Pointwise multiplication of Besov and Triebel--Lizorkin spaces

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 $B_{p,q}^{M,s}$ and $F_{p,q}^{M,s}$ with $s\in \mathbb{R}$ and $p$ and $q\in ]0,\mathrm{\infty }]$, though $p<\mathrm{\infty }$ in the $F$-case. Both generic as well as various borderline cases are treated. The spaces $B_{p_0,q_0}^{M,s_0}\oplus B_{p_1,q_1}^{M,s_1}$ and $F_{p_0,q_0}^{M,s_0}\oplus F_{p_1,q_1}^{M,s_1}$ to which $\pi (\cdot ,\cdot )$ applies are determined in the case $\max (s_0,s_1)>0$. For generic isotropic spaces $F_{p_0,q_0}^{s_0}\oplus F_{p_1,q_1}^{s_1}$ the receiving $F_{p,q}^s$ 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 $\mathrm{\Omega }$ in Euclidean space, a product ${\pi }_{\mathrm{\Omega }}(\cdot ,\cdot )$ is defined by lifting to ${\mathbb{R}}^n$. Boundedness of $\pi$ on ${\mathbb{R}}^n$ is shown to carry over to ${\pi }_{\mathrm{\Omega }}$ in general.