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We consider certain anisotropic translation invariant pseudodifferentialoperators, 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 theboundedness 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 everyelement of $\mathrm{op}(\mathcal{M}^{\lambda}_{\psi})$ be a$L^p(\mathbb{R}^n)$-multiplier. Additionally, we show that some results knownin the literature can be recovered as special cases of our necessary condition.

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