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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, can"t write it correctly) 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). However, if someone can show that this conjecture is false, I'll be very happy too :) 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)

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)

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: $$ \Gamma_{2}(c)= ( q\in \mathcal{X}_{2}^*\;:\sigma(q)\geq c),\; \Gamma_{\infty}(c)=(q\in \mathcal{X}_{2}^*\;:n_2(q)\geq c),\; \Gamma_{1}(c)=(q\in \mathcal{X}_{2}^*\;:|\mathbb{E}(q)|\geq c). $$

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, can"t write it correctly) 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). However, if someone can show that this conjecture is false, I'll be very happy too :) 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)

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, can"t write it correctly) 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). However, if someone can show that this conjecture is false, I'll be very happy too :) 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)

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: $$ \Gamma_{2}(c)= ( q\in \mathcal{X}_{2}^*\;:\sigma(q)\geq c),\; \Gamma_{\infty}(c)=(q\in \mathcal{X}_{2}^*\;:n_2(q)\geq c),\; \Gamma_{1}(c)=(q\in \mathcal{X}_{2}^*\;:|\mathbb{E}(q)|\geq c). $$

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, can"t write it correctly) 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). However, if someone can show that this conjecture is false, I'll be very happy too :) 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 Young inequality (we have an infinite number of convolutions to build q)

Do you have even an intuition of the correct exponent ? answers ?

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: $$ \Gamma_{2}(c)= ( q\in \mathcal{X}_{2}^*\;:\sigma(q)\geq c),\; \Gamma_{\infty}(c)=(q\in \mathcal{X}_{2}^*\;:n_2(q)\geq c),\; \Gamma_{1}(c)=(q\in \mathcal{X}_{2}^*\;:|\mathbb{E}(q)|\geq c). $$

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, can"t write it correctly) 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). However, if someone can show that this conjecture is false, I'll be very happy too :) 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)