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?

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?

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?

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?

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?

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?