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Consider two domains (e.g. in $\mathbb{R}^2$) $K$ and $\hat{K}$ and an affine invertible transformation $F_K\, :\, \hat{K} \rightarrow K $ s.t. $$ F_K(\hat{x})=B_K \hat{x}+b_k $$ If $v \in H^{m}(K)$, let $\hat{v}:\hat{K}\rightarrow \mathbb{R}$ be the function defined by $\hat{v}=v \circ F_K$. Then there exists a constant C such that $$ ||D^\alpha \hat{v}||_{L^2(\hat{K})} \leq C ||B_K||^m \sum_{|\beta|=m}||(D^\beta {v})\circ F_K||_{L^2(\hat{K})} $$ where $|\alpha|=m$.

In the book (Numerical models for differential problems by Quarteroni) this implication has been explained by saying it is a consequence of the chain rule for differentiation...but I actually don't understand the details of this step...

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    What book? Did you try some easy case, i.e. the identity map, or a translation, or just a rotation? – 2017-02-13
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    The book is "Numerical Models for differential problems" by Alfio Quarteroni, 2nd ed. chapter 4, page 89, lemma 4.2. Btw, no, i haven't try any particulare case... but e.g. for the identity it trivially works – 2017-02-13

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