Real interpolation of domains of sectorial operators on Lp-spaces

Let A be a sectorial operator on a non-atomic Lp-space, 1⩽p<∞, whose resolvent consists of integral operators, or more generally, has a diffuse representation. Then the fractional domain spaces D(Aα) for α∈(0,1) do not coincide with the real interpolation spaces of (Lq,D(A)). As a consequence, we obtain that no such operator A has a bounded H∞-calculus if p=1.