Some notes on endpoint estimates for pseudo-differential operators

TL;DR

This paper investigates the boundedness of pseudo-differential operators at critical symbol orders, revealing cases of boundedness and unboundedness on various function spaces, thus clarifying endpoint estimates.

Contribution

It provides new examples and results on the boundedness and unboundedness of pseudo-differential operators at critical symbol classes, especially on $L^1$ and $H^1$ spaces.

Findings

Constructed symbols in $S^{n( ho-1)}_{ ho,1}$ with unbounded $L^1$ operators

Proved boundedness from $H^1$ to $L^1$ for certain classes

Provided counterexamples for boundedness on $L^p$ spaces

Abstract

We study the pseudo-differential operator \begin{equation*} T_a f\left(x\right)=\int_{\mathbb{R}^n}e^{ix\cdot\xi}a\left(x,\xi\right)\widehat{f}\left(\xi\right)\,\textrm{d}\xi, \end{equation*} where the symbol $a$ is in the H\"{o}rmander class $S^{m}_{\rho,1}$ or more generally in the rough H\"{o}rmander class $L^{\infty}S^{m}_{\rho}$ with $m\in\mathbb{R}$ and $\rho\in [0,1]$. It is known that $T_a$ is bounded on $L^1(\mathbb{R}^n)$ for $m<n(\rho-1)$. In this paper we mainly investigate its boundedness properties when $m$ is equal to the critical index $n(\rho-1)$. For any $0\leq \rho\leq 1$ we construct a symbol $a\in S^{n(\rho-1)}_{\rho,1}$ such that $T_a$ is unbounded on $L^1$ and furthermore it is not of weak type $(1,1)$ if $\rho=0$. On the other hand we prove that $T_a$ is bounded from $H^1$ to $L^1$ if $0\leq \rho<1$ and construct a symbol $a\in S^0_{1,1}$ such that $T_a$ is unbounded from $H^1$ to $L^1$. Finally, as a complement, for any $1<p<\infty$ we give an example $a\in S^{-1/p}_{0,1}$ such that $T_a$ is unbounded on $L^p(\mathbb{R})$.