# Fundamental lemma of interpolation theory

In mathematics, particularly in functional analysis, the fundamental lemma of interpolation theory is a lemma that establishes the relationship between different methods of interpolation in Banach spaces.

## Statement

The fundamental lemma states the following:
   Fundamental lemma of interpolation theory. Let ${\bar {A}}=(A_{0},A_{1})$ be a Banach couple and let $a\in \Sigma ({\bar {A}})$ be such that $\min(1,1/t)K(t,a)\to 0$ when $t\to 0$ or $t\to \infty$. Then for each $\varepsilon >0$, there exists a representation
       $a=\sum _{n=-\infty }^{\infty }u_{n}$
   satisfying $u_{n}\in \Delta ({\bar {A}})$ (with convergence in $\Sigma ({\bar {A}})$) and
       $J(2^{n},u_{n})\leq (\alpha +\varepsilon )K(2^{n},a)$
   for all $n\in \mathbb {Z}$, where $\alpha \leq 3$ is a constant.
A stronger version of the fundamental lemma, known as the strong fundamental lemma, was developed by mathematicians Alexander Brudnyi and Krugljak. The strong fundamental lemma states that for mutually closed Banach couples, there exists a decomposition with improved estimates on the norms of the components. Specifically, for $a\in \Sigma ^{c}({\bar {A}})$, there exist elements $u_{n}\in \Delta ({\bar {A}})$ such that
    $\sum _{n=-\infty }^{\infty }\min {|u_{n}|{A_{0}},t|u_{n}|{A_{1}}}\leq (3+2{\sqrt {2}})K(t,a).$
This constant $3+2{\sqrt {2}}\approx 5.8284$ is currently the best known value, as proven by Dmitriev and later independently by Kaijser using different methods.

## History and applications

The fundamental lemma was first introduced in the context of classical interpolation theory by mathematicians Jacques-Louis Lions and Jaak Peetre in their 1964 paper Sur une classe d'espaces d'interpolation. The development of stronger versions, including the strong fundamental lemma, indicated a maturation of the theory as its applications expanded. The ongoing search for optimal constants in these results remains an active area of research, with significant contributions from mathematicians like Brudnyi, Krugljak, Cwikel, and others.
The fundamental lemma is particularly useful in establishing the equivalence of the K-method and J-method of interpolation. This equivalence is fundamental to the theory of interpolation spaces, as it allows mathematicians to choose whichever method is more convenient for a given problem. Furthermore, the lemma has found various applications in the study of K-spaces, a class of interpolation spaces defined by certain monotonicity conditions. Brudnyi and Krugljak used the strong fundamental lemma to show that K-spaces, despite their abstract definition, have a concrete structure characterized by lattice norms acting on K-functionals.
In harmonic analysis, the lemma provides essential tools for studying the behavior of various function spaces. It has been particularly useful in establishing properties of Calderón–Mityagin couples, where all interpolation spaces with respect to the couple are K-spaces. The lemma also appears in the theory of operator ideals and has applications in studying the regularity properties of solutions to partial differential equations.
Other variants of the fundamental lemma have been developed for specific applications, including versions involving the E-functional and continuous parameter formulations. These variants have proven useful in studying weighted Banach lattices and in establishing relationships between different types of interpolation spaces.