Let $\Omega\subset\mathbb{R}^2$ a bounded domain. Define triangulations $\mathcal{T}_h=\{K\}$ of $\Omega,$ with $h = \max_{\mathcal{T}_h}diam(K)$ and $\kappa:=\sup_{\mathcal{T}_h}\kappa_K,$ where $\kappa_K$ is the diameter of the inscribed circle in the triangle $K.$ We use the finite element spaces \begin{equation} S_h : = \left\lbrace \chi\in\mathcal{C}(\overline{{\Omega}})\,:\,\chi\in \mathbb{P}_1(K),\;\forall\;K\in\mathcal{T}_h\right\rbrace. \end{equation}
It is well known the $L_2-$ projection operator as the map $P_h\, :\,L_2(\Omega) \to S_h$ such that \begin{align} (P_hv, \chi) = (v, \chi),\;\;\;\;\;\forall\,\chi\in S_h, \end{align} and there holds the following estimate, \begin{align} \|v - P_hv\|_{L_2(\Omega)} \leq Ch^2\|v\|_{H^2(\Omega)},\;\;\;\;\forall\,v\in H^2(\Omega). \end{align} My question: Let $\partial K$ be the boundary of $K.$ Can we define a $L_2-$projection $P_{h,\partial K} \,:\,L_2(\partial K) \to \mathbb{P}_k(\partial K)$ with $k=0,1,$ analogous definition and estimate? Is there any reference?
Technically the answer is yes, but you have to be careful, since $K$ is a triangle, $\partial K$ is a Lipschitz hypersurface, and therefore it is not smooth enough to define the space $H^2(\partial K)$ in the same way you would for $H^2(K)$. You could either define a piecewise projection, which is the projection on each edge $e$ of $\partial K$, and then you would indeed have
$$\|v-P_ev\|_{L^2(e)}\le Ch_F^2\|v\|_{H^2(e)}$$
for $v\in H^2(e)$ (note that $H^2(e)$ is definable).
It might be the case that you are interesting in obtaining the a bound for
$$\|v-P_hv\|_{L^2(\partial K)}$$
where $P_h$ is the $L^2$-projection you mention. This is possible via a scaled trace estimate:
$$~~~~~~~~~~~~~~~~~\|v-P_hv\|_{L^2(\partial K)}^2\le Ch^{-1}\|v-P_hv\|_{L^2(K)}^2+Ch|v-P_hv|_{H^1(K)}^2$$ $$\le Ch^3\|v\|_{H^2(K)}^2.$$