Consider a pseudo-differential operator \({T_a}f(x) = \int_{{\mathbb{R}^n}} {{e^{ix \cdot \xi}}} a(x,\xi)\hat f(\xi)\,\,{\rm{d}}\xi \) where the symbol a is in the rough Hörmander class \({L^\infty}S_\rho ^m\) with m ∈ ℝ and ρ ∈ [0, 1]. In this note, when 1 ≤ p ≤ 2, if \(m < \,{{n(\rho - 1)} \over p}\) and \(a \in {L^\infty}S_\rho ^m\) , then for any f ∈ S(ℝn) and x ∈ ℝn, we prove that \(M({T_a}f)(x) \le C{(M(|f{|^p})(x))^{{1 \over p}}}\) where M is the Hardy-Littlewood maximal operator. Our theorem improves the known results and the bound on m is sharp, in the sense that \({{n(\rho - 1)} \over p}\) can not be replaced by a larger constant.
