# Calderón's Complex Interpolation: what is the corresponding classical theorem?

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?

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Doesn't the 2nd form arise naturally when you try to compute the dual space of the 1st form? That's at least some motivation, but I don't really know the history... –  Matthew Daws Dec 28 '11 at 8:59
What do you mean by "an interpolation theorem that is a concrete realization of some interpolation method" ? In addition, what do you mean by "the abstraction in some interpolation method" ? In precise terms, one can define an interpolation method to be a functor from some category of pairs of Banach (or more general topological vector) spaces to the corresponding category of spaces. Shortly, I do not understand your question. –  TaQ Dec 31 '11 at 23:28
I understand that any old functor satisfying some conditions can be an interpolation method. Nevertheless, it appears to me that the "complex interpolation" functor and the "real interpolation" functor must have arisen as generalizations of Riesz-Thorin and Marcinkiewicz interpolation theorems, respectively. This, presumably, is why Hadamard three-lines lemma is built into the first complex interpolation method described above, and this would serve as a motivation behind the first method. That is, I "get" why one would consider the first method. –  Mark Kim Jan 1 '12 at 15:40
My question, then, concerns the motivation behind the second method. I do not know of any classical interpolation theorem that makes use of such restrictions on the derivative. Therefore, if the second method is a generalization of a "concrete theorem" (like Riesz-Thorin), then I don't know what it is. If it is not a generalization of a classical one, then I would like to know why anyone would consider the second method in the first place. In short, I am looking for an answer like Matthew Daws's comment, preferably with an added historical background. –  Mark Kim Jan 1 '12 at 15:47
@ Mark Kim: In other words, if I understand correctly what you wrote, you are asking for some historical motivation for creating Calderón's second interpolation method? For short, calling it C2, hence, the question is about the historical development of thoughts associated with C2. Shouldn't there then be a tag like "history", "reference request", or so? Accidentally, I just looked at the article Vernon Williams: "Generalized interpolation spaces", Trans. AMS, 156, 1971, 309−334. –  TaQ Jan 2 '12 at 9:22