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 $M$.
