Fundamental lemma of interpolation theory

The fundamental lemma states the following:

Fundamental lemma of interpolation theory. Let $\backslash bar\{A\}\; =\; (A\_0,\; A\_1)$ be a Banach couple and let $a\; \backslash in\; \backslash Sigma(\backslash bar\{A\})$ be such that $\backslash min(1,1/t)K(t,a)\; \backslash to\; 0$ when $t\; \backslash to\; 0$ or $t\; \backslash to\; \backslash infty$. Then for each $\backslash varepsilon\; >\; 0$, there exists a representation

:$a\; =\; \backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; u\_n$

satisfying $u\_n\; \backslash in\; \backslash Delta(\backslash bar\{A\})$ (with convergence in $\backslash Sigma(\backslash bar\{A\})$) and

:$J(2^n,\; u\_n)\; \backslash leq\; (\backslash alpha\; +\; \backslash varepsilon)K(2^n,\; a)$

for all $n\; \backslash in\; \backslash mathbb\{Z\}$, where $\backslash alpha\; \backslash leq\; 3$ is a constant.{{cite journal |author=M. Cwikel and J. Peetre |title=Abstract K and J spaces |journal=J. Math. Pures Appl. |volume=60 |year=1981 |pages=1-50}}

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 $a\; \backslash in\; \backslash Sigma^c(\backslash bar\{A\})$, there exist elements $u\_n\; \backslash in\; \backslash Delta(\backslash bar\{A\})$ such that

:$\backslash sum\_\{n=-\backslash infty\}^\{\backslash infty\}\; \backslash min\{|u\_n|\{A\_0\},\; t|u\_n|\{A\_1\}\}\; \backslash leq\; (3+2\backslash sqrt\{2\})K(t,a).$

This constant $3+2\backslash sqrt\{2\}\; \backslash approx\; 5.8284$ is currently the best known value, as proven by Dmitriev and later independently by Kaijser using different methods.{{cite journal |author=M. Cwikel, B. Jawerth, M. Milman |title=On the Fundamental Lemma of Interpolation Theory |journal=Journal of Approximation Theory |volume=60 |year=1990 |pages=70-82}}

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.J. L. Lions, J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Etudes Math. 19 (1964) 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.{{cite journal |author=Ju. A. Brudnyi and N. Ja. Krugljak |title=Real interpolation functors |journal=Soviet Math. Dokl. |volume=23 |year=1981 |pages=6-8}}

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.{{cite book |author=J. Bergh and J. Löfström |title=Interpolation Spaces: An Introduction |publisher=Springer-Verlag |year=1976}} 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.{{cite journal |author=P. Nilsson |title=Reiteration theorems for real interpolation and approximation spaces |journal=Ann. Mat. Pura Appl. |volume=32 |year=1982 |pages=291-330}}

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.{{cite journal |author=M. Cwikel |title=K-divisibility of the K-functional and Calderón couples |journal=Ark. Mat. |volume=22 |year=1984 |pages=39-62}}

• Sobolev space

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Category:Lemmas

Category:Banach spaces