Commutativity of Hankel and Toeplitz operators on the Hardy space of the n-torus

We consider Hankel and Toeplitz operators on $H^2\left(T^n\right)$, the Hardy space of the n-torus $T^n$. Given symbols φ and ψ in $L^{\infty }\left(T^n\right)$ with suitable properties, we obtain necessary and sufficient conditions for the Hankel operator $H_{\psi ,n}$ and the Toeplitz operator $T_{\phi ,n}$ to commute. We then extend the study to the more general situation where no assumptions are imposed on φ, and provide new, non-trivial necessary conditions for the commutativity of $H_{\psi ,n}$ and $T_{\phi ,n}$. We also show that certain well known commutativity results between Hankel and Toeplitz operators in the one-variable case do not extend to the multivariable setting.