Relation between two different functionals: $\lVert p^{-\max}_{-\varepsilon}\rVert$ and $\kappa_{p}^{-1}(1-\varepsilon)$

Given a non-negative sequence $ p=(p_i)_{i\in\mathbb{N}}\in \ell_1$ such that $ \lVert p\rVert_1 = 1$ ,we define the two following quantities, for every $ \varepsilon \in (0,1]$ .

  1. Assuming, without loss of generality, that $ p$ is non-increasing, let $ k \geq 1$ be the smallest integer such that $ \sum_{i\geq k} \leq \varepsilon$ . Then we define $ $ \Phi(\varepsilon, p) := \left( \sum_{i=2}^{k-1} p_i^{2/3} \right)^{3/2} \tag{1} $ $ i.e., the $ 2/3$ -quasinorm $ \lVert p_{-\varepsilon}^{-\max{}} \rVert_{2/3}$ of the vector $ p^{-\max{}}_{-\varepsilon}$ obtained by removing the largest element and the $ \varepsilon$ -tail of $ p$ .
  2. Defining, for $ t>0$ , the $ K$ -functional between $ \ell_1$ and $ \ell_2$ $ $ \kappa_p(t) = \inf_{a+b=p} \lVert a\rVert_1 + t \rVert b\rVert_2 $ $ and letting $ $ \Psi(\varepsilon, p) := \kappa_p^{-1}(1-\varepsilon) \tag{2} $ $ (right inverse, IIRC)

then can we prove upper and lower bounds relating (1) and (2)? Recent works of Valiant and Valiant [1] and Blais, Canonne, and Gur [2] imply such relation in a rather roundabout way (for $ p$ ‘s nontrivially point masses, i.e., say, $ \lVert p\rVert_2 < 1/2$ ) (by showing both quantities “roughly characterize” the sample complexity of a particular hypothesis testing problem on $ p$ seen as a discrete probability distribution), but a direct proof of such a relation isn’t known (at least to me), even only a loose one.

Is there a direct proof relating (upper and lower bounds) $ \Phi(\cdot, p)$ and $ \Psi(\cdot, p)$ , of the form $ $ \forall p \text{ s.t. } \lVert p\rVert \ll 1,\forall x, \qquad x^\alpha \Phi(c x, p) \leq \Psi(c x, p) \leq x^\beta \Phi(C x, p) $ $ ?


[2] (following some previous work of Montgomery-Smith [3]) does show a relation between (2) and a third quantity interpolating between $ \ell_1$ and $ \ell_2$ norms, $ $ T\in\mathbb{N} \mapsto \lVert p\rVert_{Q(T)} := \sup\{ \sum_{j=1}^T \left( \sum_{i\in A_j p_i^2 }\right)^{1/2} A_1,\dots,A_t \text{ partition of }\mathbb{N}\} \tag{3} $ $ as, for all $ t>0$ such that $ t^2\in \mathbb{N}$ , $ $ \lVert p\rVert_{Q(t^2)} \leq \kappa_p(t) \leq \lVert p\rVert_{Q(2t^2)}. $ $


[1] Gregory Valiant and Paul Valiant. An Automatic Inequality Prover and Instance Optimal Identity Testing. SIAM Journal on Computing 46:1, 429-455. 2017.

[2] Eric Blais, Clément Canonne, and Tom Gur. Distribution testing lower bounds via reductions from communication complexity. ACM Transactions on Computation Theory (TOCT), 11(2), 2019.

[3] Stephen J. Montgomery-Smith. The distribution of Rademacher sums. Proceedings of the American Mathematical Society, 109(2):517–522, 1990.