This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of *exact interpolation functor of exponent $\theta$* is used below: it is an exact interpolation functor that gives the Riesz-Thorin-esque estimation

$$ \|T\|_{A_\theta \to B_\theta} \leq \|T\|^{1-\theta}_{A_0 \to B_0} \|T\|_{A_1 \to B_1}^{\theta}.$$

Section 4.1 of *Interpolation Spaces: an introduction* by Jöran Bergh and Jörgen Löfström contains two complex interpolation methods. The first method is as follows:

Complex Interpolation 1. Let $\bar{A} = (A_0,A_1)$ be a Banach couple and define $\mathscr{F}(\bar{A})$ to be the space of $(A_0+A_1)$-valued functions $f$ that are continuous on the closed strip $$ S = \lbrace z \in \mathbb{C} : 0 \leq \mathrm{Re}(z) \leq 1 \rbrace$$ and holomorphic on the open strip $$ S_0 = \lbrace z \in \mathbb{C} : 0 < \mathrm{Re}(z) < 1 \rbrace$$ with the added assumption that the functions $t \mapsto f(it)$ and $t \mapsto f(1+it)$ be continuous maps from $\mathbb{R}$ into $A_0$ and $A_1$, respectively, that tend to zero as $|t| \to \infty$. Given these, $\mathscr{F}(\bar{A})$ is a Banach space with the norm $$ \|f\|_{\mathscr{F}} = \max \left( \sup \|f(it)\|_{A_0} , \sup \|f(1+it)\|_{A_1} \right). $$ Furthermore, for each $0 \leq \theta \leq 1$, the space $C_\theta(\bar{A})$ consisting of $a \in A_0 + A_1$ such that $a = f(\theta)$ for some $f \in \mathscr{F}(\bar{A})$ is a Banach space with the norm $$ \|a\|_{[\theta]} = \inf \lbrace \|f\|_{\mathscr{F}} : f \in \mathscr{F}(\bar{A}) \mbox{ such that } f(\theta) = a \rbrace$$ and $C_\theta$ is an exact interpolation functor of exponent $\theta$.

The second method is as follows:

Complex Interpolation 2. Let $\bar{A} = (A_0,A_1)$ be a Banach couple and define $\mathscr{G}(\bar{A})$ to be the space of $(A_0+A_1)$-valued functions $g$ that are continuous on the closed strip $$ S = \lbrace z \in \mathbb{C} : 0 \leq \mathrm{Re}(z) \leq 1 \rbrace,$$ holomorphic on the open strip $$ S_0 = \lbrace z \in \mathbb{C} : 0 < \mathrm{Re}(z) < 1 \rbrace,$$ and have the bound $$ \|g(z)\|_{A_0 + A_1} \leq c(1+|z|) $$ for some constant $c$, with the added assumptions that (1) the differences $g(it_1) - (it_2)$ and $g(1+it_1) - g(1 + it_2)$ are in $A_0$ and $A_1$, respectively, for all $t_1, t_2 \in \mathbb{R}$ and (2) the quantity $$ \|g\|_{\mathscr{G}}= \max \left( \sup_{t_1,t_2 \in \mathbb{R}} \left\| \frac{g(it_1)-g(it_2)}{t_1 - t_2} \right\|_{A_0} , \sup_{t_1,t_2 \in \mathbb{R}} \left\| \frac{g(1+it_1) - g(1+it_2)}{t_1 - t_2} \right\|_{A_1} \right)$$ is finite. Given these, the space $\mathscr{G}(\bar{A})$ (modulo constant functions) is a Banach space with the above norm. Furthermore, for each $0 \leq \theta \leq 1$, the space $C^\theta(\bar{A})$ consisting of $a \in A_0 + A_1$ such that $a = g'(\theta)$ for some $g \in \mathscr{G}(\bar{A})$ for some $g \in \mathscr{G}(\bar{A})$ is a Banach space with the norm $$ \|a\|^{[\theta]} = \inf \lbrace \|g\|_{\mathscr{G}} : g \in \mathscr{G} \mbox{ such that } g'(\theta) = a \rbrace$$ and $C^\theta$ is an exact interpolation functor of exponent $\theta$.

Now, I can see that the first method is a generalization of Riesz-Thorin, with Hadamard's three lines lemma included in the abstraction. I am not sure, however, what the second method attempts to generalize. The setting clearly makes use of some control on the derivatives, but I do not know of any relevant classical interpolation theorem with such hypotheses or methods of proof. So:

Question.Is there an interpolation theorem that is a concrete realization of the second method? In addition, what is the motivation behind the abstraction in the second method?

Trans. AMS,156, 1971, 309−334. – TaQ Jan 2 '12 at 9:22