Abstract:

In this paper, we establish weighted $L^{p}$ inequalities for pseudo-differential operators $T$ and their commutators $[b, T]$ with smooth symbols in $S^{m}_{\rho, \delta}(\mathbb{R}^n)$, where $b\in BMO(\mathbb{R}^n)$ and the weight $\omega$ belongs to the new class $A_{p}(\varphi)$. It is well known that the Muckenhoupt class $A_{p}$ falls within the class $A_{p}(\varphi)$. This results extend the class of pseudo-differential operator to a wide range of $m$.

Key words: Pseudo-differential operator, Commutator, $BMO$, $A_{p}(\varphi)$ weights, Weighted $L^{p}$ estimate