**My question is:** 
How do I find **sharp** upper bounds on $P(|q|\leq \epsilon)$ uniformly over a set of gaussian polynomes $q$ of degree two.


**Notations and definitions** (to make the question rigorous)


* Let me define $\mathcal{X}\_{2}^\*$ as the set of real random variables $q$ that can be written $$q=c+\sum\_{i\geq 1}\beta\_i (\xi\_i^2-1)+\alpha\_i\xi\_i$$
with $c\in \mathbb{R}$, $\beta=(\beta\_i)\_i\in l\_2(\mathbb{N})$, $\alpha=(\alpha\_i)\_i\in l^2(\mathbb{N})$ and $(\xi\_i)\_{i\in \mathbb{N}}$ a sequence of iid gaussian random variables with mean $0$ and variance $1$. This set is also known as gaussian polynomial of degree two (see for example the book of Bogachev 1998).
* Let $q\in \mathcal{X}\_2^\*$ given by $q=c+\sum\_{i\geq 0}\alpha\_i\xi\_i+\sum\_{i}\beta\_i(\xi\_i^2-1)$, I use the notation $$n\_2(q)=\max\_i |\beta\_i|, \;\; \sigma(q)=\left (\sum\_{i\geq 0}2\beta\_i^2+\alpha\_i^2\right )^{1/2}$$.
* I propose to use the following **sets of polynomes**: 
$$
\Gamma\_{2}(c)= ( q\in \mathcal{X}\_{2}^\*\;:\sigma(q)\geq c),\; \Gamma\_{\infty}(c)=(q\in \mathcal{X}\_{2}^\*\;:n\_2(q)\geq c)
$$
and 
$$ \Gamma\_{1}(c)=(q\in \mathcal{X}\_{2}^\*\;:|\mathbb{E}(q)|\geq c). $$




---


**Motivation for this problem** 


It is worth noticing that this problem appears when one wants to check the so called **Noise condition** in an infinite dimensional gaussian classification problem.... Anyway, I called it small crown, even if it is not always a crown ... should be easy for expert of "small ball probabilities" ? 




---


**What I have so far**


1. There exists $C(c\_0)>0$ such that $\forall \epsilon>0\; \sup\_{q\in \Gamma\_1(c\_0)} P(|q|\leq \epsilon)\leq C(c\_0)\epsilon^{2/7}$
2. There exists $C'(c\_0)>0$ such that $\forall \epsilon>0$ $\sup\_{q\in \Gamma\_2(c\_0)} P(|q|\leq \epsilon)\leq C'(c\_0)\epsilon^{1/3}$.
3. Let $q\in \mathcal{X}\_{2}^\*$, for all $\epsilon> 0$, $P(|q|\leq \epsilon) \leq \sqrt{\frac{1}{\pi}\frac{\epsilon}{n\_2(q)}}$.


**Comments** 
Point 3 is easy but point 1 and 2 are less easy. I can provide a link to the proof if desired. 
If $n\_2(q)=\max\_{i}|\beta\_i|>c\_0$, bound of point $3$ is optimal in the sense that if $\beta=(1,0,\dots)$, $c=1$ and $\alpha=0$ we get $P(|q|\leq \epsilon)=P(|\xi^2|\leq \epsilon)\sim C\epsilon^{1/2}$ (for a constant $C$ that can be calculated explicitly). I have problem for case 1 and 2 .... 


**My analysis and conjecture:** 
 When 
$\|\beta\|2 \rightarrow 0$ ($l^2$ norm)
the behaviour of $P(|q|\leq \epsilon)$ tends to be the same behaviour that $P(|\|\alpha\|\_{l^2} \mathcal{N}(0,1)-c|\leq \epsilon)\sim C'(c\_0)\epsilon$. Also, it is possible to conjecture that point $1$ and $2$ of the Theorem can be improved (in order to obtain exponent $1/2$ instead of $2/7$ and $1/3$). The difficult cases to study are those with $\|\beta\|\_{\infty}\rightarrow 0$ but $\|\beta\|2$ doesn't tends to zero (there, in the proof of points 1 and 2 I use a gaussian approximation of q). Notice that when you restrict yourself to $\beta=0$ the answer to point 2 is that the best exponent is 1! hence a gap between linear and quadratic form...


Some of the ideas I have tryed (without success :( )


1. Using an explicit formulae of the density (if the density of $q$ is uniformly in $L^p$ for a good $p$ then we are done.. ) using characteristic funtions.
2. Using optimal Young inequality (we have an infinite number of convolutions to build q)



Small crown probabilities (and infinite dimensional margin assumption) 
Here is a solution for problem 2, with power $1/2$, using your idea 1. First some computations. Let A, B real numbers, let z ~ N(0,1), and X=$B(z^2-1) + Az$. The Fourier transform of the distribution of X (i.e.: the characteristic function of X with some $\pi$) is


$\; \; \; \;\; \; \; \;E(exp(-2\pi i\xi X) = \frac{e^{2\pi i \xi B - \frac{2\pi^2 A^2 \xi^2}{1+4\pi i B \xi}}}{\sqrt{1+4\pi i\xi B}}$ 


where the square root is the one with positive real part. Then


$\; \; \; \;\; \; \; |E(exp(-2\pi i\xi X)|^2 = \frac{e^{- \frac{4\pi^2 A^2 \xi^2}{1+16\pi^2 B^2 \xi^2} }} {1+16\pi^2 \xi^2 B^2}$ 


We also have for any pair of real numbers $a, b\ge 0$


$\; \; \; \;\; \; \; \ln(\frac{1+8b + 4a}{1+16b}) < \ln(1+\frac{4a}{1+16b})\le \frac{4a}{1+16b}$


In particular, 


$\; \; \; \;\; \; \; -\frac{4a}{1+16b} - \ln(1+16b)<-\ln(1+4(2b + a))$


thus, with $a=\pi^2\xi^2 A^2$ and $b=\pi^2\xi^2 B^2$


$\; \; \; \;\; \; \; |E(exp(-2\pi i\xi X)|^2 \le \frac{1} {1+4\pi^2 \xi^2 (2B^2+A^2)}.$ 


It follows that the Fourier transform of a Gaussian polynomial of degree two $q$ satisfies, with notation as in your definition:


$\; \; \; \;\; \; \; |E(exp(-2\pi i\xi q)|^2 \le \prod\_{j}{\frac{1} {1+4\pi^2 \xi^2 (2\beta\_j^2+\alpha\_j^2)}}\le\frac{1}{1+4\pi^2\xi^2\sigma(q)^2}.$ 


Then, the density of $q$ is bounded in $L^2$ , uniformly in $\Gamma\_2(c\_0)$, and so
there exists a constant $C'$ such that for all $\epsilon>0$


$\; \; \; \;\; \; \;\sup\_{q\in \Gamma\_2(c\_0)} P(|q|\leq \epsilon)\leq C'\epsilon^{1/2}.$


