Endpoint boundedness of toroidal pseudo-differential operators

Abstract

In this note, we prove that the toroidal pseudo-differential operator is bounded from L∞(Tn){L}^{\infty }\left({{\mathbb{T}}}^{n}) to BMO(Tn){\rm{BMO}}\left({{\mathbb{T}}}^{n}) if the symbol belongs to the toroidal Hörmander class Sρ,δn(ρ−1)∕2(Tn×Zn){S}_{\rho ,\delta }^{n\left(\rho -1)/2}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with 0<ρ≤10\lt \rho \le 1 and 0≤δ<10\le \delta \lt 1. As a corollary, we obtain a result of toroidal pseudo-differential operators on Lp{L}^{p} when 2<p<∞2\lt p\lt \infty for symbols in the class Sρ,δm(Tn×Zn){S}_{\rho ,\delta }^{m}\left({{\mathbb{T}}}^{n}\times {{\mathbb{Z}}}^{n}) with m≤n(ρ−1)12−1p+npmin{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\}.

Keywords

• torus
• toroidal pseudo-differential operator
• toroidal hörmander symbol
• bmo(𝕋n)
• 43a75
• 35s05