On Sharp Anisotropic Hardy Inequalities

Abstract

1 Introduction

Recently, Li–Yan studied in [8] the asymptotic stability of the |$(-1)$|-homogeneous axisymmetric stationary solutions to Navier–Stokes equations in |${\mathbb{R}}^{3}$|⁠. A key estimate was

where |$x^{\prime} = (x_{1}, x_{2}, 0)$|⁠. Li–Yan [8] showed (1.1) by proving the following strengthened inequality:

It is worthy to remark that (1.2) improves also the classical Hardy inequality

This motivated them to show the following general anisotropic Hardy inequalities; see [8, Theorem 1.3] and [9, pp. 6–7].

For more general |$p \geq 1$|⁠, we obtain the following partial result where the best constant is determined when |$\beta \geq 0$|⁠.

Furthermore, we give an alternative proof for the anisotropic |$L^{p}$|-Caffarelli–Kohn–Nirenberg inequalities, that is, a very special case of Li–Yan’s general result (1.4) with |$s=p=q> 1$| and |$a = \frac{1}{p}$|⁠. Let |$n\geq 2$|⁠, |$p\geq 1$|⁠, we know that |$|x^{\prime}|^\mu |x|^{\gamma _{2}}$|⁠, |$|x^{\prime}|^\beta |x|^{\gamma _{3}}$|⁠, |$|x^{\prime}|^{\alpha }|x|^{\gamma _{1}} \in L_{loc}^{p}(\mathbb{R}^{n})$| if and only if

Remark also that in our special case, the assumptions [9,(1.10)–(1.13)] are equivalent to

Our approach departs from an elementary identity: Let |$\Omega \subset{\mathbb{R}}^{n}$| be an open set, |$V \in C^{1}(\Omega )$| and |$f\in C^{2}(\Omega )$| be positive, then

The above equality can be showed by using integration by parts; or by taking |$\vec F=-\frac{\nabla f}{f}$| in the more general equality

These identities suggest to find weighted Hardy or Poincaré inequalities by testing suitable positive functions |$f \in C_{c}^{1}(\Omega )$|⁠, and provide a natural way to study Hardy type inequalities. This idea has been used in many situations in the literature, and summarized in [7]. For example, the Bessel pair with radial potential |$(V, W)$| introduced by Ghoussoub–Moradifam [6] is a special case of (1.11) for radial function |$f(x)=f(|x|)$|⁠, since

Moreover, we remark that the last integral in (1.11) is zero if and only if |$u/f$| is a constant. It is well known that the optimal Hardy inequality cannot be reached in general, that is, the best choice of |$f$| does not belong to the proper functional space. However, we can check eventually sharpness of the subsequent weight |$W:= -\frac{\textrm{div}(V \nabla f)}{f}$| by choosing appropriate functions |$u$| to approximate |$f$|⁠.

The last term in (1.11) can be interpreted also as a kind of stability, since it measures in some sense the distance between |$u$| and the eventual linear space generated by the optimal choice |$f$|⁠.

To prove Theorem 1.1, according to |$V$|⁠, we will apply (1.11) with |$f(x) = |x^{\prime}|^\theta |x|^\lambda $|⁠, and try to optimize the subsequent weight |$W$| with suitable choice of the parameters |$\theta $|⁠, |$\lambda \in{\mathbb{R}}$|⁠. As |$\theta $| or |$\lambda $| are allowed to be negative, the corresponding anisotropic Hardy inequalities are firstly proved in |$C_{c}^{1}({\mathbb{R}}^{n}\backslash \{x^{\prime} = 0\})$|⁠, then extended to |$C_{c}^{1}({\mathbb{R}}^{n})$| by density argument. Moreover, we study the sharpness by trying to approximate the optimal choice of |$f$|⁠.

An equality similar to (1.11) exists for general |$p> 1$|⁠, where we replace the last integral by a Picone-type term; see [7,section 10.2]. Let |$(\mathcal{M}, g)$| be a Riemannian manifold, consider |$V\in C^{1}({{\mathcal{M}}})$| and |$\vec F\in C^{1}({{\mathcal{M}}}, T_{g}{{\mathcal{M}}})$|⁠, then for any |$u\in C_{c}^{1}({{\mathcal{M}}})$|⁠, there holds

where

In particular, let |$\vec F=-\frac{\nabla f}{f}$| and |${{\mathcal{M}}} = \Omega $|⁠, we get that (see also [10, Theorem 3.1], [11] for |$V \equiv 1$|⁠, or [3–5] with |$f$| depending on one variable)

Hence, we obtain the |$L^{p}$|-Hardy inequality (with suitable |$V$|⁠, |$f$| and |$u$|⁠)

Here again, we need not any symmetry assumption on |$V$| or |$f$|⁠, and the residual term in (1.14) is zero if and only if |$u/f$| is constant. Therefore, we can proceed similarly as for |$L^{2}$| case to handle Theorem 1.2.

Moreover, notice that for any |$\kappa> 0$|⁠, |$(\kappa ^{-1}V, \kappa ^{\frac{1}{p-1}}\vec F)$| does not change the subsequent weight |$W = \textrm{div}(V|\vec F|^{p-2}\vec F)$| on the right-hand side of (1.13). Taking

we obtain a special weighted |$L^{p}$|-Caffarelli–Kohn–Nirenberg inequality as follows:

We will choose suitable |$\vec F$| to prove Theorem 1.3.

2 Proof of Theorem 1.1.

Let |$n\geq 2$| and |$x^{\prime}=(x_{1},\cdot \cdot \cdot ,x_{n-1},0)$| for |$x=(x_{i})\in{\mathbb{R}}^{n}$|⁠. Let |$V=|x^{\prime}|^{2\alpha +2}|x|^{2\beta }$| where |$\alpha , \beta $| satisfy (1.5) with |$p=2$|⁠. Consider |$f=f_{1} f_{2}$|⁠, then

Choose now |$f_{1}=|x^{\prime}|^\theta ,f_{2}=|x|^\lambda $|⁠, direct calculus yields that in |${\mathbb{R}}^{n}\setminus{\{x^{\prime}=0}\}$|⁠,

and

Hence,

and

Therefore,

where

and

Seeing Li–Yan’s estimate (1.3), we aim to find the maximum value of |$H_{1}$| under the constraint |$H_{2} \geq 0$|⁠. As |$\lim _{|\theta |\to \infty } H(\theta ) = -\infty $|⁠, and |$\lim _{|\lambda |\to \infty } H_{1}(\theta , \lambda ) = -\infty $| uniformly for bounded |$\theta $|⁠, |$\max _{H_{2} \geq 0}H_{1}$| exists.

It is easy to see that |$\partial _\theta H_{1} - \partial _\lambda H_{1} \equiv 1$|⁠, hence |$H_{1}$| has no critical point in |${\mathbb{R}}^{2}$| and |$\max _{H_{2} \geq 0} H_{1}$| is reached on the subset |$\{H_{2} = 0\}$|⁠, that is, when

If |$K = -4\beta (n + 2\alpha + \beta ) \leq 0$|⁠, then for any |$\theta \in{\mathbb{R}}$|⁠, there exists |$\lambda \in{\mathbb{R}}$| such that |$H_{2}(\theta ,\lambda )=0$|⁠, because the discriminant for the quadratic equation (2.2) of |$\lambda $| satisfies

This means that

Let |$K> 0$|⁠, then (2.2) holds true for |$\lambda \neq -\beta $| and

Clearly,

and

If |$K\in (0,1]$|⁠, then |$\theta (\lambda _{0}) = \theta _{0}$| for |$\lambda _{0}=-\beta -\frac{1+\sqrt{1-K}}{2}$|⁠, which yields

Consider now |$K> 1$|⁠, we can check that

By the same, for any |$K> 0$|⁠, there holds

Notice that |$H(\theta _{2}) < H(\theta _{1})$| for any |$K> 0$|⁠, hence for |$K>1$|⁠, there holds

Finally,

with |$K = -4\beta (n+2\alpha +\beta )$|⁠. Seeing (2.1) and the equality (1.11), we obtain

Under the assumption (1.5) with |$p = 2$|⁠, that is, |$2\alpha> 1-n$|⁠, |$2(\alpha + \beta )> -n$|⁠, there holds (using the spherical coordinates)

For any |$u \in C_{c}^{1}({\mathbb{R}}^{n})$|⁠, we consider |$u_\epsilon (x) = u(x) - u(x)\eta (|x^{\prime}|/\epsilon )$| for |$\epsilon \in (0, 1)$|⁠, with a standard cut-off function |$\eta \in C_{c}^{1}({\mathbb{R}})$|⁠. Applying (2.7) to |$u_\epsilon $| and sending |$\epsilon \to 0^{+}$|⁠, we can claim the estimate (1.6).

Now we show the sharpness of the constants |$C_{n, \alpha , \beta }$| in (1.7). We will use the spherical coordinates for |$n\geq 2$|⁠, that is

where |$r\in{\mathbb{R}}_{+}$|⁠, |$\varphi _{k}\in [0, \pi ]$| for |$1\leq k\leq n-2$| if |$n\geq 3$| and |$\varphi _{n-1}\in [0, 2\pi ]$|⁠, so

Let |$v(x)=h(s)g(r), s=|x^{\prime}|$| and |$r=|x|$|⁠, then

and

Hence,

Denote |$\Sigma =(0, \pi )$|⁠, we have

where |$\omega _{n-1}$| stands for the volume of the unit sphere in |${\mathbb{R}}^{n-1}$|⁠. For the estimates of |$J_{i}$|⁠, we consider three subcases.

Case |$K>1$|⁠. Seeing (2.6), we choose the test function |$v = hg$| with |$h(s)=s^{\theta _{1}},$|  |$g(r)=(r^{2}+\epsilon ^{2})^{\frac{\lambda _{1}}{2}} \eta (r)$| and |$\eta \in C_{c}^{1}(\mathbb{R})$| a standard cut-off function. Then

so

For any |$\lambda> -1$|⁠, there holds

where |$B(\cdot , \cdot )$| stands for Euler’s Beta function

which satisfy

On the other hand, for any |$\epsilon \in (0, 1)$|⁠,

Here and after, |$O(1)$| means a quantity uniformly bounded for |$\epsilon \in (0, 1)$|⁠. Indeed, we applied the following fact.

There holds then

Similarly, as

for |$\epsilon \in (0, 1)$| we have

Consequently, there hold

and also

Finally, using (2.9), we arrive at

where

Recall that |$\theta _{1} = \frac{-(n+2\alpha ) + \sqrt{K}}{2}$|⁠, |$\lambda _{1} = -\beta - \frac{\sqrt{K}}{2}$| and |$K = -4\beta (n+2\alpha + \beta )$|⁠, so |$\lambda _{1}^{2} = 2\beta \theta _{1}$| and

Moreover,

Therefore,

Notice that the function |$h(s)=s^{\theta _{1}}$| is not smooth at |$s= 0$|⁠. However, as all involved integrals converge, we may use eventually a family of smooth functions to approximate |$h$|⁠, so we omit the details. This means that for |$K> 1$|⁠, the constant |$\frac{(n-1+2\alpha )^{2}}{4} -\frac{(\sqrt K -1)^{2}}{4}$| is sharp to claim (1.6).

The analysis for other cases are similar, we will go through quickly.

Case |$K =1$|⁠. We take the test function |$v(x) = h(s)g(r)$| with |$h(s) = s^{\theta _{0} +\sigma }$|⁠, |$g(r) = (r^{2} + \epsilon ^{2})^{\frac{\lambda _{0} -\sigma }{2}}\eta (r)$|⁠, where |$s = |x^{\prime}|$|⁠, |$r = |x|$|⁠, |$\sigma> 0$| and |$\lambda _{0} = -\beta - \frac{1}{2}$|⁠. Remark that (1.5) with |$p=2$| yields

Therefore, |$K =1$| implies |$2\beta \in ({-1}, 0)$|⁠. We get then

Here |$\xi (r) = r^{2\beta + 2\sigma }(r^{2} + \epsilon ^{2})^{-\beta -\frac{1}{2}-\sigma }$| satisfies all assumptions of Lemma 2.1, and |$O_\sigma (1)$| stands for a quantity uniformly bounded for |$\epsilon \in (0, 1)$| when |$\sigma> 0$| is fixed. By the same, there holds

Taking first |$\epsilon \to 0^{+}$| and secondly |$\sigma \to 0^{+}$|⁠, we see then |$C_{n, \alpha , \beta } \leq \theta _{0}^{2}$| for |$K =1$|⁠.

Case |$K <1$|⁠. Let |$v(x) = h(s)g(r)$| with |$h(s) = s^{\theta _{0} + \sigma }$|⁠, |$g(r) = (r^{2} + \epsilon ^{2})^{\frac{\lambda _{0}}{2}}\eta (r)$|⁠. Remark that |$\beta>-\frac{1}{2}$| by (2.11), let

The above choice is motivated by (2.5). There holds then

Here |$o_\sigma (1)$| stands for a quantity tending to zero as |$\epsilon $| goes to |$0$| with fixed |$\sigma> 0$|⁠, and

Similarly,

with

Taking first |$\epsilon \to 0^{+}$| and secondly |$\sigma \to 0^{+}$|⁠, we see that |$C_{n, \alpha , \beta } \leq \theta _{0}^{2}$| for |$K <1$|⁠.

3 Proof of Theorem 1.2

Let |$V=|x^{\prime}|^{(\alpha +1)p}|x|^{\beta p}$| and |$f=|x^{\prime}|^\gamma $|⁠, then

and |$\nabla f= \gamma |x^{\prime}|^{\gamma -2}x^{\prime}$|⁠, |$\Delta f=\gamma (n-3+\gamma )|x^{\prime}|^{\gamma -2}$|⁠. There hold also

According to the general Hardy inequality (1.15), we will calculate

More precisely,

This yields

On the other hand,

Hence,

We consider respectively two cases according to the sign of |$\beta $|⁠.

Case |$\beta \geq 0$|⁠. Recall that |$p\alpha>1-n$|⁠. Let |$n-1 + p\alpha = -p\gamma _{0}$|⁠, then |$\gamma _{0} < 0$|⁠,

Thanks to (1.14) or (1.15), we have

Recall that |$|x|^\beta |x^{\prime}|^\alpha \in L^{p}_{loc}({\mathbb{R}}^{n})$| under the condition (1.5), similar to the case |$p=2$|⁠, we can extend the above estimate for |$u\in C_{c}^{1}(\mathbb{R}^{n})$| by approximation, so |$C_{n,\alpha ,\beta } \geq |\gamma _{0}|^{p}.$|

Now we prove the sharpness of the above estimate. Consider |$v=|x^{\prime}|^{\gamma }g(r)$| with |$g(r)=(r^{2}+\epsilon ^{2})^{\frac{\lambda }{2}}\eta $|⁠, a cut-off function |$\eta $| and

Then for |$\epsilon \in (0,1)$|⁠, applying Lemma 2.1 and (2.8),

where |$O_\sigma (1)$| stands for a quantity uniformly bounded for |$\epsilon \in (0,1)$| and fixed |$\sigma \in (0, 1)$|⁠. On the other hand,

In |${\mathbb{B}}^{n}$| the unit ball of |${\mathbb{R}}^{n}$|⁠, as |$\eta \equiv 1$|⁠, we have

Therefore,

Let |$\epsilon , \sigma \in (0, 1)$|⁠, clearly

By mean value theorem, as |$|x^{\prime}| \leq r$|⁠, there holds

Using spherical coordinates, we get

Moreover,

On |$2{\mathbb{B}}^{n}\setminus{\mathbb{B}}^{n}$|⁠, directly calculation gives

Consequently,

and then

Combining (3.2)-(3.4), for small enough |$\sigma> 0$| and |$\epsilon \in (0, 1)$|⁠, we can claim

Tending first |$\epsilon \to 0^{+}$|⁠, secondly setting |$\sigma \to 0^{+}$|⁠, we conclude that |$C_{n,\alpha ,\beta } \leq |\gamma _{0}|^{p}$| seeing (3.1). Hence, |$C_{n,\alpha ,\beta } = |\gamma _{0}|^{p}$| for |$p> 1$| and |$\beta \geq 0$|⁠.

Case |$\beta <0$|⁠. Here we take still |$f(x) = |x^{\prime}|^\gamma $|⁠, but rewrite

where

Denote |$\widetilde \gamma _{0} = {-\frac{n-1+(\alpha +\beta ) p}{p}}$|⁠. If |$p(\alpha + \beta )> 1-n$|⁠, there hold |$\widetilde \gamma _{0} < 0$| and

Seeing (1.15) and using approximation with functions in |$C_{c}^{1}({\mathbb{R}}^{n}\backslash \{x^{\prime}=0\})$|⁠, we claim |$C_{n,\alpha , \beta } \geq |\widetilde \gamma _{0}|^{p}$|⁠.

4 Proof of Theorem 1.3

Thanks to (1.9), we need only to prove (1.10) for

Let |$V(x) =|x^{\prime}|^{p\mu }|x|^{p\gamma _{2}}$| and |$\vec F(x)=|x^{\prime}|^{\beta -\mu }|x|^{\gamma _{3}-\gamma _{2}-1}x$|⁠, direct calculation yields that in |$\mathbb{R}^{n}\setminus{\{x^{\prime}=0}\}$|⁠,

and

Hence, we have, with (4.1),

Applying (1.16), as |$V|\vec F|^{p} = |x^{\prime}|^{\beta p}|x|^{\gamma _{3}p}$|⁠, we get immediately (1.10) for |$u \in C_{c}^{1}({\mathbb{R}}^{n}\backslash \{x^{\prime} = 0\})$| with (4.1). Recall that under the condition (1.8), |$|x^{\prime}|^{\alpha }|x|^{\gamma _{1}}, |x^{\prime}|^\beta |x|^{\gamma _{3}}, |x^{\prime}|^\mu |x|^{\gamma _{2}}\in L_{loc}^{p}(\mathbb{R}^{n})$|⁠. Similarly, as for Theorem 1.1 and 1.2, we can extend the estimate for |$u\in C_{c}^{1}(\mathbb{R}^{n})$| by approximation.

Furthermore, if |$\alpha = \beta = \mu $|⁠, then the above |$\vec F(x) = |x|^{\gamma _{3}-\gamma _{2}-1}x$|⁠. So |$u_{0}(x) = e^{-\kappa _{0}^{\frac{1}{p-1}}|x|^{\gamma _{3}-\gamma _{2}+1}}$| (with suitable value |$\kappa _{0}>0$|⁠) satisfies that the residual term in (1.16),

Hence, with |$\gamma _{3}-\gamma _{2}+1> 0$| and standard approximation, we can be convinced easily that the estimate (1.10) is sharp.

5 Further Remarks

Once our paper was posted in arXiv, we are aware about a very recent work of Musina–Nazarov [10]. By interests in degenerate elliptic equation with |$-\textrm{div}(A(x)\nabla u)$|⁠, they were motivated to establish some Hardy inequalities with anisotropic weight; see also [2]. In particular, among many other interesting results, by Theorem 1.3 and Theorem 5.4 in [10], Musina–Nazarov obtained the following theorem. Consider |$x = (y, x^{\prime\prime})\in \mathbb{R}^{k} \times \mathbb{R}^{n-k}$| with |$1 \le k \leq n-1$|⁠, assume that

Clearly, Theorem 1.1 here corresponds to the case |$k = n-1$|⁠, and our approach works for general |$k \le n-1$|⁠. Indeed, let |$V=|y|^{2\alpha +2}|x|^{2\beta }$|⁠, |$f=|y|^\theta |x|^{\lambda }$|⁠, we have

with |$G_{1}(\theta ,\lambda ) = -\theta (k+2\alpha +\theta ) - H_{2}(\theta ,\lambda )$|⁠, where |$ H_{2}(\theta ,\lambda )=\lambda (n+2\alpha +2\beta +2\theta +\lambda )+2\beta \theta $| is the same as in the proof of Theorem 1.1. Proceeding very closely to the analysis in section 2, we can claim that the best constant is |$C_{n,k,\alpha ,\beta }$| in (5.3), we skip the details.

Similar to Theorem 1.2, we get further result for general |$p \ge 1$|⁠.

Here we proceed as in section 3 by choosing |$V=|y|^{p(\alpha +1)}|x|^{p\beta }$| and |$f=|y|^{\gamma }$|⁠, which yields

Using |$k$| instead of |$n-1$|⁠, the analysis is very similar to that for Theorem 1.2, so we omit the details.

The |$L^{p}$| Caffarelli–Kohn–Nirenberg-type inequalities of Theorem 1.3 can also be extended to more general weights |$|x|^{\gamma _{i}}|y|^{\alpha _{i}}$| with |$y \in{\mathbb{R}}^{k}$|⁠, we leave it also to interested readers.

Acknowledgments

The authors would like to thank Professors Yanyan Li and Jingbo Dou for sending us respectively the references [1] and [10]. The authors are partially supported by NSFC (no. 12271164) and Science and Technology Commission of Shanghai Municipality (no. 22DZ2229014). The authors are also truly grateful to the anonymous referees for their thorough reading and valuable comments.

Data availability

Data sharing is not applicable to this work as no new data were created or analyzed in the study.

Communicated by Prof. YanYan Li

References