Bilinear pseudo-differential operators with exotic symbols, II

TL;DR

This paper proves boundedness properties of bilinear pseudo-differential operators with exotic symbols in certain function spaces, extending previous results to a full range of parameters and establishing sharp bounds.

Contribution

It establishes sharp boundedness results for bilinear pseudo-differential operators with symbols in the bilinear Hörmander class, covering a full range of Lebesgue and Hardy space exponents.

Findings

Boundedness from $H^p imes L^2$ to $L^r$ is proved.

Boundedness from $H^p imes L^{inity}$ to $L^p$ is established.

Full range of boundedness from $H^p imes H^q$ to $L^r$ or $BMO$ is achieved.

Abstract

The boundedness from $H^p \times L^2$ to $L^r$, $1/p+1/2=1/r$, and from $H^p \times L^{\infty}$ to $L^p$ of bilinear pseudo-differential operators is proved under the assumption that their symbols are in the bilinear H\"ormander class $BS^m_{\rho,\rho}$, $0 \le \rho <1$, of critical order $m$, where $H^p$ is the Hardy space. This combined with the previous results of the same authors establishes the sharp boundedness from $H^p \times H^q$ to $L^r$, $1/p+1/q=1/r$, of those operators in the full range $0< p, q \le \infty$, where $L^r$ is replaced by $BMO$ if $r=\infty$.