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

This is the authors' abstract. We don't add key points for this paper.

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 Ws,p((0,1)) for $sp=1$ and $p=2$.