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

Abstract

We consider certain anisotropic translation invariant pseudodifferential operators, belonging to a class denoted by $\mathrm{op}(\mathcal{M}^{\lambda}_{\psi})$, 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 $F_p\colon\mathbb{R}^n\to[0,+\infty)$, $1<p<\infty$, associated with $\lambda$ and $\psi$, is necessary to let every element of $\mathrm{op}(\mathcal{M}^{\lambda}_{\psi})$ 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.