Maximal regularity and Hardy spaces

In this work, we consider the Cauchy problem for $u^{\mathrm{\prime }}-Au=f$ 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 $\mathrm{\infty }$.