Let $I=[0,1]$ be the unit segment, and let $(I\_n)\_{1\leq n \leq N}$ be $N$ almost disjoint sub-intervals $I\_n=[t\_n-\delta\_n,t\_n+\delta\_n]$ of $I$ (that is, their interior are disjoint). Let $\chi(x)=\frac{1}{1+x^2}$ (which plays the role as an approximation of $1\_{[-1,1]}$).



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> How can one prove that there exists $C, c>0$ (independent of $N$ and the $I\_n$) such that for all $\lambda >0$ there holds $$\left|\left\{x\in I, \sum\_{n=1}^N \chi (\frac{x-t\_n}{\delta\_n})>\lambda\right\}\right|\leq Ce^{-c\lambda}?$$
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This results seems surprisingly difficult to prove. It looks like one should try do do some John-Nirenberg argument, but I did not succeed this way.


Maybe it is possible to regroup intervals of width $ (2K)^{-1} \leq \delta\_n < K^{-1}$ where $K\in 2^{\mathbb{N}}$. Let us denote by $A\_K$ this subset of intervals and $|A\_K|$ its cardinal. Without loss of generality one can assume that $I = \cup I\_n$. Write $1=\sum |I\_n| = \sum\_K c\_K$ where $c\_K = \sum\_{n\in A\_K} |I\_n|$, so that $$|\{x\in I, \sum\_{n=1}^N \chi (\frac{x-t\_n}{\delta\_n})>\lambda\}|\leq \sum\_K |\{x\in I, \sum\_{n\in A\_K} \chi (\frac{x-t\_n}{\delta\_n})>\lambda c\_K\}|.$$
In theory, it should be easier to handle the terms $n\in A\_K$ because all intervals have comparable sizes. However, I am not able to prove the right estimate.


Does anyone know how to prove it? I am looking for a rather elementary approach, as this looks really elementary.



Exponential integrability of a sum of approximations of disjoint intervals characteristic functions
The simplest argument I can currently come up with is the following:


Let $f\_j$ be non-negative functions on $[0,1]$ with $\sum\_j f\_j\le 1$ (characteristic functions of disjoint intervals in your case) and let $\Phi\_j$ be some even decreasing on $[0,+\infty)$ averaging kernels (Poisson kernels in your case). Let $F=\sum\_j f\_j\*\Phi\_j$. Let $E=\{x\in[0,1]:F(x)>\lambda\}$.


Then
$$
\lambda|E|\le \int F\chi\_E=\sum\_j\int (f\_j\*\Phi\_j)\chi\_E
=\sum\_j\int (\chi\_E\*\Phi\_j)f\_j
\\
\le \int (M\chi\_E)\sum\_jf\_j\le \int (M\chi\_E)\chi\_{[0,1]}\le C|E|(1+\log\tfrac1{|E|})
$$
and the desired conclusion follows. Here $M$ is the Hardy-Littlewood maximal function. Unfortunately, it is still a bit on the high tech side, though slightly less so than BMO and John-Nierenberg.


