Endpoint boundedness of toroidal pseudo-differential operators

In this note, we prove that the toroidal pseudo-differential operator is bounded from L ∞ ( T n ) {L}^{\infty }\left({{\mathbb{T}}}^{n}) to BMO ( T n ) {\rm{BMO}}\left({{\mathbb{T}}}^{n}) if the symbol belongs to the toroidal Hörmander class S ρ , δ n ( ρ − 1 ) ∕ 2 ( T n × Z n ) {S}_{\rho ,\delta }^{n\left(\rho -1)/2}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with 0 < ρ ≤ 1 0\lt \rho \le 1 and 0 ≤ δ < 1 0\le \delta \lt 1 . As a corollary, we obtain a result of toroidal pseudo-differential operators on L p {L}^{p} when 2 < p < ∞ 2\lt p\lt \infty for symbols in the class S ρ , δ m ( T n × Z n ) {S}_{\rho ,\delta }^{m}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with m ≤ n ( ρ − 1 ) 1 2 − 1 p + n p min { 0 , ρ − δ } m\le n\left(\rho -1)\left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2}-\frac{1}{p}\right)+\frac{n}{p}\min \left\{0,\rho -\delta \right\} .