On Certain New Method to Construct Weighted Hardy-Type Inequalities and Its Application to the Sharp Hardy-Poincaré Inequalities

Abstract

We apply the recent method of Drábek and the authors in order to construct the Hardy–Poincaré–type inequalities

$$\begin{aligned} \bar{C}_{\gamma ,n,p,r}\int _{\mathbb {R}^{n}}\ |\xi |^p \left( 1+r|x|^{\frac{p}{p-1}}\right) \left( 1+|x|^{\frac{p}{p-1}}\right) ^{\gamma (p-1)-p}\, dx\\ \le \int _{\mathbb {R}^{n}}|\nabla \xi |^p\left( 1+|x|^{\frac{p}{p-1}}\right) ^{\gamma (p-1)}\, dx.\end{aligned}$$

Some of the derived inequalities are proven to hold with the best constants.

Appendix

We have the following two lemmas, which we apply to prove Lemma 1.

Lemma 3

Let

$$\begin{aligned} \varPhi _q(\lambda )=|\lambda |^{q-2}\lambda ,\quad \lambda \in {\mathbb {R}^{n}},\quad q>1,\quad s\in {\mathbb {R}}, \end{aligned}$$

where the same notation is used also for $n=1$. Then we have

$$\begin{aligned} \begin{array}{cc} \begin{array}{rcl} \varPhi _q(s\lambda )&{}=&{}\varPhi _q(s)\varPhi _q(\lambda );\\ \varPhi _q(\varPhi _r(\lambda ))&{}=&{} \varPhi _{(q-1)(r-1)+1}(\lambda );\nonumber \\ \varPhi _q(s)&{}=&{}s^{q-1},\ \ \mathrm{when}\ s\ge 0;\nonumber \\ \nabla |x|^q&{}=&{}q\varPhi _q(x);\nonumber \\ \end{array} &{} \begin{array}{rcl} \nabla \varPhi _q(s)&{}=&{} (q-1)|s|^{q-2};\nonumber \\ \varPhi _2(\lambda )&{}=&{}\lambda ;\nonumber \\ \varPhi _q(x)\cdot x&{}=&{}|x|^q.\nonumber \end{array} \end{array} \end{aligned}$$

Using the above lemma, it is easy to verify the statements presented below.

Lemma 4

When $u_\alpha$ is given by (8), we have

$$\begin{aligned} \nabla u_\alpha= & {} (-\alpha p{'})u_{\alpha +1}(x)\varPhi _{p{'}}(x);\\ \varPhi _p (\nabla u_\alpha )= & {} \varPhi _p(-\alpha p{'}) u_{(\alpha +1)(p-1)}\cdot x,\ \ \varPhi _p(-\alpha p{'})=- \mathrm{sgn\alpha } (|\alpha |p{'})^{p-1} ;\\ u_\beta \varPhi _p(\nabla u_\alpha )= & {} \varPhi _p(-\alpha p{'})u_{(\alpha +1)(p-1)+\beta }\cdot x;\\ \mathrm{div}(u_\gamma \cdot x)= & {} n\cdot u_{\gamma +1}\left[ 1+\left( 1-\frac{\gamma p{'}}{n}\right) |x|^{p{'}}\right] ;\\ -\varDelta _{p,u_\beta } u_\alpha= & {} n\varPhi _p(\alpha p{'})\cdot u_{(\alpha +1)(p-1)+\beta +1}\left[ 1+ c_{(\alpha ,\beta ,p,n)}|x|^{p{'}}\right] ,\ \mathrm{where}\\ \ \ \ \ c_{(\alpha ,\beta ,p,n)}= & {} 1-\frac{\left( (\alpha +1)(p-1)+\beta \right) p{'}}{n} \end{aligned}$$