Boundary fractional Hardy’s inequality in dimension one: The critical case

We prove fractional boundary Hardy’s inequality in dimension one for the critical case [Formula: see text]. Optimality of the inequality is obtained for any [Formula: see text]. An extra logarithmic correction term appears. We also provide a concrete (workable) example of a sequence of smooth functions that converges to constant function in [Formula: see text] for [Formula: see text] and [Formula: see text].