Sharp function and weighted $L^p$ estimates for pseudo-differential operators with symbols in general Hörmander classes

The purpose of this paper is to prove pointwise inequalities and to establish the boundedness on weighted $L^p$ spaces for pseudo-differential operators $T_a$ defined by the symbol $a\in S_{\varrho ,\delta }^m$ with $0\le \varrho \le 1,$ $0\le \delta <1$. Firstly, we prove that if $m\le -n(1-\varrho )\mathrm{/}2$, then $$(T_{a}u{)}^{\mathrm{♯}}(x)\lesssim M(\mathrm{\mid }u{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}(x)$$ for all $x\in {\mathbb{R}}^n$ and all Schwartz function $u$. Secondly, it is shown that if $1\le r\le 2$ and $m\le -\frac{n}{r}(1-\varrho )$, then for any $\omega$ belongs to the class of Muckenhoupt weights $A_{p\mathrm{/}r}$ with $r<p<\mathrm{\infty }$, these operators are bounded on $L_{\omega }^p$. Moreover, these results are sharp on the bound of $m$.