On maximal LpLp-regularity

The aim of this paper is to propose weak assumptions to prove maximal LqLq regularity for Cauchy problem:dudt(t)−Lu(t)=f(t). Mainly we only require “off-diagonal” estimates on the real semigroup (etL)t>0(etL)t>0 to obtain maximal LqLq regularity. The main idea is to use one kind of Hardy space H1H1 adapted to this problem and then use interpolation results. These techniques permit us to prove weighted maximal regularity too.