A Class of Potentials in Weighted Hardy-Type Inequalities with a Finite Number of Poles

Abstract

In this paper, we discuss potentials for which we obtain multipolar weighted Hardy-type inequalities for a class of weights that are wide enough. Examples of such potentials are shown. The weighted estimates are more general than those stated in previous papers. To obtain the inequalities, we prove an integral identity by introducing a suitable vector-valued function.

1. Introduction

In this paper, we state the class of potentials V for which we obtain the weighted Hardy-type inequalities with a finite number of poles

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

(1)

for any $\phi$ in a weighted Sobolev space, where $\mu$ is a weight function satisfying suitable conditions. This class includes potentials studied in previous papers (see [1,2,3]) and it extends to the multipolar case, whose results were obtained using a different proof and are stated in [4]. To obtain the inequality of Equation (1), we state an integral identity based on the introduction of a suitable vector-valued function. The method used in the proof generalizes the technique employed in [3] and gives us the admissible potentials in the Hardy estimates for a class of weights that are wide enough.

The interest in weighted Hardy-type inequalities is also due to the applications to the study of Kolmogorov operators

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

(2)

defined based on smooth functions, where $\mu >0$ is a probability density on ${\mathbb{R}}^N$, perturbed by singular potentials and of the following related evolution problems:

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

The estimate of the bottom of the spectrum of the operators $-(L+V)$ is

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

where $H_{\mu }^1$ denotes a suitable weighted Sobolev space that is defined in the next section, and that is equivalent to the weighted Hardy-type inequality. In the case of a single pole, when $\mu =1$, Cabré and Martel [5] showed that the boundedness of the bottom is a necessary and sufficient condition for the existence of positive exponentially bounded-in-time solutions to the associated initial value problem. These results have been extended to Kolmogorov operators perturbed by the inverse square potential $V={\displaystyle \frac{c}{{\left|x\right|}^2}}$ in [6,7]. For a more general drift term, a similar result can be found in [8] and in the references therein.

When L is the Schrödinger operator with a multipolar inverse square, we can find some reference results in the literature.

In particular, for the operator

$$\mathcal{L}=-\Delta -\sum _{i=1}^n{\displaystyle \frac{c_i}{|x-a_i{|}^2}},a_1\dots ,a_n\in {\mathbb{R}}^N,$$

$n\ge 2$, $c_i\in \mathbb{R}$, for any $i\in \{1,\dots ,n\}$, V. Felli, E. M. Marchini and S. Terracini in [9] proved that the associated quadratic form

$$Q\left(\phi \right):={\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}dx-\sum _{i=1}^n{c}_i{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{{\phi }^2}{|x-a_i{|}^2}}dx$$

is positive if ${\sum }_{i=1}^n{c}_i^{+}<{\displaystyle \frac{{(N-2)}^2}{4}}$, $c_i^{+}=max\{c_i,0\}$. Conversely, if ${\sum }_{i=1}^n{c}_i^{+}>{\displaystyle \frac{{(N-2)}^2}{4}}$, there exists a configuration of poles such that Q is not positive (for some other results, see [10,11]). Later, R. Bosi, J. Dolbeaut and M. J. Esteban in [12] proved that for any $c\in \left(0,{\displaystyle \frac{{(N-2)}^2}{4}}\right]$, there exists a positive constant K such that the multipolar Hardy inequality

$$c{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\displaystyle \frac{{\phi }^2}{|x-a_i{|}^2}}dx\le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}dx+K{\int }_{{\mathbb{R}}^N}{\phi }^{2}dx$$

holds for any $\phi \in H^1\left({\mathbb{R}}^N\right)$. C. Cazacu and E. Zuazua in [13], improving a result stated in [12], obtained the inequality

$${\displaystyle \frac{{(N-2)}^2}{n^2}}\sum _{\begin{array}{c}i,j=1\\ i<j\end{array}}^n{\int }_{{\mathbb{R}}^N}{\displaystyle \frac{|a_i-a_j{|}^2}{|x-a_i{|}^2{|x-a_j|}^2}}{\phi }^{2}dx\le {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}dx,$$

for any $\phi \in H^1\left({\mathbb{R}}^N\right)$ with ${\displaystyle \frac{{(N-2)}^2}{n^2}}$ optimal constant (see also [14] for results on bounded domains).

Furthermore, for some results concerning Hardy-type inequalities with a finite number of poles involving the potential

$$V={\displaystyle \frac{{(N-2)}^2}{{(n+1)}^2}}\left(\sum _{i=1}^n{\displaystyle \frac{1}{|x-a_i{|}^2}}+\sum _{\begin{array}{c}i,j=1\\ i<j\end{array}}^n{\displaystyle \frac{|a_i-a_j{|}^2}{|x-a_i{|}^2{|x-a_j|}^2}}\right),$$

see Appendix B on multipolar potentials in the interesting paper of B. Devyver, M. Fraas and Y. Pinchover [15], in which we can also find remarks related to the optimality of the constants in some multipolar Hardy’s estimates.

In the case of mean value-type multipolar potentials, some results can be found in [16]. Finally, for the multipolar Rellich inequality, see [17], and for inequalities on Riemannian manifolds, see [18].

With regard to Ornstein–Uhlenbeck-type operators

$$Lu=\Delta u-\sum _{i=1}^{n}A(x-a_i)·\nabla u,$$

perturbed by multipolar inverse square potentials

$$V\left(x\right)=\sum _{i=1}^n{\displaystyle \frac{c}{|x-a_i{|}^2}},0<c\le {\displaystyle \frac{{(N-2)}^2}{4}},a_1\dots ,a_n\in {\mathbb{R}}^N,$$

weighted multipolar Hardy inequalities with optimal constant and related existence and nonexistence results were stated in [19], with A as a positive definite real Hermitian $N\times N$ matrix. In such a case, the invariant measure for these operators is the Gaussian measure ${\mu }_A\left(x\right)dx=\kappa e^{-{\displaystyle \frac{1}{2}}{\sum }_{i=1}^n⟨A(x-a_i),x-a_i⟩}dx$, with a normalization constant $\kappa$.

In a forthcoming paper, we will present some results on operator L defined in Equation (2) and perturbed by potentials

$$V=\sum _{i=1}^n{\displaystyle \frac{\beta (N+K_{\mu }-2)-n{\beta }^2}{|x-a_i{|}^2}}+{\displaystyle \frac{{\beta }^2}{2}}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{|a_i-a_j{|}^2}{|x-a_i{|}^2{|x-a_j|}^2}}$$

(see also [1] for $\beta ={\displaystyle \frac{N+K_{\mu }-2}{2n}}$, [3] for $\beta ={\displaystyle \frac{N+K_{\mu }-2}{n}}$). When $\beta ={\displaystyle \frac{N-2}{2n}}$, $\mu =1$, and then, $K_{\mu }=0$; see [12].

In [1,2], we can find the weighted estimate Equation (1) when V is the inverse square potential

$$V=c\sum _{i=1}^n{\displaystyle \frac{{\phi }^2}{|x-a_i{|}^2}},0<c<{\displaystyle \frac{{(N+K_{\mu }-2)}^2}{4}}.$$

We conclude the introduction to our weighted multipolar Hardy-type estimates in ${\mathbb{R}}^N$ by pointing out that, in this paper, we focus our attention only on Hardy-type estimates and on the technique to obtain them. We do not show the applications to evolution problems $\left(P\right)$ given that the existence of positive solutions can be obtained via the same reasoning as in [1] (see also [4] in the unipolar case).

In this paper, we have two sections. In Section 2, we describe weights and potentials and in Section 3, we state weighted Hardy-type inequalities.

2. Weights and Class of Potentials

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

In the next section, we state the inequalities for functions belonging to weighted Sobolev spaces.

The weight functions $\mu$ satisfy the following conditions:

(H₁)

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

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

We introduce positive functions $f_i$ related to the poles $a_i$, $i\in \{1,\dots n\}$, and $f={\prod }_{i=1}^n{f}_i$, which require the following hypotheses:

(H₂)

${\displaystyle \frac{1}{f}}{\displaystyle \frac{\partial f}{\partial x_j}}\mu \in L_{loc}^2\left({\mathbb{R}}^N\right),{\displaystyle \frac{1}{f}}{\displaystyle \frac{{\partial }^{2}f}{\partial x_j^2}}\mu \in L_{loc}^1\left({\mathbb{R}}^N\right).$

(H₃)

There exists a constant $K_{\mu }\in \mathbb{R}$ such that

$$-{\displaystyle \frac{\Delta f}{f}}+K_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2=-\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}-\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}+K_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2\ge 0,$$

(H₄)

The following holds:

$$\begin{array}{cc}\hfill W:={\displaystyle \frac{\nabla f}{f}}·{\displaystyle \frac{\nabla \mu }{\mu }}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2& =\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla \mu }{\mu }}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2\hfill \\ & =\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·\left[{\displaystyle \frac{\nabla \mu }{\mu }}+c{K}_{\mu }{\displaystyle \frac{\nabla f_i}{f_i}}\right]\le C_{\mu },\hfill \end{array}$$

(3)

where the constant c is connected to the function f.

The class of potentials V that we consider is of the following type:

$\left(H_5\right)$

$V\in L_{loc}^1\left({\mathbb{R}}^N\right)$, $0\le V\le V_f$, with

$$V_f:=-{\displaystyle \frac{\Delta f}{f}}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2=-\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}-\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2.$$

The hypothesis $\left(H_1\right)$ ensures us that the space $C_c^{\infty }\left({\mathbb{R}}^N\right)$ is dense in $H_{\mu }^1$ (see, e.g., [20]). So, we can regard $H_{\mu }^1$ as the completion of $C_c^{\infty }\left({\mathbb{R}}^N\right)$ with respect to the Sobolev norm

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

In reality, with the additional assumption

$$\underset{\delta \to 0}{lim}{\displaystyle \frac{1}{{\delta }^2}}{\int }_{B(a_i,\delta )}\mu \left(x\right)dx=0\forall i\in \{1,\dots ,n\},$$

the space $C_c^{\infty }({\mathbb{R}}^N\setminus \{a_1,\dots ,a_n\})$ is dense in $H_{\mu }^1$ (see [1]).

The condition of $\left(H_2\right)$ allows us to integrate it in parts with the proof of the main result in Section 3. The function f generalizes the functions given in previous papers and allows for a comparison with the unipolar case.

Assumption $\left(H_3\right)$ states the positivity of the function $V_f$ in $\left(H_5\right)$.

For a fixed f, $\left(H_4\right)$ is a condition on $\mu$. Conversely, for a fixed $\mu$, it represents a condition on f. For example, the function

$$f=\prod _{i=1}^n{\displaystyle \frac{1}{|x-a_i{|}^{\beta }}},\beta >0$$

(4)

satisfies $\left(H_4\right)$ with $c={\displaystyle \frac{1}{\beta }}$. In this case, ${\displaystyle \frac{\nabla f_i}{f_i}}=\beta {\displaystyle \frac{(x-a_i)}{|x-a_i{|}^2}}$, and the condition $\left(H_4\right)$ takes the form

$$W:=-\sum _{i=1}^n{\displaystyle \frac{\beta (x-a_i)}{|x-a_i{|}^2}}·\left[{\displaystyle \frac{\nabla \mu }{\mu }}-K_{\mu }{\displaystyle \frac{(x-a_i)}{|x-a_i{|}^2}}\right]\le C_{\mu }.$$

The condition $\left(H_4\right)$ extends to the multipolar case, which is a condition introduced in [4] in the case of a single pole.

For more examples of functions f when we have a single pole, see [4].

A class of weight functions satisfying hypotheses $\left(H_1\right)$ and $\left(H_4\right)$ with f in Equation (4) is as follows:

$$\mu \left(x\right)=\prod _{i=1}^n{\displaystyle \frac{1}{|x-a_1{|}^{\gamma }}}{e}^{-\delta {\sum }_{j=1}^n{|x-a_j|}^m},\delta \ge 0,m\le 2,$$

for suitable values of $\gamma$ (see [1,2] for details). We observe that for $\gamma =0$, $\delta \ne 0$ and $m=2$, we obtain the Gaussian function.

Examples of weights in the case of a single pole can be found in [8].

We observe that the potential $V_f$ in $\left(H_5\right)$ for the function f in Equation (4) is the following:

$$V_f=\sum _{i=1}^n{\displaystyle \frac{\beta (N+K_{\mu }-2)-n{\beta }^2}{|x-a_i{|}^2}}+{\displaystyle \frac{{\beta }^2}{2}}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{|a_i-a_j{|}^2}{|x-a_i{|}^2{|x-a_j|}^2}}$$

(see [1] for $\beta ={\displaystyle \frac{N+K_{\mu }-2}{2n}}$ and [3] for $\beta ={\displaystyle \frac{N+K_{\mu }-2}{n}}$).

3. Multipolar Hardy-Type Inequalities with Weight

In this section, we state an integral identity from which we deduce a weighted multipolar Hardy-type inequality with potential V. This result shows us the class the potentials for which we obtain the estimates when the weight function satisfies the conditions in Section 2. The method used in the proof is based on the introduction of a vector-valued function depending on a function f related to the poles and to the weight. Integration by parts requires the integrability conditions on f given in Section 2.

Theorem 1.

Let $N\ge 3$ and $n\ge 2$. Under hypotheses $\left(H_1\right)$–$\left(H_5\right)$, we obtain

$${\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu ={\int }_{{\mathbb{R}}^N}{V}_f{\phi }^{2}d\mu -{\int }_{{\mathbb{R}}^N}W{\phi }^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2{f}^{2}d\mu .$$

(5)

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

Proof. 

By the density argument, it is sufficient to prove Equation (5) for $\phi \in C_c^{\infty }\left({\mathbb{R}}^N\right)$.

We introduce the vector-valued function

$$F\left(x\right)=-{\displaystyle \frac{\nabla f}{f}}\mu =-\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}\mu$$

(6)

and observe that $F_j$, ${\displaystyle \frac{\partial F_j}{\partial x_j}}$, where $F_j=-{\displaystyle \frac{1}{f}}{\displaystyle \frac{\partial f}{\partial x_j}}\mu$ belongs to $\in L_{loc}^1\left({\mathbb{R}}^N\right)$. In fact, using Hölder’s inequality, for any compact set K in ${\mathbb{R}}^N$, we obtain

$${\int }_K|F_j|dx\le {\left({\int }_K{\left|-{\displaystyle \frac{1}{f}}{\displaystyle \frac{\partial f}{\partial x_j}}\right|}^{2}d\mu \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_K\mu \left(x\right)dx\right)}^{{\displaystyle \frac{1}{2}}};$$

then, we take in mind the hypotheses $\left(H_1\right)$ and $\left(H_2\right)$.

For the partial derivative of $F_j$

$${\displaystyle \frac{\partial F_j}{\partial x_j}}={\displaystyle \frac{1}{f^2}}{\left({\displaystyle \frac{\partial f}{\partial x_j}}\right)}^2\mu -{\displaystyle \frac{1}{f}}{\displaystyle \frac{{\partial }^{2}f}{\partial x_j^2}}\mu -{\displaystyle \frac{1}{f}}{\displaystyle \frac{\partial f}{\partial x_j}}{\displaystyle \frac{\partial \mu }{\partial x_j}},$$

the local integrability is achieved using hypotheses $\left(H_1\right)$ and $\left(H_2\right)$ again and by observing that we have the following through Hölder’s inequality:

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

The first step is now to consider the following integral:

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}\mathrm{div}F{\phi }^{2}dx& ={\int }_{{\mathbb{R}}^N}\left(-{\displaystyle \frac{\Delta f}{f}}\mu +{\left|{\displaystyle \frac{\nabla f}{f}}\right|}^2\mu -{\displaystyle \frac{\nabla f}{f}}·\nabla \mu \right){\phi }^{2}dx\hfill \\ & ={\int }_{{\mathbb{R}}^N}(-\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}-\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}+\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2\hfill \\ & +\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}-\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla \mu }{\mu }}){\phi }^{2}d\mu \hfill \\ & =-{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}d\mu +{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^{2}d\mu -{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla \mu }{\mu }}){\phi }^{2}d\mu .\hfill \end{array}$$

(7)

On the other hand, by integrating by parts on the left-hand side in Equation (7), we obtain

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}& \mathrm{div}F{\phi }^{2}dx=2{\int }_{{\mathbb{R}}^N}\phi \nabla \phi ·{\displaystyle \frac{\nabla f}{f}}\mu dx\hfill \\ & =-{\int }_{{\mathbb{R}}^N}{\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2{f}^{2}d\mu +{\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +{\int }_{{\mathbb{R}}^N}{\left|{\displaystyle \frac{\nabla f}{f}}\right|}^2{\phi }^{2}d\mu \hfill \\ & =-{\int }_{{\mathbb{R}}^N}{\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2{f}^{2}d\mu +{\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu +{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2{\phi }^{2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}{\phi }^{2}d\mu ,\hfill \end{array}$$

(8)

where we have used the equality

$${\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2={\displaystyle \frac{{|\nabla \phi |}^2}{f^2}}+{\left|{\displaystyle \frac{\nabla f}{f}}\right|}^2{\displaystyle \frac{{\phi }^2}{f^2}}-2\phi \nabla \phi ·{\displaystyle \frac{\nabla f}{f}}{\displaystyle \frac{1}{f^2}}.$$

Combining Equations (7) and (8), we deduce that

$$\begin{array}{cc}\hfill {\int }_{{\mathbb{R}}^N}{|\nabla \phi |}^{2}d\mu & ={\int }_{{\mathbb{R}}^N}{\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2{f}^{2}d\mu -{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}{\phi }^{2}d\mu -{\int }_{{\mathbb{R}}^N}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}{\phi }^{2}d\mu \hfill \\ & -{\int }_{{\mathbb{R}}^N}\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla \mu }{\mu }}{\phi }^{2}d\mu ={\int }_{{\mathbb{R}}^N}{\left|\nabla {\displaystyle \frac{\phi }{f}}\right|}^2{f}^{2}d\mu \hfill \\ & +{\int }_{{\mathbb{R}}^N}\left(-\sum _{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}-{\int }_{{\mathbb{R}}^N}\sum _{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2\right){\phi }^{2}d\mu \hfill \\ & -{\int }_{{\mathbb{R}}^N}\left(\sum _{i=1}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla \mu }{\mu }}+c{K}_{\mu }\sum _{i=1}^n{\left|{\displaystyle \frac{\nabla f_i}{f_i}}\right|}^2\right){\phi }^{2}d\mu \hfill \end{array}$$

from which we obtain the integral identity Equation (5) by taking in mind $\left(H_5\right)$ and W defined in Equation (3). □

Remark 1.

The theorem of Equation (5) also applies in the case $\mu =1$.

The following result is a Hardy-type inequality obtained as a consequence of the theorem of Equation (5).

Theorem 2.

Let $N\ge 3$ and $n\ge 2$. If $\left(H_1\right)$–$\left(H_5\right)$ hold, we obtain

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

(9)

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

Proof. 

The integral identity (5) implies that

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

For the class of functions $\mu$ satisfying hypothesis $\left(H_4\right)$ and for potentials in $\left(H_5\right)$, the inequality in Equation (9) follows. □

Remark 2.

In the case of a single pole, a similar weighted Hardy-type inequality was obtained in [4] with a different proof. When we have a finite number of poles in the potential $V_f$, the mixed product ${\sum }_{\begin{array}{c}i,j=1\\ i\ne j\end{array}}^n{\displaystyle \frac{\nabla f_i}{f_i}}·{\displaystyle \frac{\nabla f_j}{f_j}}$ arises from the poles. If we can estimate the product, we can obtain an inequality with $V_f=-k{\sum }_{i=1}^n{\displaystyle \frac{\Delta f_i}{f_i}}$, with k suitable constant. This is the case in [1,3], where f is exactly the same as Equation (4).