In this paper we provide sufficient conditions for regional asymptotic and nonlinear ℒ_2m gain estimation of linear systems subject to saturations and/or deadzones, based on piecewise polynomial Lyapunov functions. By using sum-of-squares relaxations, these conditions are formulated in terms of linear matrix inequalities for the global case, and bilinear ones for the regional case. The results are provided for both the cases with and without structured parametric uncertainties. Example studies are used to comparatively illustrate the proposed techniques.