Sharp heat and Poisson kernel estimates for all Fourier–Bessel type indices

The Fourier–Bessel setting involves the system $\{\psi_n^{\nu}\}$ and the heat and Poisson semigroups whose estimates are established in Theorems 1 and 2 of the paper. The type index is denoted by $\nu$.

Final conjecture. The estimates from the heat-kernel theorem and the Poisson-kernel theorem hold for all $\nu>-1$.

The conjecture proposes extending the paper's sharp estimates beyond the half-integer type indices treated by the main results. The source describes proving or disproving this extension as an important open problem in the theory of Fourier–Bessel expansions.