  The aim of this paper is to propose weak assumptions to prove maximal L^q
regularity for Cauchy problem: du/dt - Lu(t)=f(t). Mainly we only require
"off-diagonal" estimates on the real semigroup (e^{tL})_{t>0} to obtain maximal
L^q regularity. The main idea is to use a one kind of Hardy space H^1 adapted
to this problem and then use interpolation results. These techniques permit us
to prove weighted maximal regularity too.
