A pointwise estimate for pseudo-differential operators

Let T a be a pseudo-differential operator defined by the symbol a ∈ S ϱ , δ − n ( 1 − ϱ ) / 2 with 0 < ϱ ≤ 1 , 0 ≤ δ < 1 . It is shown that if δ ≤ 1 + ϱ 2 , then the operator satisfies the following pointwise estimate: ( T a u ) ♯ ( x ) ≲ M ( | u | 2 ) 1 / 2 ( x ) for all x ∈ ℝ n and all Schwartz function u . Here, u ♯ denotes the John-Nirenberg sharp function of u and M stands for the Hardy–Littlewood maximal operator.