Let $d \ge 1$ and $0 < \alpha < d \land 2$. For $p \in (1, \infty)$ and $u: \mathbb{R}^d \to \mathbb{R}$ we define the $p$-form,

${\mathbb{R}^d} \int_{\mathbb{R}^d} (u(x)-u(y)) (u(x)^{\langle p - 1 \rangle} -u(y)^{\langle p - 1 \rangle} ) \nu(x-y) \,dy \,dx$

where

$\nu(z)=\frac{2^{\alpha}\Gamma\big((d+\alpha)/2\big)\pi^{-d/2}}{|\Gamma(-\alpha/2)|} |z|^{-d-\alpha}, \quad z \in \mathbb{R}^d,$

and $a^{\langle k \rangle} := \left|a \right|^k \operatorname{sgn} a$. During the talk I will discuss the following inequality

$\mathcal{E}_p[u] \geq C \int_{\mathbb{R}^d} \frac{\left| u(x) \right|^p} {|x|^{\alpha}} \,dx, \quad u \in L^p(\mathbb{R}^d).$

The explicit formula for the best constant $C$ will be given. The talk will be based on the recent paper \verb+https://arxiv.org/abs/2103.06550