  In this work we prove optimal $W^{s,p}$-approximation estimates (with
$p\in[1,+\infty]$) for elliptic projectors on local polynomial spaces. The
proof hinges on the classical Dupont--Scott approximation theory together with
two novel abstract lemmas: An approximation result for bounded projectors, and
an $L^p$-boundedness result for $L^2$-orthogonal projectors on polynomial
subspaces. The $W^{s,p}$-approximation results have general applicability to
(standard or polytopal) numerical methods based on local polynomial spaces. As
an illustration, we use these $W^{s,p}$-estimates to derive novel error
estimates for a Hybrid High-Order discretization of Leray--Lions elliptic
problems whose weak formulation is classically set in $W^{1,p}(\Omega)$ for
some $p\in(1,+\infty)$. This kind of problems appears, e.g., in the modelling
of glacier motion, of incompressible turbulent flows, and in airfoil design.
Denoting by $h$ the meshsize, we prove that the approximation error measured in
a $W^{1,p}$-like discrete norm scales as $h^{\frac{k+1}{p-1}}$ when $p\ge 2$
and as $h^{(k+1)(p-1)}$ when $p<2$.
