem 1. These estimates represent the weighted version of well-known improved Hardy inequalities in a bounded subset of ${\mathbb{R}}^N$ (see [17,18,19,20]) and are based on examples of functions g satisfying locally the assumptions of Theorem 1.

The first result is the following.

Theorem 2.

Let $N\ge 3$ and let $B_1$ be the unit ball in ${\mathbb{R}}^N$. Then, under assumptions $\left(H_1\right)$, $\left(i\right)$ in $\left(H_2\right)$ and $\left(H_4\right)$ on μ, we get

$$\begin{array}{cc}\hfill \frac{{(N+K_2-2)}^2}{4}{\int }_{B_1}& \frac{{\phi }^2}{{\left|x\right|}^2}d\mu +\frac{1}{4}{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^2{|log|x\left|\right|}^2}d\mu \hfill \\ & \le {\int }_{B_1}{|\nabla \phi |}^{2}d\mu +K_1{\int }_{B_1}{\phi }^{2}d\mu \hfill \end{array}$$

(21)

for any functions $\phi \in C_c^{\infty }\left(B_1\right)$.

Proof. 

Reasoning as in the proof of Theorem 1, we set $g={|log|x\left|\right|}^{\beta }$, $\beta \in (0,1)$. Then, the function W in $\left(H_3\right)$ is given by

$$W=\frac{\beta (1-\beta )}{{\left|x\right|}^2{|log|x\left|\right|}^2}.$$

To get the integrability required in $\left(i\right)$ in $\left(H_2\right)$, it is sufficient that the weights $\mu$ are such that ${\displaystyle \frac{g^{\prime 2}}{g^2}\mu }$, ${\displaystyle \frac{g^{″}}{g}\mu \in L_{loc}^1\left({\mathbb{R}}^N\right)}$. More precisely, pointing out that

$$-\frac{\partial g}{\partial x_j}=-\frac{x_j}{\left|x\right|}{g}^{\prime },\frac{{\partial }^{2}g}{\partial x_j^2}=\frac{g^{\prime }}{\left|x\right|}-\frac{x_j^2}{{\left|x\right|}^2}{g}^{\prime }+\frac{x_j^2}{{\left|x\right|}^2}{g}^{″},$$

(22)

if K is a compact set in $B_1$, we get

$${\int }_K\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|d\mu \le {\left({\int }_K{\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|}^{2}d\mu \right)}^{\frac{1}{2}}{\left({\int }_K\mu \left(x\right)d\mu \right)}^{\frac{1}{2}},$$

$${\int }_K{\left|-\frac{1}{g}\frac{\partial g}{\partial x_j}\right|}^{2}d\mu \le {\int }_K{\left|\frac{g^{\prime }}{g}\right|}^{2}d\mu ={\int }_K\frac{{\beta }^2}{{\left|x\right|}^2{|log|x\left|\right|}^2}d\mu ,$$

$$\begin{array}{cc}\hfill {\int }_K& \frac{1}{\left|x\right|}\left|\frac{g^{\prime }}{g}\right|d\mu \le {\left({\int }_K\frac{\mu }{{\left|x\right|}^2}dx\right)}^{\frac{1}{2}}{\left({\int }_K{\left|\frac{g^{\prime }}{g}\right|}^{2}d\mu \right)}^{\frac{1}{2}}\hfill \\ & \le \frac{2}{N-2}{\left({\int }_K{|\nabla \sqrt{\mu }|}^{2}d\mu \right)}^{\frac{1}{2}}{\left({\int }_K\frac{{\beta }^2}{{\left|x\right|}^2{|log|x\left|\right|}^2}d\mu \right)}^{\frac{1}{2}}\hfill \end{array}$$

and, about the last term on the right-hand side in (22),

$${\int }_K\frac{x_j^2}{{\left|x\right|}^2}\left|\frac{g^{″}}{g}\right|d\mu \le {\int }_K\left|\frac{g^{″}}{g}\right|d\mu ={\int }_K\left|\frac{\beta (\beta -1)+\beta |log|x\left|\right|}{{\left|x\right|}^2{|log|x\left|\right|}^2}\right|d\mu .$$

Finally, since

$$\underset{\beta \in (0,1)}{max}\left[\beta (1-\beta )\right]=\frac{1}{4},$$

attained for $\beta =\frac{1}{2}$, we get the result. □

In the case of a weight $\mu =\frac{1}{{\left|x\right|}^{\gamma }}$, $\gamma <N-2$, inequality (21), for $K_1=0$ and $K_2=-\gamma$, is the local version of (20) with $V=\frac{1}{4}\frac{1}{{\left|x\right|}^2{|log|x\left|\right|}^2}$.

Another example of weight is given by $\mu =\frac{1}{{\left|x\right|}^{\gamma }}{e}^{-{\delta \left|x\right|}^m}$, $\gamma <N-2$ and $\delta ,m>0$. In the last case, the condition (12) in $B_1$ is satisfied for $K_2,K_3\le -\gamma -\delta m$ and $K_1\ge 0$.

For $\mu =1$, inequality (21) turns out to be the improved Hardy inequality with a Lebesgue measure in [17,18,20].

A further local inequality follows.

Theorem 3.

Let $N\ge 3$ and let $B_1$ be the unit ball in ${\mathbb{R}}^N$. Then, under assumptions $\left(H_1\right)$, $\left(i\right)$ in $\left(H_2\right)$ and $\left(H_4\right)$ on μ, we get

$$\frac{{(N+K_2-2)}^2}{4}{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^2}d\mu +{\beta }^2{\int }_{B_1}\frac{{\phi }^2}{{\left|x\right|}^{2-\beta }}d\mu \le {\int }_{B_1}{|\nabla \phi |}^{2}d\mu +K_1{\int }_{B_1}{\phi }^{2}d\mu$$

(23)

for any functions $\phi \in C_c^{\infty }\left(B_1\right)$ and $\beta \in (0,2]$.

Proof. 

It is enough to consider $g=1-{\left|x\right|}^{\beta }$, $\beta \in (0,2]$, observing that

$$V=\frac{{\beta }^2}{{\left|x\right|}^{2-\beta }}\le W=\frac{{\beta }^2}{{\left|x\right|}^{2-\beta }{(1-|x|}^{\beta })}.$$

□

Finally, we remark that for $\beta =2$ and $\mu =1$, we almost get the estimate in [16,19] in the sense that, in place of four in the left-hand side in (23), the authors obtained $z_0^2$, where $z_0$ is the first zero of the Bessel function $J_0\left(z\right)$.

Moreover, in that case, the functions $\mu =\frac{1}{{\left|x\right|}^{\gamma }}$ and more generally $\mu =\frac{1}{{\left|x\right|}^{\gamma }}{e}^{-{\delta \left|x\right|}^m}$ are good weights.

5. An Application to Evolution Problems

In the section, we give a motivation for our interest in Hardy inequalities with a weight. These estimates play a crucial role in achieving existence results for solutions to the problem

$$\left(P\right)\left\{\begin{array}{c}{\partial }_{t}u(x,t)=Lu(x,t)+\tilde{V}\left(x\right)u(x,t),x\in {\mathbb{R}}^N,t>0,\hfill \\ u(·,0)=u_0\ge 0\in L_{\mu }^2,\hfill \end{array}\right.$$

where L is the Kolmogorov operator

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

defined on smooth functions, perturbed by a potential $\tilde{V}\left(x\right)$, with $\tilde{V}\left(x\right)$ the sum of an inverse square potential and V satisfying condition $\left(H_3\right)$.

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

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

and

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

for all $\varphi \in W_2^{2,1}({\mathbb{R}}^N\times \left[0,T\right])$ having compact support with $\varphi (·,T)=0$, 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 $\left|\alpha \right|\le 2$ and weak time derivative ${\partial }_{t}u\in L^2(\Omega \times (0,T))$ equipped with the norm

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

An additional assumption on $\mu$ allows us to get a semigroup generation on $L_{\mu }^2$ (see [23], Corollary 3.7).

$\left(H_5\right)$

$\mu \in C_{loc}^{1,\lambda }({\mathbb{R}}^N\backslash \left\{0\right\})$, $\lambda \in (0,1)$, $\mu \in H_{loc}^1\left({\mathbb{R}}^N\right)$, $\frac{\nabla \mu }{\mu }\in L_{loc}^r\left({\mathbb{R}}^N\right)$ for some $r>N$, and ${inf}_{x\in K}\mu \left(x\right)>0$ for any compact set $K\subset {\mathbb{R}}^N$

We remark that condition $\left(H_5\right)$ implies $\left(i\right)$ in $\left(H_3\right)$. Indeed, if $\mu \in H_{loc}^1\left({\mathbb{R}}^N\right)$, then $\mu \in L_{loc}^1\left({\mathbb{R}}^N\right)$ and $\nabla \mu \in L_{loc}^2\left({\mathbb{R}}^N\right)$. Moreover, $\frac{\nabla \mu }{\mu }\in L_{loc}^2\left({\mathbb{R}}^N\right)$ since $r>2$. Thus, we get

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

An example of a weight function satisfying $\left(H_5\right)$ is $\mu =e^{-{\delta \left|x\right|}^m}$, $\delta ,m>0$.

In the applications to evolution problems with Kolmogorov operators, we need $C_0$-semigroup generation results, when reasoning as in [9,11,13]. Operators of a more general type for which the generation of a semigroup was stated can be found, for example, in [24], in the context of weighted spaces.

The bottom of the spectrum of $-(L+\tilde{V})$ is defined as follows

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

The authors in [11] stated the following result with a proof similar to the one given in [21]. We include hypothesis $\left(ii\right)$ in $\left(H_1\right)$ to get the density result.

Theorem 4.

Assume that μ satisfies $\left(ii\right)$ in $\left(H_2\right)$ and $\left(H_5\right)$. Let $0\le \tilde{V}\left(x\right)\in L_{loc}^1\left({\mathbb{R}}^N\right)$. Then, if ${\lambda }_1(L+\tilde{V})>-\infty$, there exists a positive weak solution $u\in C([0,\infty ),L_{\mu }^2)$ of $\left(P\right)$ satisfying the estimate

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

(24)

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

The existence result below relies on Theorems 1 and 4.

Theorem 5.

Assume hypotheses $\left(ii\right)$ in $\left(H_1\right)$ and $\left(H_2\right)$–$\left(H_5\right)$. Then, there exists a positive weak solution $u\in C([0,\infty ),L_{\mu }^2)$ of $\left(P\right)$ satisfying

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

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

Proof. 

The weighted Hardy inequality (14) implies that ${\lambda }_1(L+\tilde{V})>-\infty$. Then, the result is a consequence of Theorem 4. □

6. Conclusions

This paper fits into the context of weighted Hardy type inequalities in ${\mathbb{R}}^N$. These estimates apply to the study of Kolmogorov operators, with the weight function in the drift term, perturbed by singular potentials. The research project on this topic during the last years involved the case of inverse square potentials with a single pole and with n poles, with a Gaussian measure and with measures of a more general type. Later, the inequality was extended to the case of different potentials in the unipolar case.

In this paper, the potential was given by inverse square potentials perturbed by a nonnegative term V. As a consequence, we improved previous results in [12] concerning estimates in ${\mathbb{R}}^N$. We were interested in a suitable class of weights which included Gaussian functions and weights involving a distance to the origin.

The method used to get the results was based on the introduction of a suitable vector-valued function and on a generalized vector field method. This technique enabled us to also get some local estimates with examples of functions V, weighted versions of well-known estimates stated in the case of a Lebesgue measure.

Future developments will consider the case of n poles in a different case from potentials with multiple inverse square singularities, with attention to general methods to get the estimates.