A maximal theorem for holomorphic semigroups

Gordon Blower and Ian Doust

Email:  i.doust@unsw.edu.au

Abstract: Let  X  be a closed linear subspace of the Lebesgue space  L^p( \Omega ,μ )  for some  1 < p < \infty , and let  -A  be an invertible operator that is the generator of a bounded holomorphic semigroup   T_t  on  X.  Then for each  0 < \alpha < 1  the maximal function   sup_{t > 0} | T_tf(x) |   belongs to   L^p(\Omega ,μ)   for each  f  in the domain of  A^\alpha.  If moreover  iA  generates a bounded C₀-group and  A  has spectrum contained in  (0,\infty ) , then  A  has a bounded  H^\infty  functional calculus.