Boundary Hardy inequality is a classical result [1980s] which states that if $1 < p < \infty$ and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^d$, then

\begin{equation} \int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx, \forall \ u \in C^{\infty}_{c}(\Omega), \end{equation}

where $\delta_\Omega(x)$ is the distance function from $\partial\Omega$. B. Dyda generalised the above inequality to the fractional setting, which says, for $sp >1$ and $s\in (0,1)$

\begin{equation} \int_{\Omega} \frac{|u(x)|^{p}}{\delta_{\Omega}^{sp}(x)} dx \leq C \int_{\Omega} \int_{\Omega} \frac{|u(x)-u(y)|^{p}}{|x-y|^{d+sp}} dxdy, \ \forall \ u \in C^{\infty}_{c}(\Omega). \end{equation}

The first and the second inequality is not true for $p=1$ and $sp=1$ respectively. In this talk, I will present the appropriate inequalities for the critical cases: $p=1$ for the first and $sp= 1$ for the second inequality. If time, permits I will touch upon another related inequality called Michael-Simmon-Sobolev inequality.

This is a part of the joint work with Adimuthi, Purbita Jana and Vivek Sahu.