Fourier multipliers in Banach function spaces with UMD concavifications

Abstract

We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call ℓ r (ℓ s )-boundedness, which implies R-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.2010 Mathematics Subject Classification. Primary: 42B15 Secondary: 42B25; 46E30, 47A56.

Banach-valued multilinear singular integrals

“…Let σ be any permutation of {1, 2, 3}, m be a multiplier satisfying (1) and T m,σ denote the adjoint bilinear operator to (1) acting on pairs of X σ(1) , X σ (2) functions. Then f σ(1) L p 1 (R;X σ(1) ) f σ(2) L p 2 (R;X σ(2) ) whenever (1.5) 1 < p 1 , p 2 ≤ ∞, (p 1 , p 2 ) (∞, ∞),…”

“…Let m be a multiplier satisfying (1) and T m denote the adjoint bilinear operator to (1) acting on pairs of X, X ′ functions.…”

“…Several extensions and refinements of [30] have since appeared, see e.g. [1,4,8,9]. In general, as Corollary 1.2 demonstrates, Theorem 1.1 is outside the scope of the above references, although it does imply a strict subset of the ℓ p estimates of [30].…”

Weighted estimates for operator-valued Fourier multipliers

“…In this section we will use the following Banach space geometric property: Definition 6.1. A complex Banach space X has Fourier type t ∈ [1,2] if the vector-valued Fourier transform F : S(R n ; X) → S(R n ; X) extends for one (or equivalently all) n ∈ N to a bounded operator L t (R n ; X) → L t (R n ; X).…”

“…ˆ| w|∈2 j (k+ 1 2 ) (Q) [1,2] [K * N (w − (z − y)) − K * N (w)]v * q dw 1/q · ˆ| w|∈2 j (k+ 1 2 ) (Q) [1,2] f (z − w) q dw 1/q . (6.5)…”