Is every monomial over the UNIT OPEN BALL bounded by its L^{2} norm? Let $m\geq 2$ and  $B^{m}\subset \mathbb{R}^{m}$ be the unit OPEN ball . For  any fixed multi-index $\alpha\in\mathbb{N}^{m}$ with $|\alpha|=n$ large and $x\in B^{m}$ 
$$|x^{\alpha}|^{2}\leq \int_{B^{m}}|y^{\alpha}|^{2}dy\,??$$ 

Using the Bergman inequality, for each $K \subset B^{m}$ compact there exists $M_{K}>0$ such that 
$$|x^{\alpha}|\leq M_{K}||p_{\alpha}||_{2},\quad  \alpha\in\mathbb{N}^{m+1},\,x\in K.$$
