Sharp Pointwise Interpolation Inequalities for Derivatives

We prove new pointwise inequalities involving the gradient of a function $u\in C^1(\mathbb{R}^n)$, the modulus of continuity $\omega$ of the gradient $\nabla u$, and a certain maximal function $\mathcal{M}^{\diamond}u$ and show that these inequalities are sharp. A simple particular case corresponding to $n=1$ and $\omega(r)=r$ is the Landau type inequality
$$ |u'(x)|^2\le\frac83\,\mathcal{M}^{\diamond}u(x)\mathcal{M}^{\diamond}u''(x), $$
where the constant $8/3$ is best possible and
$$ \mathcal{M}^{\diamond}u(x)=\sup_{r>0}\frac1{2r}\bigg|\int_{x-r}^{x+r}\operatorname{sign}(y-x)u(y)\,dy\bigg|. $$