Let $P\colon U\subset \mathbb{R}^K\rightarrow \mathbb{R}^K$ be $C^\infty$, $\Omega \subset \mathbb{R}^d$ open and bounded, $m \in \mathbb{N}$, $p\in [1,\infty]$ such that $m>\frac{d}{p}$, $f \in W^{m,p}(\Omega,U)$ with $P(f)=f$, $\epsilon>0$ and $l\leq m$. I want to prove that there exist $h>0$ such that $$|P(g_1)-P(g_2)|_{W^{l,p}}\leq (1+\epsilon)|g_1-g_2|_{W^{l,p}}$$ for all functions $g_1,g_2 \in W^{m,p}(\Omega,U)$ with $|g_1-f|_{W^{m,p}}\leq h$ and $|g_2-f|_{W^{m,p}}\leq h$. In my application $P=P_M\colon U \rightarrow M$ is the shortest point projection onto a compact submanifold $M \subset \mathbb{R}^K$. I could find similar results, but not the one I need, in section 5.2.2 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 $M$.