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 $x\in B^{m}$ 
$$|x^{\alpha}|^{2}\leq \int_{B^{m}}|y^{\alpha}|^{2}dy\,??$$ 

No. For a counterexample, take $\alpha=(n,0,\ldots,0)$. Obviously, $\max_{S^m}|x^\alpha|=1$, but an easy calculation shows 
$$
\int_{S^m}|y^\alpha|^2{\mathrm{d}}\sigma(y) \to 0,
$$
as $n\to\infty$.
For the updated question, that involves the open unit ball, the answer is the same. With the same counterexample, we have
$$
\int_{B^m}|y^\alpha|^2{\mathrm{d}}y \to 0,
$$
as $n\to\infty$.
