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

$W^{s,p}$-approximation properties of elliptic projectors on polynomial  spaces, with application to the error analysis of a Hybrid High-Order  discretisation of Leray-Lions problems