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 sp =1. Optimality of the inequality is obtained for any p. The extra logarithmic correction term appears in usual fashion. We also provide a concrete (workable) example of a sequence of smooth functions that converges to constant function in W^s, p ( (0, 1) ) for sp=1 and p=2.