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In interpolation theory, given a compatible couple of Banach spaces $(X_0, X_1)$ one considers the $J$ and $K$-functionals, defined as follows:

If $x \in X_0 + X_1$ and $t > 0$ then $$K(t, x) = \inf\{\|x_0\|_{X_0} + t\|x_1\|_{X_1} : x = x_0 + x_1, x_0 \in X_0, x_1 \in X_1\}.$$

If $x \in X_0 \cap X_1$ and $t > 0$ then $$J(t, x) = \max\{\|x\|_{X_0}, t\|x\|_{X_1}\}.$$

The books I checked gave those definitions without explaining where they came from. What is the motivation for defining the $J$ and $K$-functionals?

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    $\begingroup$ I can only offer the idea that both $K(t,\cdot)$ and $J(t,\cdot)$ define equivalent norms parametrized by $t$ on $X+Y$ and $X \cap Y$, respectively, emphasizing the $X$ or $Y$ aspects of the respective vector in dependence of $t$. This is then used to determine elements of, say, $X+Y$, with a certain "ratio" of $X$ and $Y$ aspects. (For the $K$-method, one requires that the $t$-parametrized family of norms given by the $K$-functional is $p$-integrable with the weight $t^{-\theta-1/p}$. The $J$-method works by completion and works less well in this explanation.) $\endgroup$
    – Hannes
    Mar 28 at 11:18

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I found the motivation in Luc Tartar's book An Introduction to Sobolev Spaces and Interpolation Spaces. It is in the first page of Chapter 24.

I quote:

The K-method is the natural result of investigations which originated in questions of traces: if $u \in L^{p_0} (R_+; E_0)$ and $u′ \in L^{p_1} (R+; E1)$ with $1 \leq p0, p1 \leq ∞$, then $u \in C^0([0, 1]; E_0 + E_1)$, so that $u(0)$ exists and the question is to characterize the space of such values at 0 (traces). As one can change $u(t)$ in $u(tλ)$ with $λ > 0$ and not change $u(0)$, one then discovers naturally that one can consider spaces of functions such that $t^{α_0} u \in L^{p_0} (R+; E_0)$ and $t^{α_1} u′ \in L^{p_1} (R+; E_1)$, and for some set of parameters $u(0)$ exists. These ideas may have started with Emilio GAGLIARDO, and I do not know if he had first identified the traces of functions from $W^{1,p}(R^N)$ on an hyperplane before or after thinking of the general framework, but certainly Jacques-Louis LIONS and Jaak PEETRE perfected the framework, and the $K$-method is Jaak PEETRE’s further simplification, which shows that the family of interpolations spaces that they had introduced only depends upon two parameters.

If one wants to characterize the duals of the spaces obtained, then one finds easily that these dual spaces are naturally defined as integrals, and one considers then questions like that of identifying which are the elements $a ∈ E_0 + E_1$ which can be written as $∫^∞_0 v(t) dt$ where $t^{β_0} v ∈ L^{q_0} (R+; E_0)$ and $t^{β_1} v ∈ L^{q_1} (R+; E_1)$, for the range of parameters where the integral is defined. Again, looking at $v(tλ)$ shows that there are not really four parameters, but one important observation is that these spaces are (almost) the same as the ones defined by traces, and I do not know if Emilio GAGLIARDO had investigated such questions before the basic work of Jacques-Louis LIONS and Jaak PEETRE. The J-method is then the simplification by Jaak PEETRE of the preceding framework.

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