I ended up needing the following statement, which seems true but to which I couldn't find a proof nor a counterexample. Let $q \in [4/3, 2]$ and $\lambda \in \mathbb{R}^n$ with $\|\lambda\|\_q = 1$. Is it always possible to find an index decomposition $I \subseteq [n]$ such that both the following hold ($\lesssim$ means up to a universal constant):


* $\|\lambda\_I\|\_1 \lesssim n^{3/4 - 1/q}$
* $\|\lambda\_{I^c}\|\_2 \lesssim n^{1/2 - 1/q}$


In the different examples I tried (basically sparse or dense vectors) this was always possible, and a natural idea I tried was to decompose the coordinates into large and small ones, but I couldn't make the proof work like this.


Thanks a lot for any help !



Decomposition of a vector in parts with bounded $\ell^1$ and $\ell^2$ norms
$\newcommand\la\lambda\newcommand\La\Lambda\newcommand{\R}{\mathbb R}\newcommand{\cc}{\mathsf c}$As suggested by [Aleksei Kulikov](https://mathoverflow.net/questions/451595/decomposition-of-a-vector-in-parts-with-bounded-ell1-and-ell2-norms#comment1167553_451595), your desired statement does not hold in general.


Indeed, let $\La\_{n,q}:=\{\la=(\la\_1,\dots,\la\_n)\in\R^n\colon\|\la\|\_q=1\}$. You wanted to show that for each $\la\in\La\_{n,q}$ there is some $I\subseteq[n]$ such that for some universal real constant $C>0$
\begin{equation\*}
 \|\la\_I\|\_1\le Cn^{3/4-1/q} \tag{1}\label{1}
\end{equation\*}
and
\begin{equation\*}
 \|\la\_{I^\cc}\|\_2\le Cn^{1/2-1/q}, \tag{2}\label{2}
\end{equation\*}
where $\la\_I:=(\la\_i\colon i\in I)$.


If $q=2$, then \eqref{1} and \eqref{2} obviously hold for all $\la\in\La\_{n,q}$ with $C=1$ and $I=\emptyset$.


Now suppose that $q\in(0,2)$.
Let us then show that, if \eqref{2} holds for a real constant $C>1$ and all $\la\in\La\_{n,q}$ (and some $I=I(\la)\subseteq[n]$), then
\begin{equation\*}
 \|\la\_I\|\_1\ge bn^{1-1/q} \tag{1a}\label{1a}
\end{equation\*}
for some real $b>0$ depending only on $C$ and $q$, some $\la\in\La\_{n,q}$, and all $I\subseteq[n]$ -- so that \eqref{1} fails to hold.


To do this, take any $c\in(0,C^{2/(1-2/q)})\subset(0,1)$ and let $m$ be an integer varying with $n$ so that $m\sim cn$ (as $n\to\infty$). Suppose that $\la\_1=\cdots=\la\_m=m^{-1/q}>0=\la\_{m+1}=\cdots=\la\_n$. Then $\la\in\La\_{n,q}$ and for each $I\subseteq[n]$ we have
\begin{equation\*}
 \|\la\_I\|\_1=m^{-1/q}k\quad\text{and}\quad\|\la\_{I^\cc}\|\_2=m^{-1/q}(m-k)^{1/2}, \tag{3}\label{3}
\end{equation\*}
where $k:=|I\cap[m]|$. So, \eqref{2} implies
\begin{equation\*}
 k\ge m^{2/q}(m^{1-2/q}-C^2n^{1-2/q})\sim m(c^{1-2/q}-C^2)\asymp n
\end{equation\*}
and hence, by \eqref{3}, $\|\la\_I\|\_1\asymp n^{1-1/q}$, so that \eqref{1a} does hold. $\quad\Box$




---


On a small positive note, by Jensen's inequality, for any real $q\ge1$ we have $\|\la\|\_1\le \|\la\|\_q n^{1-1/q}=n^{1-1/q}$, so that, in view of \eqref{1a}, the best possible constant $p$ in the inequality
\begin{equation\*}
 \|\la\_I\|\_1\le Cn^{p-1/q} \tag{1b}\label{1b}
\end{equation\*}
given that \eqref{2} holds is $p=1$ (and then one can choose $C=1$ and $I=[n]$).


