On weighted logarithmic-Sobolev & logarithmic-Hardy inequalities

TL;DR

This paper establishes new weighted logarithmic Sobolev and Hardy inequalities in Euclidean space, characterizes the function spaces where they hold, and proves the existence of optimal constants and their attainability.

Contribution

It introduces specific Banach spaces for weighted logarithmic inequalities, identifies conditions for best constants, and extends Hardy inequalities to second order.

Findings

Identified Banach spaces where inequalities hold

Proved existence of optimal constants and their attainability

Extended Hardy inequalities to second order

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.