New Thoughts on the Vector-valued Mihlin–hörmander Multiplier Theorem

Abstract. Let X be a UMD space with type t and cotype q, and let Tm be a Fourier multiplier operator with a scalar-valued symbol m. If |∂m(ξ)| . |ξ|−|α| for all |α| ≤ ⌊n/max(t, q′)⌋ + 1, then Tm is bounded on L(R;X) for all p ∈ (1,∞). For scalar-valued multipliers, this improves the theorem of Girardi and Weis (J. Funct. Anal., 2003) who required similar assumptions for derivatives up to the order ⌊n/r⌋ + 1, where r ≤ min(t, q′) is a Fouriertype of X. However, the present method does not apply to operator-valued multipliers, which are also covered by the Girardi–Weis theorem.