On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

Abstract

For $N \geq 3$ and $p \in (1,N)$, we look for $g \in L^1_{loc}(\mathbb{R}^N)$ that satisfies the following weighted logarithmic Sobolev inequality: \begin{equation*} \int_{\mathbb{R}^N} g |u|^p \log |u|^p \ dx \leq \gamma \log \left( C_{\gamma} \int_{\mathbb{R}^N} |\nabla u|^p \ dx \right) \,, \end{equation*} for all $u \in \mathcal{D}^{1,p}_0(\mathbb{R}^N)$ with $\int_{\mathbb{R}^N} g|u|^p=1$, for some $\gamma,C_{\gamma}>0$. For each $r \in(p,\frac{Np}{N-p}]$, we identify a Banach function space $\mathcal{H}_{p,r}(\mathbb{R}^N)$ such that the above inequality holds for $g \in \mathcal{H}_{p,r}(\mathbb{R}^N)$. For $\gamma > \frac{r}{r-p}$, we also find a class of $g$ for which the best constant $C_{\gamma}$ in the above inequality is attained in $\mathcal{D}^{1,p}_0(\mathbb{R}^N)$. Further, for a closed set $E$ with Assouad dimension $=d<N$ and $a \in (-\frac{(N-d)(p-1)}{p},\frac{(N-p)(N-d)}{Np}),$ we establish the following logarithmic Hardy inequality \begin{equation*} \int_{\mathbb{R}^N} \frac{|u|^p}{|\delta_E|^{p(a+1)}} \log \left(\delta_E^{N-p-pa} |u|^p\right) \ dx \leq \frac{N}{p} \log \left(\text{C} \int_{\mathbb{R}^N} \frac{|\nabla u|^p}{|\delta_E^{pa}|} \ dx \right) \,, \end{equation*} for all $u \in C_c^{\infty}(\mathbb{R}^N)$ with $\displaystyle \int_{\mathbb{R}^N} \frac{|u|^p}{|\delta_E|^{p(a+1)}} =1,$ for some $\text{C}>0$, where $\delta_E(x)$ is the distance between $x$ and $E$. The second order extension of the logarithmic Hardy inequality is also obtained.