Background: For any fixed $t> 0$, the $K$-functional defines a norm on the space $\ell_1+\ell_2$: $$ \lVert a\rVert_{K(t)} = \inf\{\lVert a'\rVert_1+ t\lVert a''\rVert_2 : a'\in\ell_1,\ a''\in\ell_2,\ a'+a''=a\}. \tag{1} $$ For fixed $a$, the function $\kappa_a\colon t>0\mapsto \lVert a\rVert_{K(t)}$ is non-decreasing, concave, and continuous, and satisfies
- $\lim_{t\to 0^+} \kappa_a(t) = 0$
- $\frac{1}{4}\lVert a\rVert_1\leq \lim_{t\to \infty} \kappa_a(t) \leq \lVert a\rVert_1$
(See e.g. [1].) Moreover, if $a$ has finite support, then we get $\lim_{t\to \infty} \kappa_a(t) = \lVert a\rVert_1$ (actually, $\kappa_a(t)=\lVert a\rVert_1$ for any $t\geq \sqrt{\lvert \operatorname*{supp} a\rvert}$).
If $(a',a'')\in \ell_1\times\ell_2$ are such that $a=a'+a''$ and $\kappa_a(t) = \lVert a'\rVert_1+ t\lVert a''\rVert_2$, we say that $(a',a'')$ is an optimal decomposition (of $a$) for $t$.
In what follows, I focus on non-negative sequences, and even non-increasing non-negative sequences. (Actually, I only need results for non-increasing non-negative sequences with finite support, if that makes a difference,)
From what I could find and understand [2,3] (this is for functions in $L^1+L^2$, but seems to generalize to $\ell_1+\ell_2$), optimal decompositions are of the form:
$$a_k'' = \min(\lambda,a_k),\qquad a'_k = a_k - a_k'' \qquad k\in\mathbb{N}$$ for some $\lambda=\lambda(t) \geq 0$.
Seeing $(a',a'')$ (with some slight handwaving or technicalities in case of non-unicity) as functions of $t$ for $a$ fixed, this implies a specific case of a more general result [3], i.e. that $$ x\colon t\mapsto \lVert a'_t\rVert_1, \qquad y\colon t\mapsto \lVert a''_t\rVert_2 $$ are respectively non-decreasing and non-increasing.
Question: Is there more we can say about the behavior of $x,y$? Specifically, I am interested in the regime of "large $t$", when $\kappa_a(t) = \lim_\infty\kappa_a -\eta$ for small $\eta$ (and the limit is finite, i.e. $a\in\ell_1$) but I'd be interested in any structural result on $x,y$. (Convexity, upper and lower bounds as a function of $\eta$, etc.)
If the question is too general, I can narrow it down as mentioned above by enforcing that $a$ be not only non-increasing and non-negative, but additionally with finite support.
[1] Astashkin, S.V. J. Rademacher functions in symmetric spaces Math Sci (2010) 169: 725. doi:10.1007/s10958-010-0074-z
[2] Nilsson P., Peetre J., On the $K$ functional between $L^1$ and $L^2$ and some other $K$-functionals, Journal of Approximation Theory, Volume 48, Issue 3, 1986, Pages 322-327, ISSN 0021-9045, doi:10.1016/0021-9045(86)90054-7
[3] Optimal decompositions for the $K$-functional for a couple of Banach lattices. Cwikel, M. & Keich, U. Ark. Mat. (2001) 39: 27. doi:10.1007/BF02388790
