Mixed and directional derivatives

Abstract:

The estimate \[ \left \| {\frac {{{\partial ^k}f\left ( x \right )}}{{\partial {\xi _1} \cdots \partial {\xi _k}}}} \right \| \leq {\sup _\xi }\left \| {\frac {{{\partial ^k}f\left ( x \right )}}{{\partial {\xi ^k}}}} \right \|\] is proved for various spaces of functions over domains in ${R^d}$ , where $\partial f / \partial {\xi _i}$ is the directional derivative of $f$ in the ${\xi _i}$, direction and ${\xi _1}, \ldots ,{\xi _k}$ are any $k$ directions.