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 op(M_ψ^λ), where λ and ψ = (ψ_1,...,ψ_n) are the “order” and “weight” functions, defined on R^n , for the corresponding space of symbols. We prove that the boundedness of a suitable function F_p : R^n → [0,+∞), 1 < p < ∞, associated with λ and ψ, is necessary to let every element of op(M_ψ^λ) be a Lp(R^n)-multiplier. Additionally, we show that some results known in the literature can be recovered as special cases of our necessary condition.