Weighted weak-type (1, 1) inequalities for pseudo-differential operators with symbol in $S^{m}_{0,δ}$
Guangqing Wang, Suixin He, Lihua Zhang
TL;DR
The paper addresses weighted endpoint behavior of pseudo-differential operators with symbols in the Hörmander classes $S^{m}_{0,\delta}$ at critical orders, proving sharp weighted weak-type bounds when $a\in S^{-n}_{0,\delta}$ with $0\le\delta<1$ and establishing a dual result for rough symbols $a\in L^{\infty}S^{-n}_{0}$. The approach centers on a Fefferman–Stein sharp maximal function framework, establishing a pointwise bound $M^{\sharp}(T_{a}f)\lesssim Mf$ (and similarly for $T^{*}_{a}$), via a dyadic Littlewood–Paley decomposition and $L^{2}$-boundedness to control local pieces. These estimates feed into weighted weak-type $(1,1)$ results for $T_{a}$ and $T^{*}_{a}$ with $A_{1}$ weights and reveal $m=-n$ as a critical threshold. The results extend to Fourier integral operators with phase functions in $\Phi^{1}$ or $L^{\infty}\Phi^{1}$, yielding weighted weak-type bounds and $L^{p}_{\omega}$ boundedness for $1<p<\infty$, thereby enriching the endpoint weighted theory for PDOs and FIOs.
Abstract
Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in Hörmander class $S^m_{0,δ}$ with $m \in \mathbb{R} $ and $0 \leq δ\leq 1 $. It is well-known that the weak type (1,1) behavior of $T_a $ is not fully understood when the index $m $ is equal to the possibly optimal value $-\frac{n}{2} - \frac{n}{2} δ$ for $0 \leq δ< 1 $, and that $T_a $ is not of weak type (1,1) when $m = -n$ and $δ= 1 $. In this note, we prove that $T_a $ is of weighted weak type (1,1) if $a \in S^{-n}_{0, δ}$ with $0 \leq δ< 1 $. Additionally, we show that the dual operator $T_a^* $ is of weighted weak type (1,1) if $a \in L^\infty S^{-n}_0 $. We also identify $m = -n$ as a critical index for these weak type estimates. As applications, we derive weighted weak type (1,1) estimates for certain classes of Fourier integral operators.