Sharp logarithmic Sobolev constant for quantum tori

Determine the sharp logarithmic Sobolev inequality constant for the standard heat semigroups generated by the Fourier multipliers \(\Delta_\theta\) on quantum tori \(\mathbb T_\theta^d\), distinct from the modified logarithmic Sobolev constants established in the paper.

Background

The paper proves sharp complete modified logarithmic Sobolev inequalities for the heat semigroup on the circle and, by tensorization and gauge transference, obtains the sharp complete modified logarithmic Sobolev constant for classical and quantum tori. It defines the quantum torus $\mathbb T_\theta^d$ using the deformation parameter $\theta$ and the standard heat generator $\Delta_\theta(U^m)=|m|^2U^m$.

Despite these modified logarithmic Sobolev results, the authors explicitly state that the sharp logarithmic Sobolev inequality constant on quantum tori remains unresolved. The open problem therefore concerns the non-modified logarithmic Sobolev constant, rather than the complete modified logarithmic Sobolev constant computed in the paper.