On a Precise Scaling to Caffarelli-Kohn-Nirenberg Inequality

Abstract

We analyze the general form of Caffarelli-Kohn-Nirenberg inequality. Due to a new introduced parameter, this inequality presents two distinguishable ranges. One of them, the inequality is shown to be the interpolation between Hardy and weighted Sobolev inequalities. The other range, which is no more an interpolation, the positive constant in the inequality is not necessarily bounded for all value of the parameters. In both cases, the precise value of the constants were given.

Appendix

In this last section, for the sake of completeness of this paper, we provide the proofs of Hardy and also weighted Sobolev inequalities, which were used here to show the Caffarelli-Kohn-Nirenberg inequality.

Proof of Hardy’s inequality

1. First recall (9), that is

$$ \big( \int _{\mathbb{R}^{n}}\left \| x\right \| ^{(\alpha -1)p}{ |u(x)|^{p}} dx \big)^{1/p} \leq C_{H} \big( \int _{\mathbb{R}^{n}}\left \| x \right \| ^{\alpha p}\left \| \nabla u(x)\right \| ^{p} dx \big)^{1/p}. $$

The proof is divided into two steps, and we follow the main idea of convenient vector fields as introduced by Metidieri in [18]. Let $V: \mathbb{R}^{n} \setminus \left \{ 0\right \} \to \mathbb{R}^{n}$ be a smooth vector field, defined by

$$ V_{i}(x):= C \ (x_{k} \ x_{k})^{(\alpha -1)p/2} \ x_{i} \qquad (k= 1, \ldots ,n) $$

(24)

for each $i =1, \ldots ,n$, where the constant $C$ is chosen a posteriori. This vector field verifies, $(k,j= 1,\ldots ,n)$,

$$ \frac{\partial V_{i}(x)}{\partial x_{j}}= C \Big( (\alpha -1) p \ \|x \|^{(\alpha -1)p-2} \ \delta _{kj} \ x_{k} \ x_{i} + \|x\|^{(\alpha -1)p} \ \delta _{ij} \Big), $$

where $\delta _{ij}$ is the usual Kronecker delta (symbol). Taking the trace, we have

$$ {\mathrm{div}}V(x)= C \Big( (\alpha -1) p + n \Big) \left \| x\right \| ^{( \alpha -1)p}, $$

hence we take the positive constant $C= \Big ( (\alpha -1) p + n \Big )^{-1}$, recall that $\gamma = (\alpha -1)$, $r= p$ and $\gamma r + n> 0$.

2. Now, the left-hand side of (9) can be rewritten in the following way

$$ \int _{\mathbb{R}^{n}}\left \| x\right \| ^{(\alpha -1)p}{|u(x)|^{p}}dx = \int _{\mathbb{R}^{n}}\left |u(x)\right |^{p} {\mathrm{div}} V(x) \, dx $$

and applying the Gauss-Green Theorem, we obtain

$$ \begin{aligned} \int _{\mathbb{R}^{n}}\left \| x\right \| ^{(\alpha -1)p}{|u(x)|^{p}} dx &= - \int _{\mathbb{R}^{n}} V(x) \cdot \nabla (|u(x)|^{p}) dx \\ &=-C\int _{\mathbb{R}^{n}}\left \| x\right \| ^{(\alpha -1)p}(x\cdot \nabla (|u(x)|^{p})) dx \\ &\leq p C \int _{\mathbb{R}^{n}}\left \| x\right \| ^{(\alpha -1)p+1}|u(x)|^{p-1} \left \| \nabla u(x)\right \| dx. \end{aligned} $$

(25)

Applying Young’s inequality, we have

$$ \begin{aligned} \int _{\mathbb{R}^{n}} & \left \| x\right \| ^{(\alpha -1)p+1} |u(x)|^{p-1} \left \| \nabla u(x)\right \| dx \\ &\leq \frac{1}{\lambda ^{q}} \int _{\mathbb{R}^{n}} \left (\left \| x \right \| ^{(\alpha -1)(p-1)}|u(x)|^{p-1}\right )^{q} dx + \lambda ^{p} \int _{\mathbb{R}^{n}} (\left \| x\right \| ^{\alpha }\left \| \nabla u(x) \right \| )^{p} dx, \end{aligned} $$

(26)

where $\lambda$ is an arbitrary positive number, and $p, q$ are conjugate exponents. From (25) and (26), it follows that

$$ \int _{\mathbb{R}^{n}} \left \| x\right \| ^{(\alpha -1)p} |u(x)|^{p} dx \leq C(\lambda ) \int _{\mathbb{R}^{n}}\left \| x\right \| ^{\alpha p} \left \| \nabla u(x)\right \| ^{p}dx, $$

where

$$ C(\lambda )= \frac{p C \ \lambda ^{p+q}}{\lambda ^{q} - p C} $$

and taking the minimum of $C(\lambda )$ (with respect to $\lambda$), we denote this value by $(C_{H})^{p}$, which completes the proof of Hardy’s inequality. □

Proof of Weighted Sobolev inequality

1. Again we first recall (8), that is

$$ \big( \int _{\mathbb{R}^{n}} \|x\|^{\alpha p^{*}} |u(x)|^{p^{*}}dx \big)^{{1}/{p^{*}}} \leq C_{S} \; \big( \int _{\mathbb{R}^{n}} \|x\|^{ \alpha p} \left \| \nabla u(x)\right \| ^{p} dx \big)^{{1}/{p}}, $$

and divide the proof into two parts. The main idea here is to use the Hardy inequality (9) and the classical Sobolev inequality. We define a function

$$ f:\mathbb{R}^{n}\rightarrow \mathbb{R}, \quad f(x):=\left \| x\right \| ^{ \alpha }|u(x)| $$

(27)

and observe that, $f$ is smooth away from zero. Applying the product rule, we get

$$ \nabla f(x)=\alpha \left \| x\right \| ^{\alpha -2}|u(x)|x+\left \| x \right \| ^{\alpha }\nabla (|u(x)|), $$

and from this, we obtain

$$ \left \| \nabla f(x)\right \| \leq |\alpha |\left \| x\right \| ^{ \alpha -1}|u(x)| +\left \| x\right \| ^{\alpha }\left \| \nabla u(x) \right \| . $$

(28)

2. Now, let us recall the classical Sobolev inequality:

$$ \left (\int _{\mathbb{R}^{n}}|f(x)|^{p^{*}} dx \right )^{1/p^{*}}\leq C \left (\int _{\mathbb{R}^{n}}\left \| \nabla f(x)\right \| ^{p} dx \right )^{1/p}, $$

for any function $f \in W^{1,p}(\mathbb{R}^{n})$ and $1< p< n$. From this inequality and (27), (28), we obtain applying Hardy’s inequality

$$ \begin{aligned} \big(\int _{\mathbb{R}^{n}} \left \| x\right \| ^{\alpha p^{*}} & |u(x)|^{p^{*}} dx \big)^{p/p^{*}} \leq C^{p} \int _{\mathbb{R}^{n}} \big( |\alpha | \left \| x\right \| ^{(\alpha -1)}|u(x)| + \left \| x\right \| ^{\alpha } \left \| \nabla u(x)\right \| \big)^{p} dx \\ & \leq (2 C)^{p} \left ( |\alpha |^{p}\int _{\mathbb{R}^{n}}\left \| x \right \| ^{(\alpha -1)p}|u(x)|^{p}dx +\int _{\mathbb{R}^{n}}\left \| x \right \| ^{\alpha p}\left \| \nabla u(x)\right \| ^{p}dx\right ) \\ & \leq (C_{S})^{p} \int _{\mathbb{R}^{n}}\left \| x\right \| ^{\alpha p} \left \| \nabla u(x)\right \| ^{p} dx, \end{aligned} $$

where we have defined $C_{S}:= 2 C \big ( |\alpha |^{p} (C_{H})^{p} +1 \big )^{1/p}$. □