In the following $\mathbb P_p$ will denote the space of polynomials of degree $p$ with real coefficients.
Let $\hat\Omega=(0,1)^d$ be such that $\bar{\hat\Omega} =\bar{Q}\cup\bar{Q'}$, $Q$, $Q'$ open and connected and $\Omega$ be an open bounded connected set of $\mathbb R^d$.
Let $\mathbf F: \hat \Omega\to \Omega$ be a bi-Lipschitz map such that $\mathbf F_{|Q}\in \mathbb P^p(Q)$ and $\mathbf F_{|Q'}\in \mathbb P^p(Q')$. Moreover $\mathbf F\in C^{p-1}(\hat \Omega)$.
Assume now we have a Sobolev function $u\in H^k(\Omega)$, $k\ge 1$. I would like to know in which Sobolev space its pull-back is lying, namely $\hat u:=u\circ\mathbf F \in ?$
Any hints or references would be really appreciated.