Defined the real interpolation method.
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Bokuan Li
@@ -301,3 +301,11 @@
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year={2002},
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publisher={Cambridge University Press}
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}
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@book{Lunardi,
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title={Interpolation theory},
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author={Lunardi, Alessandra and others},
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volume={16},
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year={2018},
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publisher={Edizioni della normale Pisa}
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}
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@@ -64,7 +64,7 @@
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\begin{definition}[Category of Compatible Couples]
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\label{definition:compatible-category}
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Let $\catc$ be a subcategory of normed spaces over $K \in \RC$ and $(E_0, E_1)$ be a compatible couple, then $E_0 E_1$ are a \textbf{compatible couple in $\catc$} if $E_0, E_1, E_0 \cap E_1, E_0 + E_1 \in \catc$.
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Let $\catc$ be a subcategory of normed spaces over $K \in \RC$ and $(E_0, E_1)$ be a compatible couple, then $(E_0, E_1)$ is \textbf{compatible couple in $\catc$} if $E_0, E_1, E_0 \cap E_1, E_0 + E_1 \in \catc$.
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Let $(E_0, E_1)$ and $(F_0, F_1)$ be compatible couples in $\catc$ and $T \in L(E_0 + E_1, F_0 + F_1)$, then $T$ is a \textbf{morphism of compatible couples} if $T|_{E_0} \in \text{Mor}_{\catc}(E_0; F_0)$ and $T|_{E_1} \in \text{Mor}_{\catc}(E_1; F_1)$.
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\section{The Real Interpolation Method}
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\label{section:real-interpolation}
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\begin{definition}[$L_*^p$]
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\label{definition:weighted-lp-interpolation}
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Let $I \subset (0, \infty)$ be measurable and $p \in [1, \infty]$, then $L^p_*(I)$ denotes the Lebesgue space $L^p$ with respect to the measure $dt/t$ on $I$.
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\end{definition}
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\begin{definition}[$K$-Functional]
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\label{definition:k-functional}
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Let $(E_0, E_1)$ be a compatible couple of Banach spaces, then the \textbf{$K$-functional} of the couple $(E_0, E_1)$ is the mapping $(0, \infty) \times (E_0 + E_1) \to [0, \infty)$ defined by
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\[
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K(t, x) = K(t, x, E_0, E_1) = \inf_{\substack{x_0 \in E_0 \\ x_1 \in E_1 \\ x = x_0 + x_1}}\norm{x_0}_{E_0} + t\norm{x_1}_{E_1}
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\]
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\end{definition}
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\begin{lemma}
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\label{lemma:k-functional-properties}
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Let $(E_0, E_1)$ be a compatible couple of Banach spaces and $K: (0, \infty) \times (E_0 + E_1) \to [0, \infty)$ be its $K$-functional, then:
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\begin{enumerate}
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\item For each $t > 0$, the mapping $x \mapsto K(t, x)$ is a norm on $E_0 + E_1$. For any $s, t > 0$,
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\[
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K(t, x) \le \paren{1 \vee \frac{s}{t}}K(s, x)
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\]
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In particular,
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\[
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K(t, x) \le (1 \vee t)K(1, x) = (1 \vee t) \norm{x}_{E_0 + E_1}
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\]
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\item For each $t > 0$ and $x \in E_0 \cap E_1$,
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\[
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K(t, x) \le (1 \wedge t)\norm{x}_{E_0 \cap E_1}
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\]
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\item For each $x \in E_0 + E_1$, the mapping $t \mapsto K(t, x)$ is non-decreasing, concave, and hence continuous.
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\item $K$ is jointly continuous.
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\end{enumerate}
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\end{lemma}
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\begin{proof}
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(1): Let $x = x_0 + x_1 \in E_0 + E_1$, then
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\[
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K(t, x) \le \norm{x_0}_{E_0} + t\norm{x_1}_{E_1} \le \paren{1 \vee \frac{t}{s}}(\norm{x_0}_{E_0} + s\norm{x_1}_{E_1})
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\]
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As the above holds for all such decompositions of $x$,
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\[
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K(t, x) \le \paren{1 \vee \frac{t}{s}}K(s, x)
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\]
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(3): For each $x \in E_0 + E_1$, the mapping $t \mapsto K(t, x)$ is an infimum of linear and hence concave functions, so it is concave.
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(4): Let $x, y \in E_0 + E_1$ and $s, t > 0$, then
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\begin{align*}
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K(s, x) - K(t, y) &= K(s, x) - K(t, x) + K(t, x) - K(t, y) \\
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|K(s, x) - K(t, y)| &\le |K(s, x) - K(t, x)| + |K(t, x) - K(t, y)| \\
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&\le |K(s, x) - K(t, x)| + (1 \vee s \vee t)\norm{x - y}_{E_0 + E_1}
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\end{align*}
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so $K(t, y) \to K(s, x)$ as $(t, y) \to (s, x)$.
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\end{proof}
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\begin{definition}[Method of Real Interpolation]
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\label{definition:real-interpolation}
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Let $(E_0, E_1)$ be a compatible couple of Banach spaces, $K: (0, \infty) \times (E_0 + E_1) \to [0, \infty)$ be its $K$-functional, $\theta \in (0, 1)$, $p \in [1, \infty]$, and
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\[
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(E_0, E_1)_{\theta, p} = \bracsn{x \in E_0 + E_1| t \mapsto t^{-\theta}K(t, x) \in L_*^p((0, \infty))}
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\]
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with the norm
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\[
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\norm{x}_{(E_0, E_1)_{\theta, p}} = \normn{t^{-\theta}K(t, x)}_{L_*^p((0, \infty))}
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\]
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then:
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\begin{enumerate}
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\item $(E_0, E_1)_{\theta, p}$ is an intermediate Banach space between $E_0$ and $E_1$.
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\item The mapping $K_{\theta, p}$ defined by $(E_0, E_1) \mapsto (E_0, E_1)_{\theta, p}$ is an interpolation functor of exact exponent $\theta$.
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\end{enumerate}
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and the functor $K_{\theta, p}$ is the \textbf{method of real interpolation}.
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\end{definition}
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\begin{proof}[Proof, {{\cite[Theorem 1.6]{Lunardi}}}. ]
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(1): Let $x \in E_0 \cap E_1$, then by (2) of \autoref{lemma:k-functional-properties}, $K(t, x) \le (1 \wedge t)\norm{x}_{E_0 \cap E_1}$. If $p \in [1, \infty)$, then
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\begin{align*}
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\int_0^\infty |t^{-\theta} K(t, x)|^p \frac{dt}{t} &\le \norm{x}_{E_0 \cap E_1}^p\int_0^\infty |t^{-\theta}(1 \wedge t)|^p \frac{dt}{t} \\
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&\le \norm{x}_{E_0 \cap E_1}^p\braks{\int_0^1 t^{p(1 - \theta)} \frac{dt}{t} + \int_1^\infty t^{-p\theta} \frac{dt}{t}} \\
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&\le \norm{x}_{E_0 \cap E_1}^p\braks{\frac{1}{1 - \theta} + \frac{1}{p\theta}}
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\end{align*}
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If $p = \infty$, then
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\[
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\sup_{t > 0}t^{-\theta}K(t, x) \le \norm{x}_{E_0 \cap E_1} \braks{\sup_{t \in (0, 1)}t^{1-\theta} \vee \sup_{t \ge 1}t^{-\theta}} \le \norm{x}_{E_0 \cap E_1}
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\]
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so $E_0 \cap E_1$ is continuously embedded inside $(E_0, E_1)_{\theta, p}$.
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On the other hand, let $x \in (E_0, E_1)_{\theta, p}$. If $p \in [1, \infty)$, then by (3) of \autoref{lemma:k-functional-properties},
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\begin{align*}
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\int_0^\infty |t^{-\theta} K(t, x)|^p \frac{dt}{t} &\ge \int_1^2 |t^{-\theta} K(t, x)|^p \frac{dt}{t} \ge \frac{1}{2}\int_1^2 |t^{-\theta} K(t, x)|^p dt \\
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&\ge \frac{1}{2^{1+\theta p}} \int_1^2 |K(t, x)|^p dt \ge \frac{K(1, x)^p}{2^{1 + \theta p}} = \frac{\norm{x}_{E_0 + E_1}^p}{2^{1 + \theta p}}
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\end{align*}
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and if $p = \infty$, then $\norm{x}_{E_0 + E_1} = K(1, x) \le \norm{x}_{(E_0, E_1)_{\theta, p}}$.
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Finally, by (1) of \autoref{lemma:k-functional-properties}, $K(t, x + y) \le K(t, x) + K(t, y)$ for all $x, y \in (E_0, E_1)_{\theta, p}$, so $(E_0, E_1)_{\theta, p}$ is complete by \hyperref[Minkowski's inequality]{theorem:minkowski} and \autoref{lemma:banach-absolute}.
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Therefore $(E_0, E_1)_{\theta, p}$ is an intermediate Banach space between $E_0$ and $E_1$.
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(2): Let $(E_0, E_1)$ and $(F_0, F_1)$ be compatible couples of Banach spaces, and $T \in L(E_0 + E_1; F_0 + F_1)$ such that $T|_{E_0} \in L(E_0; F_0)$ and $T|_{E_1} \in L(E_1; F_1)$.
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Assume without loss of generality that $T \ne 0$. In particular, after possibly considering $(E_1, E_0)$, $(F_1, F_0)$, $(E_1, E_0)_{1 - \theta, p}$, and $(F_1, F_0)_{1 - \theta, p}$, assume without loss of generality that $\norm{T}_{L(E_0; F_0)} \ne 0$.
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Let $x = x_0 + x_1 \in E_0 + E_1$, $t > 0$, and $\lambda > \norm{T}_{L(E_1; F_1)}/\norm{T}_{L(E_0; F_0)}$,
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\begin{align*}
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\norm{Tx_0}_{F_0} + t\norm{Tx_1}_{F_1} &\le \norm{T}_{L(E_0; F_0)}\norm{x_0}_{F_0} + t\norm{T}_{L(E_1; F_1)}\norm{x_1}_{F_1} \\
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&= \norm{T}_{L(E_0; F_0)}\paren{\norm{x_0}_{F_0} + t\frac{\norm{T}_{L(E_1; F_1)}}{\norm{T}_{L(E_0; F_0)}}\norm{x_1}_{F_1}} \\
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&\le \norm{T}_{L(E_0; F_0)}({\norm{x_0}_{F_0} + \lambda t\norm{x_1}_{F_1}})
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\end{align*}
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As the above holds for all decompositions of $x$,
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\[
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K(t, Tx) \le \norm{T}_{L(E_0; F_0)}K(\lambda t, x)
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\]
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For any $p \in [1, \infty)$,
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\begin{align*}
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\int_0^\infty t^{-\theta p}K(t, Tx)^p \frac{dt}{t} &\le \norm{T}_{L(E_0; F_0)}^p\int_0^\infty t^{-\theta p}K(\lambda t, Tx)^p \frac{dt}{t} \\
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&= \norm{T}_{L(E_0; F_0)}^p\int_0^\infty \lambda^{\theta p} t^{-\theta p}K(t, Tx)^p \frac{dt}{t} \\
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\norm{Tx}_{(F_0, F_1)_{\theta, p}} &\le \norm{T}_{L(E_0; F_0)} \lambda^{\theta}\norm{x}_{(E_0, E_1)_{\theta, p}}
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\end{align*}
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Similarly, if $p = \infty$,
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\begin{align*}
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t^{-\theta}K(t, Tx) &\le \norm{T}_{L(E_0; F_0)}t^{-\theta}K(\lambda t, x) \\
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\sup_{t > 0}t^{-\theta}K(t, Tx) &\le \norm{T}_{L(E_0; F_0)}\sup_{t > 0}\lambda^{\theta}t^{-\theta}K(t, x) \\
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\norm{Tx}_{(F_0, F_1)_{\theta, p}} &\le \norm{T}_{L(E_0; F_0)} \lambda^{\theta}\norm{x}_{(E_0, E_1)_{\theta, p}}
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\end{align*}
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As the above holds for all $\lambda > \norm{T}_{L(E_1; F_1)}/\norm{T}_{L(E_0; F_0)}$,
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\begin{align*}
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\norm{Tx}_{(F_0, F_1)_{\theta, p}} &\le \norm{T}_{L(E_0; F_0)} \frac{\norm{T}_{L(E_1; F_1)}^\theta}{\norm{T}_{L(E_0; F_0)}^\theta}\norm{x}_{(E_0, E_1)_{\theta, p}} \\
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\norm{T}_{L((E_0, E_1)_{\theta, p}; (F_0, F_1)_{\theta, p})}&\le \norm{T}_{L(E_0; F_0)}^{1 - \theta}\norm{T}_{L(E_1; F_1)}^\theta
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\end{align*}
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\end{proof}
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\begin{proposition}
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\label{proposition:real-interpolation-inclusions}
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Let $(E_0, E_1)$ be a compatible couple of Banach spaces, then:
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\begin{enumerate}
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\item For each $\theta \in (0, 1)$ and $1 \le p < q \le \infty$, $(E_0, E_1)_{\theta, p} \subset (E_0, E_1)_{\theta, q}$.
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\item For each $0 < \theta < \eta < 1$ and $p, q \in [1, \infty]$, $(E_0, E_1)_{\eta, p} \subset (E_0, E_1)_{\theta, q}$.
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\end{enumerate}
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\end{proposition}
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