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