For $S \subset \mathbb{R}^d$, let us note $C_{S}^{\infty}(\mathbb{R}^d)$ the set of functions from $\mathbb{R}^d$ to $[0, +\infty)$ that are infinitely-many times continuously differentiable, and with support in $S$. Furthermore, let us note $B_d$ the closed ball of radius $1/2$ in $\mathbb{R}^d$.
Friedrichs' inequality allows writing, for any $f \in C_{B_d}^{\infty}(\mathbb{R}^d)$ and integer $k$, $\int_{\mathbb{R}^d} f^2 \leq \sum_{| \alpha | = k} \int_{\mathbb{R}^d} (\partial^\alpha f)^2$, where $\alpha$ is to be understood as a multi-index of $d$ elements, and $\partial^\alpha f$ is the patrial derivative of $f$ w.r.t. the multi-index $\alpha$. This holds for any $f$. My question is : What is the best reverse inequality that a specific function may enjoy ?
More specifically, can we build a certain $f_0 \in C_{B_d}^{\infty}(\mathbb{R}^d)$ (which depends of $d$ and $k$) and give a closed-form expression $E$ which is as high as possible such that $E(d, k, \sum_{| \alpha | = k} \int_{\mathbb{R}^d} (\partial^\alpha f_0)^2) \leq \int_{\mathbb{R}^d} f_0^2$.
I am unfamiliar with this body of literature so I'll take any pointer.
Thanks and Best Wishes