Pointwise multiplication of Besov and Triebel--Lizorkin spaces

Abstract

It is shown that para-multiplication applies to a certain product $\pi(u,v)$ defined for appropriate temperate distributions $u$ and $v$. Boundedness of $\pi(\cdot,\cdot)$ is investigated for the anisotropic Besov and Triebel--Lizorkin spaces, more precisely for $B^{M,s}_{p,q}$ and $F^{M,s}_{p,q}$ with $s\in\mathbb{R}$ and $p$ and $q\in\,]0,\infty]$, though $p<\infty$ in the $F$-case. Both generic as well as various borderline cases are treated. The spaces $B^{M,s_0}_{p_0,q_0}\oplus B^{M,s_1}_{p_1,q_1}$ and $F^{M,s_0}_{p_0,q_0}\oplus F^{M,s_1}_{p_1,q_1}$ to which $\pi(\cdot,\cdot)$ applies are determined in the case $\max(s_0,s_1)>0$. For generic isotropic spaces $F^{s_0}_{p_0,q_0}\oplus F^{s_1}_{p_1,q_1}$ the receiving $F^{s}_{p,q}$ spaces are characterised. It is proved that $\pi(f,g)=f\cdot g$ holds for functions $f$ and $g$ when $f\cdot g$ is locally integrable, roughly speaking. In addition, $\pi(f,u)=fu$ when $f$ is of polynomial growth and $u$ is temperate. Moreover, for an arbitrary open set $\Omega$ in Euclidean space, a product $\pi_\Omega(\cdot,\cdot)$ is defined by lifting to $\mathbb{R}^n$. Boundedness of $\pi$ on $\mathbb{R}^n$ is shown to carry over to $\pi_\Omega$ in general.