Intuition: think about the probability measure with density $\mathbf 1_C/\vert C\vert$. Then the expectation of the Wigner function should be smaller than 1 for a normalized $u$. Note that the Wigner function is real-valued but not positive in general and with integral 1. References: it seems to me that Philippe Jaming spoke about these problems, somewhat linked to his work on the uncertainty principle. I think that Flandrin is an engineer, specialized in signal business.

Something that I like with $(LH)$: you can try to improve the $\epsilon$, whereas $(RH)$ seems a different sort of game, all or nothing, no gradual approach.

Thanks for the answers. Let me reformulate my question: the estimate $$ \ln\bigl(\vert\zeta(\frac{1}{2}+it)\vert\bigr)=O\bigl((\ln t\ln\ln t)^{1/2}\bigr) $$ seems compatible with the remarks, but I guess that its logical relationship with $(RH)$ is not clear.

$A=P^TDP$, $P\in O(n)$, $D$ diagonal $>0$. $B=Q^T\Delta Q$, $\Delta$ diagonal, $Q\in O(n)$. Now $A\circ B=C$, with Einstein convention, $$ C_{ij}x_ix_j=p_{ki}d_kp_{kj}q_{li}\delta_l q_{lj}x_ix_j= d_k\delta_l (p_{ki}q_{li}x_i) (p_{kj}q_{lj}x_j)= d_k\delta_l y_{kl}^2, $$ with $y_{kl}=p_{ki}q_{li}x_i$. This identity gives a simple proof of Schur's result, but may also be useful for your question.

Thanks for these references, which I knew. As you point out, Bony's result with $C^m$ functions does not extend obviously to the case $m=+\infty$. Looking at the details of the proof, it is clear that the decomposition is changing drastically when $m$ increases. My problem is that these references to P. Cohen result are not supported by any real article or preprint. So somehow the core of my question is: Does anybody ever put his hand on a P.Cohen preprint tackling that counterexample of a $C^\infty$ nonnegative function which is not a finite sum of squares of $C^\infty$ functions ?

There is a famous open problem, whose solution is attributed to Paul Cohen, but no published paper seems to be available: there exists $f\in C^\infty(\mathbb R,\mathbb R_+)$ such that $f$ is not a finite sum of squares of smooth functions. That could be a relevant question for MO.