The authors present some new results on the \(L^p\)-interior regularity of the solutions of the equation with smooth coefficients \[ Pu= \sum_{\alpha\in Q} a_\alpha(x) D^\alpha_x u= f, \] where the Newton polyhedron of the coefficients \(Q\) is assumed complete, and the operator \(P\) is multi-quasi-elliptic, in the sense that \[ \Biggl| \sum_{\alpha\in Q} a(x)\xi^\alpha\Biggr|\leq C\Biggl(\sum_{\alpha\in Q} \xi^{2\alpha}\Biggr)^{1/2}, \] cf. [\textit{S. Gindikin} and \textit{L. R. Volevich}, The method of Newton's polyhedron in the theory of partial differential equations. Mathematics and Its Applications. Soviet Series. 86. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0779.35001)]. In particular, the authors obtain that \(Pu= f\in L^p\), \(1< p<\infty\), implies \(D^\alpha u\in L^p\) for every \(\alpha\in Q\). These results are deduced from a pseudodifferential calculus of the type of R. Beals, L. Hörmander, for a general class of operators including \(P\) and its parametrix \(P'\). Basic point is a theorem of \(L^p\)-boundedness for pseudodifferential operators, new with respect to the existing literature.