Global side-of-the-chord behavior for extensions larger than the self-dual extension

Establish that, for the Poisson operator and every extension parameter \(\delta>L/2\), the tail law \(r_\omega(\tau)\) lies below the chord \(-a(\omega)\tau\) throughout the entire interval \(0<\tau<\kappa_\sigma(\omega)\), equivalently that for \(\delta<L/2\) it lies above the chord; determine whether the associated sharp power law holds at all tail levels.

Background

For the one-dimensional Poisson operator, the paper identifies the exponential tail behavior of the leakage and restricted Marcinkiewicz–Zygmund constants through a constrained logarithmic-equilibrium problem. The chord $-a(\omega)\tau$ corresponds to the endpoint power-law exponent $a(\omega)$. The tail law is proved to start above this chord when $\delta<L/2$, below it when $\delta>L/2$, and to coincide with it for the self-dual extension $\delta=L/2$.

Certified interpolation configurations establish the below-chord behavior for $\delta=L$ and $\delta=2L$ over substantial portions of the tail, but not over the full interval. A full result would clarify whether a power-law bound with constants independent of both $N$ and the threshold $t$ holds for extensions larger than the self-dual one.