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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.
Adimurthi et al. (Tue,) studied this question.