A functional calculus description of real interpolation spaces for sectorial operators

For a holomorphic function V) defined on a sector we give a condition implying the identity (X,'D(A(alpha)))(theta,p) = {x is an element of X \ t(-theta Re alpha)psi(tA) is an element of L*(p)((0, infinity); X)} where A is a sectorial operator on a Banach space X. This yields all common descriptions of the real interpolation spaces for sectorial operators and allows easy proofs of the moment inequalities and reiteration results for fractional powers.