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

TL;DR

This paper establishes a fractional boundary Hardy's inequality in one dimension at the critical case where $sp=1$, demonstrating optimality and providing explicit examples of function sequences converging in the fractional Sobolev space.

Contribution

It proves the fractional boundary Hardy's inequality in the critical case $sp=1$ in one dimension, including optimality and explicit examples.

Findings

Optimal fractional boundary Hardy's inequality for $sp=1$

Logarithmic correction term in the inequality

Explicit example of function sequence converging in $W^{s,p}((0,1))$

Abstract

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 $p=2$.