Sharp function and weighted $L^{p}$ estimates for pseudo-differential operators with symbols in general H\"{o}rmander classes

TL;DR

This paper establishes pointwise inequalities and weighted $L^{p}$ boundedness for pseudo-differential operators with symbols in Hörmander classes, extending known results and proving sharp bounds based on symbol order and weight classes.

Contribution

It provides new sharp bounds for pseudo-differential operators in weighted $L^{p}$ spaces with symbols in general Hörmander classes, including pointwise inequalities and boundedness criteria.

Findings

Proved pointwise inequalities for operators with symbols in $S^{m}_{ ho, heta}$.

Established weighted $L^{p}$ boundedness under specific conditions on $m$ and weight classes.

Results are sharp regarding the bounds on the symbol order $m$.

Abstract

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 $a\in S^{m}_{\varrho,\delta}$ with $0\leq\varrho\leq1,$ $0\leq\delta<1$. Firstly, we prove that if $m\leq-n(1-\varrho)/2$, then $$(T_{a}u)^{\sharp}(x)\lesssim M(|u|^{2})^{1/2}(x)$$ for all $x\in\mathbb{R}^{n}$ and all Schwartz function $u$. Secondly, it is shown that if $1\leq r\leq2$ and $m\leq-\frac{n}{r}(1-\varrho)$, then for any $\omega$ belongs to the class of Muckenhoupt weights $A_{p/r}$ with $r<p<\infty$, these operators are bounded on $L^{p}_\omega$. Moreover, these results are sharp on the bound of $m$.