Characterization of interpolation spaces and regularity properties for holomorphic semigroups (Q916999)

Let A be a second order elliptic operator on \(L^ 1({\mathbb{R}}^ n)\), generating the holomorphic semigroup S(t). The author considers the interpolation spaces \[ D_ A(\theta,1)=\{u\in L^ 1({\mathbb{R}}^ n)| \quad \int^{\infty}_{0}\| AS(t)u\| t^{-\theta}dt<+\infty \}, \] and proves the following characterization: \[ D_ A(\theta,1)=W^{2\theta,1}({\mathbb{R}}^ n),\quad if\quad 2\theta \neq 1\text{ and } =B^{1,1}({\mathbb{R}}^ n),\quad if\quad 2\theta =1. \] Here \(W^{2\theta,1}\) is the Sobolev space and \(B^{1,1}\) is the Besov space. This extends a result of \textit{M. H. Taibleson} for the special case \(A=\Delta\) [J. Math. Mech. 13, 407-479 (1964; Zbl 0132.094)].