L^p(R^n)-continuity of translation invariant anisotropic pseudodifferential operators: a necessary condition

We consider certain anisotropic translation invariant pseudodifferential operators, belonging to a class denoted by $\mathrm{o}\mathrm{p}({\mathcal{M}}_{\psi }^{\lambda })$, where $\lambda$ and $\psi =({\psi }_1,\dots ,{\psi }_n)$ are the “order” and “weight” functions, defined on ${\mathbb{R}}^n$, for the corresponding space of symbols. We prove that the boundedness of a suitable function $F_p\text{\hspace{0.17em}⁣}:{\mathbb{R}}^n\to [0,+\mathrm{\infty })$, $1<p<\mathrm{\infty }$, associated with $\lambda$ and $\psi$, is necessary to let every element of $\mathrm{o}\mathrm{p}({\mathcal{M}}_{\psi }^{\lambda })$ be a $L^p({\mathbb{R}}^n)$-multiplier. Additionally, we show that some results known in the literature can be recovered as special cases of our necessary condition.