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 $\mathrm{\Omega }\subset {\mathbb{R}}^d$. In particular, we establish fractional version of Trudinger-type inequality with an extra singular function, namely $d$-th power of the distance function from $\mathrm{\partial }\mathrm{\Omega }$ 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.