Let $Q=(0,1)^d$ and $S\subset Q$ be a measurable set (I could ask something more!) such that $\frac{|S|}{|Q|}\ge \theta$, for a given $\theta\in (0,1]$. We consider the space of tensor product polynomials on $Q$, denoted as $\mathbb Q_p(Q)$: it consists of polynomials of degree at most $p$ with respect to every coordinate. I would like to prove $$ \|\varphi\|_{L^(Q)} \le C(\theta) \|\varphi\|_{L^2(S)}, \qquad \forall\ \varphi\in\mathbb Q_p(Q). $$ Observing that a polynomial vanishing in a subset of non-zero measure vanishes everywhere, the inequality follows by finite-dimensionality. However, I would like to find the exact dependence of $C(\theta)$ on $\theta$. Any ideas?
I believe this is related to the Remez inequality.