A pointwise estimate for pseudo-differential operators

Abstract

Let [Formula: see text] be a pseudo-differential operator defined by the symbol [Formula: see text] with [Formula: see text] [Formula: see text]. It is shown that if [Formula: see text], then the operator satisfies the following pointwise estimate: (Tau)♯(x) ≲ M(|u|2)1/2(x) for all [Formula: see text] and all Schwartz function [Formula: see text] Here, [Formula: see text] denotes the John-Nirenberg sharp function of [Formula: see text] and [Formula: see text] stands for the Hardy–Littlewood maximal operator.

Keywords

• Pseudo-differential operators
• sharp function
• pointwise estimate