2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/140109In this work, we consider the Cauchy problem for $u' - Au = f$ with $A$ the Laplacian operator on some Riemannian manifolds or a sublapacian on some Lie groups or some second order elliptic operators on a domain. We show the boundedness of the operator of maximal regularity $f\mapsto Au$ and its adjoint on appropriate Hardy spaces which we define and study for this purpose. As a consequence we reobtain the maximal $L^q$ regularity on $L^p$ spaces for $p,q$ between 1 and $\infty$.27 pagesClassical Analysis and ODEsAnalysis of PDEsFunctional Analysis34G10, 35K90, 42B30, 42B20, 47D06text