Bilinear pseudodifferential operators with symbols in Besov spaces

Abstract

Boundedness from $L^{p_1}({\mathbb {R}^n})\times L^{p_2}({\mathbb {R}^n})$ into $L^p({\mathbb {R}^n}),$ where $2\le p<\infty$ and $2\le p_1,p_2\le \infty$ are related by Hölder’s condition, is shown to hold for bilinear pseudodifferential operators with symbols in various Besov spaces of product type, extending corresponding results known for symbols in the bilinear Hörmander classes $BS^m_{0,0}.$ As a byproduct, it is shown that requiring up to $[\frac{n}{p}]+1$ derivatives in the space variable $x\in {\mathbb {R}^n}$ and up to $[\frac{n}{2}]+1$ derivatives in each of the frequency variables $\xi ,\eta \in {\mathbb {R}^n}$ of the symbol, or derivatives up to order 1 in each single real variable of the symbol, to satisfy Hörmander’s condition for the class $BS_{0,0}^m,$ $m<-n(1-\frac{1}{p}),$ is enough for such boundedness properties.