Logarithm-free critical diagonal estimate

Establish whether the refined-direction maximal operator on the finite Heisenberg group \(\mathbb H_n(\mathbb F_q)\), for \(n\geq2\), satisfies the strong diagonal estimate \(\|\mathcal M^{\mathrm{rd}}F\|_{\ell^{2n}(\mathcal D_n)}\lesssim_n q^{(2n-1)/(2n)}\|F\|_{\ell^{2n}(\mathbb H_n(\mathbb F_q))}\) without the factor \((1+\log q)^{1/(2n)}\), thereby determining whether the critical exponent \((2n-1)/(2n)\) is attained rather than merely an infimum of admissible powers.

Background

The paper determines the optimal power of $q$ in all mixed-norm estimates for the refined-direction maximal operator associated with horizontal lines in the finite Heisenberg group $\mathbb H_n(\mathbb F_q)$. At the critical pair $(u,v)=(2n,2n)$, the optimal admissible power is $(2n-1)/(2n)$, but the proved strong estimate includes an additional logarithmic factor $(1+\log q)^{1/(2n)}$.

The level-set estimate is sharp and has no logarithmic loss. The logarithm appears only when the distributional bound is summed over the ordered values of the maximal function, producing a harmonic sum. The authors state that removing this factor for $n\geq2$ is unresolved; for $n=1$, a Fourier-analytic argument is known to yield the corresponding logarithm-free estimate.