Let $M$ be a compact Riemannian submanifold of $\mathbb{R}^K$, $U\subset \mathbb{R}^K$ an open neighboorhood of $M$ such that the shortest point Projection $P_M\colon U\rightarrow M$ is well-defined and smooth, $\Omega \subset \mathbb{R}^d$ open and bounded, $m \in \mathbb{N}$, $p\in [1,\infty]$ such that $m>\frac{d}{p}$ and $f \in W^{m,p}(\Omega,\mathbb{R}^K)$ with $f(x)\in M$ a.e.. I want to prove that there exist $C,h>0$ such that
$$|P_M(g_1)-P_M(g_2)|_{W^{l,p}}\leq C|g_1-g_2|_{W^{l,p}}$$
for all $0\leq l\leq m$ and $g_1,g_2 \in W^{m,p}(\Omega,\mathbb{R}^K)$ with $\max(\|g_1-f\|_{W^{m,p}},\|g_2-f\|_{W^{m,p}})\leq h$. Here $P_M(g_1) \colon \Omega \rightarrow \mathbb{R}^K$ is defined by $P_M(g_1)(x):=P_M(g_1(x))$ for all $x\in \Omega$. I could find similar results, but not the one I need, in Section 5.5.2 of

*Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equation* 
by Thomas Runst and Winfried Sickel

My goal is to extend error estimates for an operator $Q$ to error estimates of the operator $P_M \circ Q$ for functions with values in $M$.