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 $P(f)=f$, functions $g_1,g_2 \in W^{l,p}(\Omega,U)\cap C(\Omega,U)$ and $\epsilon>0$. I want to prove that $$|P(g_1)-P(g_2)|_{W^{l,p}}\leq (1+\epsilon)|g_1-g_2|_{W^{l,p}}$$ if $|g_1-f|_{W^{l,p}}$ and $|g_2-f|_{W^{l,p}}$ is sufficiently small. In my application $P=P_M\colon U \rightarrow M$ is the shortest point projection onto a bounded submanifold $M \subset \mathbb{R}^K$. I could find similar results, but not the one I need, in

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 $M$.