Fourier Multipliers on a Vector-Valued Function Space

We study multiplier theorems on a vector-valued function space, which is a generalization of the results of Calderón and Torchinsky, and Grafakos, He, Honzík, and Nguyen, and an improvement of the result of Triebel. For 0 < p< ∞ and 0 < q≤ ∞ we obtain that if r>ds-(d/min(1,p,q)-d), then ‖{(mkfk^)∨}k∈Z‖Lp(ℓq)≲p,qsupl∈Z‖ml(2l·)‖Lsr(Rd)‖{fk}k∈Z‖Lp(ℓq),fk∈E(A2k),under the condition max (| d/ p- d/ 2 | , | d/ q- d/ 2 |) < s< d/ min (1 , p, q). An extension to p= ∞ will be additionally considered in the scale of Triebel–Lizorkin space. Our result is sharp in the sense that the Sobolev space in the above estimate cannot be replaced by Sobolev spaces Lsr with r≤ds-(d/min(1,p,q)-d).