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We characterize those pairs (, ) of smooth mappings: Rᵈ,: Rᵈᵈ for which the corresponding weighted composition operator C, f= (f) acts continuously on S (Rᵈ). Additionally, we give several easy-to-check necessary and sufficient conditions of this property for interesting special cases. Moreover, we characterize power boundedness and topologizablity of C, on S (Rᵈ) in terms of,. Among other things, as an application of our results we show that for a univariate polynomial with deg () 2, power boundedness of C, on S (R) for every M (R) only depends on and that in this case power boundedness of C, is equivalent to (C, ⁿ) ₍ converging to 0 in Lb (S (R) ) as well as to the uniform mean ergodicity of C,. Additionally, we give an example of a power bounded and uniformly mean ergodic weighted composition operator C, on S (R) for which neither the multiplication operator f f nor the composition operator f f acts on S (R). Our results complement and considerably extend various results of Fern\'andez, Galbis, and the second named author.
Asensio et al. (2024) studied this question.