$L^\infty$ error estimate for optimal $L^2$ approximants I have a function $u\in H^m(\Omega)$ for which I find $u_h \in \mathbb{P}^p(\Omega)$, the best $p$ polynomial approximant in an $L^2$ sense
$$ u_h = \arg \inf_{v\in\mathbb{P}^p(\Omega)} \| u - u_h \|_{L^2(\Omega)}$$
Is there a way to estimate $\| u - u_h \|_{L^\infty(\Omega)}$ in terms of the diameter of $\Omega$?
I'm thinking something along the lines of
$$
\exists s \in \mathbb{R} :\ 
\|u - u_h \|_{L^\infty} \lesssim h^{s} \| u - u_h \|_{L^2(\Omega)}$$
where $h$ is the diameter of $\Omega$.

I think that there is such a result given the right hypotheses (you probably want a polygonal, star shaped domain and conditions that relate the dimension of the domain and the degree of polynomial). Take a look at "The Mathematical Theory of Finite Element Methods" by Brenner and Scott (link). The answers that you desire probably lie in chapter 4 (probably the section entitled Inverse Estimates, but other things might be useful too). I can't find the exact result right now, but give it a look; if you can't find something useful, I can give it a look later.
