S​¹​​-bounded Fourier multipliers on ​​H​​¹​(R​​)​​ and functional calculus for semigroups​​​

Let T : H1(R) → H1(R) be a bounded Fourier multiplier on the analytic Hardy space H1(R) ⊂ L1(R) and let m ∈ L∞(R+) be its symbol, that is, T(h) = mh for all h ∈ H1(R). Let S1 be the Banach space of all trace class operators on 2. We show that T admits a bounded tensor extension T⊗IS1 : H1(R; S1) → H1(R; S1) if and only if there exist a Hilbert space H and two functions α, β ∈ L∞(R+;H) such that m(s+t) = α(t), β(s)H for almost every (s,t) ∈ R2+. Such Fourier multipliers are called S1-bounded and we letMS1 (H1(R)) denote the Banach space of all S1-bounded Fourier multipliers. Next we apply this result to functional calculus estimates, in two steps. First we introduce a new Banach algebra A0,S1 (C+) of bounded analytic functions on C+ = {z ∈ C : Re(z) > 0} and show that its dual space coincides with MS1 (H1(R)). Second, given any bounded C0-semigroup (Tt)t≥0 on Hilbert space, and any b ∈ L1(R+), we establish an estimate || ∞0 b(t)Tt dt LbA0,S1 , where Lb denotes the Laplace transform of b. This improves previous functional calculus estimates recently obtained by the first two authors.