A remark on the relationship $1/p = 1/p_1 + 1/p_2$ for boundedness of bilinear pseudo-differential operators with exotic symbols

TL;DR

This paper investigates the boundedness conditions of bilinear pseudo-differential operators with exotic symbols, establishing the necessity of the relation 1/p = 1/p_1 + 1/p_2 in the critical case.

Contribution

It proves that the condition 1/p = 1/p_1 + 1/p_2 is necessary for boundedness of certain bilinear operators with symbols in the bilinear Hörmander classes.

Findings

The relation 1/p = 1/p_1 + 1/p_2 is necessary for boundedness.

The study focuses on operators with symbols in the class $BS_{ ho, ho}^m$ for 0 < ρ < 1.

Results clarify the boundedness criteria for these exotic bilinear pseudo-differential operators.

Abstract

We consider the bilinear pseudo-differential operators with symbols in the bilinear H\"ormander classes $BS_{\rho, \rho}^m$, $0 < \rho < 1$. In this paper, we show that the condition $1/p = 1/p_1 + 1/p_2$ is necessary when we consider the boudnedness from $H^{p_1} \times H^{p_2}$ to $L^p$ of those operators for the critical case.