# Boundedness of bilinear pseudo-differential operators of $S_{0,0}$-type on $L^2 \times L^2$

## TL;DR

This paper extends the boundedness results of bilinear pseudo-differential operators with $S_{0,0}$-type symbols, showing they are bounded from $L^2 imes L^2$ to $L^r$ for all $1< r \,\le 2$, including wider and less smooth classes.

## Contribution

It broadens the class of symbols for which bilinear pseudo-differential operators are bounded on $L^2 \times L^2$, including wider and limited smoothness classes.

## Key findings

- Operators are bounded from $L^2 \times L^2$ to $L^r$ for all $1< r \le 2$.
- Results extend to wider symbol classes beyond $BS^{-n/2}_{0,0}$.
- Includes cases with limited smoothness symbols.

## Abstract

We extend the known result that the bilinear pseudo-differential operators with symbols in the bilinear H\"ormander class $BS^{-n/2}_{0,0}(\mathbb{R}^n)$ are bounded from $L^2 \times L^2$ to $h^1$. We show that those operators are also bounded from $L^2 \times L^2$ to $L^r $ for every $1< r \le 2$. Moreover we give similar results for symbol classes wider than $BS^{-n/2}_{0,0}(\mathbb{R}^n)$. We also give results for symbols of limited smoothness.