The Trudinger type inequality in fractional boundary Hardy inequality

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