Optimal ellipticity dependence in the one-dimensional autonomous kinetic case

Determine the optimal dependence on the ellipticity ratio of the critical exponent for the kinetic Aleksandrov estimate with autonomous one-dimensional coefficients of the form a(x,v), including whether the endpoint critical exponent is admissible.

Background

For autonomous coefficients a(x,v) in dimension one, the paper proves the estimate for every p>p_(κ), where κ=Λ/λ and p_(κ) lies in [3,4). It also establishes the uniform threshold p>4 and, through a companion result, includes p=4 for each fixed ellipticity ratio. However, the precise dependence of the optimal threshold on κ is not determined.

References

The optimal threshold in [3,4) as a function of the ratio remains open.

— Döblin--Fourier cancellation and kinetic Aleksandrov estimates  (2610.03083 - Loher et al., 2 Oct 2026) in Section 1, Subsection “The merit”

The higher-dimensional assertion in (b) and the proposed threshold in (d) remain open.

— Döblin--Fourier cancellation and kinetic Aleksandrov estimates  (2610.03083 - Loher et al., 2 Oct 2026) in Section 1, Subsection “What is known and what remains open”

We conjecture that, in the kinetic case, the answer varies with the variables on which A depends, as follows:

— Döblin--Fourier cancellation and kinetic Aleksandrov estimates  (2610.03083 - Loher et al., 2 Oct 2026) in Section 1, Subsection “The sharp exponent in the kinetic case,” Conjecture “The critical kinetic Aleksandrov exponent”

We conjecture that, in the kinetic case, the answer varies with the variables on which A depends, as follows:

— Döblin--Fourier cancellation and kinetic Aleksandrov estimates  (2610.03083 - Loher et al., 2 Oct 2026) in Section 1, Subsection “The sharp exponent in the kinetic case,” Conjecture “The critical kinetic Aleksandrov exponent”