| Abstract: |
| Boundary Hardy inequality states that if ~$1 < p < \infty$ and $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^d$, then
$$\int_{\Omega} \frac{|u(x)|^{p}}{\delta^{p}_{\Omega}(x)} dx \leq C\int_{\Omega} |\nabla u(x) |^{p}dx, \forall \ u \in C^{\infty}_{c}(\Omega),$$ 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 $s\in (0,1)$
$$
\int_{\Omega} \frac{|u(x)|^{p}}{\delta_{\Omega}^{sp}(x)} dx \leq C \int_{\Omega} \int_{\Omega} \frac{|u(x)-u(y)|^{p}}{|x-y|^{d+sp}} dxdy, \ \forall \ u \in C^{\infty}_{c}(\Omega).
$$
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.
I will also discuss the case when the weight function ($\delta_\Omega$) in the first inequality is replaced by distance function from a $k-$ dimensional sub manifold of $\Omega$ and some related applications to Moser-Trudinger inequality. |
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