The Trudinger type inequality in fractional boundary Hardy inequality

We establish Trudinger-type inequality in the context of fractional boundary Hardy-type inequality for the case $sp=d$, where $p>1, ~ s \in (0,1)$ on a bounded Lipschitz domain $Ω\subset \mathbb{R}^d$. In particular, we establish fractional version of Trudinger-type inequality with an extra singular function, namely $d$-th power of the distance function from $\partial Ω$ in the denominator of the integrand. The case $d=1$, as it falls in the category $sp=1$, becomes more delicate where an extra logarithmic correction is required together with subtraction of an average term.

💡 Research Summary

The paper investigates a Trudinger‑type exponential integrability result in the setting of fractional boundary Hardy inequalities when the parameters satisfy the critical relation (sp = d) (with (p>1) and (s\in(0,1))). The authors work on a bounded Lipschitz domain (\Omega\subset\mathbb{R}^{d}) and consider the distance function (\delta_{\Omega}(x)=\operatorname{dist}(x,\partial\Omega)). Building on B. Dyda’s fractional Hardy inequality \