Interpolation of derivatives and ultradifferentiable regularity

Interpolation inequalities for $C^m$ functions allow to bound derivatives of intermediate order $0 < j<m$ by bounds for the derivatives of order $0$ and $m$. We review various interpolation inequalities for $L^p$-norms ($1 \le p \le \infty$) in arbitrary finite dimensions. They allow us to study ultradifferentiable regularity by lacunary estimates in a comprehensive way, striving for minimal assumptions on the weights.