On Polynomial Progressions Inside Sets of Large Dimension

In this note we connect Sobolev estimates in the context of polynomial averages e.g.\ \begin{align}\label{e:savings} \| \int_0^1 \prod_{k=1}^m f_k(x-t^k) \|_{1} \leq \text{Const} \cdot 2^{-\text{const} \cdot l} \prod_{i=1}^m \| f_k \|_m \end{align} whenever some $f_i$ vanishes on $\{ |ξ| \leq 2^l