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We consider maximal regularity in the 
${{H}^{p}}$ sense for the Cauchy problem 
${{u}^{\prime }}(t)+Au(t)=f(t)(t\,\in \mathbb{R})$, where 
$A$ is a closed operator on a Banach space 
$X$ and 
$f$ is an $X$-valued function defined on 
$\mathbb{R}$. We prove that if $X$ is an AUMD Banach space, then $A$ satisfies ${{H}^{p}}$-maximal regularity if and only if $A$ is Rademacher sectorial of type 
$<\frac{\pi }{2}.$ Moreover we find an operator $A$ with ${{H}^{p}}$-maximal regularity that does not have the classical 
${{L}^{p}}$-maximal regularity. We prove a related Mikhlin type theorem for operator valued Fourier multipliers on Hardy spaces 
${{H}^{p}}(\mathbb{R};\,X)$, in the case when $X$ is an AUMD Banach space.
