Sharp Logarithmic Inequalities for Hardy Operators

Abstract

Let $\ell \ge 1$ be a fixed number. We determine, for each $K>0$, the best constant $L=L(K,\ell )\in (0,\infty ]$ such that the following holds. If $f$ is a function on $(0,1]$ with ${\int }_0^1|f(r)|\mathrm{d}r=1$, then

As an application, we derive a sharp local logarithmic estimate for $n$-dimensional fractional Hardy operator.