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 $p=2$.